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ISSN 1932-2321RELIABILITY:Theory & ApplicationsVol.2 No.3-4,December, 2007Special Issue # 1 on SSARS 2007Invited Editors E. Zio and K. KołowrockiElectronic JournalOf International Group On ReliabilitySan Diego

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e‐journal “Reliability: Theory& Applications” No 3‐4 (Vol.2), December ● Special Issue on SSARS‐2007Guest EditorialThe papers collected in these 2 issues are works presented at the 1 st Summer Safety andReliability Seminars, SSARS 2007, held in Gdansk/Sopot, Poland, from 22 nd of July 2007 until 29 thof July 2007.The Seminar was attended by 46 participants from 14 countries (Canada, Czech Republic,Germany, Greece, Italy, Lithuania, Netherlands, Poland, Portugal, Slovakia, South Africa, SouthKorea, Tunisia, United States).The motivation behind this annual event series is to provide a forum for discussing, advancingand developing methods for the safety and reliability analysis of the complex systems and processes,which form the backbone of our modern societies. The subjects of the Seminars are chosen each yearby a Program Board of selected experts in an effort to dynamically represent the methodologicaladvancements developed to meet the newly arising challenges in the field of safety and reliabilityanalysis.This year the emphasis was addressed to the following subjects:• Natural Hazards Analysis and Environment Protection Modeling;• Reliability and Safety Data Collection and Analysis;• System Safety and Reliability Modeling, Dependence, Dynamic Reliability;• Risk Assessment and Management;• Maintenance Modeling and Optimization.Both 1-2 hours lectures on advanced methods and technical presentations of 20-30 minutes onapplications of such methods were offered during the plenary sessions and the seminar sessions,respectively.The Advisory, Editorial and Organizing Boards have carried out the preliminary evaluation of the52 contributions selected for this year Seminars and sent out recommendations to the authors forimproving their work.The extended abstracts of all lectures and technical papers were collected in the SSARSProceedings, which constitute an up-to-date reference textbook for the participants to the Seminarsand all the researchers in the field.The 43 papers and lectures presented at SSARS 2007 are presented in the two special issues ofthe Journal, grouped into a System Safety, Reliability and Maintenance Modeling Section and aNatural Hazards and Risk Assessment and Management Section.Guest EditorsKrzysztof Kołowrocki, Enrico ZioSSARS 2007 Chairmen‐ 5 ‐

e‐journal “Reliability: Theory& Applications” No 3‐4 (Vol.2), December ● Special Issue on SSARS‐2007Table of ContentsIssue 1Section 1: System Safety, Reliability and Maintenance ModelingList of Papers & LecturesBlokus-Roszkowska AgnieszkaAnalysis of component failures dependency influence on system lifetime..................................8Briš RadimStochastic ageing models – extensions of the classic renewal theory ..........................................19Budny TymoteuszTwo various approaches to VTS Zatoka radar system reliability analysis...................................28Duarte Caldeira José, Soares Carlos GuedesOptimisation of the preventive maintenance plan of a series components systemwith Weibull hazard function........................................................................................................33Dziula Przemysław, Jurdziński Misrosław, Kołowrocki Krzysztof, Soszyńska JoannaOn multi-state safety analysis in shipping ....................................................................................40Elleuch Mounir, Ben Bacha Habib, Masmoudi FaouziImprovement of manufacturing cells with unreliable machines...................................................56Grabski FranciszekApplication of semi-Markov processes in reliability…………………………………………... 60Grabski FranciszekThe random failure rate……………………………..………………………………………..… 76Grabski Franciszek, Załęska-Fornal AgataThe model of non-renewal reliability systems with dependent time lengthsof components………………………………………………………………………………..… 83Guo RenkuanAn univariate DEMR modelling on repair effects…………………………………………...… 89Guze SamborNumerical approach to reliability evaluation of two-state consecutive“k out of n: F” systems..................................................................................................................98Guze Sambor, Kołowrocki KrzysztofReliability analysis of multi-state ageing consecutive „k out of n: F” systems............................104‐ 6 ‐

e‐journal “Reliability: Theory& Applications” No 3‐4 (Vol.2), December ● Special Issue on SSARS‐2007Knopik LeszekSome remarks on mean time between failures of repairable systems...........................................112Kołowrocki KrzysztofReliability modelling of complex systems - part 1 .......................................................................116Kołowrocki KrzysztofReliability modelling of complex systems – part 2.......................................................................128Kudzys AntanasTransformed conditional probabilities in the analysis of stochastic sequences............................140Kwiatuszewska-Sarnecka BożenaOn asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems ..............150Kwiatuszewska-Sarnecka BożenaOn asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: “m out of n” systems .......................159Rakowsky Uwe KayFundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling ...............................................................................................................173Soszyńska JoannaSystems reliability analysis in variable operation conditions .......................................................186Vališ DavidReliability of complex system with one shot items ......................................................................198Zio EnricoSoft computing methods applied to condition monitoringand fault diagnosis for maintenance..............................................................................................206Zio Enrico, Baraldi Piero, Popescu Irina CrengutaOptimising a fuzzy fault classification tree by a single-objective genetic algorithm ...................221‐ 7 ‐

A.Blokus-Roszkowska Analysis of component failures dependence on system lifetime - RTA # 3-4, 2007, December - Special Issuestate reliability function that is a basic characteristic ofthe multi-state system.One of basic multi-state reliability structures withcomponents degrading in time is parallel-series system.In the multi-state reliability analysis to define parallelseriessystems with degrading components we assumethat:– E ij , i = 1,2,K,kn, j = 1,2,K,li, kn , l1,l2, K,l k ,nn ∈ N, are components of a system,– all components and a system under considerationhave the state set {0,1,...,z}, z ≥ 1,– the reliability states are ordered, the state 0 is theworst and the state z is the best,– the component and the system reliability statesdegrade with time t without repair,– T ij (u), i = 1,2,K,kn, j = 1,2,K,ln, kn , l1,l2, K,l k ,nn ∈ N, are independent random variablesrepresenting the lifetimes of components Eij in thestate subset { u , u + 1,..., z},while they were in thestate z at the moment t = 0,– T(u) is a random variable representing the lifetimeof a system in the reliability state subset{ u , u + 1,..., z},while it was in the reliability state zat the moment t = 0,– e ij (t) are components E ij states at the moment t,t ∈< 0,∞),– s(t) is the system reliability state at the moment t,t ∈< 0,∞).Under above assumptions we introduce the followingdefinition of multi-state reliability function of acomponent.Definition 1. A vectorR ij (t , ⋅ ) = [R ij (t,0),R ij (t,1),...,R ij (t,z)], t ∈( −∞,∞),whereR ij (t,u) = P(e ij (t) ≥ u | e ij (0) = z) = P(T ij (u) > t), (1)for t ∈< 0,∞),u = 0,1,...,z, i = 1,2,...,k n , j = 1,2,...,l i , isthe probability that the component E ij is in thereliability state subset { u , u + 1,..., z}at the moment t,t ∈< 0,∞),while it was in the reliability state z at themoment t = 0, is called the multi-state reliabilityfunction of a component E ij .It is clear from Definition 1, that R ij (t,0) = 1.Definition 2. A vectorRkwhere( t,⋅)= [ R ( t,0),R ( t,1),K,R ( t,z)]n ,lnkn,lnkn,lnkn,lnR ( t , u)= P(s(t)≥ u | s(0)= z)= P(T ( u)> t)(2)k n ,l nfor t ∈< 0,∞),u = 0,1,...,z, is the probability that thesystem is in the reliability state subset { u , u + 1,..., z}atthe moment t, t ∈< 0,∞),while it was in the reliabilitystate z at the moment t = 0, is called the multi-statereliability function of a system.Ifp( t,⋅ ) = [ p(t,0),p(t,1),K,p(t,z)]for t ∈< 0,∞),wherep ( t,u)= P(s(t)= u | s(0)= z)for t ∈< 0,∞),u = 0,1,K,z,is the probability that thesystem is in the state u at the moment t, t ∈< 0,∞),while it was in the state z at the moment t = 0,thenandR ( t,0)= 1, R ( t , z)= p(t,z),t ∈< 0,∞),(3)k n ,l nk n ,l np( t,u)= R ( t,u)− R ( t,u + 1)(4)kn ,lnkn,lnfor t ∈< 0,∞),u = 0,1,K,z −1.Moreover, if( m)R ( t,u)= 1 for t ≤ 0,u = 1,K,z,k,n l nthen the mean lifetime of the system in the state subset{ u,u + 1, K,z}isM ( u)= ∞ R ( t,u)dt,u = 1,K,z,(5)∫0k n ,l nand the standard deviation of the system sojourn timein the state subset { u,u + 1, K,z}is2σ ( u)= N(u)− [ M ( u)], u = 1,K,z,(6)whereN(u)= 2∞ ∫ tR ( t,u)dt,u = 1,K,z.(7)Besides0k n ,l n- 9 -

A.Blokus-Roszkowska Analysis of component failures dependence on system lifetime - RTA # 3-4, 2007, December - Special IssueM ( u)= ∞ ∫ p(t,u)dt,u = 1,K,z,(8)0is the mean lifetime of the system in the state u, whilethe integrals (5), (6) and (7) are convergent. Then,according to (3)-(5) and (7), we get the followingrelationshipM ( u)= M ( u)− M ( u + 1), u = 1,2,K,z −1,M ( z)= M ( z).(9)Definition 3. A probabilityr( t)= P(s(t)< r | s(0)= z)= P(T ( r)≤ t),that the system at the moment t, t ∈< 0,∞),is in thesubset of states worse than the critical state r,r ∈ { 1,2, K,z},while it was In the state z at themoment t = 0 is called a risk function of the multistatesystem.Under this definition, from (1), we haver( t)= 1 − P(s(t)≥ r | s(0)= z)= 1 − R kn ,l n( t,r),(10)for t ∈ ( −∞,∞)and moreover, if τ is the momentwhen the risk exceeds a permitted level δ , δ ∈< 0 ,1 > ,then−τ = r 1 ( δ ),(11)where r −1 ( t ), if exists, is the inverse function of therisk function r(t).2.1. Multi-state parallel-series systemsDefinition 4. A multi-state system is called parallelseriesif its lifetime T(u) in the state subset{ u , u + 1,..., z}is given byT(u)= min {max{ T ( u)}},u = 1,2,K,z.1≤i≤kn1≤j≤liijDefinition 5. A multi-state parallel-series system iscalled homogeneous if its component lifetimes T ij (u) inthe state subset have an identical distribution functionF( t,u)= P(Tij ( u)≤ t),u = 1,2,K,z,i = ,2, K,k ,1nj = ,2, K,l ,1ii.e. if its components E ij have the same reliabilityfunctionR( t,u)= 1−F(t,u),u = 1,2,K,z.Definition 6. A multi-state parallel-series system iscalled regular ifl= l = K = l k l , l n ∈N,1 2 =n ni.e. if the numbers of components in its parallelsubsystems are equal.2.1.1. Parallel-series systems with independentcomponentsThe results presented below can be found in [4].Proposition 1. If in a homogeneous regular multi-stateparallel-series system(i) components failure independently,(ii) components have reliability functions R(t),then the multi-state system reliability function is givenby the formulaRkwhere( t,⋅)= [1, R ( t,1),K,R ( t,z)],n ,lnkn,lnkn,lnl n k nR ( t , u)= [1 − [1 − R(t)]] for t ∈ ( −∞,∞),k n ,l nu = 1,K,z,where knis the number of its parallel subsystemslinked in series and l nis the number of components ineach subsystem.In the case when components of a system haveexponential reliability functions i.e.R( t,⋅ ) = [1, R(t,1),K,R(t,z)],(12)whereR ( t,u)= 1 for t < 0,R( t,u)= exp[ −λ(u)t]for t ≥ 0,λ ( u)> 0,(13)then the multi-state system reliability function takesformRkwhere( t,⋅ ) = [1, R ( t,1),K,R ( t,z)],(14)n ,lnkn,lnkn,ln- 10 -

A.Blokus-Roszkowska Analysis of component failures dependence on system lifetime - RTA # 3-4, 2007, December - Special IssueR ( t,u)= 1 for t < 0,k n ,l nl n k nR ( t , u)= [1 − [1 − exp[ −λ(u)t]]] (15)k n ,l nfor t ≥ 0,u = 1,K,z.2.1.2. Parallel-series systems with dependentcomponent failuresA multi-state parallel-series system is in the reliabilitystate subset { u , u + 1,..., z}if all of its parallelsubsystems are in this state subset. Taking into accountthis fact, a multi-state parallel-series system withdependent components is considered as a system oflinked independently in series multi-state parallelsubsystems composed of components with failuredependency. In each of these subsystems we assumethe following model of failure dependency. Aftergetting out of the reliability state subset { u , u + 1,..., z}v, v = 0,1,2, K , li−1,of components in a subsystem theincreased load is shared equally among others so astheir load increase with the same scale. Then theirreliability is getting worse so that the mean values ofthe i-th, i = 1,2,K,kn, subsystem component lifetimesT ij '(u)in the state subset { u , u + 1,..., z}are of the formv li− vE[ Tij'( u)]= E[Tij( u)]− E[Tij( u)]= E[Tij( u)],llij = ,2, K,l , v 0,1,2,K , l −1,u = 1,2,K,z.(16)1i=iThen the rates of getting out from the reliability statesubset { u , u + 1,..., z}of remaining components of thei-th, i = ,2, K,k , multi-state subsystem are given by1n( v)li− vλ ( u)= λ(u),v = 0,1,2,K , lili−1,(17)u = 1,2,K,z.Proposition 2. If in a homogeneous regular multi-stateparallel-series system(i) components failure in dependent way according to(16),(ii) components have exponential reliability functionsgiven by (12)-(13),then the multi-state system reliability function is givenby the formulaRkwhere( t,⋅ ) = [1, R ( t,1),K,R ( t,z)],(18)n ,lnkn,lnkn,lniR ( t,u)= 1 for t < 0,k n ,l n⎡ln∑−1v( lnλ(u)t)R kn ,l n( t,u)= ⎢exp[ − lnλv=0⎢⎣v!⎤( u)t]⎥⎦for t ≥ 0,u = 1,K,z.(19)3. Asymptotic approachThe investigation of exact reliability functions of largesystems often leads to complicated formulae. Thus,from practical point of view, the asymptotic approachto large systems reliability evaluation is veryimportant. The suggested method allows us to obtainformulae that simplify optimising calculations.In the paper in the asymptotic approach to thereliability evaluation of multi-state parallel-seriessystems the linear standardization of their lifetimes inthe reliability state subsets is used. This approach relieson an investigation of limit distribution of astandardized random variable ( T(u)− bn( u)) / an( u),u = 1,2,K,z,where T (u)is the system lifetime in thestate subset { u,u + 1, K,z}and ( u)> 0,b n( u)∈ ( −∞,∞),are called normalizing constants. Forthat reason, we assume the following definition [4].Definition 7. A reliability functionR ( t,⋅ ) = [1, R ( t,1),K,R ( t,z)],t ∈ ( −∞,∞),is called a multi-sate limit reliability function of asystem with reliability functionRk( t,⋅)= [1, R ( t,1),K,R ( t,z)],n ,lnkn,lnkn,lnif there exist normalizing constants a n( u)> 0,b n( u)∈ ( −∞,∞)such aslim R ( a ( u)t + b ( u),u)= R ( t,u)for t ∈ C ,n→∞k n ,l nnnkna nR (u)where C R (u)is the set of continuity points ofR ( t,u),u = 1,2,K,z.From Definition 7 it follows that the knowledge thelimit reliability function of a system R ( t,⋅)forsufficiently large n allows us to estimate the multi-statereliability function of a system Rk n ,l n( t,⋅)accordingthe following approximate formulaR ( t,⋅)≅ R (( t − b ( u)) / a ( u),⋅),(20)k n ,l nnn- 11 -

A.Blokus-Roszkowska Analysis of component failures dependence on system lifetime - RTA # 3-4, 2007, December - Special Issuei.e.[ 1, R ( t,1),K,R ( t,z)]kn ,lnkn,lnt − bn(1) t − bn( z)≅ [ 1, R ( ,1), K , R ( , z)],t ∈ ( −∞,∞).a (1)a ( z)nnthenRwheret,⋅ ) = [1, R ( t,1),K,R ( t,)], t ∈ ( −∞,∞),10(1010z(23)3.1. Reliability evaluation of large parallelseriessystems with independent componentsIn this point the possibilities of multi-state asymptoticapproach to the reliability evaluation of large parallelseriessystems are presented. There are formulated twopropositions that allow us to find multi-state limitreliability functions of homogeneous parallel-seriessystems with independent components under theassumption that they have exponential reliabilityfunctions.Proposition 3. If in a homogeneous regular multi-stateparallel-series system(i) components failure independently,(ii) components have exponential reliability functionsgiven by (12)-(13),(iii) k n= n,l n− c log n >> s,c > 0,s > 0,1(iv) a ( u)= 1 lnn, bn( u)= logλ(u) log n λ(u)log n,u = 1,2,K,z,thenRwheret,⋅ ) = [1, R ( t,1),K,R ( t,)], t ∈ ( −∞,∞),(21)3(33zR ( t,u)= exp[ − exp ] for u = 1,K,z,(22)3tis the multi-state limit reliability function of consideredsystem.Proposition 4. If in a homogeneous regular multi-stateparallel-series system(i) components failure independently,(ii) components have exponential reliability functionsgiven by (12)-(13),(iii) k n→ k,k > 0,l n→ ∞,1(iv) a ( u)= log lnn , bn( u)=λ(u)λ(u), u = 1,2,K,z,[ 1 − exp[ − exp[ t ] kR ( t , u)=]] for u = 1,K,z,(24)3−is the multi-state limit reliability function of consideredsystem.3.2. Reliability evaluation of large parallelseriessystems with dependent componentfailuresThe proofs of presented below propositions are givenin [3].Proposition 5. If in a homogeneous regular multi-stateparallel-series system(i) components failure in dependent way according to(16),(ii) components have exponential reliability functionsgiven by (12)-(13),(iii) R ( t,⋅)is a multi-state system reliability(iv)k n ,l nfunction given by (18)-(19),k n→ k = constant as n → ∞,(v) l n= n,1(vi) a n( u)= ,λ(u)nthenRwhereb n1( u)= λ(u)for u = 1,K,z ,t,⋅ ) = [1, R ( t,1),K,R ( t,)], t ∈ ( −∞,∞),(25)2(22z= (26)kR 2 ( t , u)= [1 − FN(0,1)( t,u)]for u 1,K,z,and1ttFN( 0,1) ( t,u)= ∫ exp[ − ] dt,2π−∞ 22t ∈( −∞,∞),- 12 -

A.Blokus-Roszkowska Analysis of component failures dependence on system lifetime - RTA # 3-4, 2007, December - Special Issuer t)= 1 − R 10( t,2)(, 22= 1 − exp[ − exp[(0.2041t− 1)44log10+ log( 50 / π log10)]] for t ∈ ( −∞,∞).The moment when the system risk exceeds thepermitted level e.g. δ = 0.05,according to (11), isτ = r −1( δ ) ≅ 2 years and 11 days.Comparing the expected values of the elevatorlifetimes in the state subset and the elevator meanlifetimes in the particular states in the case whenstrands failure in dependent and independent way wecan conclude that these values are lower in the firstcase for about 68% percent for exact reliabilityfunctions and for about 67% and 69% for approximatereliability functions.The obtained results illustrate that the increased load ofremaining un-failed components causes shortening thelifetime of these components in a significant way. Thatfact can be interpreted as a decrease of their reliabilitymuch faster then for the systems with independentcomponents. Taking into account the presented shipropeelevator we can notice that the lifetime in thereliability state subset of the elevator under theassumption that strand failure in dependent way is evenabout 70% shorten then in the case when strands areindependent.5. ConclusionIn the paper the exact reliability analysis andasymptotic approach to the reliability evaluation ofhomogeneous multi-state parallel-series systems arepresented. For these systems the exact and limitreliability functions and other characteristics both inthe case when their components are independent andwhen they are dependent are determined under theassumption that components of systems haveexponential reliability functions.Introduced in the paper the method of reliabilityevaluation of large systems relies on application ofsome approximate methods based on classicalasymptotic approach to this issue. The obtained resultsare concerned with typical systems with regularstructure. Applied in the paper analytical methods aresuccessful rather for not very complex systems. In thisbackground it seems to be justified the extension ofthis issue for systems with less regular structures anduse of any other reliability analysis methods.References[1] Blokus, A. (2006). Reliability analysis of largesystems with dependent components. InternationalJournal of Reliability, Quality and SafetyEngineering 13(1): 1-14.[2] Blokus-Roszkowska, A. (2006). Reliability analysisof multi-state rope transportation system withdependent components. Proc. InternationalConference ESREL’06, Safety and Reliability forManaging Risk, A. A. Balkema, Estoril: 1561-1568.[3] Blokus-Roszkowska, A. (2007). Reliability analysisof homogenous large systems with componentdependent failures (in Polish). Ph.D. Thesis,Maritime University, Gdynia – System ResearchInstitute, Warsaw.[4] Kolowrocki, K. (2004). Reliability of LargeSystems. Amsterdam - Boston - Heidelberg -London - New York - Oxford - Paris - San Diego -San Francisco - Singapore - Sydney - Tokyo:Elsevier.[5] Kolowrocki, K. et al. (2002). Asymptotic approachto reliability analysis and optimisation of complextransport systems (in Polish). Gdynia: MaritimeUniversity. Project founded by the PolishCommittee for Scientific Research.[6] Smith, R. L. (1982). The asymptotic distribution ofthe strength of a series-parallel system with equalload-sharing. Annals of Probability 10: 137-171.[7] Smith, R. L. (1983). Limit theorems andapproximations for the reliability of load sharingsystems. Advances in Applied Probability 15:304-330.[8] Smith, R. L. & Phoenix, S. L. (1981). Asymptoticdistributions for the failure of fibrous materialsunder series-parallel structure and equal loadsharing.Journal of Applied Mechanics 48: 75-81.- 17 -

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special Issueavailable systems has been carried out by a number ofresearchers. A survey is given by Gertsbakh [7], withemphasis on results related to the convergence of thedistribution of the first system failure to theexponential distribution.If the lifetimes are distributed arbitrarily, then thesystem can be described by a semi-Markov process orMarkov renewal process. Semi-Markov processes andMarkov renewal processes are based on a marriage ofrenewal processes and Markov chains. Pyke [8] gavea careful definition and discussion of Markov renewalprocesses in detail. In reliability, these processes areone of the most powerful mathematical techniques foranalysing maintenance and random models. Adetailed analysis of the non-exponential case (nonregenerativecase) is however outside of the scope ofthe introduction part. Further research is needed topresent formally proved results for the general case.Presently, the literature covers only some particularcases, what is also the case of this presentation.2. Models with a negligible renewal periodIn some cases we can take into account a renewalperiod equal to zero. For example the situation when atime to a renewal is substantially smaller than a timeto a failure and its implementation would notinfluence an expected result. This case was intensivelystudied in [6]. Basic relationships for Poisson processare derived in [4]:Renewal function and renewal density are given asfollowsH ( t)= EN t= ∑∞ ( λt)enn=1 n!= λte−λt∞ ( λt)∑n=1 n!nn−λt= λt.Basic definitions from renewal theory see inAppendix. Renewal density is constanth( t)= H'( t)= λ.Methodology based on Laplace transforms wasdramatically extended in [6] for the case when a timeto failure is modelled by the Erlang distribution,which has a probability density functionaλ(λt)ef ( t)=Γ(a)−λt,t ≥ 0, λ f 0,a f 0.After the backward transformation a renewal densityis equal toλ a−1λ( ) = + ∑ + sh ta k = 1 ak e s ktin the expression there is, t f 0,which is k th nonzero root of the equation(s + ) a = aFor example for a = 4 nonzero roots are equal tosiπ(21= λ e −1)= ( −1+ i)λiπs2 = λ(e − 1) = −2λ,si3π(23= λ e −1)= ( −1− i)λand a renewal densityλ 3 λ= + ∑ + sh(t)k e s kt4 k = 1 4=λ[14− e−2λt− e−λt,,sin( λt)].Figure 1. Renewal density for Erlang distributionAnother calculation method was applied for Weibulldistribution of time to failure, which has a probabilitydensityα −1−(λt)αf ( t)= αλ(λt)e , t ≥ 0, > 0 is a parameter of the shape, > 0 is a parameterof a scale.A probability density f n (t), n = 2,3,…., or a probabilitydensity of time to n th failure can be calculated as aconvolution of the function ( f n-1 *f ). We can expressit numerically e.g. with the help of discrete Fouriertransformation [6].By a numerical integration we can determine a vectorof a distribution function F n .A formula (1) for a calculation of the renewal function= ∞ n=0s ∈ C,s k∈ C,H ( t ) ∑ F n( t )(1)- 20 –

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special Issueis necessary to substitute by a finite sum of the first Kcomputed terms at the numerical calculation. It can beconducted because these terms converge quickly to azero at a definite interval [0, T]. Equally, S n = X 1 + X 2+…+ X n has an asymptotic normal distribution N(n, 2 ) where and 2 are definite expected value and adispersion of X i.Considering that Weibull distribution of time tofailure for > 1 has an increasing failure rater(t)= λα(λt)α −1a distribution function F n (t) can be upper estimated bythe functiont i( )n−1µFn( t)≤1− ∑ ei=0 i!−µt= G n(t), t p nµ ,where is an expected value of a time to failure [3].Γ(1+µ =λ1)α.We can estimate in this way an error of a finite sumK( t ) = ∑ F n( t ) ,Hn=0because a remainder is limited∞∞∑ Fn( t )≤ ∑Gntn=K + 1 n=K + 1( ), t p nµ.Exact relationship for the reminder is derived in [6].The behaviour of the renewal function estimated fordifferent number of members in the above finite sumwe can see in Figure 2.it may be wise to replace it before it has aged toogreatly. In this section we shall concentrate on theoperating characteristics of some commonlyemployed replacement policies.A commonly considered replacement policy is thepolicy based on age (age replacement). Such a policyis in force if a unit is always replaced at the time offailure or c hours after its installation, whicheveroccurs first; c is a constant unless otherwise specified.If c is a random variable, we shall refer to the policyas a random age replacement policy. Under a policy ofblock replacement the unit is replaced at times c (k =1,2,…), and at failure. This replacement policyderives its name from the commonly employedpractice of replacing a block or group of units in asystem at prescribed times c (k = 1,2,…) independentof the failure history of the system.3.1. Replacement based on ageA unit is replaced c hours after its installation or atfailure, whichever occurs first; c is consideredconstant. Let R(t) denote the probability that an itemdoes not fail in service before time t. ThennR( t)= R(τc) R(t −nτc),n ∈ N ∪{0}: nτ ≤ t < n + 1) τ .c(cThe distribution function of a time to failure X isnF( t)= 1 − R(t)= − R(τ ) R(t −nτ),1cct ≥ 0, n ∈ N ∪{0} : nτ ≤ t < n + 1) τ .Expected time to failure E(X) is= ∞ 0E( x)∫ R(x)dxc(c= ∞ n=0( n+1) τcn∑ ∫ R(τ ) R(t − nτnτccc) dtτcn= ∑∞ R(τc) ∫ R(t)dtn=0 0=F1(τc∫ R(t)dt,)τ c 0Figure 2. Renewal function for Weibull distribution3. Models with maintenanceIn many situations, failure of a unit during actualoperation is costly or dangerous. If the unit ischaracterized by a failure rate that increases with age,- 21 -

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special Issue= ∞ k = 0( k + 1) τc∑ ∫ R((k + 1) τ − x)R(τkτc⋅ R( x + y − lτc) fi−1(x)dxc)l−(k + 1)cFigure 3. The Weibull distribution function for a unitwith the replacement based on age.When the time to failure is exponentially distributed X~ exp(1/µ), then we havenF( t)= 1 − R(τc) R(t −nτc)=−nµτ m ( t n )1c − µ − τc− ee= 1 −−e µt,z = x − kτc, n = [( z + y) / τc]= ∑ ∞ τckn−1Ri−1( τc) ∫ R(τc− z)R(τc)k = 0 0⋅ R( z + y − nτc) fi−1( z)dzn−1R(τ )τcc= ∫ R(τ c − z)R(z + y − nτc ) f i−1 − R ( τ )i−1c01( z)dz.which means that distribution function is independenton replacements. Other words, the unit does not age.3.2. Block replacement policyUnder a policy of block replacement all componentsof a given type are replaced simultaneously at times c (k = 1,2,…) independent of the failure history ofthe system. If X i is time of i-th failure of a unit whichhas distribution function F i and probability density f iand R i (t)=1- F i (t), thenR ( t)= R(τ)R(t −nτn1 ccn ∈ N ∪{0}: nτ ≤ t < n + 1) τ ,f ( t)= R(τ )c),(cd[1 − R(t − nτdtn1 ccDistribution of time to i-th failure for i > 1 we canderive on the basis of conditional probability:x > 0, y > 0,k = [ x / τc], l = [( x + y) / τc], l > k,Pr( X y \ X1x)i>i−== R((k + 1) τThenc− x)R(τ)( n−(k + 1))cPr( X i> y) ∫ R((k + 1) τ − x)R(τ)].R(x + y − nτ= ∞ ( l−(k + 1)cc )0⋅ R( x + y − nτc) fi−1(x)dxc) .Figure 4. The Weibull distribution function for a unitwith block replacementIn Figure 4, we can see time dependencies for R i (t) forWeibull distribution of time to failure.3.3. Periodical preventive maintenanceMay a device goes through a periodical maintenanceafter a time interval of the operation c , whoseintention is a detection of possible dormant flaws andtheir possible elimination. The period of devicemaintenance is d and after this period the device startsoperating again. F(t) is here a time distribution to afailure X. Then in the interval [0, c + d ) there is aprobability that the device appears in the not operatingstate equal toP ( t)= F(t),t pτc= 1, t ≥τ c.The state of a failure is considered then both a time tothe maintenance after a possible failure and the timewhen the device is under maintenance. Theprobability P(t) (also a coefficient of unavailability)for t ∈[ 0, ∞)is generated by following system ofequations:P( t)= P ( t),nn ∈ N ∪{0}: n τ + τ ) p t ≤ ( n + 1)( τ + τ ),(c dc d- 22 –

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special Issue…Pi( t)Pi( t)− Pi1(iτ c)[1 − P0( t − i(τc+ τ=−1 −di = 1,2,...,n −1,…))]P ( t)= F(), t f 0.0tHere P i (t) stands for a probability that a device existsin the failure state provided that before it had gonethrough i inspections. A term P i-1 (t) - P i-1 (i c )represents a probability that a device was all right atthe previous inspection and it failed in the interval (i c ,t), P i-1 (i c ) P 0 (t - i( c + d )) is a probability that it hadfailed in the previous inspection and since then itfailed again.3.3.1. Exponential distribution of time tofailureMayF(t)= 1 − e−λt.For the time t ∈[ 0, ∞)is then a probability P(t) equaltoP( t)= F(t − n(τc+ τd)),t ∈[ n(τc+ τd), n(τc+ τd) + τc),P ( t)= 1, t ∈ n(τ + τ ) + τ , ( n + 1)( τ + τ )),[c d cc dIf d = 0, the expression for P(t) can be furthersimplified into the formP( t)= F(t − nτc),n ∈ N ∪{0}: n τ + τ ) ≤ t p ( n + 1)( τ + τ ).(c dc d3.3.2. Weibull distribution of time to failureLet the intensity of failures of the given distribution oftime to failure is not constant, it is then a function oftime past since the last renewal. In this case it isnecessary for the given t and n, related with it whichsets a number of done inspections to solve abovementioned system of n equations and the solution ofthe given system is not eliminated anyhow as it is atan exponential distribution.In the Figures 5 and Figure 6 a slightly marked curvedraws points of the local extremes in the case ofshortening a time to the first inspection.Figure 5. Coefficient of unavailability for exponentialdistributionFigure 6. Coefficient of unavailability for Weibulldistribution.4. Alternating renewal modelsAlternating models are those where two of thesignificantly diverse states appear, between which amodel converts from one to another. A faulty device isthe example of the alternating model where a time to arepair is compared with a time to failure and it cannotbe neglected.In the case that both a time to failure and a time to arepair follows an exponential distribution, generalsolution for a calculation of a coefficient ofavailability can be found in [4].4.1. Lognormal distribution of a time to failureIf a distribution to a failure X f has a lognormaldistribution, then a probability density is in the form1 ln t − λf f( t)= ϕ(), t f 0σtδwhere (x) is standard normal density.In this case a numerical calculation is offered againfor the computation of the coefficient of availability.We can compute a probability density of a sum ofrandom quantities X f and X r (X r is an exponential time toa repair) from a discrete Fourier transformation [6],equally as a convolution in the equationK(t)= Rft( t)+ ∫ h(x)R ( t − x)dx.0fThe calculation of a renewal density is substituted bya finite sumN( t ) = ∑ f n( t ).hn=1- 23 -

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special IssueAn example: In the following example a calculationfor parameter values =1/4, =8, =1/2, is done. Inthe Figure 7 there is a renewal density. Theasymptotic value is marked by dots, which is in thiscase equal tolim h(t)=t→∞EXf1+ EXr=e1≅1−λ+ σ22+ τ0.123A failure occurrence in the renewal time is not takeninto account, after the renewal both the parts areconsidered to be new.May X f1 and X f2 are independent random valuesdescribing time of failures with probability densitiesf f1 (t) and f f2 (t), further a time to a repair is X r with adensity f r (t). A probability that no failure occurs in theinterval [0,t) is equal toRf( t)= P(X f ≥ t ∧ X f 2 ≥1 t= 1 − F ( t)][1− F ( )] = R t)R ( t).[ f 1 f 2 t)f (1 f 2and is a reliability function of the time to failure X f ofthe whole device. Then X f has a probability densityFigure 7. A renewal density for lognormal distributionFigure 8 shows a procedure of the coefficient ofavailability K(t), the asymptotic value is marked bydots again and it is given by the following formula:K = lim K ( t)=t→∞EXf1+ EXr≅ 0.938ffd( t)= − Rf( t).dtWith the knowledge f f (t) we can calculate thefunctions describing this alternating process.If the time to failure has an exponential distributionwith mean values 1/ and 1/, thenff( t)= −ddte= ( λ + µ ) e−( λ + µ ) t−(λ+µ ) tThen f f (t) has an exponential distribution with a meanvalue 1/() and the coefficient of availability isequal.( ) tK(t)= τ λ + µ − λ+µ + τ+ e , t f 0λ + µ + τ λ + µ + τ.Figure 8. Coefficient of availability for a lognormaldistribution5. Alternating renewal models with two typesof failuresThe following part presents models, which consideran appearance of two different independent failures.These failures can be described by an equaldistribution with different parameters or by differentdistributions.5.1. Common repairA device composed of two serial elements can be anexample whereas a failure of one of them causes afailure of the whole device. A time to a renewal iscommon for both the failures and begins immediatelyafter one of them. It is described by an exponentialdistribution with a mean value 1/.If the analytical procedure is uneasy or impossible, anumerical calculation can be used. For a renewaldensity computation is desirable instead of theequation∞h( t)= f ( t)+ ∫ h(x)f ( t − x)dxf0use a renewal equation for a renewal densityh ( t )= ∞ n=1∑ f n( t )fand conduct a sum of the only definite number ofelements with a fault stated above. f n (t) is aprobability density of time to n th failure. Then for thecalculation of convolutions is used for example aquick discrete Fourier’s transformation.In the Figure 9 there is a graph of a coefficient ofavailability in the case that a time to a failure X f1 and- 24 –

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special IssueX f2 have Weibull distribution. Expected value to thefailure EX f is equal toEXf2f12f 12f 2EX EX2 22 6= = = 3. 62 2EX + EX 2 + 62f 2and that is why an asymptotic coefficient ofavailability is equal toEXfK =EX + EXfr= 0.79Figure 9. Coefficient of availability for Weibulldistribution5.2. Two independent partsSupposing the device consists of two independentparts. The behaviour of each one is described by itsalternating model with a given time to a failure and atime to a repair. Maintenance proceeds for bothdifferently and independently. Equally, the failure ofone of them can appear regardless of the state of theother part, even in the state of a failure.Let us consider the whole device to be in the state of afailure when at least one of the parts is in the state of afailure. K a (t) is a coefficient of availability of the firstpart and K b t) is a coefficient of availability of thesecond one. For the whole device K(t) is equal toK( t)= K ( t)K ( t).abMay dormant faults occur in the first part, withWeibull’s distribution and with an expected value EX= 2 and a parameter of the form = 2 which areeliminated by periodical inspections with a period c =2 (See Models with periodical preventivemaintenance) and the second part is equal as in theprevious model. The course of the coefficient ofavailability as the product of already computed partialones is designed in the Figure 10.Figure 10. Coefficient of availability for independentparts6. ConclusionIn this paper a few types of renewal processes, whichdifferentiate in a renewal course and a type ofprobability distribution of a time to failure, weredescribed. These processes were mathematicallymodelled by the means of a renewal theory and thesemodels were subsequently solved.In the cases, when the solving of integral equationswas not analytically feasible, numerical computationswere successfully applied. It was known from thetheory that the cases with the exponential probabilitydistribution are analytically easy to solve.With the gained results and gathered experience itwould be possible to continue in modelling andsolving more complex mathematical models whichwould precisely describe real problems. For exampleby the involvement of certain relations which wouldspecify the emergence, or a possible renewal ofindividual types of failures which in reality do nothave to be independent on each other. Equally, itwould be practically efficient to continue towards thecalculation of optimal maintenance strategies with theset costs connected with failures, exchanges andinspections of individual components of the systemand determination of the expected number of theseevents at a given time interval.AcknowledgementsThis research is supported by The Ministry ofEducation, Youth and Sports of the Czech Republic.Project CEZ MSM6198910007.References[1] Aldemir, T. (2004). Characterization of timedependent effects due to ageing. Proc. of the Kickoff meeting, JRC Network on Incorporating AgeingEffects into PSA Applications, EuropeanCommission, DG JRC – Institute for Energy,Nuclear Safety Unit, Probabilistic Risk &Availability Assessment Sector. Edited by M.Patrik and C. Kirchsteiger.[2] Aven, T. & Jensen, U. (1999). Stochastic models inreliability. Springer-Verlag , ISBN 0-387-98633-2.[3] Barlow, R. & Proschan, F. (1965). MathematicalTheory of Reliability. Wiley.- 25 -

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special Issue[4] Bartlett, M.S. (1970). Renewal Theory. Meten &Co. Ltd. Science paperbacks, ISBN 412 20570 X.[5] Bri, R., Châtelet, E. & Yalaoui, F. (2003). Newmethod to minimize the preventive maintenancecost of series–parallel systems. In ReliabilityEngineering & System Safety, ELSEVIER, Vol.82, Issue 3, p.247-255.[6] Bri, R. & epin, M. (2006). Stochastic ageingmodels under two types of failures. Proc. of the31st ESReDA SEMINAR on Ageing, SmoleniceCastle, Slovakia.[7] Gertsbakh, I.B. Asymptotic methods in reliability:A review. Adv. Appl. Prob. 16, 147-175.[8] Pyke, R. Markov renewal processes: Definitionsand preliminary properties. Ann Math Statist 32,1231-1242.AppendixRenewal ProcessRenewal process serves for example to modelmathematically a device behaviour which ismaintained in such a way that it stays running as mosteffectively and longest as possible. May a file ofcomponents or the whole device with a time to afailure X (non-negative absolutely continuous randomvariable) with a dispersion given by a probabilitydensity f(t) exists and may a symbol t denotes forclearness a time. The first component is put intooperation at time t = 0. Further, X 1 is a period whenthe first component comes to the failure and at thesame time it is substituted by a new identicalcomponent from a given file. It means that a renewalperiod (in this case a change period) is negligible, orequal to zero. This second component breaks downafter the period X 2 since it started to operate. At thetime X 1 + X 2 the second component is renewed by theexchange for the third one and the process continuesfurther in such a way. The r-th renewal will happen atthe time S r = X 1 + X 2 +…+X r .If X 1, X 2… are independent non-negative equallydistributed random variables with a finite expectedvalue and dispersion,Sn0= 0, Sn= ∑ Xi, n∈N,i=1then a random process { } ∞ n n=0S is called a renewalprocess in a renewal theory. Sometimes an order ofstated random variables {X n } n=0 is denoted in thisway. In the case when a time distribution until thefailure is exponential, we speak about Poissonprocess. A function F n (t) indicates a distributionfunction of a random variable S n. . There are a fewother random variables connected with the renewalprocess, which describe its behaviour (at time). Let wecall N t a number of renewals in the interval [0, t] for afirm t 0, it meansNt= max{ n : S p t}nFrom this we also get that S Nt t < S Nt+1. Regardingthe fact that the interval [0, t] contains n failures (aswell as renewals) only if n th failure happens at thelatest at the time tP{ N ≥ n} = P{ S p t} F (t)t n=and the probability that at the time t there are nrenewals in the given renewal process can bedescribed in the following wayP{ N n} = P{ S p t ∧ S ≥ t}t=n n+1= Fn[ − F ( t)] = F ( t)F ( )n( t) 1n+ 1 n−n+1tProvided that X 1, X 2… are independent non-negativeequally distributed random variables and P r (X 1 = 0)

R. Bri Stochastic ageing models – extensions of the classic renewal theory - RTA # 3-4, 2007, December - Special IssueThis equation can be easily derived from the previousequation with help of its integral transformation (e.g.Laplace).An asymptotic behaviour of a renewal process issubstantial. An asymptotic behaviour of a renewalprocess is discussed in an Elementary theorem about arenewal: if a time distribution to a renewal has a finiteexpected value , thenH ( t)lim =t→∞t1µ.It is a Blackwell theorem, which testifies about alimited behaviour of an expected number of renewalsat a finite interval (t, t+t]: if a time to a renewal hasa non-lattice distribution with a definite positiveexpected value , then ∀ h f 0 ishlim[ H ( t + h)− H ( t)] = .t→0 µIf a derivation of a renewal function exists (i.e. X 1,X 2… are absolutely continuous random variables), thenfor the arbitrary time t > 0 a function h(t) that isdefined by a relationh(t)=lim∆t→0+H ( t)− H ( t + ∆t)= H '(t)∆tis a renewal density. Then with a help of a probabilitydensity f n (t) = F’ n (t) we haveh ( t )= ∞ n=1∑ f n( t ).A renewal density most often appears in the followingintegral equationth(t)= f ( t)+ ∫ h(t − u)f ( u)du ,0so called a renewal equation for a renewal density.Here f(t) is a probability density of a absolutelycontinuous non-negative time to the renewal X.We can describe the equation approximately by wordsin such a way that for t0 renewal probabilityh(t)t in the interval (t, t + t ] is equal to aprobability sum f(t)t that in the interval (t, t + t ]the first renewal happens and the sum of probabilitiesfor ∀u ∈( 0, t)that the renewal happens at the time t –u followed by a time to the failure of the length u.Alternating Renewal ProcessProvided that there are two kinds of components withvarious independent time to a failure X,Y ,respectively adequate distributional functions F(t),G(t) (densities f(t), g(t)), at the time t = 0 thecomponent of the first type is activated and every timeat the time of failure is substituted by the componentof the opposite type, resulting process is namedAlternating renewal process.We can simulate a renewal process with a definitetime to a renewal with such a model. At the time t =0 the component begins to work to the moment offailure X 1 . The final time to the renewal Y 1 follows. Atthe moment X 1 + Y 1 the renewal ends and a new (orrepaired) component is activated with a time to afailure X 2 . X 1, X 2 …resp. Y 1 ,Y 2 …are independent nonnegativerandom variables with a distr. function F(t)resp. G(t).The n th failure happens at the momentSn= X + Y1+ ... + Xn−1+ Yn−1+ X1 nfor n th renewal we haveTnX + Y1+ ... + Xn 1+ Yn1+ Xn+ Y=1 − −nA random process {S 1 , T 1 , S 2 , T 2 …..} is then analternating renewal process. A coefficient ofavailability K(t) (or also A(t) - availability) is a basiccharacteristic of a renewal process with a finite timeto a renewal. It determines a probability that at thetime t the component will work. It is consequentlyequal to a sum of probabilities that X 1 > t, it meansthat the first component has a time to a failure greaterthan t, and that the renewal happens in the interval (u,u + u], u 0, 0 < u < t and a renewed componentwill have a time to a failure greater than t - u.Written by an integral equation:K(t)= 1 − F(t)+ ∫ h(x) 1tt0[ − F(t − x)]= R(t)+ ∫ h(x)R(t − x)dx,0h(x) is a renewal process density of a renewal{T n } n=0, F(t) is a distribution function of the time to afailure, resp. 1 – F(t) = R(t) is reliability function.In particular, an asymptotic coefficient of availabilityof the alternating renewal process is importantpractical reliability characteristics,K = lim K(t).t ∞→It describes behaviour of the alternating renewalprocess in the situation when the system is stabilizedin “a distant time moment t”, i.e. a stationary case,when the influence of the beginning configurationsubsides.,dx.- 27 -

T.BudnyTwo various approaches to VTS Zatoka radar system reliability analysis - RTA # 3-4, 2007, December - Special IssueBudny TymoteuszGdynia Maritime University, Faculty of Navigation, Gdynia, PolandTwo various approaches to VTS Zatoka radar system reliability analysisKeywordsVTS system, system reliability, shipping safetyAbstractIn the paper we propose two ways of reliability calculation of radar system in Vessel Traffic Services Zatoka.Reliability and availability of the system were calculated on the base of reliability of the system components. Inthe first approach there was assumed that system is series, in the second approach system is treated as a series-“mout of n”. We obtain different results. Conclusion is that choosing proper method of approach to system reliabilityand availability analysis is decisive in appropriate evaluation of those properties.1. IntroductionOne of the most important properties of the devicesand technical systems is their reliability. Reliability isextremely important when concerns systems, whichassure people safety or/and natural environmentprotection. Vessel Traffic Services System – VTSZatoka is that type of system. Its main task is to assuresafe navigation for all ships that sails to ports ofGdynia and Gdansk. The most important part of theVTS Zatoka system are shore based maritime radars.Reliability of that system can be evaluated in differentways. In the paper there are proposed two possibleapproaches to calculate that reliability [1].2. Systems’ definitionsWe assume that [2]E i , i = 1,2,...,n, n ∈ N,are two-state components of the system havingreliability functionsR i (t) = P(T i > t), t ∈( −∞,∞),i = 1,2,...n,whereT i , i = 1,2,...,n,are independent random variables representing thelifetimes of components E i with distribution functionsF i (t) = P(T i ≤ t), t ∈ ( −∞,∞),i = 1,2,...,n.Definition 1. A two-state system is called series if itslifetime T is given byT = min{ T }.1≤i≤niFigure 1. The scheme of a series systemThe above definition means that the series system isnot failed if and only if all its components are notfailed, and therefore its reliability function is given bynR( t)= ∏ R i( t), t ∈ ( −∞,∞).(1)i=1Definition 2. A two-state series system is called nonhomogeneousif it is composed of a, 1 ≤ a ≤ n,different types of components and the fraction of theith type component in the system is equal to q i , whereaq ii=1q i > 0, ∑ = 1 . MoreoverR( i)E 1 E 2 … E n( i)( t)= 1−F ( t),t ∈ ( −∞,∞),i = 1,2,...,a, (2)is the reliability function of the ith type component.- 28 -

T.BudnyTwo various approaches to VTS Zatoka radar system reliability analysis - RTA # 3-4, 2007, December - Special IssueThe scheme of a non-homogeneous series system isgiven in Figure 2.Figure 2. The scheme of a non-homogeneous seriessystemIt is easy to show that the reliability function of thenon-homogeneous two-state series system is given byR 'k n l na( i)qin( t)= ∏( R ( t)), t ∈ ( −∞,∞).(3)i = 1A two-state system is called an “m out of n” system ifits lifetime T is given byT= T( n −m+1),q 1 q 2 q aE 1 E 2. . . E nm = 1,2,...,n,where T( n−m+1)is the mth maximal order statistic in thesequence of component lifetimes T1, T2,..., Tn.The scheme of a non-homogeneous “m out of n”system is given in Figure 3, wherei , 1i2, …, i n∈ { 1,2,..., n}and ij≠ ikfor j ≠ k.The reliability function of the non-homogeneous twostate“m out of n” system is given either by'( m)R n= 1−( t)an∑ ∏ ( )q i0≤ri≤qini=1r1+ r2+ ... + ra≤m−1t ∈ ( −∞,∞),or by'( m )R n=( t)an∑ ∏ ( )q i0≤ri≤qini=1r1+ r2+ ... + ra≤mt ∈ ( −∞,∞),where m = n − m.riri( i)( i)r( i)qin−rii[ R ( t)][ F ( t)], (7)ri( i)qin−ri[ F ( t)][ R ( t)], (8)The above definition means that the two-state “m outof n” system is not failed if and only if at least m outof its n components are not failed. The two-state “mout of n” system becomes a parallel system if m = 1,whereas it becomes a series system if m = n. Thereliability function of the two-state “m out of n” systemis given either by( m)ri 1−riR ( t)= 1−∑ ∏[R ( t)][ F ( t)],(4)nt ∈ ( −∞,∞),or by1r1,r2 ,..., rn = 0 i = 1r1+ r2+ ... + rn ≤m−1niiE i1E i2. . .E im. . .E inq 1q 2q a( m )ri 1−riR ( t)= ∑ ∏[F ( t)][ R ( t)], (5)n1nr1,r2 ,..., rn = 0 i = 1r1+ r2+ ... + rn ≤mt ∈ ( −∞,∞),m = n − m.Definition 3. A two-state “m out of n” system is callednon-homogeneous if it is composed of a, 1 ≤ a ≤ n,different types of components and the fraction of theith type component in the system is equal to q i , whereaq ii = 1qi > 0, ∑ = 1 . MoreoverR( i)i( i)( t)= 1−F ( t),t ∈ ( −∞,∞),i = 1,2,...,a, (6)iFigure 3. The scheme of a non-homogeneous “m out ofn” systemDefinition 4. A multi-state system is called series- “mout of k ” if its lifetime T is given bynT , m = ,2,..., k ,= T( kn −m+1)1nwhere T( k n −m+1)is m-th maximal statistics in therandom variables setT i = min { T }, i =1,2,...,k n .1≤j≤liij- 29 -

T.BudnyTwo various approaches to VTS Zatoka radar system reliability analysis - RTA # 3-4, 2007, December - Special IssueThe above definition means that series-“ m out ofkn”system is composed of k n series subsystems and it isnot failed if and only if at least m out of its k n seriessubsystems are not failed.The reliability of the series-“m out ofgiven either by( )R m, 1, 2 ,...,tkn l l lkn1( ) = 1−kn li likn” system isri1−ri∑ ∏[∏ R ( t)][1 − ∏ R ( t)], (9)ijr1, r2,..., r = 0 kn i= 1 j = 1 j = 1r1+ r2+ ... + r ≤m−1knt ∈ ( −∞,∞),or by( )R mk , l1 , l2,..., ltn k n1( ) =kn li liri1−ri∑ ∏[1− ∏ R ( t)][ ∏ R ( t)], (10)ijr1, r2,..., r = 0 kn i= 1 j = 1 j = 1r1+ r2+ ... + r ≤mknt ∈ ( −∞,∞),where m = k m .n −ijij• Lighthouse Hel, (radar height 42,5 m a.s.l.);• Port of Gdynia Harbourmaster Office building(HMO), (32,5 m a.s.l.);• Northern Port of Gdansk HarbourmasterOffice building, (66 m a.s.l.);• Western Hills?, (17,5 m a.s.l.);• Lighthouse Krynica Morska, (26 m a.s.l.).Radars work permanently and their range cover wholeresponsibility area assigned to VTS Zatoka. Two, oreven three, radars cover most part of Gulf of Gdansksimultaneously. That situation has great matter in caseof failure of single radar.For the VTS Zatoka systems’ radars, apart fromstandard equipment, additional Radar Data Processor(RDP) has been installed. RDP changes radar datafrom analogue to digital form. This digital informationis next transferred to VTS centre by wireless line orlight cable, which connect two Harbourmaster’s officesof Ports in Gdynia and Gdansk. Signals from radars,after preliminary treatment, are transferred to the VTSCentre. Then after final processing signals are sendingout and visualized (with use of computer programARAMIS) at VTS Centre itself, Harbourmaster’soffices of Gdynia and Gdansk ports and atHarbourmaster’s office of Krynica Morska port.Scheme of the radar’ subsystem and datatransmission is showed on Figure 5.Radar HMOGdyniaVTSCenterRadar HMOGdansk2. System of VTS Zatoka radarsRadars’ system is the basic subsystem of whole VTSsystem and also part of identification and watchingsystem at the Gulf of Gdask region. The purpose ofLighthouse„Hel” RadarWesternHills Radar- light cable- wireless line- net cableHMO – Harbormaster Office buildingLighthouse„Krynica Morska”RadarFigure 5. Scheme of VTS Zatoka radars’ systemFigure 4. Positions of VTS Zatoka shore based radars(dots)that system is assuring real time information aboutships’ traffic in that region [3]. VTS Zatoka systemworks involving five shores based radars, which areput in following places:3. VTS Zatoka radar system reliabilityIn order to analyse the considered system reliability wewill firstly calculate reliability parameters of singleradar.3.1. Single radar reliabilityVTS Zatoka system has been designed and constructedby Holland Institute of Traffic Technology (HITT). It- 30 -

T.BudnyTwo various approaches to VTS Zatoka radar system reliability analysis - RTA # 3-4, 2007, December - Special Issuecame into service on the 1 st of May 2003. Used radarswere produced by Danish corporation TermaElectronic AS (frequency 9735 MHz). Mean timebetween failure and mean time to repair given by HITThave been used to construct VTS Zatoka radars’subsystem reliability model (Table 1). Times to repairgiven in Table 1, concern the situation in which serviceteam is near damaged radar. In fact, sustaining serviceTable 1. MTBF i – mean time between failure andMTTR i – mean time to repair [5]Device/component MTBF i [h] i MTTR i [h]Antenna 50 000 0,00002 3Single radartransmitter13 000 0,000077 0,5Receiver 17 390 0,000058 0,34Video processor 40 000 0,000025 0,25Radar processor 20 000 0,00005 0,25Data transmitter 87 500 0,000011 0,5team near each radar is very expensive. For furtherconsideration we assume that service team is in Tricityi.e. in Gdynia, Sopot or Gdansk. Taking intoaccount access time, time needed to fix what device isdamaged and mean time of device’ exchange, meantime to repair of the single radar (MTTR S ) is about 3and a half hours. After considering frequency offailures of particular parts of radars, the mean time ofexchanging damaged part is about 40 minutes.To find single radar reliability we assume that theradar is a series system. This means that the failure ofone component of radar causes the failure of the wholeradar. Two radars placed at Harbourmaster officebuildings have five elements (antenna, single radartransmitter, receiver, video processor, radar processor),three others additionally have data transmitter. Weassume that component reliability functions areexponential and given by the equation( − ⋅ t)R ( t)= exp λ , t ≥ 0 . (11)iiWhen we put data from Table 1 to (11), and then toequation (1), as a result we obtain the single radarreliability function for radars at harbourmaster officebuildingsR H( t)= exp( −0,000229⋅t), t ≥ 0 , (12)and for three others radarsR O( t)= exp( −0,000235⋅ t), t ≥ 0 . (13)= ∞ 0MTBF ∫ R( t)dt . (14)According to equations (12)-(14) and to Table 1, weobtain mean time between failures of single radar atharbourmaster office buildings1MTBF H = = 4366h≈ 182 days,0,000229and mean time between failures of three other radars1MTBF O = = 4255h≈ 177 days.0,0002353.2. Reliability of radar systemIn order to evaluate radars system reliability, we canuse different approaches. First, we can assume thatsubsystem is series. VTS Zatoka radars’ subsystem isworking when all five radars are working. Accordingto equation (3) with parameters2n = , q = , q551 2=3,5and equations (12) and (13), we obtain the systemreliability function2R S( t)= [exp[ −0,000229t]][exp[ −0,000235t= exp[ −0,001163t], t ≥ 0 . (15)Mean time between failures of that system is given byequationMTBFs= ∞ ∫ R ( t)dt =0S10,001163= 860h ≈ 36 days. (16)System availability is given by equation [1]GS= (17)MTBFMTBFS+ MTTRSand after substituting MTBF S = 860h, MTTR S = 3,5h,amountsG = 0,9959.]]3Mean time between failures of single radar is given byequation- 31 -

T.BudnyTwo various approaches to VTS Zatoka radar system reliability analysis - RTA # 3-4, 2007, December - Special Issue10,90,80,70,6R 0,50,40,30,20,100 500 1000 1500 2000t [h]Figure 6. Radars’ subsystem’s reliability functionsAnother way of describing reliability of radars’subsystem is assumption that system is „m out of n”.We can assume that system is working if at least fourout of five radars are working. If we take into accountthat particular radars are series systems we obtain anon-homogenous series-„4 out of 5” system.The above assumption is acceptable because rangesof any four radars covered fairways to ports in Gdanskand Gdynia and most traffic (nearly entire) isconcentrated in those fairways.According to equations (10) and Table 1, reliabilityfunction of such system is given by equationR SMN= R(1)5,5,5,6,6,6t( t)= 2 exp( −0,000934)+ 3exp(−0,000928t)− 4 exp( −0,001163t).(18)R SMN (t) function is showed on Figure 4.Mean time to failure of series-„m out of n” systemMTBF SMN is given by equationMTBF = ∞ SMN ∫ RSMN( t)dt.(19)0R S (t)R SMN (t)The mean time to failure of above described systemaccording to equations (18) and (19) equalsGMTBFSMNSMN = (21)MTBFSMN+ MTTRSand henceG SMN1935= = 0,9982.1935 + 3,5As we can see system defined as series-„m out of n”has both higher reliability and availability than a seriessystem.4. ConclusionAs we can see from the performed analysis evaluationof the system reliability depends on taken assumptions.Reliability functions are significantly different onefrom the other (Figure 6), so choosing proper methodof describing of system reliability structure is veryimportant.Whatever method was chosen, thanks to reliabilityof components of radars, VTS Zatoka radars system ishighly reliable. Access to spare parts and organizationof service has significant matter for availability of thesubsystem. In order to sustain acceptable availability itis necessary to provide the service support located inTri-City. It allows for quick reaction in case of failuresand for repairing damaged parts of radars.References[1] Budny, T. (2006). Radar subsystem reliability inthe VTS Zatoka. Navigational Faculty Works 18,Gdynia.[2] Kolowrocki, K., Blokus, A. Milczek, B., Cichocki,A., Kwiatuszewska-Sarnecka, B. & Soszynska, J.,(2005). Asymptotic approach to complex systemsreliability analysis, Two-state non-renewalsystems. Gdynia Maritime University PublishingHouse, Gdynia.[3] Stupak, T., Wawrach, R., Dziewicki, M. &Mrzygód, J. (2004). Research on availability ofVTS Zatoka radars. Maritime Office, Gdynia.(1)MTBF SMN0001163= ∞ ∫ R5,5,5,6,6,6( t)dt = 2 ⋅10,0009341+ 3 ⋅ − 4 ⋅0,000928 0,1= 1935h ≅ 81days. (20)The availability of the series-„m out of n” system isgiven by equation- 32 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special IssueDuarte José CaldeiraMathematical Department, Instituto Politécnico de Setúbal/Escola Superior de Tecnologia de Setúbal, PortugalIST – Unit of Marine Technology and Engineering, Lisbon, PortugalSoares Carlos GuedesIST – Unit of Marine Technology and Engineering, Lisbon, PortugalOptimisation of the preventive maintenance plan of a series componentssystem with Weibull hazard functionKeywordsavailability, hazard function, Weibull distribution, preventive maintenance, series componentsAbstractIn this paper we propose an algorithm to calculate the optimum frequency to perform preventive maintenance inequipment that exhibits Weibull hazard function and constant repair rate in order to ensure its availability. Basedon this algorithm we have developed another one to solve the problem of maintenance management of a seriessystem based on preventive maintenance over the different system components. We assume that all components ofthe system still exhibit Weibull hazard function and constant repair rate and that preventive maintenance wouldbring the system to the as good as new condition. The algorithm calculates the interval of time between preventivemaintenance actions for each component, minimizing the costs, and in such a way that the total downtime, in acertain period of time, does not exceed a predetermined value1. IntroductionThroughout the years, there has been tremendouspressure on manufacturing and service organizations tobe competitive and provide timely delivery of qualityproducts. In many industries, heavily automated andcapital intensive, any loss of production due toequipment unavailability strongly impairs the companyprofit. This new environment has forced managers andengineers to optimise all sectors involved in theirorganizations.Maintenance, as a system, plays a key role in achievingorganizational goals and objectives. It contributes toreducing costs, minimizing equipment downtime,improving quality, increasing productivity, andproviding reliable equipment that are safe and wellconfigured to achieve timely delivery of orders tocostumers. In addition, a maintenance system plays animportant role in minimizing equipment life cycle cost.To achieve the target rate of return on investment,plant availability and equipment effectiveness have tobe maximized.Grag and Deshmukh [38] had recently review theliterature on maintenance management and points outthat, next to the energy costs, maintenance costs can bethe largest part of any operational budget.A brief bibliographic review, (Andrews & Moss [1],Elsayed [5], McCormick [9] and Modarres et al [10]),is enough to conclude that the discipline known asreliability was developed to provide methods that canguarantee that any product or service will functionefficiently when its user needs it. From this point ofview, reliability theory incorporates techniques todetermine what can go wrong, what should be done inorder to prevent that something goes wrong, and, ifsomething goes wrong, what should be done so thatthere is a quick recovery and consequences areminimal.So, reliability has several meanings. However it isusually associated to the ability of a system to performsuccessfully a certain function. To measurequantitatively the reliability of a system it is used aprobabilistic metric, which we state next.- 33 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special IssueReliability of a system is the probability that a systemwill operate without failure for a stated period of timeunder specified conditions.Another measure of the performance of a system is itsavailability that reflects the proportion of time that weexpect it to be operational. Availability of a system isthe probability to guarantee the intended function, thatis, the probability that the system is normal at time t.The availability of a system is a decreasing function ofthe failure rate and it is an increasing function of therepair rate.According to Elsayed [5], reliability of a systemdepends mainly in the quality and reliability of itscomponents and in the implementation andaccomplishment of a suitable preventive maintenanceand inspection program. If failures, degradation andaging are characteristics of any system, however, it ispossible to prolong its useful lifetime and,consequently, to delay the wear-out period carrying outmaintenance and monitoring programs.This type of programs leads necessarily to expensesand so we are taken to a maintenance optimisationproblem.Basic maintenance approaches can be classified as:• Unplanned (corrective): this amounts to thereplacement or repair of failed units;• Planned (preventive):• Scheduled: this amounts to performinginspections, and possibly repair, following apredefined schedule;• Conditioned: this amounts to monitor thehealth of the system and to decide on repairactions based on the degradation levelassessed.In the unplanned, corrective strategy, no maintenanceaction is carried out until the component or structurebreaks down. Upon failure, the associated repair timeis typically relatively large, thus leading to largedowntimes and high costs. In this approach, efforts areundertaken to achieve small mean times to repair(MTTRs).To avoid failures at occasions that have high costconsequences preventive maintenance is normallychosen. This allows that inspections and upgrading canbe planned for periods, which have the lowest impacton production or availability of the systems.The main function of planned maintenance is to restoreequipment to the “as good as new” condition;periodical inspections must control equipmentcondition and both actions will ensure equipmentavailability. In order to do so it is necessary todetermine:• Frequency of the maintenance, substitutionsand inspections• Rules of the components replacements• Effect of the technological changes on thereplacement decisions• The size of the maintenance staff• The optimum inventory levels of spare partsThere are several strategies for maintenance; the onewe have just described and that naturally frames inwhat has been stated is known as Reliability CenteredMaintenance - RCM. Gertsbakh [7] reviews some ofthe most popular models of preventive maintenance.In theory, maintenance management, facing theproblems stated above, could have benefited from theadvent of a large area in operations research, calledmaintenance optimisation. This area was founded inthe early sixties by researchers like Barlow andProschan. Basically, a maintenance optimisation modelis a mathematical model in which both costs andbenefits of maintenance are quantified and in which anoptimal balance between both is obtained. Well-knownmodels originating from this period are the so-calledage and the block replacement models.Valdez-Flores & Feldman [12] presents acomprehensive review of these approaches. Dekker [2]gives an overview of applications of maintenanceoptimisation models published so far and Duffuaa [4]describes various advanced mathematical models inthis area that have “high potential of being applied toimprove maintenance operations”.More recently, Nakagawa [11] summarizes the resultsof maintenance policies and Garg and Deshmukh [6]describe the existence of 24 papers on the quantitativeapproach to maintenance optimisationAs we have already mentioned, one of the most criticalproblems in preventive maintenance is thedetermination of the optimum frequency to performpreventive maintenance in equipment, in order toensure its availability.The Preventive Maintenance policies are adapted toslow the degradation process of the system while thesystem is operating and to extend the system life. Anumber of Preventive Maintenance policies have beenproposed in the literature. These policies are typicallyto determine the optimum interval between preventivemaintenance tasks to minimize the average cost over afinite time span. But in many areas one complainsabout the gap between theory and practice.Practitioners say maintenance optimisation models aredifficult to understand and to interpret [2]. Vatn et al[13] claim there exists a vast number of academicpapers describing narrow maintenance models, whichare rarely, if ever, used in practice. Most of thesepapers are too abstract, and the models that could beuseful are difficult to identify among this large numberof suggested models.- 34 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special IssueIn this paper we propose an algorithm to solve theprevious problem for equipment that exhibits Weibullhazard function and constant repair rate. Based on thisalgorithm we have developed another one to optimisemaintenance management of a series system based onpreventive maintenance over the different systemcomponents.This is a problem with many applications in realsystems and there are not many practical solutions forit. The main objective of this paper is to present anoptimisation model understandable by practitioners,simple and useful for practical applications.Duarte and Craveiro [3] have already outlined asolution for this problem for equipment that exhibitlinearly increasing hazard rate and constant repair rate.We assume that all components of the system stillexhibit Weibull hazard function and constant repairrate and that preventive maintenance would bring thesystem to the as good as new condition. We define acost function for maintenance tasks (preventive andcorrective) for the system. The algorithm calculates theinterval of time between preventive maintenanceactions for each component, minimizing the costs, andin such a way that the total downtime, in a certainperiod of time, does not exceed a predetermined value.2. Previous concepts and resultsIn this section we present the classical concept ofavailability, while describing how to calculate it.Point-wise availability of a system at time t, A(t), is theprobability of the system being in a working state(operating properly) at time t. The unavailability of thesystem, Q(t), is Q(t) = 1 – A(t).The pointwise availability of a system that has constantfailure rate λ and constant repair rate µ isµ λA( t)= + exp[ − ( µ + λ ) t].(1)µ + λ µ + λand the limiting availability isµA = lim A(t)= .(2)t →+∞ µ + λThe second parcel in formula (1) decreases rapidly tozero as time t increases; so, we can stateµA(t) ≈µ + λand this means that the availability of such a system isalmost constant.Example. A system is found to exhibit a constantfailure rate of 0,000816 failures per hour and aconstant repair rate of 0,02 repairs per hour.Using formula 1, the availability of such a system (seeFigure 1) is obtained as-2A ( t)= 0,9608 + 3.9201×10 exp( - 2,0816×10and the limiting availability islim A(t)= 0,9608.t→∞10.990.980.970.960.950 100 200 300 400tFigure 1. The availability function-2t).It should be notice that, in this in case, we do not havealmost any variation in the value of component’savailability for t > 200.We can therefore conclude that, to guarantee a value ofavailability A, known the constant repair rate, µ, thevalue of the constant failure rate of the system it willhave to satisfy the relationshipµA ≈µ + λ( − A)µ 1⇔ λ ≈A3. Model and assumptions. (3)Suppose a system is found to exhibit an increasinghazard rate,h(t)=βθ⎛ t⎜⎝θ⎞⎟⎠β −1, θ > 0, β > 0, t ≥ 0 ,and a constant repair rate µ.Our goal is to determine the interval time betweenpreventive maintenance tasks (we assume that thesystem is restored to the “as good as new” conditionafter each maintenance operation) in such a way thatthe availability of the system is no lesser than A.The main idea for the solution of this problem consistsof determining the time interval during which the- 35 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special Issueincreasing hazard rate can be substituted by a constantfailure rate in order to guarantee the pre-determinateavailability level.Applying the mean value theorem of integral calculusto the functionh(t)=βθwe obtainx0∫θββx⇒θββtβ −1⎛ t⎜⎝θ= λx⎞⎟⎠⇒ x = 0 ∨ x =β −1β −1, θ > 0, β > 0, t ≥ 0 ,⎡ tdt = λx⇒ ⎢⎣θλθβ.ββ⎤⎥⎦x0= λx(4)Substituting λ in (4) by its approximate value given informula 3, we havexµ( 1 − A)= β − 1βAθ.We can therefore conclude that in the time intervaland a constant repair rate µ. To guarantee anavailability for the system equal or greater than A theinterval of time between two consecutive preventivemaintenance tasks must be equal or lesser thanβ −1µ( 1−A)Aθβ.Example. A system is found to exhibit an− 8 1, 25increasing hazard rate, h ( t)= 5 × 10 × t , and a−2constant repair rate m ( t)= 4 × 10 .What should be the greatest time interval betweenpreventive maintenance tasks (we assume that thesystem is restored to the “as good as new” conditionafter each maintenance operation) in such a way thatthe availability of the system is at least 98%?If the system had a constant failure rate, to guaranteesuch availability it should be−24 × 10λ =0,98( 1−0,98)= 0,0008163.We want to calculate the instant x in order to satisfythe following conditionx0−8∫ 5 × 10 × t1,25dt = 0,0008163x⎡ µ0,1⎢β −⎢⎣( 1 − A)Aθβ⎤⎥,⎥⎦−85 × 10 × x⇒2,252,25= 0,0008163xthe hazard functions,andh(t)=βθµh(t)=⎛ t⎜⎝θ⎞⎟⎠β −1( 1−A)A, θ > 0, β > 0, t ≥ 0 ,guarantee approximately the same value of availability.What we have just demonstrated can formally be statedon the following form:Proposition: Let S be a system exhibiting an increasinghazard rate,⇒ x = 0 ∨ x = 4488.We can therefore conclude that the system must berestored to the “as good as new” condition after eachmaintenance task every 4488 hours in order to achievethe availability target of 98%.Figure 2 illustrates this example.0.00180.00160.00140.00120.0010.00080.00060.00040.0002h(t)=βθ⎛ t⎜⎝θ⎞⎟⎠β −1, θ> 0, β> 0, t ≥ 001000 2000 3000 4000 5000tFigure 2. Hazard functions h(t)=5×10 -8 t 1,25 andh(t)=0,0008163 over the interval [0, 4488]- 36 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special Issue4. Optimisation of the preventive maintenanceplan of a series components systemIn this section we will present a model for thepreventive maintenance management of a seriessystem.The system is composed by a set of n components inseries as Figure 3 shows.Component 1Figure 3. A series system of n componentsLet τ 1 , τ 2 , …, τ n be the time units between preventivemaintenance tasks on components 1, 2, …, n,respectively (Figure 4); assuming that these actionswill restore periodically the components to the "asgood as new" condition, they will have, therefore,consequences at the reliability and availability levels ofthe system.0Component 2 ...Component nt 1t 2t 3t 4t 5t 6t 7t 8t 9t 10t 11t 12The cost of each preventive maintenance task is cmp iand the cost of each corrective maintenance task iscmc i .Since the availability of the system consisting of ncomponents in series requires that all units must beavailable (assuming that components’ failures areindependent), system availability A iswhere A i is the availability of component i.Applying proposition presented in section 3 we canwrite that the availability of each component i is A iover the interval⎡ µ⎢0,βi−1⎢⎣i( 1 − A )Aiiθβii⎤⎥,⎥⎦and its hazard function can be approximated by theconstant functionµh ( t)=inA ii=1A = ∏i( 1−A )AiiComponen t 1Componen t 2Componen t 3Component nτ 1τ2τnτ3Then, the expected number of failures in that timeinterval isβi−1µi( 1−A ) µ ( 1 − A )Aiiθβii×iAiiFigure 4. A preventive maintenance plan.Our goal is to calculate the vector[ τ τ τ L ] T123τ nin such a way that the total down time in a certainperiod of time does not exceed a predetermined value,that is to say, that it guarantees the specified servicelevel and simultaneously minimizes the maintenancecosts.We assume that each component has a Weibull hazardfunction,βhi( t)=θii⎛ t⎜⎝θi⎞⎟⎠βi−1, θand a constant repair ratei> 0, βi> 0, t ≥ 0The objective function (defined as a cost function perunit time) is⎡⎢n⎢c(A1, A2, K,An) = ∑ ⎢i=1⎢⎢⎣subject toβi−1n⎧⎪∏Ai≥ A,⎨i=1⎪⎩0< Ai< 1, i = 1,2, K,n.5. Numerical exampleµicmp+( 1−A ) µ ( 1−Ai)AiiiθβiicmcThe model described on section 4 was implemented toa three components series system.Aii⎤⎥⎥⎥⎥⎥⎦m t)= µ .i(i- 37 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special IssueWe assume that each component has a Weibull hazardfunction and a constant repair rate. Components aremaintained preventively at periodic times.Data is presented on Table 1.First we present the nomenclature.θ i , β i – parameters of hazard function.TTR – Mean Time to Repair (corrective maintenance).TTP – Time of one preventive maintenance action.PMC – Preventive maintenance cost.CMC – Corrective maintenance cost.τ - time between two consecutive preventivemaintenance tasks.Table 1. Initial conditionsComponentsθ i β i TTR TTP PMCostCMCost1 4472,136 2 100 10 2000 4000 20002 1873,1716 2 50 40 2500 5000 15003 500,94 2 80 10 1000 2000 250With this preventive maintenance plan the availabilityachieved is about 90,30% and the life cycle cost is122055,79.The target for availability is 90%.The objective function was slightly modified in orderto include the cost of down time.MATLAB was used to optimise the objective function.Table 2 shows the results. With this new preventivemaintenance policy we have a reduction of 5,5% inLife Cycle Cost (LCC) and simultaneously theavailability A achieved (92,70%) is greater than theexisting one (90,30%).Table 2. Results of MatLab optimisationMatLabOptimisationτ 1 1600.2τ 2 1246.8τ 3 170.7535A - % 92.70LCC 115345.22∆ LCC - % -5.5With these results as initial conditions we have appliedthe tool “SOLVER” of Excel and we got a bettersolution (Table 3).Table 3. Results of MatLab + Excel optimisationMatLab + ExcelOptimisation5. Conclusionτ 1 1606.498τ 2 1255.498τ 3 175.4996A - % 93.02LCC 113809.75∆ LCC - % -6.8This paper deals with a maintenance optimisationproblem for a series system. First we have developedτ ian algorithm to determine the optimum frequency toperform preventive maintenance in systems exhibitingWeibull hazard function and constant repair rate, inorder to ensure its availability. Based on this algorithmwe have developed another one to optimisemaintenance management of a series system based onpreventive maintenance over the different systemcomponents. We assume that all components of thesystem still exhibit Weibull hazard function andconstant repair rate and that preventive maintenancewould bring the system to the as good as newcondition. We define a cost function for maintenancetasks (preventive and corrective) for the system. Thealgorithm calculates the interval of time betweenpreventive maintenance actions for each component,minimizing the costs, and in such a way that the totaldowntime, in a certain period of time, does not exceeda predetermined value. The maintenance interval ofeach component depends on factors such as failurerate, repair and maintenance times of each componentin the system. In conclusion, the proposed analyticalmethod is a feasible technique to optimise preventivemaintenance scheduling of each component in a seriessystem.Currently we are developing a software package for theimplementation of the algorithms presented in thispaper.References[1] Andrews, J.D. & Moss, T.R. (1993). Reliability andRisk Assessment, Essex, Longman Scientific &Technical.[2] Dekker, R. (1996). Applications of maintenanceoptimisation models: a review and analysis,Reliability Engineering and System Safety, 51, 229-240.[3] Duarte, J.C., Craveiro, J.T. & Trigo, T. (2006).Optimisation of the preventive maintenance plan ofa series components system, International Journalof Pressure Vessels and Piping 83, 244-248.[4] Duffuaa, S. A. (2000). Mathematical Models inMaintenance Planning and Scheduling. In M. BenDaya et al (eds.), Maintenance, Modelling andOptimisation, Massachusetts, Kluwer AcademicPublishers.[5] Elsayed, E. A. (1996). Reliability Engineering,Massachusetts, Addison Wesley Longman, Inc.[6] Garg, A. & Deshmukh, S.G. (2006). Maintenancemanagement: literature review and directions.Journal of Quality in Maintenance Engineering. 12No. 3, 205–238.[7] Gertsbakh, I. (2000). Reliability theory: withapplications to preventive maintenance, Berlin,Springer.- 38 -

J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special Issue[8] Henley; Ernest J. & Kumamoto, Hiromitsu (1992).Probabilistic risk assessment: reliabilityengineering, design, and analysis, New York, IEEEPress.[9] McCormick, N. J. (1981). Reliability and RiskAnalysis, San Diego, Academic Press Inc.[10] Modarres, M., Kaminskiy, Mark and Krivtsov,Vasily (1999). Reliability and risk analysis, NewYork, Marcel Dekker, Inc.[11] Nakagawa, T. (2005). Maintenance Theory ofReliability, London, Springer-Verlag[12] Valdez-Flores, C. & Feldman, R. M. (1989). ASurvey of Preventive Maintenance Models forStochastically Deteriorating Single-Unit Systems,Naval Research Logistics, 36: 419-446.[13] Vatn, J., Hokstad, P. & Bodsberg, L. (1996). Anoverall model for maintenance optimisation,Reliability Engineering and System Safety, 51, 241-257.- 39 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special IssueDziula PrzemysawJurdziski MirosawKoowrocki KrzysztofSoszyska JoannaMaritime University, Gdynia, PolandOn multi-state safety analysis in shippingKeywordssafety, multi-state system, operation process, complex system, shippingAbstractA multi-state approach to defining basic notions of the system safety analysis is proposed. A system safetyfunction and a system risk function are defined. A basic safety structure of a multi-state series system ofcomponents with degrading safety states is defined. For this system the multi-state safety function is determined.The proposed approach is applied to the evaluation of a safety function, a risk function and other safetycharacteristics of a ship system composed of a number of subsystems having an essential influence on the shipsafety. Further, a semi-markov process for the considered system operation modelling is applied. The paper alsooffers a general approach to the solution of a practically important problem of linking the multi-state system safetymodel and its operation process model. Finally, the proposed general approach is applied to the preliminaryevaluation of a safety function, a risk function and other safety characteristics of a ship system with varying intime its structure and safety characteristics of the subsystems it is composed of.1. IntroductionTaking into account the importance of the safety andoperating process effectiveness of technical systems itseems reasonable to expand the two-state approach tomulti-state approach in their safety analysis [2]. Theassumption that the systems are composed of multistatecomponents with safety states degrading in timegives the possibility for more precise analysis anddiagnosis of their safety and operational processes’effectiveness. This assumption allows us to distinguisha system safety critical state to exceed which is eitherdangerous for the environment or does not assure thenecessary level of its operational process effectiveness.Then, an important system safety characteristic is thetime to the moment of exceeding the system safetycritical state and its distribution, which is called thesystem risk function. This distribution is strictly relatedto the system multi-state safety function that is a basiccharacteristic of the multi-state system. Determiningthe multi-state safety function and the risk function ofsystems on the base of their components’ safetyfunctions is then the main research problem. Modellingof complicated systems operations’ processes isdifficult mainly because of large number of operationsstates and impossibility of precise describing ofchanges between these states. One of the usefulapproaches in modelling of these complicatedprocesses is applying the semi-markov model [3].Modelling of multi-state systems’ safety and linking itwith semi-markov model of these systems’ operationprocesses is the main and practically importantresearch problem of this paper. The paper is devoted tothis research problem with reference to basic safetystructures of technical systems [9], [10] andparticularly to safety analysis of a ship series system[5] in variable operation conditions. This new approachto system safety investigation is based on the multi-- 40 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issuestate system reliability analysis considered for instancein [1], [4], [6], [7], [8], [11] and on semi-markovprocesses modelling discussed for instance in [3].2. Basic notionsIn the multi-state safety analysis to define systems withdegrading components we assume that:- n is the number of system's components,- E i , i = 1,2,...,n, are components of a system,- all components and a system under considerationhave the safety state set {0,1,...,z}, z ≥ 1,- the safety state indexes are ordered, the state 0 is theworst and the state z is the best,- T i (u), i = 1,2,...,n, are independent random variablesrepresenting the lifetimes of components E i in thesafety state subset {u,u+1,...,z}, while they were in thestate z at the moment t = 0,- T(u) is a random variable representing the lifetime ofa system in the safety state subset {u,u+1,...,z} while itwas in the state z at the moment t = 0,- the system and its components safety states degradewith time t,- E i (t) is a component E i safety state at the moment t,t ∈< 0,∞).- S(t) is a system safety state at the moment t,t ∈< 0,∞).The above assumptions mean that the safety states ofthe system with degrading components may bechanged in time only from better to worse. The way inwhich the components and the system safety stateschange is illustrated in Figure 1.transitionsfor t ∈< 0,∞),u = 0,1,...,z, i = 1,2,...,n,is theprobability that the component E i is in the state subset{ u , u + 1,..., z}at the moment t, t ∈< 0,∞),while it wasin the state z at the moment t = 0, is called the multistatesafety function of a component E i .Similarly, we can define a multi-state system safetyfunction.Definition 2. A vectors n (t , ⋅ ) = [s n (t,0), s n (t,1),..., s n (t,z)], t ∈< 0,∞),(3)wheres n (t,u) = P(S(t) ≥ u | S(0) = z) = P(T(u) > t) (4)for t ∈< 0,∞),u = 0,1,...,z, is the probability that thesystem is in the state subset { u , u + 1,..., z}at themoment t, t ∈< 0,∞),while it was in the state z at themoment t = 0, is called the multi-state safety functionof a system.Definition 3. A probabilityr(t) = P(S(t) < r | S(0) = z) = P(T(r) ≤ t), (5)t ∈< 0,∞),that the system is in the subset of states worse than thecritical state r, r ∈{1,...,z} while it was in the state z atthe moment t = 0 is called a risk function of the multistatesystem.Under this definition, considering (4) and (5), we haveworst statebest stateFigure 1. Illustration of a system and componentssafety states changingThe basis of our further considerations is a systemcomponent safety function defined as follows.Definition 1. A vectors i (t , ⋅ ) = [s i (t,0), s i (t,1),..., s i (t,z)], t ∈< 0,∞),(1)i = 1,2,...,n,where0 1 u-1 u z-1 z. . . . . . . .s i (t,u) = P(E i (t) ≥ u | E i (0) = z) = P(T i (u) > t) (2)r(t) = 1 - P(S(t) ≥ r | S(0) = z) = 1 - s n (t,r), (6)t ∈< 0,∞),and, if τ is the moment when the risk exceeds apermitted level δ, δ ∈< 0 ,1 > , thenτ = r −1 ( δ ),(7)where r −1 ( t), if it exists, is the inverse function of therisk function r(t) given by (6).3. Basic system safety structuresThe proposition of a multi-state approach to definitionof basic notions, analysis and diagnosing of systems’safety allowed us to define the system safety functionand the system risk function. It also allows us to definebasic structures of the multi-state systems ofcomponents with degrading safety states. For these- 41 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issuebasic systems it is possible to determine their safetyfunctions. Further, as an example, we will consider aseries system.Definition 4. A multi-state system is called a seriessystem if it is in the safety state subset { u , u + 1,..., z}ifand only if all its components are in this subset ofsafety states.Corollary 1. The lifetime T(u) of a multi-state seriessystem in the state subset { u , u + 1,..., z}is given byT(u) = min{ ( u)}, u = 1,2,...,z.T i1≤i≤nThe scheme of a series system is given in Figure 2.Figure 2. The scheme of a series systemIt is easy to work out the following result.Corollary 2. The safety function of the multi-stateseries system is given by⎺s n (t , ⋅ ) = [1,⎺s n (t,1),...,⎺s n (t,z)], t ∈< 0,∞),(8)wheren⎺s n (t,u) = ∏ s ( t,u), t ∈< 0,∞),u = 1,2,...,z. (9)i=1iCorollary 3. If components of the multi-state seriessystem have exponential safety functions, i.e., ifs i (t , ⋅ ) = [1, s i (t,1),..., s i (t,z)], t ∈< 0,∞),wheresi( t,u)= exp[ −λi( u)t]for t ∈< 0,∞),λi( u)> 0 ,u = 1,2,...,z, i = 1,2,...,n,then its safety function is given by⎺s n (t , ⋅ ) = [1,⎺s n (t,1),...,⎺s n (t,z)], (10)whereE 1 E 2 . . . E nn⎺s n (t,u) = exp[ −∑λ ( u)t]for t ∈< 0,∞),(11)u = 1,2,...,z.i=1i4. Basic system safety structures in variableoperation conditionsWe assume that the system during its operation processhas v different operation states. Thus we can defineZ (t), t ∈< 0 , +∞ > , as the process with discreteoperation states from the setZ = { z1 , z2, . . ., zv},In practice a convenient assumption is that Z(t) is asemi-markov process [3] with its conditional lifetimesθ at the operation state z when its next operationblstate is zl, b , l = 1,2,..., v,b ≠ l.In this case theprocess Z(t) may be described by:- the vector of probabilities of the process initialoperation states [ p b(0)]1xν,- the matrix of the probabilities of the processtransitions between the operation states [ p ] ,bbl νxνwhere p bb( t)= 0 for b = 1,2,...,v.- the matrix of the conditional distribution functions[ H ( t)]of the process lifetimes θ , b ≠ l,in theblνxνoperation state zbwhen the next operation state iszl, where Hbl( t)= P(θbl< t)for b , l = 1,2,..., v,b ≠ l,and H bb( t)= 0 for b = 1,2,...,v.Under these assumptions, the lifetimes θblmeanvalues are given byM= θ ] ∫ tdH bl(t), b , l = 1,2,..., v,b ≠ l.(12)blE[ bl= ∞ 0The unconditional distribution functions of thelifetimes θbof the process Z (t)at the operation statesz b = 1,2,...,v,are given byb ,H b(t) = ∑ pvl = 1blHbl( t),b = 1,2,...,v.The mean values E[ θb] of the unconditional lifetimesθbare given byM= θ ] = ∑ p blM blbE[ bvl=1, b = 1,2,...,v,where Mblare defined by (12).Limit values of the transient probabilities at theoperation statesp b(t) = P(Z(t) = z b) , t ∈< 0 , +∞),b = 1,2,...,v,are given bybl- 42 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special IssueπbMbpb= lim p b( t)= ,vt→∞∑πMl=1llb = 1,2,...,v,(13)where the probabilities πbof the vector [ πb ] satisfy1xνthe system of equations⎧[π b ] = [ π⎪⎨ v⎪∑ π l = 1.⎩l=1b][ pbl]We assume that the system is composed of ncomponents E , i = 1,2,...,n,the changes of theiprocess Z(t) operation states have an influence on thesystem components E safety and on the system safetyistructure as well. Thus, we denote the conditionalsafety function of the system component E while thesystem is at the operational state zb, b = 1,2,...,v,bys b( )( )( t,⋅)i = [1, s b( )( t,1),s b( )( t,2),i i ..., s b( t,z)i ],where( b)i( b)is ( t,u)= P(T ( u)> t Z(t)= zfor t ∈< 0,∞),b = 1,2,...,v,u = 1,2,...,z,and theconditional safety function of the system while thesystem is at the operational state zb, b = 1,2,...,v,by( b)n b( b)n b( b)n bs ( t,⋅)= [1, s ( t,1),s ( t,2),..., s ( t,z)],n b∈{ 1,2,..., n},b)( b)n bwhere nbare numbers of components in the operationstates z and( b)n bb( b)s ( t,u),= P ( T ( u)> t Z(t)= z )for t ∈< 0,∞),n b∈ { 1,2,..., n},b = 1,2,...,ν ,u = 1,2,...,z.The safety function( )s b( t,u)is the conditionali( )probability that the component E lifetimeiT bi( u)inthe state subset { u , u + 1,..., z}is not less than t, whilethe process Z(t) is at the operation state zb. Similarly,( b)n bthe safety function s ( t,u)is the conditional( )probability that the system lifetime T b ( u)in the statebisubset { u , u + 1,..., z}is not less than t, while theprocess Z(t) is at the operation state z b.In the case when the system operation time is largeenough, the unconditional safety function of the systemis given bys ( t,⋅)= [1, s (t,1),s (t, 2),..., s ( t,z)], t ≥ 0,nwherens ( t,u)= P ( T(u)> t)≅ ∑ pbs( t,u)nnνb=1( b)nbn(14)for t ≥ 0,n b∈ { 1,2,..., n},u = 1,2,...,z,and T (u)is theunconditional lifetime of the system in the safety statesubset { u , u + 1,..., z}.The mean values and variances of the system lifetimesin the safety state subset { u , u + 1,..., z}areν( b)m ( u)= E[T(u)]≅ ∑ p m ( u),u = 1,2,...,z,(15)where [2]m( b)b=1b( u)= ∞ ∫ s ( t,u)dt,n b∈ { 1,2,..., n},(16)0u = 1,2,...,z,and( b)( b)n b2= ∞ 0( b)[ σ ( u)]2∫ts( t,u)dt −[m ( u)], (17)u = 1,2,...,z,nbfor b =1,2,...,ν , and2= ∞ 0[ σ ( u)]2∫ts( t,u)dt −[m(u)], u = 1,2,...,z.(16)nThe mean values of the system lifetimes in theparticular safety states u , are [2]m ( u)= m(u)− m(u + 1), u = 1,2,...,z −1,m ( z)= m(z).(19)5. Ship safety Model in constant operationconditionsWe preliminarily assume that the ship is composed of anumber of main technical subsystems having anessential influence on its safety. There aredistinguished her following technical subsystems:22- 43 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special IssueS 1 - a navigational subsystem,S - a propulsion and controlling subsystem,2S - a loading and unloading subsystem,3S - a hull subsystem,4S - a protection and rescue subsystem,5S - an anchoring and mooring subsystem.6According to Definition 1, we mark the safetyfunctions of these subsystems respectively by vectorss i (t , ⋅ ) = [s i (t,0), s i (t,1),..., s i (t,z)], t ∈< 0,∞),(20)i =1,2,...,6,with co-ordinatess i (t,u) = P(S i (t) ≥ u | S i (0) = z) = P(T i (u) > t) (21)for t ∈< 0,∞),u = 0,1,...,z, i =1,2,...,6,where T i (u), i =1,2,...,6, are independent random variables representingthe lifetimes of subsystems S i in the safety state subset{u,u+1,...,z}, while they were in the state z at themoment t = 0 and S i (t) is a subsystem S i safety state atthe moment t, t ∈< 0,∞).Further, assuming that the ship is in the safety statesubset {u,u+1,...,z} if all its subsystems are in thissubset of safety states and considering Definition 4, weconclude that the ship is a series system of subsystemsS , S , S , S , S , S with a scheme presented in1 2 3 4 5 6Figure 3.S 1Figure 3. The scheme of a structure of ship subsystemsTherefore, the ship safety is defined by the vectors6( t,⋅)= [ s ( t 6,0), s ( t 6,1),..., s6( t,z)], (22)t ∈< 0,∞),with co-ordinatess ( t,) = P(S(t) ≥ u | S(0) = z) = P(T(u) > t) (23)6ufor t ∈< 0,∞),u = 0,1,...,z, where T(u) is a randomvariable representing the lifetime of the ship in thesafety state subset {u,u+1,...,z} while it was in thestate z at the moment t = 0 and S(t) is the ship safetystate at the moment t, t ∈< 0,∞),according toCorollary 2, is given by the formulas ( t,) = [1, s ( ,1),..., s ( t,) ], t ∈< 0,∞),(24)6⋅S 2 S 3 S 4 S 5 S 66 t6zwhere6s6( t,u)= ∏ si( t,u), t ∈< 0,∞),u = 1,2,...,z. (25)i=16. Ship operation processTechnical subsystems S , S , S , S , S , S are1 2 3 4 5 6forming a general ship safety structure presented inFigure 3. However, the ship safety structure and theship subsystems safety depend on her changing in timeoperation states.Considering basic sea transportation processes thefollowing operation ship states have been specified:z1- loading of cargo,z - unloading of cargo,2z3- leaving the port,z - entering the port,4z5- navigation at restricted water areas,z6- navigation at open sea waters.In this case the process Z(t) may be described by:- the vector of probabilities of the initial operationstates [ p b (0)] 1x6,- the matrix of the probabilities of its transitionsbetween the operation states [ pbl] 6 x6, wherep bb( t)= 0 for b = 1,2,...,6,- the matrix of the conditional distribution functions[ Hbl( t)] 6 x6of the lifetimes θbl, b ≠ l,whereHbl( t)= P(θbl< t)for b , l =1,2,...,6,b ≠ l,andH bb( t)= 0 for b = 1,2,...,6.Under these assumptions, the lifetimes θblmeanvalues are given byM= θ ] ∫ tdH bl(t), b , l =1,2,...,6,b ≠ l.(26)blE[ bl= ∞ 0The unconditional distribution functions of thelifetimes θbof the process Z (t)at the operation statesz b = 1,2,...,6,are given byb ,6H b(t) = ∑ p H ( t),b = 1,2,...,6.l=1blblThe mean values E[ θb] of the unconditional lifetimesθbare given byMwhere= θ ] = ∑ p , b = 1,2,...,6,(27)bE[ b6l=1bl M blMblare defined by (26).- 44 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special IssueLimit values of the transient probabilities at theoperation statesp b(t) = P(Z(t) = z b), t ∈< 0 , +∞),b = 1,2,...,6,are given byπbMbpb= lim p b( t)= ,6t→∞∑πMl= 1llb = 1,2,...,6, (28)where the probabilities πbof the vector [ πb] 1 x6satisfythe system of equations⎧[π b ] = [ π b ][ p⎪⎨ 6⎪∑ π l = 1.⎩l=1bl]7. Safety model of ship in variable operationconditions(29)We assume as earlier that the ship is composed ofn = 6 subsystems Si, i =1,2,...,6,and that thechanges of the process Z(t) of ship operation stateshave an influence on the system subsystems Sisafetyand on the ship safety structure as well. Thus, wedenote the conditional safety function of the shipsubsystem S while the ship is at the operational statez b = 1,2,...,6,by,bs bi( )( )( t,⋅)i = [1, s b( )( t,1),s b( )( t,2),i i ..., s b( t,z)i ],where( b)i( b)is ( t,u)= P(T ( u)> t Z(t)= zfor t ∈< 0,∞),b = 1,2,...,6,u = 1,2,...,z,and theconditional safety function of the ship while the ship isat the operational state zb, b = 1,2,...,6,by( b)n b( b)n b( b)n bs ( t,⋅)= [1, s ( t,1),s ( t,2),..., s ( t,z)],where( b)n b( b)s ( t,u),= P ( T ( u)> t Z(t)= z )b)b( b)n bfor t ∈< 0,∞),b = 1,2,...,6,n b∈{1,2,3,4,5,6 },u = 1,2,...,z.( )The safety function s b ( t,u)probability that the subsystemiis the conditional( )S ilifetime T bi( u)inthe state subset { u , u + 1,..., z}is not less than t, whilethe process Z(t) is at the ship operation state z b.( b)n bSimilarly, the safety function s ( t,u)is the( )conditional probability that the ship lifetime T b ( u)in the state subset { u , u + 1,..., z}is not less than t,while the process Z(t) is at the ship operation state z b.In the case when the ship operation time is largeenough, the unconditional safety function of the systemis given bys ( t,) = [1, s ( ,1),s ( , 2),..., s ( t,) ], t ≥ 0,6⋅where6 t6 t6zs ( t,) = P ( T(u)> t)≅ ∑ pbs( t,u)6u6b=1( b)nb(30)for t ≥ 0,nb∈{1,2,3,4,5,6 }, u = 1,2,...,z,and T (u)isthe unconditional lifetime of the ship in the safety statesubset { u , u + 1,..., z}.The mean values and variances of the ship lifetimes inthe safety state subset { u , u + 1,..., z}are6( b)m ( u)= E[T(u)}≅ ∑ p m ( u),u = 1,2,...,z,(31)wherem( b)b= 1b( u)= ∞ ∫ s ( t,u)dt,(32)0( b)n bfor b = 1,2,...,6,n ∈{1,2,3,4,5,6 }, u = 1,2,...,z,and2b∞[ σ ( u)]= D[T(u)]= 2∫ts( t,u)dt −[m(u)], (33)u = 1,2,...,z,06The mean values of the system lifetimes in theparticular safety states u , arem ( u)= m(u)− m(u + 1), u = 1,2,...,z −1,m ( z)= m(z).(34)8. Preliminary application of general safetymodel of ship in variable operation conditionsAccording to expert opinions [5] in the ship operationprocess, Z(t),t ≥ 0 , we distinguished seven operation2- 45 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issuestates: z1, z2, z3, z4, z5, z 6 . On the basis of datacoming from experts, the probabilities of transitionsbetween the operation states are approximately givenbyWhereas, by (27), the unconditional mean lifetimes inthe operation states areM 1= E[ θ 1] = p13M13+ p15M15+ p16M16[ ] 6 6p blx⎡0.00⎢⎢0.48⎢0.00= ⎢⎢0.49⎢0.02⎢⎢⎣0.020.000.000.000.490.020.020.960.480.000.020.000.000.000.000.020.000.480.010.020.020.960.000.000.950.02⎤0.02⎥⎥0.02⎥⎥,0.00⎥0.48⎥⎥0.00⎥⎦= 0 .96 ⋅ 2 + 0.02 ⋅1+ 0.02 ⋅1= 1.96,M 2= E[ θ 2]= p + + +21M21p23M23p25M25p26M26= 0 .48 ⋅ 2 + 0.48 ⋅ 2 + 0.02 ⋅1+ 0.02 ⋅1=1.96,and the distributions of the ship conditional lifetimes inthe operation states are exponential of the followingforms:HHHHHHHHH( t)= 1 − exp[ −0.5], H ( t)= 1 − exp[ −1.0],13t15t( t)= 1 − exp[ −1.0], H ( t)= 1 − exp[ −0.5],16t21t( t)= 1 − exp[ −0.5], H ( t)= 1 − exp[ −1.0],23t25t( t)= 1 − exp[ −1.0], H ( t)= 1 − exp[ −25.0],26t34t( t)= 1 − exp[ −25.0], H ( t)= 1 − exp[ −12.5],35t36t( t)= 1 − exp[ −0.33], H ( t)= 1 − exp[ −0.33],51t52t( t)= 1 − exp[ −0.5], H ( t)= 1 − exp[ −0.5],54tfor t ≥ 0.56t( t)= 1 − exp[ −0.2], H ( t)= 1 − exp[ −0.2],61t62t( t)= 1 − exp[ −0.25], H ( t)= 1 − exp[ −0.25]64t65tHence, by (26), the conditional mean values oflifetimes in the operation states areM 13 = 2, M , M 16 = 1,15 =1M 21 = 2, M 23 = 2,M 25 =1,M 26 = 1,M 34 = 0.04, M 35 = 0.04,M 36 = 0.08,M 41 = 0.08, M 42 = 0.08,M 43 = 0.04,M 51 = 3, M 52 = 3,M 54 = 2,M 56 = 2,M 61 = 5, M 62 = 5,M 64 = 4,M 65 = 4.M 3= E[ θ 3] = p34M34+ p35M35+ p36M36= 0 .02 ⋅ 0.04 + 0.96 ⋅ 0.04 + 0.02 ⋅ 0.08 = 0.0408,M 4= E[ θ 4] = p41M41+ p42M42+ p43M43= 0 .49 ⋅ 0.08 + 0.49 ⋅ 0.08 + 0.02 ⋅ 0.04 = 0.0792,M 5= E[ θ 5]= p + + +51M51p52M52p54M54p56M56= 0 .02 ⋅ 3 + 0.02 ⋅ 3 + 0.48 ⋅ 2 + 0.48 ⋅ 2 = 2.04,M 6= E[ θ 6]= p + + +61M61p62M62p64M64p65M65= 0 .02 ⋅ 5 + 0.02 ⋅ 5 + 0.01⋅4 + 0.95 ⋅ 4 = 4.04.Since from the system of equations⎧[π 1,π 2 , π 3,π 4 , π 5 , π 6 ]⎪⎨=[ π 1,π 2 , π 3,π 4 , π 5,π 6 ]⎪⎩π1+ π2+ π3+ π4+ π5+ πwe get[ ]p bl 6x66= 1,π 1 = 0.126, π 2 = 0.085,π 3 = 0.165,π 4 = 0.155, π 5 = 0.312,π 6 = 0.157,then the limit values of the transient probabilitiesp b(t) at the operational states zb, according to (28),are given byp 1 = 0.145, p 2 = 0.098,p 3 = 0.004,p 4 = 0.007,- 46 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issuep 5 = 0.374, p 6 = 0.372.(35)We assume that the ship subsystems S ,ii = 1,2,...,6,are its five-state components, i.e. z = 4, with the multistatesafety functions( )s b( )( t,⋅)= [1, s b( )( t,1),s b( )( t,2),..., s b( t,z)],i i i ib =1,2,...,6, i = 1,2,...,6,with exponential co-ordinates different in various shipoperation states z , b =1,2,...,6.bAt the operation states z1and z 2 , i.e. at the cargoloading and un-loading state the ship is built ofn1 = n2= 4 subsystems S3,S4, S5and S6forminga series structure shown in Figure 4.Figure 4. The scheme of the ship structure at theoperation states z1and z 2We assume that the ship subsystems S ,ii = 3,4,5,6,are its five-state components, i.e. z = 4, having themulti-state safety functions( )s b( )i( t,⋅)= [1, s b ( )i( t,1),s b( )i( t,2),s b ( t,3),i = 3,4,5,6, b = 1,2,( )is b ( t,4)i],with exponential co-ordinates, for b = 1,2,respectively given by:- for the loading subsystem S 3( )s b ( )( ,1) = exp[−0.06t], s b ( ,2)= exp[−0.07t],3t3t( )s b ( )( ,3) = exp[−0.08t], s b ( ,4)= exp[−0.09t],3t- for the hull subsystem S 43t( )s b ( )( ,1) = exp[−0.03t], s b ( ,2)= exp[−0.04t],4t4t( )s b ( )4( t,3)= exp[−0.06t], s b4( t,4)= exp[−0.07t],- for the protection and rescue subsystem S 5( )s b ( )( ,1) = exp[−0.10t], s b ( ,2)= exp[−0.12t],5t5t( )s b ( )( ,3) = exp[−0.15t], s b ( ,4)= exp[−0.16t],5tS 3 S 4 S 5 S 65t- for the anchor and mooring subsystem S 6( )s b ( )( ,1) = exp[−0.06t], s b ( ,2)= exp[−0.08t],6t6t( )s b ( )( ,3) = exp[−0.10t], s b ( ,4)= exp[−0.12t].6t6tAssuming that the ship is in the safety state subsets{ u , u + 1,..., z}, u = 1,2,3,4 , if all its subsystems are inthis safety state subset, according to Definition 1 andDefinition 4, the considered system is a five-stateseries system. Thus, by Corollary 3, after applying(10)−(11), we have its conditional safety functions inthe operation states z1and z2respectively forb = 1,2, given by( b)s4 ( t,⋅)( b)( b)= [1, s4( t,1),s4( t,2),( b)( b)s4( t,3),s4( t,4)],t ≥ 0, b = 1,2,wheressss( b)4t( b)4t( ,1) = exp[−(0.06 + 0.03 + 0.10 + 0.06)t]= exp[ −0.25t],( ,2) = exp[-(0.07 + 0.04 + 0.12 +0.08)t]( b)4t( b)4t= exp[ −0.31t],( ,3) = exp[−(0.08 + 0.06 + 0.15 + 0.10)t]= exp[ −0.39t],( ,4) = exp[−(0.09 + 0.07 + 0.16 + 0.12)t]= exp[ −0.44t]for t ≥ 0, b = 1, 2 .The expected values and standard deviations of theship conditional lifetimes in the safety state subsetscalculated from the above result, according to (16)-(17), for b = 1,2,are:(b)(b)m (1) ≅ 4.00, (2)(b)m (4)≅ 2.27 years,(b)(b)σ (1) ≅ 4.00, (2)(b)m ≅ 3.26, m (3)≅ 2.56,(b)σ ≅ 3.26, σ (3)≅ 2.56,- 47 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issue(b)σ (4)≅ 2.27 years,- for the protection and rescue subsystem S 5and further, from (10), the ship conditional lifetimes inthe particular safety states, for b = 1,2,are:(b)m (1) ≅ 0.74, (b)(2)(b)m (4)≅ 2.27 years.(b)m ≅ 0.70, m (3)≅ 0.29,At the operation states z 3 and z 4 , i.e. at the leavingand entering state the ship is built of n = n 53 4=subsystems S1,S2, S4, S5and S6forming a seriesstructure shown in Figure 5.Figure 5. The scheme of the ship structure at theoperation states z 3 and z 4We assume that the ship subsystems S ,ii = 1,2,4,5,6,are its five-state components, i.e. z = 4, having themulti-state safety functionss bS 1 S 2 S 4 S 5 S 6( )( )i( t,⋅)= [1, s b ( )i( t,1), s b( )i( t,2), s b( )i( t,3), s bi( t,4)],i = 1,2,4,5,6, b = 3,4,with exponential co-ordinates, for b = 3,4,respectivelygiven by:- for the navigational subsystem S 1( )s b ( ,1) = exp[−0.15t], ( ,2)= exp[−0.20t],1 t( )1t( )s b1t( )s b1ts b ( ,3) = exp[−0.22t], ( ,4)= exp[−0.25t],- for the propulsion and controlling subsystem S 2( )2t( )s b2ts b ( ,1) = exp[−0.05t], ( ,2)= exp[−0.06t],( )2t( )s b2ts b ( ,3) = exp[−0.07t], ( ,4)= exp[−0.08t],- for the hull subsystem S 4( )4t( )s b4ts b ( ,1) = exp[−0.04t], ( ,2)= exp[−0.05t],( )4t( )s b4ts b ( ,3) = exp[−0.07t], ( ,4)= exp[−0.08t],( )s b ( )( ,1) = exp[−0.12t], s b ( ,2)= exp[−0.14t],5t5t( )s b ( )( ,3) = exp[−0.16t], s b ( ,4)= exp[−0.18t],5t5t- for the anchor and mooring subsystem S 6( )s b ( )( ,1) = exp[−0.02t], s b ( ,2)= exp[−0.04t],6t6t( )s b ( )( ,3) = exp[−0.06t], s b ( ,4)= exp[−0.08t].6t6tAssuming that the ship is in the safety state subsets{ u , u + 1,..., z}, u = 1,2,3,4 , if all its subsystems are inthis safety state subset, according to Definition 1 andDefinition 4, the considered system is a five-stateseries system. Thus, by Corollary 3, after applying(10)−(11), we have its conditional safety functions inthe operation states z3and z4respectively forb = 3,4, given by( b)s5 ( t,⋅)= [1,( b)( b)( b)( b)s5( t,1),s5( t,2),s5( t,3),s5( t,4)],t ≥ 0, b = 3,4,wheressss( b)5t( b)5t( b)5t( b)5t( ,1) = exp[−(0.15 + 0.05 + 0.04 + 0.12 + 0.02)t]= exp[ −0.38t],( , 2) = exp[-(0.20 + 0.06 + 0.05 +0.14+ 0.04)t]= exp[ −0.49t],( ,3) = exp[−(0.22 + 0.07 + 0.07 + 0.16 + 0.06)t]= exp[ −0.58t],( , 4) = exp[−(0.25 + 0.08 + 0.08 + 0.18 + 0.08)t]= exp[ −0.67t]for t ≥ 0, b = 3, 4 .The expected values and standard deviations of theship conditional lifetimes in the safety state subsetscalculated from the above result, according to (16)-(17), for b = 3,4,are:- 48 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issue(b)(b)m (1) ≅ 2.63, (2)(b)m (4)≅ 1.49 years,(b)(b)σ (1) ≅ 2.63, (2)(b)σ (4)(b)m ≅ 2.04, m (3)≅ 1.72,≅ 1.49 years,(b)σ ≅ 2.04, σ (3)≅ 1.72,and further, from (10), the ship conditional lifetimes inthe particular safety states, for b = 1,2,are:(b)m (1) ≅ 0.59, (b)(2)(b)m ≅ 0.32, m (3)≅ 0.23,(5)4t(5)4ts ( ,3) = exp[−0.09t], s ( ,4)= exp[−0.10t],- for the protection and rescue subsystem S 5(5)5t(5)5ts ( ,1) = exp[−0.14t], s ( ,2)= exp[−0.15t],(5)5t(5)5ts ( ,3) = exp[−0.17t], s ( ,4)= exp[−0.20t],- for the anchor and mooring subsystem S 6(5)6t(5)6ts ( ,1) = exp[−0.02t], s ( ,2)= exp[−0.03t],(b)m (4)≅ 1.49 years.(5)6t(5)6ts ( ,3) = exp[−0.04t], s ( ,4)= exp[−0.05t].At the operation state z5, i.e. at the navigation atrestricted areas state the ship is built of n 5 = 5subsystems S1,S2, S4, S5and S6forming a seriesstructure shown in Figure 6.Figure 6. The scheme of the ship structure at theoperation state z5We assume that the ship subsystems S ,ii = 1,2,4,5,6,are its five-state components, i.e. z = 4, having themulti-state safety functions(5)(5) (5) (5) (5)s i( t,⋅)= [1, s i( t,1), s i( t,2), s i( t,3), s i( t,4)],i = 1,2,4,5,6,with exponential co-ordinates respectively given by:- for the navigational subsystem S 1(5)s ( ,1) = exp[−0.18t], s ( ,2)= exp[−0.22t],1 t(5)1t(5)1t(5)1ts ( ,3) = exp[−0.24t], s ( ,4)= exp[−0.26t],- for the propulsion and controlling subsystem S 2(5)(5)s ( ,1) = exp[−0.06t], s ( ,2)= exp[−0.07t],2t(5)2tS 1 S 2 S 4 S 5 S 62t(5)2ts ( ,3) = exp[−0.08t], s ( ,4)= exp[−0.09t],- for the hull subsystem S 4Assuming that the ship is in the safety state subsets{ u , u + 1,..., z}, u = 1,2,3,4 , if all its subsystems are inthis safety state subset, according to Definition 1 andDefinition 4, the considered system is a five-stateseries system. Thus, by Corollary 3, after applying(10)−(11), we have its safety function given by(5)s5 ( t,⋅)= [1,(5) (5)(5) (5)s ( ,1),s ( ,2),s ( ,3),s ( , 4)], t ≥ 0,wheressss(5)5t(5)5t(5)5t(5)5t5t5t5t5t( ,1) = exp[−(0.18 + 0.06 + 0.06 + 0.14 + 0.02)t]= exp[ −0.46t],( , 2) = exp[-(0.22 + 0.07 + 0.08 +0.15+ 0.03)t]= exp[ −0.55t],( ,3) = exp[−(0.24 + 0.08 + 0.09 + 0.17 + 0.04)t]= exp[ −0.62t],( , 4) = exp[−(0.26 + 0.09 + 0.10 + 0.20 + 0.05)t]= exp[ −0.70t]for t ≥ 0.The expected values and standard deviations of theship lifetimes in the safety state subsets calculatedfrom the above result, according to (16)-(17), are:(6)(6)m (1) ≅ 2.17, (2)(6)m ≅ 1.82, m (3)≅ 1.61,(5)4t(5)4ts ( ,1) = exp[−0.06t], s ( ,2)= exp[−0.08t],- 49 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issue(6)m (4)≅ 1.43 years,- for the protection and rescue subsystem S 5(6)(6)σ (1) ≅ 2.17, (2)(6)σ ≅ 1.82, σ (3)≅ 1.61,(6)5t(6)5ts ( ,1) = exp[−0.15t], s ( ,2)= exp[−0.16t],(6)σ (4)≅ 1.43 years,(6)5t(6)5ts ( ,3) = exp[−0.18t], s ( ,4)= exp[−0.22t].and further, from (10), the ship lifetimes in theparticular safety states are:(6)m (1) ≅ 0.35, (6)(2)(6)m (4)≅ 1.43 years.(6)m ≅ 0.21, m (3)≅ 0.18,At the operation state z6, i.e. at the navigation at opensea state the ship is built of n 6 = 4 subsystems S1, S 2,S4 , and S5forming a series structure shown in Figure7.S 1 S 2 S 4 S 5Figure 7. The scheme of the ship structure at theoperation state z6We assume that the ship subsystems S ,ii = 1,2,4,5,are its five-state components, i.e. z = 4, having themulti-state safety functions(6)(6) (6) (6) (6)s i( t,⋅)= [1, s i( t,1), s i( t,2), s i( t,3), s i( t,4)],i = 1,2,4,5,with exponential co-ordinates respectively given by:- for the navigational subsystem S 1(6)s ( ,1) = exp[−0.18t], s ( ,2)= exp[−0.22t],1 t(6)1t(6)1t(6)1ts ( ,3) = exp[−0.24t], s ( ,4)= exp[−0.26t],- for the propulsion and controlling subsystem S 2(6)2t(6)2( t(6)2ts ( ,1) = exp[−0.06t], s ( ,2)= exp[−0.07t],(6)2ts ,3) = exp[−0.08t], s ( ,4)= exp[−0.09t],- for the hull subsystem S 4(6)4t(6)4ts ( ,1) = exp[−0.05t], s ( ,2)= exp[−0.06t],(6)4t(6)4ts ( ,3) = exp[−0.07t], s ( ,4)= exp[−0.08t],Assuming that the ship is in the safety state subsets{ u , u + 1,..., z}, u = 1,2,3,4 , if all its subsystems are inthis safety state subset, according to Definition 1 andDefinition 4, the considered system is a five-stateseries system. Thus, by Corollary 3, after applying(10)−(11), we have its safety function given by(7)s4 ( t,⋅)= [1,(7) (7)(7)(7)s ( ,1),s ( , 2),s ( ,3),s ( , 4)], t ≥ 0,wheressss(6)4t(6)4t(6)4t(6)4t4t4t4t4t( ,1) = exp[−(0.18 + 0.06 + 0.05 + 0.15)t]= exp[ −0.44t],( , 2) = exp[-(0.22 + 0.07 + 0.06 +0.16)t]= exp[ −0.51t],( ,3) = exp[−(0.24 + 0.08 + 0.07 + 0.18)t]= exp[ −0.57t],( , 4) = exp[−(0.26 + 0.09 + 0.08 + 0.22)t]= exp[ −0.67t]for t ≥ 0.The expected values and standard deviations of theship lifetimes in the safety state subsets calculatedfrom the above result, according to (16)-(17), are:(6)(6)m (1) ≅ 2.27, (2)(6)m (4)≅ 1.49 years,(6)(6)σ (1) ≅ 2.27, (2)(6)σ (4)(6)m ≅ 1.96, m (3)≅ 1.75,≅ 1.49 years,(6)σ ≅ 1.96, σ (3)≅ 1.75,and further, from (18), the ship lifetimes in theparticular safety states are:- 50 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issue(6)m (1) ≅ 0.31, (6)(2)(6)m ≅ 0.21, m (3)≅ 0.26,+ 0.004⋅ exp[ −0.67t]+ 0.007⋅ exp[ −0.67t](6)m (4)≅ 1.49 years.+ 0.374⋅ exp[ −0.70t]+ 0.372⋅ exp[ −0.67t]for t ≥ 0.In the case when the system operation time is largeenough, the unconditional safety function of the ship isgiven by the vectors6 ( t,⋅)= [1, s ( ,1),s ( , 2),s ( ,3),s ( ,4)], t ≥ 0,6 t6 t6 t6 twhere, according to (14), the co-ordinates ares ( ,1) = p s ( ,1)+ p s ( ,1)+ p s ( ,1)6 t(4)4 5t(1)1 4t(5)5 5t(2)2 4t(6)6 4t+ p s ( ,1) + p s ( ,1)+ p s ( ,1)(3)3 5tThe mean values and variances of the systemunconditional lifetimes in the safety state subsets,according to (31) and (33), respectively are(1)m (1) = p 1 m (1)+ p 2 m (1)+ p 3 m (1)(2)(4)+ p 4 m (1) + p 5 m (1)+ p 6 m (1),(5)≅ 0.145⋅ 4.00 + 0.098⋅ 4. 00 + 0.004⋅ 2. 63+ 0.007⋅ 2.63 + 0.374⋅ 2. 17 + 0.372⋅ 2. 27 = 2.66.2[σ (1)]≅ 2[0.145⋅[4.00]2(6)(3)+ 0.098⋅[4.00]2= 0.145⋅ exp[ −0.25t]+ 0.098⋅ exp[ −0.25t]+ 0.004⋅[2.63]2+ 0.007⋅[2.63]2+ 0.374⋅[2.17]2+ 0.004⋅ exp[ −0.38t]+ 0.007⋅ exp[ −0.38t]+ 0.374⋅ exp[ −0.46t]+ 0.374⋅ exp[ −0.44t],s ( , 2) = p s ( ,2)+ p s ( , 2)+ p s ( , 2)6 t(4)4 5t(1)1 4t(5)5 5t(2)2 4t(6)6 4t+ p s ( , 2) + p s ( , 2)+ p s ( , 2)(3)3 5t= 0.145⋅ exp[ −0.31 t]+ 0.098⋅ exp[ −0.31t]+ 0.004⋅ exp[ −0.49t]+ 0.007⋅ exp[ −0.49t]+ 0.3.74⋅ exp[ −0.55t]+ 0.372⋅ exp[ −0.51t],s ( ,3) = p s ( ,3)+ p s ( ,3)+ p s ( ,3)6 t(4)4 5t(1)1 4t(5)5 5t(2)2 4t(6)6 4t(3)3 5t+ p s ( ,3) + p s ( ,3)+ p s ( ,3)= 0.145⋅ exp[ −0.39t]+ 0.098⋅ exp[ −0.39t]+ 0.004⋅ exp[ −0.58t]+ 0.007⋅ exp[ −0.58t]+ 0.0374⋅ exp[ −0.62t]+ 0.372⋅ exp[ −0.57t],s ( ,4) = p s ( , 4)+ p s ( , 4)+ p s ( , 4)6 t(1)1 4t(2)2 4t(3)3 5t22+ 0.372 ⋅[2.27]] − [2.66]2 = [2.87] , σ ( 1) ≅ 2.87,(1)m (2) = p 1 m ( 2)+ p 2 m ( 2)+ p 3 m ( 2)(2)(4)+ p 4 m ( 2) + p 5 m ( 2)+ p 6 m ( 2),(5)≅ 0.145⋅ 3.26 + 0.098⋅ 3. 26 + 0.004⋅ 2. 04+ 0.007⋅ 2.04 + 0.374⋅1.82 + 0.372⋅1.96 = 2.22,2[σ (2)]+ 0.004⋅[2.04]≅ 2[0.145⋅[3.26]2+ 0.007⋅[2.04]2(6)(3)+ 0.098⋅[3.26]22+ 0.374⋅[1.82]22+ 0.372 ⋅[1.96]] − [2.22]2 = [2.38] , σ ( 2) ≅ 2.38,(1)m (3) = p 1 m (3)+ p 2 m (3)+ p 3 m (3)(4)+ p 4 m (3) + p 5 m (3)+ p 6 m (3),(5)≅ 0.145⋅ 2.56 + 0.098⋅ 2. 56 + 0.004⋅1.72+ 0.007⋅1.72+ 0.374⋅1.61 + 0.372⋅1.75 =1.89,(2)(6)(3)2(4)4 5t(5)5 5t(6)6 4t+ p s ( , 4) + p s ( ,4)+ p s ( , 4)2[σ (3)]≅ 2[0.145⋅[2.56]2+ 0.098⋅[2.56]2= 0.145⋅ exp[ −0.44t]+ 0.098⋅ exp[ −0.44t]+ 0.004⋅[1.72]2+ 0.007⋅[1.72]2+ 0.374⋅[1.61]2- 51 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issue22+ 0.372 ⋅[1.75]] − [1.89]2 = [1.97] , σ ( 3) ≅1.97,(1)m (3) = p 1 m ( 4)+ p 2 m ( 4)+ p 3 m ( 4)(2)(4)+ p 4 m ( 4) + p 5 m ( 4)+ p 6 m ( 4),(5)≅ 0.145⋅ 2.27 + 0.098⋅ 2. 27 + 0.004⋅1.49+ 0.007⋅1.49+ 0.374⋅1.43 + 0.372⋅1.49 =1.66,2[σ (4)]+ 0.004⋅[1.49]≅ 2[0.145⋅[2.27]22+ 0.007⋅[1.49](6)(3)+ 0.098⋅[2.27]22+ 0.374⋅[1.43]22+ 0.372 ⋅[1.49]] − [1.66]2 = [1.73] , σ ( 4) ≅ 1.3.The mean values of the system lifetimes in theparticular safety states, by (34), arem( 1) = m(1)− m(2)= 0.44,m( 2) = m(2)− m(3)= 0.33,m( 3) = m(3)− m(4)= 0.23,m( 4) = m(4)= 1.66.If the critical safety state is r = 2, then the system riskfunction, according to (6), is given byR(t) = 1 − s 6( t , 2)= 0.145⋅ exp[ −0.31t] + 0.098⋅ exp[ −0.31t]+ 0.004⋅ exp[ −0.49t]+ 0.007⋅ exp[ −0.49t]+ 0.3.74⋅ exp[ −0.55t]+ 0.372⋅ exp[ −0.51t] for t ≥ 0.Hence, the moment when the system risk functionexceeds a permitted level, for instance δ = 0.05, from(7), isτ = r −1 (δ) ≅ 0.11 years.29. ConclusionIn the paper the multi-state approach to the safetyanalysis and evaluation of systems related to theirvariable operation processes has been considered.Theoretical definitions and preliminary results havebeen illustrated by the example of their application inthe safety evaluation of a ship transportation systemwith changing in time its operation states. The shipsafety structure and its safety subsystemscharacteristics are changing in different states whatmakes the analysis more complicated but also moreprecise than the analysis performed in [2]. However,the varying in time ship safety structure used in theapplication is very general and simplified and thesubsystems safety data are either not precise or not realand therefore the results may only be considered as anillustration of the proposed methods possibilities ofapplications in ship safety analysis. Anyway, theobtained evaluation may be a very useful example insimple and quick ship system safety characteristicsevaluation, especially during the design and whenplanning and improving her operation processes safetyand effectiveness.The results presented in the paper suggest that it seemsreasonable to continue the investigations focusing onthe methods of safety analysis for other more complexmulti-state systems and the methods of safetyevaluation related to the multi-state systems in variableoperation processes [9], [10] and their applications tothe ship transportation systems [5].References[1] Aven, T. (1985). Reliability evaluation of multistatesystems with multi-state components. IEEETransactions on Reliability 34, 473-479.[2] Dziula, P., Jurdzinski, M., Kolowrocki, K. &Soszynska, J. (2007). On multi-state approach toship systems safety analysis. Proc. 12 thInternational Congress of the InternationalMaritime Association of the Mediterranean,IMAM 2007. A. A. Balkema Publishers: Leiden -London - New York - Philadelphia - Singapore.[3] Grabski, F. (2002). Semi-Markov Models of SystemsReliability and Operations. Warsaw: SystemsResearch Institute, Polish Academy of Science.[4] Hudson, J. & Kapur, K. (1985). Reliability boundsfor multi-state systems with multi-statecomponents. Operations Research 33, 735- 744.[5] Jurdzinski, M., Kolowrocki, K. & Dziula, P. (2006).Modelling maritime transportation systems andprocesses. Report 335/DS/2006. Gdynia MaritimeUniversity.- 52 -

J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special Issue[6] Kolowrocki, K. (2004). Reliability of large Systems.Elsevier: Amsterdam - Boston - Heidelberg - London- New York - Oxford - Paris - San Diego - SanFrancisco - Singapore - Sydney - Tokyo.[7] Lisnianski, A. & Levitin, G. (2003). Multi-stateSystem Reliability. Assessment, Optimisation andApplications. World Scientific Publishing Co., NewJersey, London, Singapore , Hong Kong.[8] Meng, F. (1993). Component- relevancy andcharacterisation in multi-state systems. IEEETransactions on reliability 42, 478-483.[9] Soszynska, J. (2005). Reliability of large seriesparallelsystem in variable operation conditions. Proc.European Safety and Reliability Conference, ESREL2005, 27-30, Tri City, Poland. Advances in Safety andReliability, Edited by K. Kolowrocki, Volume 2,1869-1876, A. A. Balkema Publishers: Leiden -London - New York - Philadelphia - Singapore.[10] Soszynska, J. (2006). Reliability evaluation of a portoil transportation system in variable operationconditions. International Journal of Pressure Vesselsand Piping, Vol. 83, Issue 4, 304-310.[11] Xue, J. & Yang, K. (1995). Dynamic reliabilityanalysis of coherent multi-state systems. IEEETransactions on Reliability 4, 44, 683-688.- 53 -

M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special IssueElleuch MounirBen Bacha HabibMasmoudi FaouziNational school of engineering, Sfax, TunisiaImprovement of manufacturing cells with unreliable machinesKeywordsmanufacturing cell, intercellular transfer, markov chains, availability, simulation, performancesAbstractThe performance of cellular manufacturing (CM) is conditioned by disruptive events, such as failure of machines,which randomly occur and penalize the performance of the cells and disturb seriously the smooth working of thefactory. To overcome the problems caused by the breakdowns, we develop a solution, based on the principle ofvirtual cell (VC) and the notion of intercellular transfer that can improve performances of the system. In thiscontext, we use an analytical method based on Markov chains to model the availability of the cell. The foundresults are validated using simulation. The proposed solution in this paper confirmed that it is possible to reducethe severity of breakdowns in the CM system and improve the performances of the cells through an intercellulartransfer. Simulation allowed a validation of the analytical model and showed the contribution of the suggestedsolution.1. IntroductionGroup technology (GT) is a manufacturing philosophythat has attracted a lot of attention because of itspositive impacts in the batch-type production. Cellularmanufacturing (CM) is an application of GT tomanufacturing. It has emerged in the last two decadesas an innovative manufacturing strategy that collectsthe advantages of both product and process orientedsystem for a medium-volume and medium-varietyproduction. By applying the GT concept and CMsystem, manufacturing companies can achieve manybenefits including reduced set-up times, reduced workin-process,less material handling cost, higherthroughput rates.The performance of a cellular manufacturing system isconditioned by disruptive events (e.g., failures ofmachines) that randomly occur and penalize theperformance of the system. Therefore, equipment thatfalls in breakdown generates eventually theinterruption of the whole cell. Consequently, failure ofthe machine implies total loss of cell capabilities and itleads to the partial deterioration of the performance ofthe total system [1]. Therefore, the application of anefficient strategy against these perturbations permits toimprove the performance of those production systems.Few researches can be found related to the effect of thefailure on the operation of CM system. Some of thesediscussed the efficient maintenance politics to improvethe performance of the cellular manufacturing [1], [4]and [5]. Others developed new coefficients ofsimilarity that consider a number of alternative waysduring the machine breakdown [3].This paper is concerned with problems of theavailability of production cells facing random eventdue to an internal disruption of breakdown-machinetype. It uses intercellular transfer as a policy tosurmount this type of disruption. The proposedsolution is based on the external routing flexibility: theability to release parts to alternative cell. This policy isassessed through modelling of the production cell andits simulation with Arena software.The remainder of this paper is organized as follows. Insection 2 and 3, we formulate a comprehensive idea ofintercellular transfer policy and we give the method formodelling the availability of the cell. Section 4presents a comprehensive simulation model to validatethe analytical model and evaluate the policy ofintercellular transfer. Finally, we recapitulate in section5 the main conclusion of this work and we makerecommendations for future research.- 54 -

M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special Issue2. Description of the manufacturing cellThe shop consists of several machines that are groupedinto different group technology cells operating in astatic environment. Each cell is characterized by aclassically structured flow line with (m) machines inseries. These machines are unreliable, with operationdependent failures, and have a constant failure andrepair rates λ i and µ i (i = 1… m).In this section, we introduce the parameters thatcharacterize the behaviour of machines in a materialflow model. Each machine M i is characterized by threeparameters:- Average utilization rate Tu i : This is the rate at whichmaterial flows gets processed through the machine M iin the absence of failure.- Average failure rate λ i : This is the rate at whichmachine M i fails when working at its maximumprocessing rate (100%).- Average repair rate µ i : This is the rate at whichmachine Mi gets repaired if it is down for a failure.Using theses parameters Tu i , λ i and µ i , we can definethe basic parameters of an isolated machine M i :- There is the average failure rate λ i.Fc : This is the rateat which machine Mi fails when working at itsutilization rate Tu i .λwhere= λ ⋅i. Fc i F c(1)F c = Tui.(2)- Isolated efficiency A i (): that is the averageproportion of time during which machine Mi would beoperational. It is equal to the steady-state availability.A i () is defined in terms of parameters as follows:µiAi( ∞ ) = .(3)λ i.Fc + µiIf the single machine, whose parameters are defined asabove, were part of a production cell, additionalparameters would be needed to characterize it. Theseparameters are given below.- Average utilization rate Tu i * : in a production cellenvironment, a cell allows the manufacturing of familyproducts that made of several types, which formsvarious batches, having different sequences ofoperations. We regard Tu i.k as the utilisation rate of themachine M i during the manufacture of the batch k.Therefore to determine the machine utilisation rate, itis enough to calculate its utilisation rate during themanufacturing of family products (various batchesallocated to cell) without taking account of the effectof the breakdowns of the other machines. This rate isgiven by the following expression:Tuwith:i=n∑k = 1Tuik . ik .tti⋅t(4)- t i.k : the time put by the machine (i) tomanufacture the batch k,- tt i : the total time to manufacture familyproducts on the machine (i),- n: number of batches treated in manufacturingcell.Taken account of the condition which says that thebreakdown of a machine generates the interruption ofthe cell, i.e. that we do not have a breakdown at thesame time inside a cell; the utilization rate will beapproximated by the following expression:*iTu = Tu ⋅ E, (5)E =1i- where E is the efficiency of the cell given bythe following expression [2]m+∑i=11λµi.Fci.- Failure rate λ i Fc* : This is the rate at which machineM i fails when it work in a cell.λwherei . Fc *λ i F c *(6)= ⋅ (7)1F c*= Tui ⋅mµ1+ ∑λjj=1 j.Fcj≠i, (8)with: m represents the number of machine in the cell.Then the stationary availability will be given by theexpression (9).- 55 -

M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special IssueAcell( )∞ =11λm+∑i=1*i.Fcµi.3. Problem definition and solution technique(9)The failure of only one machine in the cellularmanufacturing system can disrupt the product flow inthe whole system. Indeed, this failure is going togenerate the interruption of different machine in thecorresponding cell. It implies a reduction of themachine utilization rate, a reduction of the productioncapacity and a dissatisfaction of customers. In thisstudy we are interested in a solution based on theexternal routing flexibility. This solution has thetendency to apply a strategy that permits to reduce theseverity of the failure by the application of anintercellular transfer policy in case of breakdown of amachine of the cell. For a production cell treating atype of product, the breakdown of a machine doesn'timply the interruption of the production in this cell.Sometimes the continuity of the production will beassured by the transfer of the product flow toward aneighbouring cell admitting an inactive machinecapable to treat this product type. By this action, it willbe possible to continue the process of manufacturing inpresence of the fault. The cell will be formed bymachines of the first cell and the standby machine ofthe second cell. The creation of this intercellulartransfer is the origin of the formation of the virtualcells. Then virtual cells are created periodically, forinstance at machine breakdown, depending on thepresence of the failure and the standby machine. It isnecessary to note that the realization of intercellulartransfer can bring advantages at the level ofperformance of the system taking into account thetransfer duration, the inactivity delay of the standbymachine and the repair duration.Therefore, for the studied system the strategy consistsin applying the intercellular transfer of the cell (a)toward the cell (b) in case of failure of one of machinesof the first cell (see Figure 1).The production cell can be assimilated to a repairablesystem operating according to a set structure composedof (M) independent modules. The number of modulesis equal to the number of machines constituting thestudied cell (a). Then the block diagram of cellreliability (a) is given by Figure 2.We study the system in steady state; that is where theprobability of the system being in a given state doesnot depend on the initial conditions. In the case of anapplication of a transfer policy, the availability of thecell is determined from different module availability.The availability of module (2) will be determined withthe process of Markov chains.Figure 1. Manufacturing system with intercellulartransferFigure 2. Function block diagram of cell reliability (a)For this raison, we are considering the modulerepresented by both machines: the main machine andthe standby one. The main machine M 1 belonging tothe studied cell, exhibits breakdown and repair rates λ 1and µ 1 respectively. The standby machine M 2 ischaracterized respectively by breakdown and repairrates λ 2 and µ 2 . In case of a breakdown of the mainmachine, the probability of transfer toward thereplacement machine M 2 is equal to Pt 2 with a transferrate equal to δ 12 .The Markov process describing the evolution of thestochastic behaviour of the module is given by the statediagram depicted in Figure 3.Given a set of differential equations developed fromthe diagram, we can determine the module availabilityrepresented in the following operational states 1, 2 1 ,and 2 2 . Therefore, the stationary availability is valuedby the expression (10).AModuleM 1aλ 1a, µ1a( ∞)M 2aM 3aM 2b M 1bCell aPrincipal cellVirtual cellM1aModule 1Pt 2b2( µ ⋅ µ ⋅( µ + λ1+ µ + λ 2) + λ1⋅Pt 2 ⋅ ( µ ⋅λ2+ µ ⋅ µ + λ1⋅µ + µ)δ 12 ⋅1 2 1=⎛λ1⋅λ2 ⋅δ12 ⋅ Pt 2⎜⎝+µ1⋅µ2⋅λ1⋅Pt2λ 2a,µ 2aM2aλ 2b,µ 2bM2bδ2a2bModule 2M 3b1Cell b( µ + ) + ⋅ ⋅( ⋅( + ) + ⋅( + ))1 λ1δ 12 µ2 λ1µ2 λ1λ 2 µ1 λ1⎞⋅( λ + µ + µ + λ ) + µ ⋅ µ ⋅δ⋅( µ + ⋅ λ + µ ) ⎟⎟ 2 2 2 1 1 1 2 12 12 1 2 ⎠(10)12M 4bIntercellular transferλ3a,µ 3aM3aModule 322- 56 -

M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special Issueδ12λ1. Pt212Pt2µ 2µ 122λ121µ 2λ231M 1aM 2aM 3aM 2b M 1bM 3b M 4bλ1.(1-Pt2)Figure 3. Diagram of module stateBesides, the expression (10) shows that intercellulartransfer in case of failure permits to improve thestationary availability of the module, only in the casewhere the transfer rate is superior to a limit value(δ 12 *). This value is given by the expression (11).δ*122µ1⋅( µ1+ λ1+µ2+ λ 2)= . (11)µ212+ 2⋅µ ⋅λ+ λ + µ ⋅ µ + µ ⋅λ1111With the help of a good team of maintenance arrangingsome necessary logistical means, the time ofpreparation of an intercellular transfer can be reduced.In these conditions, times of transfer preparation canbe disregarded in front of times between failings andrepair machine times. Therefore, the expression of theavailability of the module will be given by theexpression (12).AModule( ∞)=22( µ ⋅ µ ⋅( µ + λ 1 + µ + λ 2) + λ 1⋅Pt 2 ⋅( µ ⋅λ2 + µ ⋅ µ + λ 1⋅µ + µ)12µ1321⎛λ1⋅λ2 ⋅ Pt⎜⎝+µ ⋅ µ ⋅1 222112 ( µ1+ λ 1) + µ2⋅( λ 1⋅( µ2+ λ 1) + λ 2 ⋅( µ1+ λ 1))( ) ⎟ ⎞µ + 2⋅λ1 +1µ2⎠(12)4. Simulation of manufacturing cellsTo conduct our simulation, we defined first theproblem and stated our objectives. The problem facingcellular manufacturing is the effect of breakdownmachines. The objectives of this simulation were toexamine the performance of the system with andwithout the policy of intercellular transfer in the eventof breakdowns and to validate the analytical model.We consider the system shown in the Figure 4. Thecell (a) is constituted of three different machinesdedicated to the manufacturing of two product types.The cell (b) is formed of four different machinescapable to manufacture three product types.µ1Operational stateTransitory stateNot-operational state1222Figure 4. Manufacturing system with intercellulartransferTable 1 summarizes the routing and the processingtime information of each partTable 1. The routing and processing times of each partCellaCellbCell aProducttypeBatchsizeCell bMachine (Processing times)1 19 M 1 (19) M 2 (12) M 3 (13)2 30 M 2 (10) M 3 (9)3 16 M 1 (14) M 2 (11) M 4 (13)4 25 M 1 (16) M 3 (14) M 4 (11)5 20 M 2 (17) M 3 (13) M 4 (7)The mean time to failure (MTTF) and the mean time torepair (MTTR) for all machines are shown in Table 2.Table 2. MTTF and MTTR of each machine in thesystemMachine M 1a M 2a M 3a M b1 M b2 M b3 M b4MTTF 6000 3500 3000 5500 4000 3500 6500MTTR 500 420 350 400 400 360 380In this paper, the batches arrival is considered cyclic.Indeed, the manufacturing of a new batch is onlypermitted if the previous batch is finished. In addition,products are generated in a cyclic manner. For a givenbatch, the time between the arrivals of two products isequal to the time of execution of the first operation.We perform discrete flow simulation using simulationsoftware called ARENA. We simulate the systemduring a 12 years horizon; during the first 60000minutes, statistics are not collected. This warm-upperiod (the first 60000 minutes) is used to avoidtransient effects on the final results.- 57 -

M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special Issue4.1. Study of the system without intercellulartransferIn this section we simulate our system without theintercellular transfer policy in order to validate ourmodel of cellular system and the developedexpressions. The following table summarizes thevalues of the parameters developed according to theanalytical model and that of simulation.Table 3. Results from the approximate method andsimulation1009590858075M1a M2a M3a Cell (a)A()analytic (%)A()analytic_tr (%)Machines M 1a M 2a M 3a M b1 M b2 M b3 M b4Tu* analytic (%) 42.53 62.21 60.91 50.77 41.98 49.63 50.69Tu* simulation (%) 42.56 62.26 60.96 50.61 41.85 49.47 50.53Error (%) < 1 %A() analytic (%) 96.46 92.54 92.89 96.31 95.8 94.9 97.04A() simulation (%) 96.53 92.67 92.84 96.01 95.68 94.79 97.37Error (%) < 0.5 %The obtained results show that the error between thevalues given by the analytical formulation andsimulation is relatively small. Then the approximationof the cell to a production line working under acontinuous and constant load is sufficiently robust toestimate the availability of the autonomous cells.4.2. Study of the system with intercellulartransferFor the studied system, the strategy consists inapplying the intercellular transfer of the cell (a) towardthe cell (b) in case of failure of one of machines of thefirst cell (see Figure 4). We assume that preparationtimes of intercellular transfer are negligible comparedto times between failings and repair machine times.The values of the availability governed by the twotypes of policy with and without transfer are calculatedaccording to the analytical model. The obtained resultsshow the improvement made by the application of theintercellular transfer policy in term of cell availability(see Figure 5).To evaluate the performance of our policy withsimulation tool and to support the results of theanalytical model, we select the productivity of the cell,presented in the number of produced pieces, andmachine utilization rate as performance criteria (seeTable 4).Figure 5. Improvement of the availability by theintercellular transfer policy predicted by the analyticalmodelTable 4. Performance of the cell (a) with and withoutintercellular transfer policy from simulationUtilizationrate (%)Number ofmanufacturedproductsMachineCell (a)M 1a M 2a M 3aWithout transfer 42.56 62.26 60.96With transfer 44.12 62.95 62.40Without transfer 346650With transfer 359337The simulation results show that intercellular transferpolicy improves the machines utilisation rate and theproductivity of the cell. This improvement reflects theaugmentation of cell availability.5. ConclusionIn this paper, an analysis of cellular manufacturingsystem is presented. The notion of virtual cells andintercellular transfer allowed the development of asolution, which overcomes the effect of failures bycontinuing the process with the machine of theadjacent cell. Analytical modelling makes possible todetermine the expression of the availability of the celland to explain the improvement obtained by applyingthe intercellular transfer policy. The simulation resultsvalidated our analytical model and proved theeffectiveness of the applied policy in the improvementof the system performance.References[1] Banerjee, A. & Flynn B. (1987). A SimulationStudy of Some Maintenance Policies in a GroupTechnology Shop. International Journal ofProduction Research. 25, 1595-1609.- 58 -

M.Elleuch, B.Ben, F.MasmoudiImprovement of manufacturing cells with unreliable machines - RTA # 3-4, 2007, December - Special Issue[2] Buzacott, J.A. (1968). Prediction of the efficiencyof production systems without internal storage.International Journal of Production Research. 6,173-188.[3] Geonwook, J., Herman, R.L. & Hamid, R.P. (1998).A cellular manufacturing system based on newsimilarity coefficient which considers alternativeroutes during machine failure. Computers andIndustrial Engineering. 35, 73-76.[4] Lahoti, A. & Kennedy W.J. (1991). Analyzing theEffect of Installing Diagnostic Equipment in aManufacturing Cell. Proceeding of the IEEEAnnual Reliability and Maintainability Symposium.10-14.[5] Savsar, M. (2006). Effects of maintenance policieson the productivity of flexible manufacturing cells.OMEGA the International Journal of ManagementScience. 34, 274-282.- 59 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueGrabski FranciszekNaval University, Gdynia, PolandApplications of semi-Markov processes in reliabilityKeywordssemi-Markov processes, reliability, random failure rate, cold standby system with repairAbstractThe basic definitions and theorems from the semi-Markov processes theory are discussed in the paper. The semi-Markov processes theory allows us to construct the models of the reliability systems evolution within the timeframe. Applications of semi-Markov processes in reliability are considered. Semi-Markov model of the coldstandby system with repair, semi-Markov process as the reliability model of the operation with perturbations andsemi-Markov process as a failure rate are presented in the paper.1. IntroductionThe semi-Markov processes were introducedindependently and almost simultaneously by P. Levy,W.L. Smith, and L.Takacs in 1954-55. The essentialdevelopments of semi-Markov processes theory wereproposed by Cinlar [3], Koroluk & Turbin [13],Limnios & Oprisan [14]. We would apply only semi-Markov processes with a finite or countable statespace. The semi-Markov processes are connected tothe Markov renewal processes.The semi-Markov processes theory allows us toconstruct many models of the reliability systemsevolution through the time frame.2. Definition of semi-Markov processes with adiscrete state spaceLet S be a discrete (finite or countable) state space andlet R+= [ 0, ∞), N = {0,1,20,...}. Suppose, thatξ , , n = 0,1,2,... are the random variables definednϑ non a joint probabilistic space ( Ω , F, P ) with valueson S and R+respectively. A two-dimensional randomsequence {( ξn,ϑ n), n = 0,1,2 ,...} is called a Markovrenewal chain if for alli ,...., i , i S,t ,..., t ∈ R n∈N0 n− 1∈0 n +,0:1. P{ ξ j, ϑ ≤t| ξ = i,ϑ = t ,..., ξ = i t }n + 1 = n+1 n n n 0 0,ϑ0={ = j ϑ ≤ t | = i} Q ( ),= P n + 1 , n+1 ξ n = ij tξ (1)02.P ξ =(2){ 0 = io,ϑ 0=0} = P{ξ0= i0}p iohold.From the above definition it follows that a Markovrenewal chain is a homogeneous two-dimensionalMarkov chain such that the transition probabilities donot depend on the second component. It is easy tonotice that a random sequence { ξn: n = 0,1,2 ,...} is ahomogeneous one-dimensional Markov chain with thetransition probabilitiespij= P{ ξ ! = j | ξ = i}lim Q ( t).(3)The matrixn+ n =t→∞Q ( t)= [ Qij ( t):i,j ∈ S],(4)Is called a Markov renewal kernel. Both Markovrenewal kernel and the initial distribution define theMarkov renewal chain. This fact allows us toconstruct a semi-Markov process.Letττ0n= ϑ0= ϑ1= 0,+ ... + ϑn, τ ∞= sup{ τnij: n∈NA stochastic process { X ( t) : t ≥ 0}following relation0}given by the- 60 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueX ( t)= ξ for t τ , τ )(5)n∈[ n n+ 1is called a semi-Markov process on S generated bythe Markov renewal chain related to the kernelQ( t),t ≥ 0 and the initial distribution p.Since the trajectory of the semi-Markov process keepsthe constant values on the half-intervals [ τ n, τ n+ 1)andit is a right-continuous function, fromequality X ( τ ) = ξ , it follows that the sequence{ X ( ) : n = 0,1,2,... }nnτ nis a Markov chain with thetransition probabilities matrixP = [ : i,j ∈ S](6)p ijτ nis called anembedded Markov chain in a semi-Markov process{ X ( t) : t ≥ 0}.The functionThe sequence { X ( ) : n = 0,1,2,... }{ τ −τ≤ t | X ( τ ) = i,X ( j}Fij ( t)= Pn+ 1 nnτn+1) =Qij( t)= (7)pijis a cumulative probability distribution of a randomvariable T that is called holding time of a state i , ifijthe next state will be j . From (11) we haveQij ( t)= pijFij( t). (8)The function{ −τ≤ t | X ( ) = i} = ∑G ( t)= P τ1τ Q ( t)(9)in+nnj∈Sis a cumulative probability distribution of a randomvariable T that is called waiting time of the state i .iThe waiting time Timeans the time being spent instate i when we do not know the successor state.N ( t) : t ≥ 0 defined byA stochastic process { }N ( t)= n for t τ , τ )(10)∈[ n n+ 1is called a counting process of the semi-Markovprocess { X ( t) : t ≥ 0}.The semi-Markov process { X ( t) : t ≥ 0}is said to beregular if for all t ≥ 0P{ N(t)< ∞}= 1ijX has the finitenumber of state changes on a finite period.Every Markov process { X ( t) : t ≥ 0}with the discretespace S and the right-continuous trajectories keepingconstant values on the half-intervals, with thegenerating matrix of the transition ratesIt means that the process { ( t) : t ≥ 0}Α=[ α : i,j ∈ S], 0 −α ii= α < ∞ is the semi-ij

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issueq~ik( s)st= ∞ −∫ e0dQik( t),g~( s)ist= ∞ −∫ e0dG ( t)iwhere π = [ π , j ∈ S]jis the unique stationarydistribution of the embedded Markov chain thatsatisfies system of equations.are given.Passing to matrices we obtain the following equationp~( s)= [ I −~ g(s)]+ q~ ( s)~p(s), (14)wherep~( s)= [ ~ pij ( s) : i,j ∈ S], q~( s)= [ q ~ij( s) : i,j ∈ S],~g ( s)=δ [ (1 − g~( s)):i,j ∈ S].ijiIn many cases the transitions probabilities P ij(t)andthe states probabilitiesP j{ X ( t)= j} , j ∈ ,( t)= PS(15)approach constant values for large tPij= lim P ( t),P = lim P ( t). (16)t→∞ijjt→∞To formulate the appropriate theorem, we have tointroduce a random variable{ n ∈ N : X ( j}j=n=j∆ min τ ) , (17)That denotes the time of first arrival at state j. Anumberf= P{ ∆ < ∞ | X ( τ ) i}(18)ij jn=is the probability that the chain that leaves state i willsooner or later achieve the state j.As a conclusion of theorems presented by Korolyukand Turbin [13], we have obtained following theoremTheorem 1.Let { X( t ) : t ≥ 0}be a semi-Markov process with adiscrete state space S and continuous kernelQ ( t)= [ Qij ( t):i,j ∈ S].If the embedded Markovτ n, contains one positiverecurrent class C, such that for each statei ∈ S, j ∈ C,f = 1 and 0 < E(T i) < ∞,i ∈ S,thenchain { X ( ) : n = 0,1,2,... }Pij= lim Pt→∞ijij( t)= Pjπ j E(T j )= lim Pj( t)=t→∞∑πE(T )i∈Sjj(19)∑π p = π , j ∈ S,∑π= 1.(20)i∈Siijji∈S4. First passage time from the state i to thestates subset A.The random variableΘwhere∆= τA ∆ AA,= min{ n ∈ N : X ( τ ) ∈ A},ndenotes the time of first arrival of semi-Markovprocess, at the set of states A.The functioniΦ ( t)= P{ Θ ≤ t | X (0) i}.(21)iA A =is the cumulative distribution of the random variableΘiAthat denotes the first passage time from the state ito the states subset A.Theorem 2. [4], [13]For the regular semi-Markov processes such that,fiA = P{ ∆A< ∞ | X (0) = i}= 1,i ∈ A′, (22)the distributions Φ ( t), i ∈ A are proper and theyare the unique solutions of the system of equationsΦiAi ∈ A'( t)=∑ Qj∈AijiA′( t)+ ∑ ∫ Φk∈St0kA( t − x)dQik( x),Applying Laplace-Stieltjes transformation we obtainthe system of linear equations~~φ ( s)= ∑ q~( s)+ ∑φ( s)q~( s),i ∈ A'(23)iAj∈Awith unknown transforms~φ ( s)iAst= ∞ −∫ e0ijdΦiA( t).kAk∈A'Generating matrix form we get equationik- 62 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issue~ ~ ~− , (24)( I q ( s)) ( s)= b(s)whereA'A'I = [ δ : i,j ∈ A'], q~( ) [~' s = q ( s) : i,j ∈ A]ijare the square matrices and~A'( s)=~ ⎡b(s)= ⎢ ∑⎣A[~ ( s):i∈A']j∈AiATTq~⎤ij( s):i ∈ A'⎥⎦,ij5. Semi-Markov model of the cold standbysystem with repairThe problem is well known in reliability theory(Barlow & Proschan [1]). The model presented here issome modification of the model that was consideredby Brodi & Pogosian [2].5.1. Description and assumptionsA system consists of one operating component, anidentical stand-by component and a switch, (Figure1).1are the one-column matrices of transforms. Theformal solution of the equation is~ ~( s)= .~ 1A'A'−( I − q ( s)) b(s)To solve this equation we use any computer programs,for example MATHEMATICA. Obtaining the inverseLaplace transform is much more complicated.It is essentially simpler to find the expected valuesand the second moment of the random variablesΘiA, i ∈ A′. If the second moments of the waitingtimes T i, i ∈ A′are positive and commonly bounded,and f iA= 1 , i ∈ A′, then the expected values of therandom variablesof equation( I PA') A'= TA'whereΘiA, i ∈ A′are the unique solution− , (25)[ δ : i,j ∈ A'] ,'= [ p : i,j ∈ A'],I = PA'ijAT[ E Θ ) : i ∈ A'] , = [ E(T ) : i ∈ A'] T= T(iAA'and the second moments of the time to failure are theunique solution of equation2( I − PA' ) A′= BA'where[ δ : i,j ∈ A'],' = [ p : i,j ∈ A'],I = PijAijiji(26)Figure 1. Diagram of the systemWhen the operating component fails, the spare is putin motion by the switch immediately. The failed unit(component) is repaired. There is a single repairfacility. The repairs fully restore the components i.e.the components repairs means their renewals. Thesystem fails when the operating component fails andcomponent that was sooner failed in not repaired yetor when the operating units fail and the switch fails.We assume that the time to failure of the operatingcomponents are represented by the independent copiesof a non-negative random variable ς with distributiongiven by a probability density function (pdf)f ( x),x ≥ 0 . We suppose that the lengths of the repairperiods of the components are represented by theidentical copies of the non-negative random variablesγ with cumulative distribution function (CDF)G( x)= P(γ ≤ x).Let U be a random variable havingbinary distributionb(k)= P(U = k)= ak(1 − a)21−k, k = 0,1, 0 < a < 1,where U = 0 , when a switch is failed at the momentof the operating component failure, and U = 1, whenthe switch work at that moment. We suppose that thewhole failed system is replaced by the new identicalsystem. The replacing time is a non-negative randomvariable η with CDF H( x)= P(η ≤ x).2T[ E Θ iA ) : i ∈ A'] , = [ b : i∈A'] ,2 TA′ = ( B A'ib = E(T ) + 2 ∑ p E(T ) E(Θ ) .iiikk∈ A'ikkA- 63 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueQ10( t)= P(ς≤ t,γ> ζ )1z 1 g 1 z 3 g 3 z 5+ P(U = 0, ς ≤ t,γ< ζ )2z 2 g 2 z 4 g 4t0= ∫ [1 − G(x)]dF(x)+ (1 − a)∫ G(x)dF(x)t0t 1 t 2 t 3 t 4tFigure 2. Reliability evolution of the standby systemMoreover we assume that the all random variables,mentioned above are independent.5.2. Construction of the semi-Markov modelTo describe reliability evolution of the system, wehave to define the states and the renewal kernel. Weintroduce the following states:0 - the system is failed1 - the failed component is repaired, spare is operated2 - both operating component and spare are “up”.∗ ∗ ∗Let 0 = τ0, τ1, τ2,... denote the instants of the stateschanges, and { Y ( t) : t ≥ 0}be a random process withthe state space S = {0,1,2 } , which keeps constant∗ ∗values on the half-intervals [ τn, τ n+1),0,1,...and isright-continuous. The realization of this process isshown in Figure 1. This process is not semi-Markov,because the condition (1) of definition (2) is notsatisfied for all instants of the state changes of theprocess.Let us construct a new random process a followingway. Let 0 = τ0and τ , τ 1 2,... denote the instants ofthe system components failures or the instants ofwhole system renewal. The random process{ X ( t): t ≥ 0} defined by equationX ( 0) = 0, X ( t)= Y ( τn) for t ∈[τn, τn+1)is the semi-Markov process.To have semi-Markov process as a model we mustdefine its initial distribution and all elements of itskernel⎡ 0Q(t)=⎢⎢Q10( t)⎢⎣Q20( t)QQFor t ≥ 0 we obtain01121( t)( t)Q ( t)= P(η ≤ t)= H(),02tQ02( t)⎤0⎥⎥0 ⎥⎦t0= F(t)− a∫G(x)dF(x),QQt11 ∫( t)= P(U = 1, ς ≤ t,γ < ζ ) = a0G(x)dF(x),( t)= P(U = 0, ς ≤ t)= (1 − a)F(),20tQ ( t)= P(U = 1, ς ≤ t)= aF().21tWe assume that, the initial state is 2. It means that aninitial distribution is[ 0 0 1]p ( 0) = .Hence, the semi-Markov model is constructed.5.3. The reliability characteristicsThe random variable ΘiA, that denotes the firstpassage time from the state i to the states subset A,for i = 2 and A = {0}in our model, represents thetime to failure of the system. The functionR t)= P(Θ > t)= 1− Φ ( t),t 0 (27)(20 20≥is the reliability function of the considered coldstandby system with repair.System of linear equation (23) for the Laplace-Stieltjes transforms of the functionsΦ( t),t ≥ 0, ii0 =in this case is1,2,~~φ ( ) ~ ( ) ( ) ~10s = q10s + φ10s q11(s)~~φ ( s)= q~( s)+ φ( s) q ~20 20 10 21sThe solution is~ q~10( s)φ10(s)= ~ ,1−q11(s)~~~ q ) ~21(s q10( s)φ20( s)= q20(s)+ ~ .(28)1−q ( s)11()- 64 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueHence, we obtain the Laplace transform of thereliability function~ ~1−φ( )( )20 sR s = .(29)sThe transition probabilities matrix of the embeddedMarkov chain in the semi-Markov processX ( t) : t ≥ 0 is{ }⎡ 0 0 1⎤P =⎢⎥⎢p 10 p110⎥(30)⎢⎣p 0⎥20 p21⎦Wherep10 = 1−p 11p∞11 ∫= P( U = 1, γ < ζ ) = a0p20= 1−a,p21= P(U = 1) = a.G(x)dF(x),Using formula (9) we obtain the CDF of the waitingtimes of T i , i = 0,1,2 .G t)= H ( t),G ( t)= F(t),G ( t)= F() .Hence0(12tE T ) = E(η),E(T ) = E(ς ), E(T ) = E() .(0 13ςThe equation (25) in this case has form⎡1⎢⎣− p− a11The solution isE(ΘE(Θ10200⎤⎡E(Θ⎥⎢1⎦⎣E(ΘE(ς )) =1−p11,1020) ⎤ ⎡E(ς ) ⎤⎥ =)⎢ ⎥⎦ ⎣E(ς ) ⎦a E(ς )) = E(ς ) + . (31)1−p11We will apply theorem 1 to calculate the limitprobability distribution of the state. Now, the systemof linear equation (20) isπ =1p10+ π2p20π0 ,π =1p12+ π2p21π1 ,π0= π 2,π0+ π1+ π2=1.Since, the stationary distribution of the embeddedMarkov chain isπππ012===p112 p11+ p21p212 p11+ p21p112 p11+ p21,,.Using formula (19) we obtain the limit distribution ofsemi-Markov processp11E(η)P0= p E(η)+ p E(ς ) + p E(ς )(32)P1P2==pp111111pE(η)+ ppE(η)+ p5.4. Conclusion2121211121E(ς )E(ς ) + pE(ς )E(ς ) + p111111E(ς )E(ς )The expectation E ( Θ20) denoting the mean time tofailure isE(Θwherep20∞11= a∫ 0a E(ς )) = E(ς ) + ,1−pG(x)dF(x).11Let us notice, that the cold standby determinesaincrease the meantime to failure 1+ times.1−p11The limiting availability coefficient of the system isp21E(ς ) + p11E(ς )A = P1+ P2=.p E(η)+ p E(ς ) + p E(ς )112111- 65 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issue6. Semi-Markov process as the reliabilitymodel of the operation with perturbationSemi-Markov process as the reliability model ofmulti-stage operation was considered by F. Grabski in[8] and [10]. Many operations consist of someelementary tasks, which are realized in turn. Durationof the each task realization is assumed to be positiverandom variable. Each elementary operation may beperturbed or failed. The perturbations increase thetime of operation and the probability of failure aswell.6.1. Description and assumptionsSuppose, that the operation consists of n stages whichfollowing in turn. We assume that duration of an i-thstage, (i = 1, ... , n) is a nonnegative random variableξi; i = 1, L,n with a cumulative probabilitydistributionFti i = ∫ i,0( t) P( ≤ t) f ( x)dx,i = 1, K n= ξ ,where f i(x)denotes its probability density functionin an extended sense.Time to failure of the operation on the i-th stage(component) is the nonnegative random variable η ,ii = 1,L,nwith exponential distribution6.2. Semi-Markov modelTo construct reliability model of operation, we have tostart from definition of the process states.Let e , i=1,...,n, j=0,1 denotes j-th reliability statei jon i-th step of the operation where, j=0 denotesperturbation and j=1 denotes successe - failure (un-success) of the operation2 n+1e11- an initial state.For convenience we numerate the statese i ↔ i,i 1,...,n1=e i ↔ i+n,i 1,...,n0=e2 n+ 1↔ 2n+ 1,Under the above assumptions, stochastic processdescribing of the overall operation in reliabilityaspect, is a semi-Markov process { X ( t):t ≥ 0}witha space of states S = { 1,2,...,2n,2n+ 1}and flowgraph shown in Figure 3.1 2 n-1 n−λ( ≤ t) = 1−eit; i = 1, K nP η .i,n+1n+22n-12nThe operation on each step may be perturbed. Weassume that no more then one event causingperturbation on each stage of the operation may occur.Time to event causing of an operation perturbation oni-th stage is a nonnegative random variableζ ; i = 1, K n with exponential distributioni,−α( ≤ t) = 1−eit; i = 1, K nP ζ .i,The perturbation degreases the probability of theoperation fail. We suppose that time to failure of theperturbed operation on the i-th stage is thenonnegative random variable νi, i = 1,2....,n thathas the exponential distribution with a parameterβ >iλ i−β( ν ≤ t) = 1 − eit; i = 1, K,n.PiWe assume that the operation is cyclical.We assume that random variablesξ i , , ηi, ν i , ζ i , i = 1,...,n are mutually independent.2n+1Figure 3. Transition graph for n-stage cyclic operationTo obtain a semi-Markov model we have to define allnonnegative elements of semi- Markov kernel[ Q ( t) : i j S]Q( t)=ij, ∈{ X ( τ ) = j,τ −τ≤ t | X ( i}Qij t)Pnn nn=( =+ 1 + 1τ )First, we define transition probabilities from the state ito the state j for time not greater than t for i=1,…,n-1.Q t = P(ξ ≤t,η > ξ ζ > ξ=where( ) )i i+1 i i i i i−eα i y −λeiz∫∫∫αiλifi( x)DD = {( x,y,z):x ≤ t,z > x,x > y}dx dy dzx ≥ 0, y ≥ 0, z ≥ 0,- 66 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueSince, we havet xi i+1 i ) i0 0Q−αiy( t) = ∫f( x dx ∫αe dy ∫=t∫0e− ( λi + α i ) xfi( x )∞−λizλiexdx .For i = n +1,...,2n−1we obtaindzx = u + v, y = v,z = w.This mapping assigns to points from set∆ = {( u , v,w) : 0 ≤ u ≤ t,v ≥ 0, w > u}the points from plane region D. The Jacobian of thismapping isJ ( u,v,w)= 1.Qi i+n( t)= P(ζi≤ t,ηi> ζi, ζi> ξi)Since, we getForti i= ∫αe [1 − F ( u)]du , i = 1,..., n −1.0i−(λ + α ) ui = 1,...,n we geti−βi zei∫∫∫αiβief ( x)dx dy dzD− α y− α ivi−βiw= ∫∫∫αie βiefi( u + v)J ( u,v,w)du dv dw∆Qi 2n+1( t)= P(ηi≤ t,ηi< ζi, ηi< ξi)= ∞ ∞ t−αi v −βi wiii0 u0∫αe dv∫βe dw∫ f ( u + v)dut= ∫ λ e0i = 1,...,n −1.i−(λi+ αi) u[1 − F ( u)]du ,If on i-th stage a perturbation has happened thetransition probability to next state for time less then orequal to t isQn + i n+i+1( t)= P(ξi = 1,...,n −1,whereD = {( x,y,z) :0 ≤ x − y ≤ t,i− ζ−αy≤ t,ν∫∫∫αiieβieD=−αi∫∫αeEz > x − y,iii−βi zyi> ξx ≥ 0, y ≥ 0, z ≥ 0,ii− ζi| ξi> ζfi( x)dx dy dz,f ( x)dx dyx > y}E = {( x,y):x ≥ 0, y ≥ 0, x > y}.i)t−βu∞−αiv= ∫eidu ∫ α e f ( u + v)dv .0 0Let us notice that∫∫αeE∞i−αiy= ∫αe0i∞= 1 − ∫αe0i−αiyiii∞f ( x)dx dy = ∫αe0[1 − F ( y)]dy−αiyFinally, we obtainQn + i n+i+1F ( y)dy.i = 1,..., n −1.In the same way we getti∫ e0( t)=i−βiui∞−αiy∫ fyi( x)dx[ ∫ αiefi( u + v)dv]du0,∞.−αy∫ α ei[1 − F ( y)]dy0∞i−αiviTo find the triple integral over the region D, we applychange of coordinates:u = x − y, v = y,w = z.Hence- 67 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueQn + i 2n+1 (t)= P(νi≤ t,ν−αi y∫∫∫ αieβD=∫∫ α eEii< ξie−αiyi− βiz− ζf ( x)dx dy dzf ( x)dx dyiii| ξi> ζi),⎡ 0⎢⎢Q 21(t)Q(t)= ⎢ 0⎢⎢Q41(t)⎢⎣ 0Q120000( t)Q130000( t)QQ0243400( t)( t)QQQQQ1525354555( t)⎤⎥( t)⎥( t)⎥,⎥( t)⎥( t)⎥⎦i = 1,..., nwherewhereD = {( x,y,z) :0 ≤ z ≤ t,z < x − y,x ≥ 0, y ≥ 0,x > y}E = {( x,y):x ≥ 0, y ≥ 0, x > y}.HenceQn + i 2n+1i = 1,..., n.β( t)=Similar way we obtainQ( t)= P(ξt= ∫ eti w ∞i v ∞−β−αii0 w0∞−αyei∫αii0∫edw∫αe dv∫ f ( u + v)du≤ t,η−(λn+ αn) uf> ξn( u)du[1 − F ( y)]dy> ξn1 n n n n n )0ζiQQQQQQQt12 ∫0−(λ()1+α=1 ) ut e dF ( u),t13 ∫ 1 −0−(λ( ) =1i+ α1)ut α e [1 F ( u)]du ,t15 ∫ 1 −0−(λ( )1i+ α1)ut = λ e [1 F ( u)]du,t21(t)= ∫0e−(λ2+ α2 ) u1211dF ( u),t2 4 ∫ 2 −0−(λ( )2 + α=2 ) ut α e [1 F ( u)]du ,t25 ∫ 2 −034−(λ( )2 + α=2 ) ut λ e [1 F ( u)]du,t∫ e0( t)=−βu1[1∫ α 1ef1( u + v)dv]du0,∞−αy∫ α ei[1 − F ( y)]dy0∞1−αv122Q2n1( t)= P(ξtn−βn u≤ t,ν| ξ∫edu ∫ α nef n ( u + v)dv0 0=.∞−αenx∫α[1 − F ( x)]dx0− ζnn∞n−αi v> ξnn− ζnn> ζTherefore the semi-Markov reliability model ofoperation has been constructed.n)β( t)=1∫edw∫α1edv∫ f1(u + v)du0 w0,∞−αe1x∫α[1 − F ( x)]dx6.3. Two-stage cyclical operation t∞∞ We will consider 3−β2w−α2vβ2 ∫edw∫α2edv∫ f2( u + v)duWe will investigate particular case of that model,0 w0Q4 5( t)=,assuming n = 2 . A transition matrix for the semi-∞−α2x2e[1 F2( x)]dxMarkov model of the 2-stage cyclic operation in∫α−0reliability aspect takes the following formQQ3 54 1t1∫ e0( t)=t−βu−βiw01∞−αv2du2∫α2ef2( u + v)] dv0,∞−αe2x∫α[1 −F( x)]dx0Q ( t)= U ( ).55t2∞−αv21∞- 68 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueThat model allows us to obtain some reliabilitycharacteristics of the operation. The randomvariableΘ 15denoting the first passage time from state1 to state 5 in our model, means time to failure of theoperation. The Laplace-Stieltjes transform for thecumulative distribution function of that randomvariable we will obtain from a matrix equation (25).In this case we have A ′ ={ 1,2,3,4}, A = {5}and~⎡~φ15( s)⎤ ⎡q⎢~⎥ ⎢~~ ( ) ~( ) ⎢φ25 s ⎥' ~ , ( ) ⎢q A s = b s =⎢~φ35( s)⎥ ⎢q⎢~⎥ ⎢( )~⎢⎣φ45s ⎥⎦⎣q15253545( s)⎤( s)⎥⎥,( s)⎥⎥( s)⎦⎡ 0 q~~12( s)q13( s)0 ⎤⎢~~ ⎥~= ⎢q21( s)0 0 q24( s)q ⎥A'( s)⎢ 0 0 0~ .q ( ) ⎥34 s⎢~⎥⎣q41( s)0 0 0 ⎦From the solution of equation (24) we obtain Laplace-Stieltjes transform of the cumulative distributionfunction of the random variable Θ15denoting time tofailure of the operation~ a~ (~ sφ ( ))15s = , (33)b ( s)a~( s)= q~15+ q~− q~12~b ( s)= 1 − q~13( s)+ q~( s)q~12( s) q ~2412( s)q~34( s)q~( s)q~21( s) q ~4525( s)+ q~( s)+ q~( s)− q~41( s).121313( s)q~( s)q~( s)q~243435( s)q~( s)( s)q~4145( s)( s)The Laplace transform of the reliability function isgiven by the formula~ ~1−φ( )( )15 sR s = .(34)s6.4. ExamplesExample 1We suppose thatThenQ12 κ ⎛ −λ1+ α1+ κ1⎝⎞⎟⎠1−(λ1+ α + κ1 ( ) =11⎜) tte ,α ⎛ ⎞Q13 − ⎟λ + α + κ ⎝⎠1−(λ1+ α 1)( ) =11+ κ tt⎜ e ,111λ ⎛ ⎞Q15 −⎟λ + α + κ ⎝⎠QQQQQQQ212 42 534354 14 51−(λ1+α 1)( ) =11+ κ tt⎜ e ,1( t)=λ2( t)=λ( t)=λ221212−(λ( 12 + α2κ2 ) t− e ),κ 2++ α2+ κ2−(λ( 12 + α2κ2 ) t− e ),α 2++ α2+ κ2−(λ( 12 + α2κ2 ) t− e ),λ 2++ α+ κ1 −(β( ) ( 11 κ1 ) tt = −e),β + κ1κ +11 −(β( ) ( 11 κ1 ) tt = −e),β + κ1( t)=ββ +2( t)=β12−(β( 12 κ2 ) t− e ),κ2+2+ κ2−(β( 12 κ2 ) t− e ),β2++ κLaplace-Stieltjes transform of these functions are:q~q~1213q~q~κ1( s)= ,s + λ1+ α1+ κ1α1( s)= ,s + λ1+ α1+ κ1λ1( s)= ,s + λ + α + κ1521( s)= s + λ12κ2+ α12+ κ12,Fi−κit( t)= 1 − e , t ≥ 0, i = 1,2 .q~24( s)= s + λ2α2+ α2+ κ2,- 69 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issueq~25( s)= s + λ2λ2+ α2+ κ2,This function is shown on Figure 5.1q~q~q~q~34354145κ( s)= s + β1β( s)= s + β1κ( s)= s + β21β( s)= s + β21+ κ+ κ211+ κ2+ κ22,,,.For κ 1 = 0, 1; κ 2 = 0, 12 ; λ 1 = 0, 002; λ 2 = 0, 001;α = 0,021 ; α 2 = 0, 04 ; β 1 = 0, 01; β 1 = 0, 01,applying (33) and (34), with help ofMATHEMATICA computer program, we obtain thedensity function and the reliability function as inverseLaplace transforms .The density function is given by the formulaφ15( t)= 0.00712201e− 0.000144434 e−0.213357t−0.125901t− 0.00872613 eThis function is shown in Figure 4.The reliability function isR(t)= 3.79823×10−16+ 0.00374856 e+ 0.0333808 e−0.180053t−0.213357t−0.00368869t0.80.60.40.2200 400 600 800 1000Figure 5. The reliability function of 2-stage cyclicoperationMean time to failure we can find solving the matrixequation( I P A ') ΘA'= TA'where− (35)⎡ 0 p12p130 ⎤⎢⎥⎢p210 0 p24P =⎥A',⎢ 0 0 0 p ⎥34⎢⎥⎣ p410 0 0 ⎦⎡E(T1) ⎤⎢ ⎥⎢E(T2)T = ⎥,⎢E(T3) ⎥⎢ ⎥⎣E(T4) ⎦ΘA'⎡E(Θ⎢⎢E(Θ=⎢E(Θ⎢⎣E(Θ15253545) ⎤)⎥⎥.) ⎥⎥) ⎦From this equation we obtain the mean time to failureE( Θ 15) = 275, 378Example 2Now we assume that− 0.0484641e+ 1.01623 e0.0030.00250.0020.00150.0010.0005−0.180053t−0.00368869t− 0.0011472 e−0.125901tF⎧0dla t ≤ Li( t)= ⎨, i = 1,2⎩1dla t > Lii.It means that the duration of the stages are determinedand they are equal ξi= Lifor i = 1,2.In this case the elements of Q(t) are:Q12( t)⎪⎧0⎨ −⎪⎩ e=( λ1+ α1) L1forfort ≤ L1t > L1200 400 600 800 1000Figure 4. The density function the time to failure of2-stage cyclic operation- 70 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issue- 71 -( )⎪⎪⎩⎪⎪⎨⎧>⎟⎠⎞⎜⎝⎛ −+≤⎟⎠⎞⎜⎝⎛ −+=+−+−11)11(1111)11(11113for1Ltfor1LteetQLtαλαλαλααλα( )⎪⎪⎩⎪⎪⎨⎧>⎟⎠⎞⎜⎝⎛ −+≤⎟⎠⎞⎜⎝⎛ −+=+−+−11)11(1111)11(11115for1Ltfor1LteetQLtαλαλαλλαλλ( )⎩⎨⎧>≤= +−222 )2(221Ltforfor0LeLttQαλ( )⎪⎪⎩⎪⎪⎨⎧>⎟⎠⎞⎜⎝⎛ −+≤⎟⎠⎞⎜⎝⎛ −+=+−+−22)22(2212)22(22224for1Ltfor1LteetQLtαλαλαλααλα( )⎪⎪⎩⎪⎪⎨⎧>⎟⎠⎞⎜⎝⎛ −+≤⎟⎠⎞⎜⎝⎛ −+=+−+−22)22(2212)22(22225for1Ltfor1LteetQLtαλαλαλλαλλ()( )( ),,1))(1(0,1))(1(1111111111111)(1111)(11134⎪⎪⎩⎪⎪⎨⎧>−−−≤≤−−−=−−−−−−−−LteeeLteeetQLa La Lta La Lαβαβαβααβα( )( ) ( )( )⎪⎪⎩⎪⎪⎨⎧>⎥⎦⎤⎢⎣⎡−−⎟−⎠⎞⎜⎝⎛ −−≤≤⎥⎦⎤⎢⎣⎡−−−−−=−−−−−−−−−−111)1(111111 11111 )1(1111111 135,1)(1)(110,1)(1)(11LteeeeLteeeetQLa LLa Lta Lta Lαββαββαββαββ( )( )( )⎪⎪⎩⎪⎪⎨⎧>−−−≤≤−−−=−−−−−−−−222 )2(222222222 )2(222222241,1))(1(0,1))(1(LteeeLteeetQLLaLatLaLaαβαβαβααβα( )( ) ( )( )⎪⎪⎩⎪⎪⎨⎧>⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛ −−−−−≤≤⎥⎦⎤⎢⎣⎡−−−−−=−−−−−−−−−−212 )2(22222122222 )2(2222222245,1)(1)(110,1)(1)(11LteeeeLteeeetQLLaLLatLatLaαββαββαββαββThe Laplace’a-Stieltjes transform of these functionsare:1)11(12 )(~ Lsesq++−=αλ111)11(111113 )(~αλααλααλ++−++=++−sessqLs111)11(111115 )(~αλλαλλαλ++−++=++−sessqLs2)22(21 )(~ Lsesq++−=αλ,22)(222224222)(~αλααλααλ++−++=++−sessqLs2221)22(222225 )(~αλλαλλαλ++−++=++−sessqLs⎟⎟⎠⎞⎜⎜⎝⎛−+−−=+−−−−111)11(111113411)(~αβααβααseeesqLsLL⎟⎟⎠⎞⎜⎜⎝⎛−+−−+−−=+−−−+−−111)11(1 111)1(1 1135)(111)(~αβββαβαβαseeseesqLsLLsL⎟⎟⎠⎞⎜⎜⎝⎛−+−−=+−−−−222)22(222224111)(~αβααβααseeesqLsLL⎟⎟⎠⎞⎜⎜⎝⎛−+−−+−−=+−−−+−−222)2(2222)2(22245)(111)(~αβββαβαβαseeseesqLsLLsLThe mean time to failure we can find solving thematrix equation (35), where11)1(12Lepλ+α−=1111)1(113)(1αλααλ+−=+−Lep1111 )1(115)(1αλλαλ+−=+−Lep22 )2(21Lepλ +α−=2222 )2(224)(1αλααλ+−=+−Lep22212 )2(225)(1αλλαλ+−=+−Lep⎟⎟⎠⎞⎜⎜⎝⎛−−−=−−−−1111)1(111113411 αβααβααLLLeeep

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issueppp354145β=1 − ee−1−α1L1−α1L11((1 − eβ −α1− eβ−(β1−α1)L11−β1L1−α2L2−(β −⎛ −2 αα2e1 e= ⎜−α1 −2Le2⎝ β2−αβ=1 − ep 51 = 1⎛⎜1− e⎜⎝βE(T1) =αE(T2E(T3E(T42−α2L21) =α−β2L2221+ λ11+ λd(q~) = −d(q~) = −34412e−e−α1−α2L21)2 ) L22⎞⎟⎠−(β(1 − eβ − α12−(α1+ λ1 ) L1e−α+ λ−(α2+λ2 ) L22( s)+ q~ds( s)+ q~ds3545+ λ2( s))( s))s=0s=02−α2 ) L2) ⎞⎟⎟⎠For the same parameters κ 1 = 0, 1; κ 2 = 0, 12 ;λ 1 = 0,002; λ 2 = 0, 001; α 1 = 0, 02; α 2 = 0, 04 ;β 1 = 0,01; β 1 = 0, 01,1 1and L 1 = i L 2 = , the mean time to failure ofκ1κ1the operation isE ( Θ15 ) = 366.284 .In previous case the mean time to failure isE ( Θ15 ) = 275,378 .6.5. ConclusionIt means that for the determined duration of the stagesmean time to failure of the operation is essentiallygreater than for exponentially distributed duration ofthe stages with the same expectations.To assess reliability of the many stage operation wecan apply a semi-Markov process. Construction of thesemi-Markov model consist in defining a kernel ofthat process. A way of building the kernel for thesemi-Markov model of the many stage operation ispresented in this paper. From Semi-Markov model wecan obtain many interesting parameters andcharacteristics for analysing reliability of theoperation.From presented examples we get conclusion that forthe determined duration of the stages, mean time tofailure of the operation is essentially greater than forexponentially distributed duration of the stages withthe same expectations.7. Semi-Markov process as a failure rateThe reliability function with semi-Markov failure ratewas considered by Kopociski & Kopociska [11],Kopociska [12] and by Grabski [4], [6], [9]. Supposethat the failure rate { λ ( t): t ≥ 0}is the semi-Markovprocess with the discrete state space S = { λj : j ∈ J},J = { 0,1,..., m}or J = {0,1,2 ,...}, 0 ≤ λ0 < λ1< ...with the kernelQ( t)= [ Qij ( t) : i,j ∈ J ]and the initial distribution p = [ pi : i ∈ J].We define a conditional reliability function as⎡ t⎛ ⎞ ⎤Ri( t)= E⎢exp⎜− ∫ (u)du⎟λ(0)= λi⎥ , t ≥ 0,⎣ ⎝ 0 ⎠ ⎦i∈JIn [6] it is proved, that for the regular semi-Markovprocess { λ ( t): t ≥ 0}the conditional reliabilityfunctions R i( t), t ≥ 0,i ∈ J defined by (17), satisfythe system of equationsR(t)= ei−λit[1−G(t)]+∑∫eitj 0−λixR ( t − x)dQ ( x),i ∈JApplying the Laplace transformation we obtain thesystem of linear equations~ 1 ~ ~Ri( s)= −Gi( s + λi) +∑Rj(s) q ~ij(s + λi), i ∈J,s + λjwherei~s tRi ( s)= ∞ − ~s t∫ e Ri( t)dt , G ( s)= ∞ −∫ e G ( t dt ,0jiji i )0- 72 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issueq~( s)ijs t= ∞ −∫ e0dQ ( t).In matrix notation we have~ ~[ I − q~λ( s)]R(s)= H(s),where[ I −~q λ( s)]⎡1− q~⎢⎢− q~=⎢ − q~⎢⎣001020( s + λ )( s + λ )( s + λ )M~⎡R⎤0( s)⎢ ~ ⎥~ ⎢R1( s)R(s)= ⎥⎢ ~R ( ) ⎥2 s⎢ ⎥⎢⎣M ⎥⎦120ij− q~01(s + λ0)1−q~11(s + λ1)− q~( s + λ )21M2− q~− q~1−q~⎡ 1⎢s + λ⎢0⎢ 1~H(s)= ⎢ s + λ1⎢ 1⎢⎢ s + λ2⎢⎣0212( s + λ )( s + λ )22( s + λ )M012L⎤L⎥⎥L⎥⎥O⎦~ ⎤− G0( s + λ0) ⎥⎥~− G + ⎥1(s λ1)⎥~ ⎥− G1( s + λ2) ⎥⎥M ⎥⎦The conditional mean times to failure we obtain fromthe formula~µ = lim ( p),p ∈(0,∞),i ∈ J(21)iR ip→0+The unconditional mean time to failure has a formNµ = ∑ P ( λ(0)= λ ) µ .i=1ii7.1. Alternating random process as a failurerateAssume that the failure rate is a semi-Markov processwith the state space S = λ 0, λ } and the kernel⎡ 0 G0( t)⎤Q ( t)= ⎢⎥ ,⎣G1( t)0 ⎦{1where G0( t),G1( t),are the cumulative probabilitydistribution functions with nonnegative support.Suppose that at least one of the functions is absolutelycontinuous with respect to the Lebesgue measure. Letp = [ p 0, p1] be an initial probability distribution ofthe process. That stochastic process is called thealternating random process. In that case the matricesfrom the equation (20) are⎡− ~ + ⎤−~1 g0(s λ0)[ I qλ( s)]=,⎢⎥⎣−g~1(s + λ1) 1 ⎦whereg~( s)= q~∞s t( s)= e −∫ dG ( ) ,0 010 t0g~( s)= q~∞s t( s)= e −∫ dG ( ) ,1 101 t0~~ ⎡R⎤0 ( s)R ( s)= ⎢ ~ ⎥ ,⎣R1(s)⎦⎡ 1~~⎤s+λ− G0(s + λ )00H ( s)= ⎢ ~ ⎥ .1⎢⎣s+λ− G1( s + λ1) ⎥1⎦The solution of (20) takes the form~R ( s)=01s+λ0~R ( s)=11s+λ0~−G~0(s + λ0) + g0(s + λ0)1−g~( s + λ ) g~( s + λ )0~−G~1(s + λ1)+ g1(s + λ1)1−g~( s + λ ) g~( s + λ )00011[ ] 1 −G~1(s + λ1)s+λ1,1[ ] 1 ~− G0( s + λ0)s+λ0.The Laplace transform of the unconditional reliabilityfunction is~ ~ ~R ( s)= p R ( s)+ p R (Example 3.Assume that0 0 1 1sG ( t)= t ∫ g ( x dx , G ( t)= t ∫ g ( x dx ,wheregg0 0 )0).11 1 )0α0β 0 α0−1−β0x0 ( x)= x e , x ≥Γ(α 0 )α1β1α 1( )1−−β1x1 t = x e , x ≥Γ(α 2 )0.0 ,- 73 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special IssueSuppose that an initial state is λ0. Hence the initialdistribution is p ( 0) = [1 0]and the Laplacetransform of the unconditional reliability function~ ~is R ( s ) = R 0( s ) . Now the equation (20) takes theform of⎡⎢⎢ 1⎢⎢ α⎢ β 11⎢−⎢⎣( s + + 11 1 ) αβ λFor⎡⎢ 1⎢ s + λ0= ⎢⎢ 1⎢⎢⎣s + λ1−( s + λ0αβ00−α( s + β0+ λ0))( s + ββ11−( s + λ ) ( s + β1βα01α011+ λα11+ λ )α00 )⎤⎥ ⎡R~ ⎤⎥ 0( s)⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎣ R~1( s)⎦⎥⎦⎤⎥⎥⎥⎥⎥⎥⎦α = , α = 3, β = 0.2 , β = 0.5, λ = 0, λ 0.2 ,021 010 1=we have1 0.04 0.04 ⎡ 1 0.125 ⎤− +22 ⎢ −3⎥~ s s(s+0.2) ( s+0.2) 0.2 ( 0.2)( 0.7)0()⎣s+s+s+R s =⎦0.04 0.1251−2 3( s+0.2) ( s+0.7)Using the MATHEMATICA computer program weobtain the reliability function as the inverse Laplacetransform.R(t)= 1.33023exp( −0.0614293t)+ exp( −0.02t)(1.34007⋅10−14+ 9.9198 ⋅10−15t)− 2 exp( −0.843935t)[0.0189459cos(0.171789t)+ 0.00695828sin( 0.171789 t)]− 2 exp( −0.37535t)[0.146168cos(0.224699t)+ 0.128174sin(0.224699 t)]Figure 6 shows the reliability function.10.80.60.40.220 40 60 80 100Figure 6. The reliability function from example 2The corresponding density functionf ( t)= −R'(t)is shown in Figure 70.040.030.020.0120 40 60 80 100Figure 7. The density function from example 28. ConclusionThe semi-Markov processes theory is convenient fordescription of the reliability systems evolutionthrough the time. The probabilistic characteristics ofsemi-Markov processes are interpreted as thereliability coefficients of the systems. If A representsthe subset of failing states and i is an initial state, therandom variable ΘiAdesignating the first passagetime from the state i to the states subset A, denotesthe time to failure of the system. Theorems of semi-Markov processes theory allows us to find thereliability characteristic, like the distribution of thetime to failure, the reliability function, the mean timeto failure, the availability coefficient of the systemand many others. We should remember that semi-Markov process might be applied as a model of thereal system reliability evolution, only if the basicproperties of the semi-Markov process definition aresatisfied by the real system.References[1] Barlow, R. E. & Proshan, F. (1975). Statisticaltheory of reliability and life testing. Holt, Reinhartand Winston Inc. New York.[2] Brodi, S. M. & Pogosian, I. A. (1973). Embeddedstochastic processes in queuing theory. Kiev,Naukova Dumka.- 74 -

F.GrabskiApplications of semi-Markov processes in reliability - RTA # 3-4, 2007, December - Special Issue[3] Cinlar, E. (1969). Markov renewal theory. Adv.Appl. Probab.1,No 2, 123-187.[4] Grabski, F. (2002). Semi-Markov models ofreliability and operation. IBS PAN, v. 30.Warszawa.[5] Grabski, F. & Koowrocki, K. (1999). Asymptoticreliability of multistate systems with semi-Markovstates of components. Safety and Reliability, A.A.Balkema, Roterdam, 317-322.[6] Grabski, F. (2003). The reliability of the objectwith semi-Markov failure rate. AppliedMathematics and Computation, Elsevier.[7] Grabski, F. & Jawiski, J. (2003). Some problemsof the transport system modelling. ITE.[8] Grabski, F. (2005). Reliability model of the multistageoperation. Advances in Safety and Reliability.Koowrocki (ed.). Taylor & Fracis Group, London.ISBN 0 415 38340 4, 701-707[9] Grabski, F. (2006). Random failure rate process.Submitted for publication in Applied Mathematicsand Computation.[10] Grabski, F. (2006). Semi-Markov Model of anOperation. International Journal of Materials &Structural Reliability. V. 4, 99-113.[11] Kopociska, I. & Kopociski, B. (1980). Onsystem reliability under random load of elements.Aplicationes Mathematicae, XVI, 5-15.[12] Korolyuk, V. S. & Turbin, A.F. Semi-Markovprocesses end their applications, Naukova Dumka,Kiev.[13] Limnios, N. & Oprisan, G. (2001). Semi-MarkovProcesses and Reliability. Boston, Birkhauser.- 75 -

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special IssueGrabski FranciszekNaval University, Gdynia, PolandThe random failure rateKeywordsreliability, random failure rate, semi-Markov processAbstractA failure rate of the object is assumed to be a stochastic process with nonnegative, right continuous trajectories. Areliability function is defined as an expectation of a function of a random failure rate process. The properties andexamples of the reliability function with the random failure rate are presented in the paper. A semi-Markovprocess as the random failure rate is considered in this paper.1. IntroductionOften, the environmental conditions are randomlychangeable and they cause a random load of an object.Thus, the failure rate depending on the random load isa random process. The reliability function with semi-Markov failure rate was considered in the followingpapers Kopociski & Kopociska [5], [6], Grabski[3], [4].2. Reliability function with random failurerateLet { ( t): t ≥ 0}be a random failure rate of anobject. We assume that the stochastic process has thenonnegative, right continuous trajectories. Thereliability function is defined as⎡ t⎛ ⎞⎤R(t)= E⎢exp⎜− ∫ ( x)dx⎟⎥,t ≥ 0.(1)⎣ ⎝ 0 ⎠⎦It means that the reliability function is an expectationof the process { ( t) : t ≥ 0},whereLett⎛ ⎞ ( t)= exp⎜− ∫ ( x)dx⎟,t ≥ 0.(2)⎝ 0 ⎠(t⎛⎞R(t)= exp⎜− ∫ E[(x)]dx⎟, t ≥ 0.⎝ 0 ⎠(3)From Jensen’s inequality we get very important result⎡ t⎛ ⎞⎤R(t)= E⎢exp⎜− ∫ (x)dx⎟⎥⎣ ⎝ 0 ⎠⎦t⎛⎞ (≥ exp⎜− ∫ E[(x)]dx⎟= R(t),t ≥ 0.⎝ 0 ⎠(4)The above mentioned inequality means that thereliability function defined by the stochastic process{ (t): t ≥ 0} is greater than or equal to thereliability function with the deterministic failure rate,equal to the expectation λ ( t)= E[(t)].It is obvious, that for the stationary stochastic process{ ( t): t ≥ 0} , that has a constant mean valueλ ( t)= E[ ( t)]=λ , the reliability function defined by(3) is(t⎛ ⎞R(t)= exp⎜−λ∫dx⎟= exp( −λt),t ≥ 0.⎝ 0 ⎠(5)Hence, we come to conclusion: for each stationaryrandom failure rate process, the according reliabilityfunction for each t ≥ 0 , has values greater than orequal to the exponential reliability function withparameter λ .- 76 -

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special IssueExample 1.Suppose that, the failure rate of an object is astochastic process { ( t) : t ≥ 0}, given by ( t)= C t,t ≥ 0, where C is a nonnegative randomvariable. Trajectories of the process { (t) : t ≥ 0},aretξ ( t)= exp( −c), t ≥ 0,22where c is a value of the random variable C. Assumethat the random variable C has the exponentialdistribution with parameter β :−βuP( C ≤ u)= 1 − e , u ≥ 0.Then, according to (1), we compute the reliabilityfunction⎡ t⎛ ⎞⎤R(t)= E⎢exp⎜− ∫ Cxdx⎟⎥⎣ ⎝ 0 ⎠⎦⎛ 2 ⎞β2= β∞ ⎜ t−u+⎜⎝⎟⎟ ⎠∫ e0du =tt2∞ −u2 uβe−β0= ∫ eFigure 1 shows that function.10.80.60.40.222β+ 2βdu0.5 1 1.5 2Figure 1. Reliability function R(t)In that case the function (3) is(t2⎛ ⎞ ⎛ t ⎞R(t)= exp⎜− ∫ E[Cx]dx ⎟ = exp⎜⎟− , ≥ 0.⎝ 0 ⎠ 2t⎝ β ⎠Figure 2 shows that function.Suppose that a failure rate process { ( t) : t ≥ 0}is alinear function of a random load process { u ( t): t ≥ 0}: ( t)= ε u(t).10.80.60.40.20.5 1 1.5 2Figure 2. Reliability function R ( (t)Assume that the process { u(t): t ≥ 0}has an ergodicmean, i.e.T1lim u x dx = E u t = uTT ∫ ( ) [ ( )] .→∞0Then, [2], [3]lim R(t ) = exp[ −ut].ε →0εIt means, that for small εR( x)≈ exp[ −εux].3. Semi-Markov process as a random failurerateThe semi-Markov process as a failure rate and thereliability function with that failure rate wasintroduced by Kopociski & Kopociska [5]. Someextensions and developments of the results from [3]were obtained by Grabski [3], [4].3.1. Semi-Markov processes with a discretestate spaceThe semi-Markov processes were introducedindependently and almost simultaneously by P. Levy,W.L. Smith, and L.Takacs in 1954-55. The essentialdevelopments of semi-Markov processes theory wereachieved by Cinlar [1], Koroluk & Turbin [8],Limnios & Oprisan [7], Silvestrov [9]. We will applyonly semi-Markov processes with a finite or countablestate space. The semi-Markov processes are connectedto the Markov renewal processes.Let S be a discrete (finite or countable) state spaceand let R+= [ 0, ∞), N 0 = {0,1,2 ,...} . Suppose, thatξ , , n = 0,1,2,... are the random variables definedn ϑ non a joint probabilistic space ( Ω , F, P ) with valueson S and R+respectively. A two-dimensional randomsequence {( ξn, ϑ n), n = 0,1,2 ,...} is called a Markovrenewal chain if for alli ,...., i , i S,t ,..., t ∈ R n∈N0 n− 1∈0 n +,0.- 77 -

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special IssueThe equalities1. P{ ξ j, ϑ ≤ t | ξ = i,ϑ = t ,..., ξ = i t }2.n + 1=n+1 n n n 0 0, ϑ 0={ = j ϑ ≤ t | = i} Q ( )= P n + 1 , n+1 ξ n = ij tξ (6)P ξ =(7){ 0 = io , ϑ 0=0} = P{ξ 0 = i0}p i ohold.It follows from the above definition that a Markovrenewal chain is a homogeneous two-dimensionalMarkov chain such that the transition probabilities donot depend on the second component. It is easy tonotice that a random sequence { ξn: n = 0,1,2 ,...} is ahomogeneous one-dimensional Markov chain with thetransition probabilitiespijA matrix= P{ ξ ! = j | ξ = i}lim Q ( t).(8)n+ n =t→∞[ Q ( t):i j S]Q(t)= ij , ∈Is called a Markov renewal kernel. The Markovrenewal kernel and the initial distributionp = [ pi : i ∈ S]define the Markov renewal chain.That chain allows us to construct a semi-Markovprocess.Letτ= ϑ0= 0,τn= ϑ1+ ... + ϑn,τ∞= sup{ τn: n∈0}0NA stochastic process { X ( t): t ≥ 0}following relationnij0given by theX ( t)= ξ for t τ , τ )(9)∈[ n n+ 1is called a semi-Markov process on S generated bythe Markov renewal chain related to the kernelQ ( t),t ≥ 0 and the initial distribution p.Since the trajectory of the semi-Markov process keepsthe constant values on the half-intervals [ τ n, τ n+ 1)andit is a right-continuous function, fromequality X ( τn) = ξn, it follows that the sequence{ X ( τ n) : n = 0,1,2,... } is a Markov chain with thetransition probabilities matrixP = [ pij : i,j ∈ S].(10)The sequence { X ( ) : n = 0,1,2,... }τ nis called anembedded Markov chain in a semi-Markov processX ( t) : t ≥ 0 .{ }The function{ τ −τ≤ t | X ( τ ) = i,X ( j}Fij ( t)= P n+ 1 nn τ n+1)=Q ij ( t)= (11)pijis a cumulative probability distribution of a holdingtime of a state i , if the next state will be j . From(11) we haveQij ( t)= pijFij( t). (12)The function{ −τ≤ t | X ( ) = i} = ∑G ( t)= P τ 1 τ Q ( t)(13)in+nnj∈Sis a cumulative probability distribution of anoccupation time of the state i.N( t): t ≥ 0 defined byA stochastic process { }N ( t)= n for t τ , τ )(14)∈[ n n+ 1is called a counting process of the semi-Markovprocess { X ( t): t ≥ 0}.The semi-Markov process { X ( t): t ≥ 0}is said to beregular if for all t ≥ 0P { N(t)< ∞}= 1. (15)It means that the process { X ( t): t ≥ 0}has the finitenumber of state changes on a finite period.Every Markov process { X ( t): t ≥ 0}with the discretespace S and the right-continuous trajectorieskeeping constant values on the half-intervals, with thegenerating matrix of the transition ratesΑ=[ α : i,j ∈ S], 0 −α ii= α < ∞ is the semi-ij

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special Issuep = 0 .ii3.2. Semi-Markov failure rateSuppose that the random failure rate { λ ( t): t ≥ 0}isthe semi-Markov process with the discrete state spaceS = { λ j : j ∈ J},J = { 0,1,..., m}or J = {0,1,2 ,...},00 1

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special Issue⎡ 1~− G0( s + λ ) ⎤s+λ0~ ⎢01~ ⎥H ( s)= ⎢ − G1( s + λ1)⎥.(23)s+λ1⎢ 1~ ⎥⎢2 ( 2 )⎣− G s + λs+λ2⎥⎦The Laplace transform of unconditional reliabilityfunction is~ ~ ~ ~R ( s)= p R ( s)+ p R ( s)+ p R (Example 2.Assume thatand0 0 1 1 1 1sp0 = 1,p1= 0, p2= 0G ( t)= 1 − (1 + α t)eG01 (t)= 1 − e−β t,−αt−γtG2 ( t)= 1 − (1 + γ t)e , t ≥ 0.The corresponding Laplace transforms are~ αG0( s)=s(s + α)~G1(s)=s(sβ+22,β )~ γG2( s)=s(s + γ )~ αg 0 ( s)=( s + α)~ βg1(s)= s + βLet~ γg 2 ( s)=( s + γ )222,222,.,,,)α = .4, β = 0.04, γ = 0.02, λ = 0, λ = 0.1, λ 0.2 .00 12=Since the matrices (22) and (23) are[ I − ~ ( s)]=q λ⎡ 1⎢= ⎢−0.4⎢⎢ 0⎣⎡~ ⎢H(s)= ⎢⎢⎢⎣0.040.14+s1s1s+0.11s+λ2−−−−−0.0025(0.05+s)210.0004(0.22+s)20−0.60.0025s(0.05+s)20.04( s+0.1)(0.14+s)0.0004( s+0.2)(0.22+s)20.040.14+sFrom solution of equation (19), in this case, we obtain~ ~ a~ ( ( ) ~ sR ( s)= R)0s =b ( s)where1⎤⎥⎥.⎥⎥⎦a~ ( s)= (0.01623 + 0.23349s+ s⋅ (0.05002 + 0.44655s+ s⋅ (0.04882 + 0.44138s+22)⎤⎥⎥,⎥⎥⎦~b ( s)= (0.03083 + s)(0.07486+ s)(0.13292+ s)Using the MATHEMATICA computer program weobtain the reliability function as the inverse LaplacetransformR(t)= 0.51646e+ 2.28565e− 2 ⋅ 0.01539e−0.13292t−0.13292t−0.22069t2s+ 0.23349e))−0.07486tcos(0.01075t)−0.22069t− 2 ⋅ 0.01343ecos(0.01075t).Figure 3 shows this reliability function.p = [1, 0,0] , a = 0. 4and- 80 -

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special Issue10.80.60.40.220 40 60 80 100 120 140Figure 3. The reliability function from example 2The corresponding density function is shown inFigure 4.0.0140.0120.010.0080.0060.0040.00220 40 60 80 100 120 140Figure 4. The density function from example 23.4. The Poisson process as a failure rateSuppose that the random failure rate { λ(t): t ≥ 0}is thePoisson process with parameter λ > 0 . Of course, thePoisson process is the Markov process with thecounting state space S = {0,1,2 ,...}. That process canbe treated as the semi-Markov process defined on bythe initial distribution p = [1,0,0 ,...] and the kernel⎡0G0( t)0 0⎢⎢0 0 G1( t)0Q( t)= ⎢00 0 G2( t)⎢⎢...... ... ...⎢⎣...... ... ...where−λtG ( t)= 1 − e , t ≥ 0, i = 0,1,2,...i... ⎤...⎥⎥0⎥,⎥... ⎥... ⎥⎦The Poisson process is of course a Markov processtoo.Applying equation (19), Grabski [3] proved thefollowing theorem:If the random failure rate { λ ( t): t ≥ 0}is the Poissonprocess with parameter λ > 0 , than the reliabilityfunction defined by (16) takes formR( t)= exp{ −λ[ t −1+ exp( −t)]},t ≥ 0.The corresponding density function is given by theformulaf ( t)= λ exp{ −λ[ t −1+exp( −t)] }[1 − exp( −t)],t ≥ 0.Those functions with parameter λ = 0. 2 are shown inFigure 5 and Figure 6.10.80.60.40.25 10 15 20Figure 5. The reliability function for the Poissonprocess0.140.120.10.080.060.040.025 10 15 20Figure 6. The density function for the Poisson process3.5. The Furry-Yule process as a failure rateThe Furry-Yule is the semi-Markov process on thecounting state space S = {0,1,2 ,...} with the initialdistribution p = [1,0,0 ,...] and the kernel similar to thePoison process⎡0G0( t)0 0⎢⎢0 0 G1( t)0Q( t)= ⎢00 0 G2( t)⎢⎢...... ... ...⎢⎣...... ... ...whereG ( t)= 1 − ei−λ(i+1) t... ⎤...⎥⎥0⎥,⎥... ⎥... ⎥⎦, t ≥ 0, i = 0,1,2,...The Furry-Yule process is also the Markov process.Assume that the random failure rate { λ ( t): t ≥ 0}isthe Furry-Yule process with parameter λ > 0 . Thefollowing theorem is proved by Grab ski [4]:If the random failure rate { λ ( t): t ≥ 0}is the Furry-Yule process with parameter λ > 0 , then thereliability function defined by (1) is given by- 81 -

F.Grabski The random failure rate - RTA # 3-4, 2007, December - Special Issue( λ + 1) exp( −λt)R( t)= , t ≥ 0.1 + λ exp[ −(λ + 1) t]The corresponding density function isλ(λ + 1) exp[1 − ( λ + 1) t]f ( t)= , t ≥ 0.2{1 + λ exp[ −(λ + 1) t]}Those functions with parameter λ = 0. 2 are shown inFigure 7 and Figure 8.10.8[5] Kopociska, I. & Kopociski, B. (1980). Onsystem reliability under random load of elements.Aplicationes Mathematicae, XVI, 5-15.[6] Kopociska, I. (1984). The reliability of anelement with alternating failure rate. AplicationesMathematicae, XVIII, 187-194.[7] Limnios, N. & Oprisan, G. (2001). Semi-MarkovProcesses and Reliability. Boston, Birkhauser.[8] , . . & , . (1976). . «», .[9] , . . (1969). . . .0.60.40.25 10 15 20Figure 7. The reliability function for the Furry-Yuleprocess0.140.120.10.080.060.040.025 10 15 20Figure 8. The density function for the Furry-Yuleprocess4. ConclusionFrequently, because of the randomly changeableenvironmental conditions and tasks, the assumptionthat a failure rate of an object is a random processseems to be proper and natural. We obtain the newinteresting classes of reliability functions for thedifferent stochastic failure rate processes.References[1] Cinlar, E. (1961). Markov renewal theory. Adv.Appl. Probab. 1, No 2, 123-187.[2] Grabski, F. 2002. Semi-Markov models ofreliability and operation. Warszawa: IBS PAN.[3] Grabski, F. (2003). The reliability of the objectwith semi-Markov failure rate. AppliedMathematics and Computation, 135, 1-16.Elsevier.[4] Grabski, F. (2006). Random failure rate process.Submitted for publication in Applied Mathematicsand Computation.- 82 -

F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special IssueGrabski FranciszekZaska-Fornal AgataNaval University, Gdynia, PolandThe model of non-renewal reliability systems with dependent time lengthsof componentsKeywordsreliability, dependent components, series systems, parallel systemsAbstractThe models of the non-renewal reliability systems with dependent times to failure of components are presented.The dependence arises from some common environmental stresses and shocks. It is assumed that the failureoccurs only because of two independent sources common for two neighbour components. The reliability functionof series and parallel systems with components depending on common sources are computed. The reliabilityfunctions of the systems with dependent and independent life lengths of components are compared.1. IntroductionThe problem of determining the reliability function ofthe system with dependent components is importantbut difficult to solve. Many papers are devoted to it i.e.[1], [2], [3], [4]. In the Barlow & Proshan book [1975],there is defined, based on the reliability theory,multivariate exponential distribution as a distributionof a random vector, the coordinates of which aredependent random variables defining life lengths of thecomponents. Their dependence arises from someenvironmental common sources of shocks. Using thatidea we are going to present some examples of systemswith dependent components, giving up the assumptionthat the joint survival probability is exponential andaccepting the assumption that the failure occurs onlybecause of two independent sources common for twoneighbour components.Assume that due to reliability there are n orderedcomponentsE = ( e1,e2,..., en).Assume also that n+1 independent sources of shocksare present in the environmentZ( z1 , z2,..., z n, z+ 1)=nand each component e i can be destroyed only becauseof shocks from two sources z i and z i+1 .Let U i be non-negative random variable defining thetime to failure of the component caused by the shockfrom the source z i . Thus the life length of the objectdepends on the random vector= (1 2n+1U U , U ,..., U n, U ) . (1)Admit that the coordinates of the vector areindependent random variables with distributionsdefined as followsGi ( ui) = P(Ui≤ ui), i = 1, 2,..., n + 1.(2)The life length of the component e i is a randomvariable satisfyTi = min( Ui, Ui+ 1), i = 1,2,...,n . (3)Notice that two neighbour components in the sequence( e1,e2,..., en) have one common source of shock –depend on the same random variable. The randomvariables T 1 , T 2 ,…,T n are dependent. Their jointdistribution is expressed by means of the multivariatereliability function and it can be easily determined:- 83 -

F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special IssueR(t1, t2,..., tn) = P(T1> t , T12> t ,..., T2n> tn)R ( t , tijij) = P(Ti> t , Tij> tj)= P(min(U..., min(= P(U..., Un= P(U... P(UThusR(tn11, U, U> max( t1, tn2Un111n + 1> t , U22n−1> t ) P(U> max( t,..., tG (max( twhereG ( uiinn−1, t) = P(Ui = 1, 2,..., n + 1.) > t) > t> max( t, t2nn−1) G1), U> max( t, t, min( Un) = G ( tin1> un1)n+1i1, tn + 121, t)) P(U) G( tn2)2),> t, U2n)))n + 1(max( t) = 1−G ( uii3) > t> t ) .1n, t22,)) ...) = P(T ≤ ui),(4)The reliability functions of the components can beobtained as marginal distributions computing the limitof the function (4), when++++t1→ 0 ,..., ti−1→ 0 , ti+1→ 0 ,..., tn→ 0 .= G ( t ) Giii + 1 =j,i+1(max( t , tit could be proved thatP(T1= P(T> t | T111ii+1)) Gi,j = 1,2,..., n −1,2> t | Tand generallyP(T > ti1= P(T > ti| Ti| T> t ,..., T2i+1i+12> t> t2)i+1> tn,..., Ti+1> tnni+2)> tn)( ti+1),), i = 1,2,..., n −1.(7)(8)(9)That property asserts that the life length of e i dependsonly on the life length of the next component e i+1 , doesnot depend on the life lengths of the rest of thecomponents. That is a certain kind of Markov property.2. Reliability of the object with the seriesstructureIf the object has a series reliability structure then itslife length T is the random variable defined by theformulaT = min(T1 , T2,..., Tn) . (10)Using (4) we can determine the reliability function:R ( t ) = P(T > t ) = G ( t ) G+ 1(t ), i 1,2,..., n.(5)i i i i i i i i=R(t)= P(T > t)= P(T1> t,T2> t,...,Tn> t)The bivariate reliability functions can be determinedby computing the limit of (4), when= R(t,t,...,t)(11)tt1→ 0j+1+→ 0R ( t , tiji + 1 t , Tt) Gi+1> tj,i,j = 1,2,..., n − 1,jijj→ 0j)j+1( tj+),,..., tj−1→ 0+,(6)= G ( t)G12( t)... Gn( t)Gn+1( t).Let us compare the function with the reliabilityfunction of a series system in which the life lengths ofthe components T 1 , T 2 , …,T n are independent andtheir reliability functions are defined by (5). Let~R ( t),t ≥ 0 be a reliability function of that system. ItsatisfiesR ~( t)= P(T > t)= P(T > t,T2> t,K , Tn >)1t)= R1 ( t)R2( t)KRn( t- 84 -

F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special Issue= G1 ( t)G2( t)G2( t)G3(t)KGn ( t)Gn( t)Gn+1(t)= G t)G ( t)KGn ( t)R()(12)2( 3tThus, for t ≥ 0~R(t)≤ R(t)holds.The inequality means that the reliability of a seriessystem with dependent (in the considering sense) lifelengths of components is greater than (or equal) to thereliability of that system with independent life lengthsof components and the same distributions as themarginals of T 1 , T 2 , …, T n .Accepting the assumption about independence of thelife lengths of the components even though the randomvariables describing the life lengths are dependent, wemake an obvious mistake but that error is „safe”because the real series system has a greater reliability.That estimation is very conservative.Example 1.Assume that a non-negative random variable Ui,describing time to failure of the component caused bythe shock from source z i has a Weibull distributionwith parametersα i, λ i, i = 1,2,..., n + 1for u i > 0−λG ( u ) = P(U > u ) = ei u i, i = 1, 2,..., n + 1.iiiiThe reliability function of a series system withdependent components satisfiesR(t)= P(T > t)= G= eFor n = 3 andG−(λ1ta1+ ... + λn+1t a n+1 )ttαiGG1( )2( ) ...n( )n+1()ttR(t)10.80.60.40.20.5 1 1.5 2 tFigure 1. The graph of the series reliability functionwith dependent components~The reliability function R ( t),t ≥ 0 of the series systemwith independent life lengths of components, the samemarginals satisfies~ −(0.1t1.2+ 0.4t2+ 0.2t2.2+ 0.2 t 3)R(t)= P(T> t)= e3. Reliability of the object of the parallelstructureThe life length of the object of a parallel structure is arandom variable defined byT = max(T1 , T2,..., Tn) . (13)Let us compute the reliability function of the object:R(t)= P(T > t)= 1 − P(T ≤ t)= 1 − P(T= P({T11≤ t,T2> t}∪{T≤ t,...,T2n≤ t)> t}∪ ... ∪{Tn.> t}).(14)Using the formula of probability of a sum of events weobtainR(t)= P(T > t)= ∑ P(Tni=1in> t)− ∑ P(T > t,T > t)i,j=1i t)−...we get+ ( −1)n+1P(T1> t,T2> t,...,Tn> t) .R(t)= P(T> t)= e−(0.1t1.2+ 0.2t2+ 0.1t2.2+ 0.2t3)Hence and from (4), (6), (7) we getThe graph of the function is presented in Figure 1.- 85 -

F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special IssuenR( t)= ∑ G ( t)G + 1 ( t)− ∑ G ( t)G + 1(t)G ( t)G + 1(t)i=1n− − 1i=1iini,j=1i+1

F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special IssueSoA ∪ A ⊃ A ∩ A ) ∪ ( A ∩ ) ,PThus1 4(1 2 3A4( A1 41 2 3A4∪ A ) ≥ P((A ∩ A ) ∪ ( A ∩ )) .(R( t)≥ R(t).The inequality can be proved for any n by theinduction. It assures that the reliability of the parallelsystem with the independent components is greaterthan (or equal) to the reliability of that system withdependent components.Computing the reliability of the real systems we oftenassume that the components life lengths areindependent even though the random variablesdescribing the life lengths are dependent. That exampleshows that such assumption leads towards carelessconclusions. The real parallel system may havesignificantly lower reliability. Moreover, we come tothe similar conclusions if we take under considerationmore general assumption about the association of therandom variables T 1 , T 2 , …,T n [1].Example 2.Assume as previously that a non-negative randomvariable U i , describing time to failure of thecomponent caused by the source z i has a Weibulldistribution with parametersα i, λ i, i = 1,2,..., n + 1.Let n = 3. Then for u i > 0−λαiG ( u ) = P(U > u ) = ei u i, i = 1, 2,3,4 .iAs previouslyαii= 1.2, λ1= 0.1, α2= 2, λ21=i0.2,α3= 2.2,λ3= 0.1, α4= 3, λ4= 0.2.Using (16) we obtain the reliability function of theparallel system with dependent components. For t > 0it satisfiesR(t)10.80.60.40.21 2 3 4 5 6 tFigure 2. The graph of the reliability function of theparallel system with dependent componentsR( t ) = G1(t)G2( t)+ G2( t)G3( t)+ G3(t)G4( t)+ G1( t)G2( t)G3(t)− G2( t)G3(t)G4( t)= e+ e− e−( 0.1t1.2+ 0.2t2) −(0.2t2+ 0.1t2.2)+ e−( 0.1t2.2+ 0.2t3) −(0.1t1.2+ 0.2t2+ 0.1t2.2)− e−( 0.2t2+ 0.1t2.2+ 0.2t3).Figure 2. Presents its graph.4. ConclusionThe reliability of a series system with dependent (inthe considered sense) life lengths of components isgreater than (or equal) to the reliability of that systemwith independent life lengths of components.Assuming the independence of the life lengths of thecomponents even though the random variablesdescribing the life lengths are dependent, we make amistake but that error is „safe” because the real seriessystem has a greater reliability. The estimation of thereliability function is very conservative.The reliability of the parallel system with theindependent components is greater than (or equal) tothe reliability of that system with dependentcomponents.Computing the reliability of the real systems we oftenassume that the components life lengths areindependent even though the random variablesdescribing the life lengths are dependent. Theexamples presented here show that such assumptionleads towards careless conclusions. The real parallelsystem may have significantly lower reliability.- 87 -

F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special IssueReferences[1] Barlow, R.E & Proshan, F. (1975). Statisticaltheory of reliability and life testing. Holt, Reinhartand Winston Inc. New York.[2] Limnios, N. Dependability analysis of semi-Markovsystem. Reliab. Eng. and Syst. Saf.,55:203-207.[3] Navarro, J.M., Ruiz, C.J. & Sandoval. (2005). Anote on comparisons among coherent systems withdependent components using signatures, Statist.Probab. Lett. 72 179-185.[4] Rausand, M. A. & Høyland. (2004). SystemReliability Theory; Models, Statistical Methods andApplications", 2nd edition, Wiley, New York.- 88 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special IssueGuo RenkuanUniversity of Cape Town, Cape Town, South AfricaAn univariate DEMR modelling on repair effectsKeywordsgrey theory, DEMR, repair effect, credibility measure theory, random fuzzy variableAbstractRepairable system analysis is in nature an evaluation of repair effects. Recent tendency in reliability engineeringliterature was estimating system repair effects or linking repair to certain covariate to extract repair impacts byimposing repair regimes during system reliability analysis. In this paper, we develop a differential equationmotivated regression (abbreviated as DEMR) model with a random fuzzy error term based on the axiomaticframework of self-dual fuzzy credibility measure theory proposed by Liu [5] and grey differential equationmodels. The fuzzy variable indexes the random fuzzy error term will be used to facilitate the evaluation of repaireffects. We further propose a parameter estimation approach for the fuzzy variable (repair effect) under themaximum entropy principle.1. IntroductionRepairable system analysis is in nature an evaluationof repair effects. Recent tendency in reliabilityengineering literature was estimating system repaireffects or linking repair to certain covariate to extractrepair impacts by imposing repair regimes to thesystem. Guo [3], [4] proposed an approach to isolaterepair effects in terms of grey differential equationmodelling, particularly, the one-variable first orderdifferential equation model, abbreviated as GM (1,1)model, initiated by Deng [2]. The efforts of modellingof system repair effects in terms of grey differentialequation models has attracted attention from becauseit is easy to calculated, for example, in MicrosoftExcel. However, there were two fundamentalproblems necessary to be addressed. The first issue isthe nature of the GM(1,1) model. In The secondfundamental problem is GM(1,1) model is adeterministic approach and is just a delicateapproximation approach and in nature ignores theregression error structure, which may be veryreasonable if the sample size is too small, however, ingeneral, Deng's approach results in information loss,particularly he used the adjective word "grey",implying grey uncertainty involved, but there was notuncertainty structure build up to describe "greyuncertainty". In other words, the existing GM(1,1)model has a good idea without a convincinglyrigorous mathematical foundation yet.In this paper, we will review the coupling principlematerialization in GM(1,1) model in section 2. Insection 3, will propose a families of first orderdifferential equation motivated regression modelsunder unequal-gaped data, which is suitable for theusages in system functioning time analysis. In section4, we argue that the differential equation motivatedregression model is a coupling regression model withrandom fuzzy error terms in nature. In section 5,review Liu's [5] fuzzy credibility measure theory andthen discuss the random fuzzy variable theory in orderto establish the differential equation motivatedregression models as a coupling regression withrandom fuzzy error terms. In section 6, we willdiscuss the parameter estimation for the fuzzy variablerepair effect indexing the random fuzzy error terms ofthe differential equation motivated regressionmodelling on system functioning time sequence undermaximum entropy principle. Section 7 concludes thepaper.2. An univariate DEMR modelThe success of GM(1,1) model lies on the followingtwo aspects: data accumulative generation operator(abbreviated as AGO), which is the partial sumoperation in algebra, and a simple regression model- 89 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issuecoupled with a first-order linear constant coefficientdifferential equation model, which Deng [2] called isas whitening differential equation or the shadowdifferential equation. Let X (0) =(x (0) (1), x (0) (2),…, x (0)(n)) be a data sequence, and the partial sum withThe estimate for regression parameter pair ( β)denoted as ( a , b), can be calculated by,( a,b)T − T( X X ) X YT 1α, ,= (7)respect to X (0) k(1)(0)x ( k)= ∑ x ( i),k = 2,3,4,K,n , (1)i=1and the mean of two consecutive partial sums, whichis used as an approximation to the primitive function1of x t( )( )where⎡1⎢1X = ⎢⎢M⎢⎣1(1)− z (2) ⎤(1) ⎥− z (3) ⎥,M ⎥(1)⎥− z ( n)⎦Y⎡x⎢⎢x=⎢⎢⎣x(0)(0)(0)(2) ⎤⎥(3) ⎥ . (8)M ⎥⎥( n)⎦(1) (1)[ x ( k)+ x ( −1)](1) 1z ( k)= k . (2)2Definition 1. Given a (strictly positive) discrete realvalueddata sequence X (0) =(x (0) (1), x (0) (2),…, x (0) (n)),the equation(1)( − z ( k)) ε ,(0)x ( k)= α + β +k(3)k = 2,3,4,K,n ,“coupled” with the first-order constant coefficientlinear ordinary differential equation.(1)⎧dx( t)(1)⎪ + βx( t)= αdt⎨⎪(0)⎪⎩x ( k)= α + β(1)( − z ( k))+ ε, k = 2,3,4, K nk ,(4)is called a univariate DEMR model with respect to thedata sequence X (0) = (x (0) (1), x (0) (2), …, x (0) (n)) .Parameter is called the developing coefficient,parameter a is the grey input, term x (0) is called a greyderivative and term x (1) (k) is called the k th 1-AGO ofX (0) value (partial sum in fact). Furthermore, thedifferential equation dx (1) /dt + x (1) = in Eq. (4) iscalled the whitening differential equation or theshadow equation of the grey differential equation Eq.(3) by Deng [2]. The unknown parameter values ()can be estimated in terms of a standard regression.Note that Eq. (3) can be re-written as in a simpleregression form,ywhere= α + β + ε , k = 2,3,4,K,n , (5)kx kk(0)(1)y k= x ( k),x k= −z ( k),k = 2,3,4,K,n . (6)The grey filtering-prediction equation is thus(0)(1) (1)x € ( k)= x€( k)− x€( k −1), (9)wherex⎡( k + 1) = ⎢x⎣(1)(0)€a ⎤(1) − eb ⎥⎦− bk +ab. (10)Note that Eq. (10) is the discrete version of thesolution to the differential equation (Eq. (4))(1) ⎡ (1) α ⎤ −⎢β tx = x (0)− +β⎥e⎣ ⎦αβ. (11)The typical goodness-of-fit measure of GM(1,1)model is the (absolute) relative error described byDeng [2], i.e.(0) (0)x ( k)− x€( k)e( k)= , k = 2,3,4,K,n , (12)(0)x ( k)and the model efficiency is defined as1E =n −1n∑i=2e(i). (13)The nature of the univariate DEMR model can beidentified as that the model couples a differentialequation model and a simple regression modeltogether organically. The form of the motivateddifferential equation (i.e., Deng’s whiteningdifferential equation) in Eq. (4) determines the formof the coupling regression (i.e., CREG) in Eq. (3). Theab , in CREGdata assimilated parameter pair ( )determines the system parameter pair ( αβ , ) . Thecoupling translation rule is listed in Table 1.- 90 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special IssueTable 1. Coupling Rule in Univariate DEMT ModelTerm Motivated DE Coupling REGTranslation between MDE and CREGIntrinsic ContinuousDiscretefeatureIndependenttkVariable1 st -order(1)(0)dx ( t)/ dt x ( k)Derivative2 st -order (2) (1) 2( 1)d x ( t)/ dt x − ( k)DerivativePrimitivefunctionModelFormation(1)(1)x ( t)z ( k)( 1 )( t) ( 1βx)( t)dxdt+ = αParameter ( )Dynamics(Solution)Filtering(Prediction)( 1 )( t)x⎡⎢x⎣x( )( )( )( )0 1x k + bz k = aParameter Couplingα, β( a,b)=α⎤β⎥⎦(1) t(0) − e −β+(0)() t =⎡⎣α−βx(0)(1) ⎤⎦αβte −β(1)€ ( 1)x⎡⎢x⎣(0)k+ =(0)€ ()xa⎤−bka(1) − eb⎥+⎦ bt=€ ( + 1) −€()(1) (1)x k x kIn DEMR modelling, the motivated differentialequation and the coupling regression model are notseparable but are organic integration. The DEMRmodels are differential equation motivated but definedby system data. A DEMR model starts with amotivated differential equation, then the couplingregression model is specified in the form “translated”from the form of the motivated differential equation,in return, in terms of coupling regression model, theparameters specifying the motivated differentialequation are estimated under L 2 -optimality, andfinally, the solution to the motivated differentialequation (or the discredited solution) equipped withdata-assimilated parameters is used for systemanalysis or prediction. In nature a DEMR model is acoupling of a motivated differential equation and aregression formed by the discredited version of themotivated differential equation. We call the“translation” rule in grey differential equationmodelling as a coupling principle.3. Unequal-gapped differential equationmotivated regression model with term ofproduct of exponential and sine functionThe basic form of the first order linear differentialequation with constant term in right side isdx δ+ βx= αet ( ωt+ ϖ )dtsin (14)Note here, the proposal of the motivated differentialequation in Eq. (14) is featured by the termδtαesin ω t to replacing the constant term α in Eq. (4)with an intention that the fluctuating pattern ofδte ω t+ϖ will help the model goodness-of-fit.sin ( )Then the solution to Eq. (14) isx = x h+ x p(15)wherex−βth= c 0e , (16)is the solution to the homogeneous equationdx+ α x = 0(17)dtwhile a particular solution to the motivateddifferential equation Eq. (14) takes a formx( A ( ωt+ ϖ ) + B cos( ω + ϖ ))δtp= e sin0t0. (18)Note that xpsatisfies Eq. (14), thus substitute theparticular solution into Eq. (14), we obtaindxdtp+ βxpδt( ωt+ ϖ ) + B δecos( ω + ϖ )δt= A0δesin0tδt( ωt+ ϖ ) + B ωesin( ω + ϖ )δt+ A0ωecos0tδt( ωt+ ϖ ) + B βecos( ω + ϖ )δt+ A0βesin0t( ω ϖ )= αe δ t sin t + , (19)which leads to an equation system by comparing theδte sin ω t+ϖ and termcoefficients of term ( )δte ( ω t+ϖ)respectively,⎧A0( δ + β ) − B0ω= α⎪⎨⎪⎩ A0ω+ B0( δ + β ) = 0(20)Solving the linear equation Eq. (19), we obtainthe coefficients A0and B 0respectively as follows- 91 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issue⎧ ( δ + β ) α⎪ A0=22⎪ω + ( β + δ )⎨⎪αω⎪B0= −2⎩ ω + ( β + δ )2(21)In theory, the expressions of A0and B0willdetermine the particular solutionxpδt= A0esin( ωt+ ϖ )x pδ+ B et cos( ωt+ )(22)0ϖwhich will result in the general solution to Eq. (14) as−βtδtx = c1e+ A0esin( ωt+ ϖ )δ+ B et cos( ωt+ )(23)0ϖNote that for the unequal-gapped data sequence,( ) ( ) (( ) ) (X 0 = ( x 0 t 0 ( ) 0)1 , x t2, L , x ( tn)), the coupling (ortranslation) rule is slightly different from the equalgappeddata sequence.Table 2. Coupling Principle in unequal gappedGM(1,1) Model.Term Motivated DE Coupling REGModel FormationIntrinsic ContinuousDiscretefeatureIndependenttVariablet kResponse ( 0x) ( t)1 st -orderDerivative2 nd -orderDerivativePrimitivefunction( 0x ) ( t k )dx (1) ( t)/dt ( 0 ) ( )x t k0 02 (1) 2d x ( t)/dt x( )( ) − x( ) ( t )x(1) () tParameter ( )DynamiclawDynamics(Solution)Filtering(Prediction)t ktk− tk−1(1) ( ) z t kData Assimilation in Modelα, β( a,b)dxδt+β x =αe sin( ω t+ϖ)dt( 1 ) ( )δt0δt0−βt1x t = ce( t )( t )+ A e sin ω +ϖ+ B e cos ω +ϖ( 0 ) −βtx ( t)= −βce1δt+ ( A0ω+ B0δ) e cos( ω t+ϖ)δt+ ( A δ−B ω) e sin( ω t+ϖ)0 0The coupling regression isx(0)( tk)=kα δ tke sin( ωt+ ϖ )k −1( )( 0x )( tk) =αe ( 1sin( ω t ) )k+ϖ +β −z ( tk)δt k( 1 ) ( )kδtk0δtk0−βtk1x t = ce( tk)( t )+ A e sin ω +ϖ+ B e cos ω +ϖ( 0 ) −βt x ( t )kk = −βce1δt+ ( A0ω+ B0δ) e cos( ω tk+ϖ)δt+ ( A δ−B ω) e sin( ω t +ϖ)0 0kkwherez =z(1)( −z( )) ε ,+ β + k = 2,3,4,K,n ,(24)(1)(0)( t1)z ( t1)t1t kk( t − t )(1)(1)(0)( tk) = z ( tk−1) + z ( tk)k k −1k = 2,3,4,K,n . (25)The parameter pair ( αβ , ) is obtained by least-squareestimation ( , ) ( ) −1⎡e⎢⎢eX =⎢⎢⎢⎣e⎡z⎢⎢zY =⎢⎢⎢⎣zδt2δt3δtn(0)(0)(0)T T Ta b = X X X Y , wheresin( ωtsin( ωtMsin( ωt( t( tM(12t n) ⎤⎥) ⎥⎥⎥) ⎥⎦12n+ ϖ )+ ϖ )+ ϖ )− z− z− z(1)(1)M(1)( t( t( t12n) ⎤⎥) ⎥,⎥⎥) ⎥⎦(26)since δ and ω are given (in a manner by trials anderrors).Formally, we have a DEMR model as⎧dx⎪ + βx= αedt⎨⎪(0)⎪⎩x ( tk) = αeδtδtsin( ωt+ ϖ )sin( ωt+ ϖ ) + β4. Fuzzy repair effect structure(1)( − z ( t ))k+ ε .k(27)In standard regression modelling exercises, it is oftento assume that the error terms εi, i= 1,2, L , n arerandom with zero mean and constant variance, i.e.,2E ε = 0 and VAR [ ε ] =σ , i= 1,2, L , n. It is[ ]iitypically assuming a normal distribution with zero2mean and constant variance, i.e., ( 0, )N σ .Furthermore, as we pointed out that a grey differentialequation model is a motivated differential equationmotivated regression, which takes the form translatedfrom the motivated differential equation, as shown inTable 1 for GM(1,1) case. However, we should befully aware that translation back and forward betweenthe motivated differential equation and the coupling- 92 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issueregression will bring in new error which is different2from the random sampling error ( 0, )N σ . The errorsbrought in come from the steps of the usage of0 1 1difference x k = x k −x k −1to replace the( )( )derivative ( dx dt ) t = k( )( )accumulated partial sum( )( )and the usage of the average( )( )( 1 )( )z t to replace the1primitive function x tkduring the translationbetween the motivated differential equation and thecoupling regression.Our simulation studies have shown that the couplingintroducederror is dependent upon the grids size ∆ , orequivalent to the total number of approximation N.The simulation evidences have shown that the largerthe number of approximating grid, or equivalently, thesmaller the approximating grid, the couplingtranslation error is smaller. However, the couplingtranslation error and the approximating grid do nothold a deterministic functional relation. What we cansee is the functional relation has a certain degree ofbelongingness. In other words, the couplingtranslation process induces a fuzzy error term, denotedas ς with a membership function.We perform a simulation study of the error occurrencecos π 2 byfrequencies of approximating ( )( sin ( π 2) − sin ( π 2 +∆x)) ∆ x.frequencyerror's frequency Chart0.60.40.20-1 -0.5 -0.2 0 0.5 1errorFigure 1. Error occurrence frequencyTherefore, in general the error terms of a differentialequation motivated regression model (i.e., greydifferential equation in current grey theory literature)is fuzzy because the vague nature of the erroroccurrences.As a standard exercise, the fuzzy error componentmay be assumed as triangular fuzzy variable with amembership functionkei⎧s+ o if − o ≤ s < 0⎪⎪o⎪o− s if 0 ≤ s ≤ oµe( s)= ⎨(28)⎪ o⎪⎪0otherwise⎩which has a fuzzy mean zero.However, in the modelling of system functioningtimes, we further note that the repair will reset thesystem dynamic rule so that the repair impact may beunderstood as a fuzzy variable having a triangularmembership⎧ z − a⎪if a ≤ z < bb − a⎪c− zµr( z)= ⎨if b ≤ z < c (29)⎪c− b⎪0otherwise⎪⎩The fuzzy mean of the fuzzy repair effect is thus1Eµ( r)= ( a + 2b+ c), (30)4which provides a repair effect structure. Therefore, the“composite” fuzzy “error” term appearing in thedifferential equation motivated regression formodelling a system function time will beζi= e + r ,iii = 2,3,K,n , (31)with a triangular membership function, i.e.,⎧w− a + ϖ⎪if a −ϖ≤ w < bb − a + ϖ⎪c+ ϖ − wµ ζ ( w)= ⎨if b ≤ w < c + ϖ (32)⎪ c + ϖ − b⎪0otherwise⎪⎩because the sum of two triangular fuzzy variables isstill a triangular fuzzy variable. The total errorξ = ζ + ε = r + e ) + ε , i = 2,3,K,n , (33)iii(i i iwhich is a sequence of random fuzzy variablesbecause the summation nature of a random fuzzyvariable and a fuzzy variable according to Liu [5].Now, we reach a point that the random fuzzy variableconcept is involved and therefore it is necessary to- 93 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issuehave a quick review on the relevant theoreticalfoundation.5. A random fuzzy variable foundationFirst we need to review the fuzzy credibility measuretheory foundation proposed by Liu [5], then we willestablish the normal random fuzzy variable theory fora facilitation of error analysis in the differentialequation motivated regression models.Let Θ be a nonempty set, and 2 Θ the power set onΘΘ . Each element, let us say, A ⊂Θ, A∈ 2 is calledCr A ,an event. A number denoted as { }Θ0≤Cr{ A}≤ 1, is assigned to event A∈2, whichindicates the credibility grade with which eventΘA∈ 2 occurs. Cr{ A}satisfies following axiomsgiven by Liu [5]:Axiom 1. Cr{ Θ } = 1.Axiom 2. Cr{}⋅ is non-decreasing, i.e., wheneverA⊂ B, Cr{ A} ≤ Cr{ B}.Axiom 3. Cr{}⋅ is self-dual, i.e., for anyΘcA∈ 2 , Cr{ A} + Cr{ A } = 1.Axiom 4. Cr{ U A} ∧ 0.5 = sup ⎡Cr{ A}⎤ for any { }with { }Cr A ≤ 0.5 .ii i iiΘAxiom 5. Let set functions Cr { ⋅}:2 k→ [0,1]satisfyAxioms 1-4, and Θ=Θ1×Θ 2× L ×Θp, then:Cr{ θ θ , θ } = Cr{ θ } ∧Cr{ θ } ∧L∧Cr{ θ }1 2K,p12, (34)θ1, θ2,Kθ p∈Θfor each { , } 2 .Definition 5.1. Liu [5] Any set functionCr : 2 Q ® [ 0,1]satisfies Axioms 1-4 is called a ( ∨, ∧ ) -credibility measure (or classical credibility measure).ΘThe triple ( Θ ,2 ,Cr)is called the ( ∨, ∧)-credibilitymeasure space.Definition 5.2. Liu [5] A fuzzy variable ξ is aΘmapping from credibility space ( ,2 ,Cr)Θof real numbers, i.e., ξ :( Θ,2 ,Cr)→R .k⎣⎦pA iΘ to the setDefinition 5.3. Liu [5] The (induced) membershipΘΘ ,2 ,Cr is:function of a fuzzy variable ξ on ( )( 2Cr{ ξ = }) 1,µ ( x)= x ∧ x ∈ R(35)Conversely, for given membership function thecredibility measure is determined by the credibilityinversion theorem.Theorem 5.4. Liu [5] Let ξ be a fuzzy variable withmembership function m. Then for ∀B ⊂ R ,1 ⎛⎞Cr { ξ ∈ B} = ⎜supµ ( x)+ 1−sup µ ( x)⎟ (36)2 ⎝ x∈BCx∈B⎠As an example, if the set B is degenerated into a pointx, thenCr1 ⎛2 ⎝{ ξ = x} = ⎜ µ ( x)+ 1−sup µ ( y)⎟,y ≠ x⎞⎠∀x ∈ R (37)Definition 5.5. Liu [5] The credibility distributionΘΦ: → 0,1Θ ,2 ,Cr is[ ]R of a fuzzy variable ξ on ( ){ ∈ Θξ( ≤ x}Φ ( x)= Cr θ θ ) . (38)The credibility distribution Φ ( x)is the accumulatedcredibility grade that the fuzzy variable ξ takes avalue less than or equal to a real number x ∈ R .Generally speaking, the credibility distribution isneither left-continuous nor right-continuous.Theorem 5.6. Liu [5] Let ξ be a fuzzy variable onΘ( ,2 ,Cr)Θ with membership function . Then itscredibility distribution,1 ⎛⎞Φ( x)= ⎜supµ ( y)+ 1−sup µ ( y)⎟,∀x ∈ R (39)2 ⎝ y≤xy>x ⎠Definition 5.7. Liu [5] Let be the credibilitydistribution of the fuzzy variable ξ . Then functionφ : R → [0, +∞)of a fuzzy variable ξ is called acredibility density function such that,xΦ( x)= ∫φ(y)dy,∀x ∈ R . (40)−∞Now we are ready to state the normal random fuzzyvariable theory for the error analysis in the repairablesystem modelling.Liu [5] defines a random fuzzy variable as a mappingΘfrom the credibility space ( Θ ,2 ,Cr)to a set ofrandom variables. We would like to present adefinition similar to that of stochastic process inprobability theory and expect readers who are familiar- 94 -

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issuewith the basic concept of stochastic processes canunderstand the comparative definition.Definition 5.8. A random fuzzy variable, denoted as{ X βθ ( ),}ξ= θ∈Θ , is a collection of random variablesX βdefined on the common probability space( Ω, A, Pr)and indexed by a fuzzy variable βθ ( )Θdefined on the credibility space ( ,2 ,Cr)Θ .Similar to the interpretation of a stochastic process,+X = X , t∈R , a random fuzzy variable is a{ t }bivariate mapping from ( Ω×Θ , × 2 Θ)A to the space( R,B ). As to the index, in stochastic process theory,index used is referred to as time typically, which is apositive (scalar variable), while in the random fuzzyvariable theory, the “index” is a fuzzy variable, say,β . Using uncertain parameter as index is not startingin random fuzzy variable definition. In stochasticprocess theory we already know that the stochastic{ }process X = X τω ( ),ω∈Ω uses stopping timet( w), w W, which is an (uncertain) random variableas its index.In random fuzzy variable theory, we may say that thataverage chance measure, denoted as ch , plays asimilar role similar to a probability measure, denotedas Pr , in probability theory.Definition 5.9. Liu and Liu [6] Let x be a randomfuzzy variable, then the average chance measuredenoted by ch{}, of a random fuzzy event { ξ ≤ x},is{ ≤ x}=1{ ≥ α}ch ξ θ ∈ Θ Pr{ ξ ( θ ) ≤ x}∫Cr dα. (41)0Then function () is called as average chancedistribution if and only if{ ξ ≤ x}Ψ( x)= ch . (42)Liu [5] stated that if a random variable η has zeromean and a fuzzy variableζ , then the sum of the two,η + ζ , results in a random fuzzy variable ξ . Now, itis time to find the average chance distribution for anormal random fuzzy variable x N ( zs ,2)d: , where ζ is2a triangular fuzzy variable and σ is a given positivereal number. Note that fuzzy eventθ ∈ Θ :Pr ξ ( θ)≤ x ≥ α{ { } }⎧ ⎛ x − ζ⇔ ⎨θ∈ Θ : Φ⎜⎩ ⎝ σ⇔(43)≥ ζ ( θ )( θ )⎞ ⎫⎟ ≥ α ⎬⎠ ⎭{ : x−θ ∈ Θ + σΦ1 ( α )}−{ θ ∈ Θ : ζ ( θ ) ≤ −σΦ1 ( α)}⇔ x (43)The fuzzy mean is assumed to have a triangularmembership functionand⎧ w − aζ⎪aζ≤ w ≤ bζ⎪ bζ− aζ⎪ cν− wµζ( w ) = ⎨bζ≤ w ≤ aζ(44)⎪ cν− bν⎪⎪ 0 otherwise⎩⎧⎪ 0 w

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issue( α )−∞< g

R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issuerepair effect is zero for estimation parameter ο forspecifyingε i, the translation error because undertriangular membership assumption, the empiricalmembership can be defined and satisfies theasymptotical requirements.7. ConclusionIn this paper, we argue that a differential equationmotivated regression model will result in a regressionmodel with random fuzzy error terms and thuscomplete our mission for solidifying a rigorousmathematical foundation for the grey modelling onsystem repair effects proposed by Guo [3], [4]. Themaximum entropy principle facilitates a way for fuzzyparameter estimation. However, the average changedistribution is also providing a way for parameterdata-assimilation.AcknowledgementsThis work was supported partially by South AfricaNatural Research Foundation GrantFA2006042800024. We are grateful to ProfessorB.D. Liu for his constant encouragements of applyinghis fuzzy credibility measure theory. The errorfrequency in Figure 1 is provided by my MSc student,Mr Li Xiang.References[1] De Luca, A. & Termini, S. (1972). A Definition ofNon-probabilistic Entropy in the Setting of FuzzySets Theory. Information and Control 20, 301-312.[2] Deng, J. L. (1985). Grey Systems (Social Economical). The Publishing House of DefenceIndustry, Beijing (in Chinese).[3] Guo, R. (2005 a ). Repairable System Modelling ViaGrey Differential Equations. Journal of GreySystem, 8 (1), 69-91.[4] Guo, R. (2005 b ). A Repairable System Modellingby Combining Grey System Theory with Interval-Valued Fuzzy Set Theory. International Journal ofReliability, Quality and Safety Engineering, 12 (3),241-266.[5] Liu, B. (2004). Uncertainty Theory: AnIntroduction to Its Axiomatic Foundations.Springer-Verlag Heidelberg, Berlin.[6] Li, P. & Liu, B. Entropy of CredibilityDistributions for Fuzzy Variables. IEEEETransactions on Fuzzy Systems,( to be published).- 97 -

S.Guze Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems - RTA # 3-4, 2007, December - Special IssueGuze SamborMaritime University, Gdynia, PolandNumerical approach to reliability evaluation of two-state consecutive “kout of n: F” systemsKeywordstwo-state system, consecutive “ k out of n : F” system, reliability, algorithmAbstractAn approach to reliability analysis of two-state systems is introduced and basic reliability characteristics for suchsystems are defined. Further, a two-state consecutive “ k out of n : F” system composed of two-statecomponents is defined and the recurrent formulae for its reliability function are proposed. The algorithm fornumerical approach to reliability evaluation is given. Moreover, the application of the proposed reliabilitycharacteristics and formulae to reliability evaluation of the system of pump stations composed of two-statecomponents is illustrated.1. IntroductionThe assumption that the systems are composed of twostatecomponents gives the possibility for basicanalysis and diagnosis of their reliability. Thisassumption allows us to distinguish two states ofsystem reliability. The system works when itsreliability state is equal to 1 and is failed when itsreliability state is equal to 0. In the stationary case thesystem reliability is the independent of time probabilitythat the system is in the reliability state 1. The mainresults determining the stationary reliability and thealgorithms for numerical approach to this reliabilityevaluation for consecutive “k out of n: F” systems aregiven for instance in [1], [5]-[6]. An exemplarytechnical consecutive “k out of n: F” system can befound in [3]. There is considered the ordered sequenceof n relay stations E1, E2,K,En,, which have toreroute a signal from a source station E0to a targetstation En+ 1. A range of each station is equal to k. Itmeans, when the station Ei, i = 0,1,...,n,is operating,it sends a signal directly to a stationEi+ 1,..., Emin(i+k,n).The failed station does not send anysignal. The probability of efficiency of the stations E0and En+ 1is equal to 1. The signal from E 0to E n+ 1cannot be sent, if at least k consecutive stations out ofE1, E2,K,En,are damaged.The paper is devoted to extension of these stationaryresults to the non-stationary case and applying them intransmitting then for two-state consecutive “ k out ofn : F” systems with dependent of time reliabilityfunctions of system components ([3]). Then, thereliability function, the lifetime mean value and thelifetime standard deviation are basic characteristics ofthe system.2. Reliability of two-state consecutive “k out ofn: F” systemsIn the non-stationary case of two-state reliabilityanalysis of consecutive “k out of n: F” systems weassume that ([3]):– n is the number of system components,– Ei, i = 1,2,...,n,are components of a system,T are independent random variables representing–ithe lifetimes of components Ei, i = 1,2,...,n,– Ri ( t)= P(Ti> t),t ∈< 0, ∞),is a reliabilityfunction of component Ei, i = 1,2,...,n,– F ( t)= 1−R ( t)= P(T ≤ t),t ∈< 0, ∞),is theiii- 98 -

S.Guze Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems - RTA # 3-4, 2007, December - Special Issuedistribution function (unreliability function) ofcomponent Ei, i = 1,2,...,n.Definition 1. A two-state system is called a two-stateconsecutive “ k out of n : F” system if it is failed if andonly if at least its k neighbouring components out ofn its components arranged in a sequence of E1,E 2,..., En,are failed.The following auxiliary theorem is proved in [3], [6].Lemma 1. The stationary reliability of the two-stateconsecutive “ k out of n : F” system composed ofcomponents with independent failures is given by thefollowing recurrent formula⎧⎪1for n < k,⎪n⎪⎪1−∏qjfor n = k,⎪j=1Rk , n= ⎨ pnRk,n−1(1)⎪ k −1⎪+∑ pn−iRk,n−i−1⎪ i=1⎪ n⎪⋅>⎪∏ qjfor n k,⎩ j=n−i+1where– piis a stationary reliability coefficient ofcomponent Ei, i = 1,2,...,n,q is a stationary unreliability coefficient of–icomponent Ei, i = 1,2,...,n,R ,is the stationary reliability of consecutive “k–k nout of n: F” system.After assumption that:is a random variable representing the lifetime– Tk , nof a consecutive “k out of n: F” system,– Rk , n( t)= P(Tk,n> t),t ∈< 0, ∞),is the reliabilityfunction of consecutive “k out of n: F” system,– F t)= 1−R ( t)= P(T ≤ t),t ∈< 0, ), isk , n(k,nk , n∞the distribution function of consecutive “k out of n:F” system,we can formulate the following result.Lemma 2. The reliability function of the two-stateconsecutive “ k out of n : F” system composed ofcomponents with independent failures is given by thefollowing recurrent formula⎧⎪1for n < k,⎪n⎪⎪1−∏Fj( t)for n = k,⎪j=1Rk , n(t)= ⎨Rn(t)Rk,n−1(t)(2)⎪ k −1⎪+∑ Rn−i( t)Rk, n−i−1(t)⎪ i=1⎪ n⎪⋅>⎪∏ Fj( t)for n k,⎩ j=n−i+1for t ∈< 0,∞).Motivation. When we assume in formula (1) thatpi ( t)= Ri( t),qi ( t)= Fi( t)for t ∈< 0,∞),i = 1,2,...,n,we get formula (2).From the above theorem, as a particular case for thesystem composed of components with identicalreliability functions, we immediately get the followingcorollary.Corollary 1. If components of the two-stateconsecutive “ k out of n : F” system are independentand have identical reliability functions, i.e.R i( t)= R(t),F i( t)= F(t)for t ∈< 0,∞),i = 1,2,...,n,then the reliability function of this system is given by⎧⎪1for n < k,⎪n⎪1− [ F(t)]for n = k,⎪Rk,n( t)= ⎨R(t)R, −1( t)(3)k n⎪ k −1i⎪+R(t)∑ F ( t)⎪ i=1⎪⎩⋅Rk,n−i−1( t)for n > k,for t ∈< 0,∞).In further considerations we will used the followingreliability characteristics:- 99 -

S.Guze Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems - RTA # 3-4, 2007, December - Special Issue- the mean value of the system lifetime,E [ T k ,n]∞= ∫0R k,n (t)dt, (4)- the second order ordinary moment of the systemlifetime,∞2, n] = 2∫0E[ Tk t R k,n (t)dt, (5)- the standard deviation of the system lifetime,σ = D[ T k ,n],(6)whereD[T22)] = E[T ] − ( E[T ]) .(7)k, nk , nk,n3. Algorithm for reliability evaluation of a twostateconsecutive „k out of n: F” systemFor numerical approach to evaluation of the reliabilitycharacteristics, given by (3)-(6), we use the trapeziumrule of numerical integration.In particular situation, for t = 0 0, step h , we have2. If k > n then ( t)13. else if k = nR k , n=Rk, n( ) = 1−t [ F(t)]4. else5. for i = 0 to t do6. {7. for j = 1 to k – 1 doj8. temp := temp + F(i)]⋅ R ( );[k, n−j−1i( i)⋅ Rk, n− 1(i)temp;9. R ( ) = R +10. }k, niwhere- k is a length of the sequence of consecutivecomponents,- n is a number of all components in sequence,- t is an end of the time interval,- F(t) is a distribution function of components,- R(t) is a reliability function of components.Example implementation of Algorithm 1 and formulas(3), (8)-(9) in the D programming language is given inAppendix.4. ApplicationFrom Corollary 1, in a particular case, substitutingk = 3 in (3), we get:- for n = 1nE [ T k ,nh=2]n 1∑ −i=0∞= ∫0R k,n (t)dt[ R ( t + ih)+ R ( t + ( i + 1) ⋅ h)],k , n0k , n0(8)R t)1 for t ∈< 0,∞),(10)( 3 ,1=- for n = 2R ( t)1, for t ∈< 0,∞),(11)3 ,2=∞2k , n] = 2∫0E[ T t R k,n (t)dt= hn∑ − 1i=0{( t0 + ih)⋅ Rk,n( t0+ ih)+ t + ( i + 1) ⋅ h)⋅ R ( t + ( i + 1) ⋅ )}. (9)(0 k,n 0hNecessary in (7)-(8) values of function R k,n (t) arecalculated from (2) using the following algorithm.- for n = 33R ( t)= 1− F ( ) for t ∈< 0,∞),(12)3,3t- for n ≥ 4R ( ) = R(t)R ) + R( t)F(t)R ( )3,nt( 3,n−1 t2+ R( t)[F(t)]( 3,n−3 t)3,n−2tR for t ∈< 0,∞),(13)Algorithm 1.1. Given: t, k, n, F(t), R(t);Example 1. Let us consider the pump stationssystem with n = 20 pump stations E1, E2,..., E20.We assume that this system fails when at least 3consecutive pump stations are down. Thus, the- 100 -

S.Guze Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems - RTA # 3-4, 2007, December - Special Issueconsidered pump stations system is a two-stateconsecutive “3 out of 20: F” system, andaccording to (9)-(12), its the reliability function isgiven byR ( ) = R(t)R 3( t)3,20tfor t ∈< 0,∞).,19+ R( t)F(t)R 3( t),182+ R( t)[F(t)]3,17( t)R (14)In the particular case when the lifetimes Ti, of thepump stations Ei,i = 1,2,K,20 have exponentialdistributions of the formF(t)−0.01t= 1−e for t ≥ 0,i.e. if the reliability functions of the pump stations E i,i = 1,2,K,20are given byR(t)−0.01t= e for t ≥ 0,considering (9)-(12), (13) we get the followingrecurrent formula for the reliability R ( ) of pumpstations system3,20tR t)1 for t ∈< 0,∞),(15)( 3 ,1=R t)1 for t ∈< 0,∞),(16)( 3 ,2=0.01t3R 3t)= 1 − [1 − e − ] for t ∈< 0,∞),(17)R,3 (( ) = e −0. 01t R )3,nt+ e −0. 01t.01[1 e −0 t ]+ e −0. 01t[1n = 4,5,...,20.( 3,n−1 t− R ( )3,n−2t0.01t2− e − ] ( 3,n−3 t)R for t ∈< 0,∞),(18)The values of reliability function of the system ofpump stations given by (14), calculated by thecomputer programme based on the formulae (10)-(18)and Algorithm 1, are presented in Table 1 andillustrated in Figure 1.Table 1. The values of the two-state reliability functionof the pump stations system for λ = 0. 01t R ( )3,20t2 t R 3( t)0.0 1.0000 0.00005.0 0.9980 9.980010.0 0.9859 19.718915.0 0.9583 28.749920.0 0.9137 36.547425.0 0.8535 42.674330.0 0.7811 46.865735.0 0.7008 49.056140.0 0.6170 49.361445.0 0.5337 48.034750.0 0.4541 45.411755.0 0.3805 41.858460.0 0.3144 37.728265.0 0.2564 33.333170.0 0.2066 28.927475.0 0.1647 24.702480.0 0.1299 20.789385.0 0.1016 17.266290.0 0.0787 14.168895.0 0.0605 11.5004100.0 0.0462 9.2416105.0 0.0350 7.3588110.0 0.0264 5.8107115.0 0.0198 4.5531120.0 0.0148 3.5426125.0 0.0109 2.7385130.0 0.0081 2.1044135.0 0.0060 1.6082140.0 0.0044 1.2229145.0 0.0032 0.9255150.0 0.0023 0.6974155.0 0.0017 0.5235160.0 0.0012 0.3916165.0 0.0009 0.2918170.0 0.0006 0.2168175.0 0.0004 0.1607180.0 0.0003 0.1188185.0 0.0002 0.0876190.0 0.0002 0.0644195.0 0.0001 0.0473200.0 0.0000 0.0347205.0 0.0000 0.0253210.0 0.0000 0.0185215.0 0.0000 0.0135220.0 0.0000 0.0098225.0 0.0000 0.0072230.0 0.0000 0.0052235.0 0.0000 0.0038240.0 0.0000 0.0027245.0 0.0000 0.0020,20- 101 -

S.Guze Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems - RTA # 3-4, 2007, December - Special Issue10.90.80.70.60.50.40.30.20.10250.0 0.0000 0.0014255.0 0.0000 0.0010260.0 0.0000 0.0007265.0 0.0000 0.0005270.0 0.0000 0.0004275.0 0.0000 0.0003280.0 0.0000 0.0002285.0 0.0000 0.0001R 3,20 (t)0 50 100 150 200Figure 1. The graph of the pump stations systemreliability functionUsing the values given in the Table 1, the formulae(4)-(9) and numerical integration we find:- the mean value of the pump stations system lifetimeE[ T 3, 20]∞= ∫0R 3,20 (t)dt ≅ 50.8639,- the second order ordinary moment of the pumpstations system lifetimeE[2T 3,20(1)] =∞2∫t R 3,20 (t)dt ≅ 3246.69,0- the standard deviation of the pump stations systemlifetimeσ = D[ T 3, 20] = 659.558 ≅ 25.6819.5. ConclusionTwo recurrent formulae for two-state system reliabilityfunctions, a general one for non-homogeneous and itssimplified form for homogeneous two-stateconsecutive “ k out of n : F” systems have beenproposed. The algorithm for reliability evaluation ofttwo-state consecutive “k out of n: F” system has beenshown as well. The formulae and algorithm for twostatereliability function of a homogeneous two-stateconsecutive “ k out of n : F” system have been appliedto reliability evaluation of the pump stations system.The considered pump stations system was a two-stateconsecutive “3 out of 20: F” system composed ofcomponents with exponential reliability functions. Onthe basis of the recurrent formula and the algorithm fortwo-state pump stations system reliability function theapproximate values have been calculated and presentedin table and illustrated graphically. On the basis ofthese values the mean value and standard deviation ofthe pump stations system lifetime have been estimated.The input structural and reliability data of theconsidered pump stations system have been assumedarbitrarily and therefore the obtained its reliabilitycharacteristics evaluations should be only treated as anillustration of the possibilities of the proposed methodsand solutions.The proposed methods and solutions and the softwareare general and they may be applied to any two-stateconsecutive “k out of n: F” systems.AppendixWe present the D programming language code forformulas (3), (8)-(9) and Algorithm 1.import std.stdio;import std.stream;import std.math;import std.string;const real LAMBDA1 = 0.01;real Ft(real t) {return (1-exp(-(LAMBDA1)*t));}real Rt(real t) {return exp(-(LAMBDA1)*t);}real SigmaFi(real ii, real k, real t, real n) {real result = 0;for(real i = ii; i < k; i++) {result += pow(Ft(t),i)*Rkn(t,k,n-i-1);}return result;}real Rkn(real t, real k, real n) {if (n < k)return 1;- 102 -

S.Guze Numerical approach to reliability evaluation of two-state consecutive “k out of n: F” systems - RTA # 3-4, 2007, December - Special Issue}if (n == k)return 1 – pow(Ft(t),n);return Rt(t)*(Rkn(t, k, n-1) + SigmaFi(1, k, t, n));real trapeziumT(real k, real n, uint p, real t){real integ = 0;real step = 0;}step=t/p;for(real i = 0; i < p; i = i + step){integ += (((Rkn(i, k, n) +Rkn(i + step, k, n))*step)/2);}return integ;real trapezium2T(real k, real n, uint p, real t){real integ = 0;real step = 0;}step=t/p;for(real i=0; i < p; i = i + step){integ += (((i*Rkn(i, k, n) +(i + step)*Rkn(i + step, k, n))*step));}return integ;int main(char[][] args) {real integral = 0;real integral1 = 0;real dif = 0;real sq = 0;if (args.length < 3) {writefln("Usage:\n "~ args[0] ~" t k n\n");return 0;}for(real i = 0; i < atoi(args[1]); i = i + 5){writefln("%s\t%4s\t%4s\t%s", i, Rkn(i,atoi(args[2]), atoi(args[3])), 2*i*Rkn(i,atoi(args[2]),atoi(args[3])), 1 - Rkn(i,atoi(args[2]), atoi(args[3])) );}integral=trapeziumT(atoi(args[2]), atoi(args[3]),atoi(args[4]) );}diff=(integral1)-pow(integral,2);sq=sqrt(diff);writefln(“The mean value of the system lifetime”);writefln("%s", integral );writefln(“The second order ordinary moment of thesystem lifetime”);writefln("%s", integral1 );writefln("%s", diff);writefln(“The standard deviation of the systemlifetime”);writefln("%s",sq);return 0;References[1] Antonopoulou, J. M. & Papstavridis, S. (1987). Fastrecursive algorithm to evaluate the reliability of acircular consecutive-k-out-of-n: F system. IEEETransactions on Reliability, R-36, 1, 83 – 84.[2] Barlow, R. E. & Proschan, F. (1975). StatisticalTheory of Reliability and Life Testing. ProbabilityModels. Holt Rinehart and Winston, Inc., NewYork.[3] Guze, S. (2007). Wyznaczanie niezawodnocidwustanowych systemów progowych typu„kolejnych k z n: F”. Materiay XXXV SzkoyNiezawodnoci, Szczyrk.[4] Kossow, A. & Preuss, W. (1995). Reliability oflinear consecutively connected systems with multistatecomponents. IEEE Transactions onReliability, 44, 3, 518-522.[5] Malinowski, J. & Preuss, W. (1995). A recursivealgorithm evaluating the exact reliability of aconsecutive k-out-of-n: F system. Microelectronicsand Reliability, 35, 12, 1461-1465.[6] Malinowski, J. (2005). Algorytmy wyznaczanianiezawodnoci systemów sieciowych o wybranychtypach struktur. Wydawnictwo WIT, ISBN 83-88311-80-8, Warszawa.[7] Shanthikumar, J. G. (1987). Reliability of systemswith consecutive minimal cut-sets. IEEETransactions on Reliability, 26, 5, 546-550.integral1=trapezium2T(atoi(args[2]), atoi(args[3]),atoi(args[4]) );- 103 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special IssueGuze SamborKoowrocki KrzysztofMaritime University, Gdynia, PolandReliability analysis of multi-state ageing consecutive „k out of n: F”systemsKeywordsmulti-state system, ageing system, consecutive “ k out of n : F” system, reliabilityAbstractA multi-state approach to reliability analysis of systems composed of ageing components is introduced and basicreliability characteristics for such systems are defined. Further, a multi-state consecutive “ k out of n : F” systemcomposed of ageing components is defined and the recurrent formulae for its reliability function are proposed.Moreover, the application of the proposed reliability characteristics and formulae to reliability evaluation of thesteel cover composed of ageing sheets is illustrated.1. IntroductionTaking into account the importance of the safety andoperating process effectiveness of technical systems itseems reasonable to expand the two-state approach tomulti-state approach in their reliability analysis. Theassumption that the systems are composed of multistatecomponents with reliability states degrading intime [4]-[5], [10] gives the possibility for more preciseanalysis and diagnosis of their reliability andoperational processes’ effectiveness. This assumptionallows us to distinguish a system reliability criticalstate to exceed which is either dangerous for theenvironment or does not assure the necessary level ofits operational process effectiveness. Then, animportant system safety characteristic is the time to themoment of exceeding the system reliability criticalstate and its distribution, which is called the systemrisk function. This distribution is strictly related to thesystem multi-state reliability function that is a basiccharacteristic of the multi-state system. The mainresults determining the multi-state reliability functionsand the risk functions of typical series, parallel, seriesparallel,parallel-series, series-“k out of n” and “k outof n”- series systems with ageing components aregiven in [4]-[5]. The paper is devoted to transmittingthese results on the multi-state ageing consecutive “ kout of n : F” systems [1], [2]-[3], [6], [7]-[8], [9].2. Multi-state system with ageing componentsIn the multi-state reliability analysis to define systemswith degrading components we assume that [4]-[5],[10]:– Ei, i = 1,2,...,n,are components of a system,– all components and a system under considerationhave the reliability state set {0,1,...,z}, z ≥ 1,– the state indexes are ordered, the state 0 is the worstand the state z is the best,– T i(u),i = 1,2,...,n,are independent randomvariables representing the lifetimes of componentsEiin the state subset {u,u+1,...,z}, while they werein the state z at the moment t = 0,– T i(u),is a random variable representing thelifetime of a system in the state subset {u,u+1,...,z}while it was in the state z at the moment t = 0,– the system state degrades with time t without repair,– e i(t)is a component Eistate at the moment t,t ≥ 0,– s (t)is a system state at the moment t, t ≥ 0.The above assumptions mean that the reliability statesof the system with degrading components may bechanged in time only from better to worse. The way inwhich the components and the system reliability stateschange is illustrated in Figure 1.- 104 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special Issueworst statetransitions0 1 u-1 u z-1 z. . . . . . . . .best stateFigure 1. Illustration of reliability states changing insystem with ageing componentsThe basis of our further consideration is a systemcomponent reliability function defined as follows.Definition 1. A vectorR j (t , ⋅ ) = [R i (t,0),R i (t,1),...,R i (t,z)], t ≥ 0,whereR i (t,u) = P(e i (t) ≥ u | e i (0) = z) = P(T i (u) > t)for t ≥ 0,u = 0,1,...,z, i = 1,2,...,n,is the probabilitythat the component Eiis in the reliability state subset{ u , u + 1,..., z}at the moment t, t ≥ 0,while it was inthe reliability state z at the moment t = 0, is called themulti-state reliability function of a component E i.Similarly, we can define a multi-state system reliabilityfunction.Definition 2. A vectorR n(t , ⋅ ) = [1,R n(t,0),R n(t,1),...,R n(t,z)], t ≥ 0,whereR n(t,u) = P(s(t) ≥ u | s(0) = z) = P(T(u) > t),for t ≥ 0,u = 0,1,...,z, is the probability that the systemis in the reliability state subset { u , u + 1,..., z}at themoment t, t ≥ 0,while it was in the reliability state z atthe moment t = 0, is called the multi-state reliabilityfunction of a system.Under this definition we haveR n (t,0) ≥ R n (t,1) ≥ . . . ≥ R n (t,z), t ≥ 0,and ifp(t) = [p(t,0), p(t,1),..., p(t,z)], t ≥ 0,wherep(t,u) = P(s(t) = u | s(0) = z),for t ≥ 0,u = 0,1,...,z, is the probability that the systemis in the state u at the moment t, t ≥ 0,while it was inthe state z at the moment t = 0,thenandR n (t,0) = 1, R n (t,z) = p(t,z), t ≥ 0,(1)p(t,u) = R n (t,u) - R n (t,u+1), u = 0,1,...,z-1, t ≥ 0.(2)Moreover, ifR n (t,u) =1 for t < 0,u = 1,2,...,z,then∞M(u) = E [ T ( u)]= ∫0R n (t,u)dt, u = 1,2,...,z, (3)is the mean lifetime of the system in the state subset{ u , u + 1,..., z},2σ ( u)= D[T ( u)]= N ( u)− [ M ( u)], (4)u = 1,2,...,z,where= ∞ 0N( u)2∫t R n (t,u)dt, u = 1,2,...,z, (5)is the standard deviation of the system lifetime in thestate subset { u , u + 1,..., z}and moreover∞M (u) = ∫0p( t,u)dt,u = 1,2,...,z, (6)is the mean lifetime of the system in the state u whilethe integrals (3), (4) and (5) are convergent.Additionally, according to (1), (2), (3) and (6), we getthe following relationshipsM (u) = M(u) - M(u+1), u = 1,2,...,z-1, (7)M (z) = M(z).Close to the multi-state system reliability function itsbasic characteristic is the system risk function definedas follows.Definition 3. A probability- 105 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special Issuer(t) = P(s(t) < r | s(0) = z) = P(T(r) ≤ t), t ≥ 0,that the system is in the subset of states worse than thecritical state r, r ∈{1,...,z} while it was in the reliabilitystate z at the moment t = 0 is called a risk function ofthe multi-state system.Considering Definition 3 and Definition 2, we haver( t)= 1− R ( t,r),t ≥ 0,(8)nand if τ is the moment when the system risk functionexceeds a permitted level δ, thenτ = r −1 ( δ ),(9)where r −1 ( t ) , if it exists, is the inverse function of therisk function r(t).3. Reliability of a multi-state ageing consecutive„k out of n: F” systemDefinition 4. A multi-state system is called an ageingconsecutive “ k out of n : F” system if it is out of thereliability state subset {u,u+1,...,z} if and only if atleast its k neighbouring components out of n itscomponents arranged in a sequence of ,1E , nare out of this reliability state subset.E E , ...,In our further analysis, we denote by s k , n( t)thereliability state of the ageing consecutive “ k out of n :F” system at the moment t, t ∈< 0,∞),and by ( )2T k , nuthe lifetime of this system in the reliability subset{u,u+1,...,z}. Moreover, we denote byRk, n( t,u)= P( sk, n( t)≥ u | s(0) = z) = P( T,( u)t)k n>for t ≥ 0,u = 0,1,...,z, the probability that theageing consecutive “ k out of n : F” system is in thereliability state subset { u , u + 1,..., z}at the momentt, t ≥ 0,while it was in the reliability state z at themoment t = 0 and byFk, n( t,u)= 1 − R ( t,) = P( T,( u)≤ t)k , nuk nfor t ≥ 0,u = 0,1,...,z, the distribution function ofthe lifetime ( ) of this system in the reliabilityT k , nustate subset {u,u+1,...,z} while it was in the state z atthe moment t = 0.Theorem 1. The reliability function of the ageingconsecutive “ k out of n : F” system composed ofcomponents with independent failures is given by thefollowing recurrent formulaR ( t,) = [1, R ( ,1),R ( ,2),..., R ( t,) ],wherek, n⋅k , ntk, ntk , nz⎧⎪1for n < k,⎪⎪1− ∏ F j ( t,u)for n = k,⎪⎪R k , n ( t,u)= ⎨Rn( t,u)Rk, n−1( t,u)(10)⎪ k −1⎪+∑ Rn−i( t,u)Rk, n−i−1(t,u)⎪ i=1⎪ n⎪⋅∏ Fj( t,u)for n > k,⎩ j=n−i+1for t ∈< 0 , ∞ > , u = 1,2,...,z.Motivation. Since for each fixed u , u = 1,2,...,z,theassumptions of this theorem as the same as theassumptions of Theorem 2 proved in [2] and theformula (10) is equivalent with the formula (12) from[2], then after considering Definition 4, we concludethat this theorem is valid.From the above theorem, as a particular case for thesystem composed of components with identicalreliability, we immediately get the following corollary.Corollary 1. If components of the ageing consecutive“ k out of n : F” system are independent and haveidentical reliability functions, i.e.R i( t,u)= R(t,u),F i( t,u)= F(t,u)for t ∈< 0,∞),u = 1,2,...,z, i = 1,2,...,n ,then the reliability function of this system is given byR ( t,) = [1, R ( ,1),R ( ,2),..., R ( t,) ],k, n⋅k , ntk, ntk , nzwhere⎧⎪1for n < k,⎪n⎪1− [ F(t,u)]for n = k,⎪R k , n ( t,u)= ⎨R(t,u)Rk, n−1( t,u)(11)⎪k −1i⎪+R(t,u)∑ F ( t,u)⎪i=1⎪⎩⋅Rk,n−i−1(t,u)for n > k,for t ∈< 0,∞),u = 1,2,...,z.- 106 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special IssueFrom Corollary 1, in a particular case, substitutingk = 2 in (11), we get:- for n = 1R t,) = [1, R ,1),R ,2),..., R t,) ], (12)where2,1 ( ⋅2,1 ( t2,1 ( t2,1 ( zR t,u)1 for t ∈< 0,∞),u = 1,2,...,z,(13)2 ,1 ( =- for n = 2R t,) = [1, R ( ,1),R ,2),..., R ( t,) ], (14)where2,2 ( ⋅2,2t2,2 ( t2,2z2R2,2( t,u)= 1 − F ( t,u)for t ∈< 0,∞),(15)u = 1,2,...,z,- for n ≥ 3R ( t,) = [1, R ( ,1),R ( ,2),..., R ( t,) ], (16)where2,n⋅2,nt2,ntR ( t,) = R( t,u)R t,)2,nu2,n−1 ( u2,nz+ R( t,u)F ( t,u)R ( t,) for t ∈< 0,∞),(17)u = 1,2,...,z.4. Application2,n−2uExample 1. Let us consider the steel covercomposed of n = 24 arranged identical sheetsE1, E2,..., E24. We assume that z = 4,i.e. the coverand the sheets it is composed of may be in the oneof the reliability states from the set { 0,1,2,3,4 }. Thecover is out of the reliability state subset{ u , u + 1,...,4} if at least k = 2 of its neighbouringsheets is out of this reliability state subset. If theconsidered steel cover critical reliability state isr = 2 , then this steel cover is failed if at least 2neighbouring sheets from 24 sheets are out of thereliability state subset { 2,3,4}.Thus, theconsidered steel cover is a five-state ageingconsecutive “2 out of 24: F” system, andaccording to (16)-(17), its the reliability functionis given by[1, R ( ,1),R ( ,2),R ( ,3),R ( ,4)], (18)where2,24t2,24t2,24tR ( t,) = R( t,u)R t,)2,24u2,23 ( u2,24t+ R( t,u)F ( t,u)R ( t,) for t ∈< 0,∞),(19)u = 1,2,3,4.2,22uIn the particular case when the lifetimes T i(u),u = 1,2,3,4, of the sheets Ei,i = 1,2,3,4,5 , in thereliability state subsets have Weibull distributions ofthe form2−λ ( u)tF(t,u)= 1 − e for t ≥ 0,u = 1,2,3,4 ,whereλ ( 1) = 0.01, λ ( 2) = 0.02,λ ( 3) = 0.05,λ( 4) = 0.10,i.e. if the reliability function of the sheets ,i = 1,2,3,4,5, is given byR(t , ⋅ ) = [1,R(t,1), R(t,2), R(t,3), R(t,4)], t ∈

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special Issue2.01e −0 t2.01+ [1 − e −0 t ] R2,n−2( t,1)for t ∈< 0,∞),(23)n = 3,4,...,24,- R ( ,2)is determined by the formulae2,24tR t,2)1 for t ∈< 0,∞),(24)2 ,1 ( =0.02t2R 2t,2)= 1 − [1 − e − ] for t ∈< 0,∞),(25)R(,22,nt2.02e −0 t( , 2) = R , 2)2.02e −0 t222,n−1 ( t.02+ [1 − e −0 t ] R2,n−2( t,2)for t ∈< 0,∞),(26)n = 3,4,...,24,- R ( ,3)is determined by the formulae2,24tR t,3)1 for t ∈< 0,∞),(27)2 ,1 ( =0.05t2R 2t,3)= 1 − [1 − e − ] for t ∈< 0,∞),(28)R(,22,nt2.05e −0 t( ,3) = R ,3)2.05e −0 t222,n−1 ( t.05+ [1 − e −0 t ] R2,n−2( t,3)for t ∈< 0,∞),(29)n = 3,4,...,24,- R ( ,4)is determined by the formulae2,24tR t,4)1 for t ∈< 0,∞),(30)2 ,1 ( =0.10t2R 2t,4)= 1 − [1 − e − ] for t ∈< 0,∞),(31)R(,22,nt2.10e −0 t( , 4) = R , 4)2.10e −0 t222,n−1 ( t.10+ [1 − e −0 t ] R ( , 4)for t ∈< 0,∞),(32)2,n−2tn = 3,4,...,24.The values of the particular vector components of themulti-state reliability function of the steel cover givenby (20), calculated by the computer programme basedon the formulae (21)-(32), are presented in the Tables1-4 and illustrated in Figure 1. As earlier we haveassumed that r = 2 is the cover critical reliability state,then according to (8) and (26) its risk function is givenby0.02tr(t) = 1 - R 2( t,2)= 1 − e − R 2t,2)2.02e −0 t,2422,23 (.02− [1 − e −0 t ] R ( , 2)for t ∈< 0,∞).(33)2,22tThe values of the steel cover risk function are given inTable 5 and illustrated in Figure 2.Table 1. The values of the steel cover multi-statereliability function vector component u = 1t R ( ,1)2,24t2 t R 2( t,1)0.0 1.0000 0.00001.0 0.9978 1.99552.0 0.9664 3.86573.0 0.8531 5.11834.0 0.6362 5.08895.0 0.3750 3.74996.0 0.1664 1.99577.0 0.0538 0.75348.0 0.0125 0.20019.0 0.0021 0.037410.0 0.0002 0.0049Table 2. The values of the steel cover multi-statereliability function vector component u = 2t R ( ,2)2,24t,242 t R 2( t,2)0.0 1.0000 0.00000.5 0.9994 0.99941.0 0.9912 1.98241.5 0.9580 2.87422.0 0.8802 3.52072.5 0.7479 3.73983.0 0.5731 3.43883.5 0.3876 2.71314.0 0.2275 1.82004.5 0.1145 1.03075.0 0.0491 0.49055.5 0.0178 0.19586.0 0.0055 0.06556.5 0.0014 0.01847.0 0.0003 0.0044Table 3 The values of the steel cover multi-statereliability function vector component u = 3t R ( ,3)2,24t,242 t R 2( t,3)0.0 1,0000 0,00000.2 0.9999 0.39990.4 0.9986 0.79880.6 0.9928 1.19140.8 0.9781 1.56491.0 0.9489 1.89781.2 0.9005 2.16131.4 0.8302 2.32461.6 0.7385 2.3632,24- 108 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special Issue1.8 0.6299 2.26752.0 0.5122 2.04892.2 0.3953 1.73922.4 0.2883 1.38372.6 0.1980 1.02982.8 0.1278 0.71583.0 0.0774 0.46423.2 0.0438 0.28063.4 0.0233 0.15813.6 0.0115 0.08303.8 0.0053 0.04064.0 0.0023 0.0185Table 4. The values of the steel cover multi-statereliability function vector component u = 4t R ( ,4)2,24t2 t R 2( t,4)0.0 1.0000 0.00000.1 0.9999 0.03990.2 0.9996 0.15990.3 0.9982 0.35930.4 0.9943 0.63640.5 0.9864 0.98640.6 0.9725 1.40040.7 0.9508 1.86360.8 0.9195 2.35400.9 0.8775 2.84331.0 0.8244 3.29751.1 0.7605 1.67311.2 0.6875 1.64991.3 0.6076 1.57991.4 0.5242 1.46771.5 0.4406 1.32171.6 0.3602 1.15281.7 0.2862 0.97311.8 0.2207 0.79441.9 0.1650 0.62692.0 0.1195 0.47792.1 0.0838 0.35192.2 0.0569 0.25022.3 0.0373 0.17182.4 0.0237 0.11382.5 0.0146 0.07282.6 0.0086 0.04502.7 0.0050 0.02682.8 0.0028 0.01542.9 0.0015 0.00863.0 0.0008 0.0046,24R 2,24 (t,u)10.80.60.40.20u=4u=3u=2u=10 2 4 6 8 10Figure 1. The graphs of the steel cover multi-statereliability function vector componentsTable 5. The values of the steel cover multi-statereliability function vector component u = 2 and its riskfunctiont R ( ,2)r(t) = 1- R 2( t,2)2,24t0.0 1.0000 0.00000.5 0.9994 0.00061.0 0.9912 0.00881.5 0.9581 0.04192.0 0.8802 0.11982.5 0.7480 0.25203.0 0.5731 0.42693.5 0.3876 0.61244.0 0.2275 0.77254.5 0.1145 0.88555.0 0.0490 0.95105.5 0.0178 0.98226.0 0.0055 0.99456.5 0.0014 0.99867.0 0.0003 0.9997,24t- 109 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special Issuer(t)10.90.80.70.60.50.40.30.20.100 2 4 6 8Figure 2. The graphs of the steel cover risk functionUsing the values given in these Tables 1-4, theformulae (3)-(7) and numerical integration we find:- the mean values of the cover lifetimes in thereliability state subsetsM 1) = E[(1)] ∫( T 2, 24= ∞ 0M 2) = E[(2)] ∫( T 2, 24= ∞ 0M 3) = E[(3)] ∫( T 2, 24= ∞ 0M 4) = E[(4)] ∫( T 2, 24= ∞ 0R 2,24 (t,1)dt ≅ 4.5634,R 2,24 (t,2)dt ≅ 3.2268,R 2,24 (t,3)dt ≅ 2.0408,R 2,24 (t,4)dt ≅ 1.4431,- the second ordinary moments of the cover lifetimes inthe reliability state subsetsN(1)= E[T 2,24 (1)]=N(2)= E[T 2,24 (2)] =22∞2∫t R 2,24 (t,1)dt ≅ 22.9715,0∞2∫t R 2,24 (t,2)dt ≅ 11.4879,0∞2N(3)= E[T 2,24 (3)] = 2∫t R 2,24 (t,3)dt ≅ 4.5944,0∞2N(4)= E[T 2,24 (4)] = 2∫t R 2,24 (t,4)dt ≅ 2.2967,0- the standard deviations of the cover lifetimes in thereliability state subsets2σ ( 1) = N (1) − [ M (1)] ≅ 1.4651,2σ ( 2) = N(2)− [ M (2)] ≅ 1.0370,2σ ( 3) = N (3) − [ M (3)] ≅ 0.6553,t2σ ( 4) = N(4)− [ M (4)] ≅ 0.4628,- the mean values of the cover lifetimes in thereliability particular statesM ( 1) = M (1) − M (2)≅ 4.5634 - 3.2268 = 1.3366,M ( 2) = M (2) − M (3)≅ 3.2268 - 2.0408 = 1.1860,M ( 3) = M (3) − M (4)≅ 2.0408 - 1.4431 = 0.5977,M ( 4) = M (4)≅ 1.4431.Using the values given in these Tables 5 and theformula (9) we find the approximate value of themoment when the system risk function exceeds anexemplary permitted level δ = 0.05, namely−1τ = r (0.05) ≅ 1.58.5. ConclusionTwo recurrent formulae for multi-state reliabilityfunctions, a general one for non-homogeneous and itssimplified form for homogeneous multi-stateconsecutive “ k out of n : F” systems composed ofageing components have been proposed. The formulaefor multi-state reliability function of a homogeneousmulti-state consecutive “ k out of n : F” system hasbeen applied to reliability evaluation of the steel covercomposed of ageing components. The considered steelcover was a five-state ageing consecutive “2 out of 24:F” system composed of components with Weibullreliability functions. On the basis of the recurrentformula for steel cover multi-state reliability functionthe approximate values of its vector components havebeen calculated and presented in tables and illustratedgraphically. On the basis of these vales the meanvalues and standard deviations of the steel coverlifetimes in the reliability state subsets and the meanvalues of the steel cover lifetimes in particularreliability states have been estimated. Moreover, thecover risk function and the moment when the riskfunction exceeds the permitted risk level have beendetermined.The input structural and reliability data of theconsidered steel cover have been assumed arbitrarilyand therefore the obtained its reliability characteristicsevaluations should be only treated as an illustration ofthe possibilities of the proposed methods and solutions.The proposed methods and solutions and the softwareare general and they may be applied to any multi-stateconsecutive “k out of n: F” system of ageingcomponents.- 110 -

S.Guze, K.Koowrocki Reliability analysis of multi-state ageing consecutive „k out of n: F” systems - RTA # 3-4, 2007, December - Special IssueReferences[1] Barlow, R. E. & Proschan, F. (1975). StatisticalTheory of Reliability and Life Testing. ProbabilityModels. Holt Rinehart and Winston, Inc., NewYork.[2] Guze, S. (2007). Wyznaczanie niezawodnocidwustanowych systemów progowych typu„kolejnych k z n: F”. Materiay XXXV SzkoyNiezawodnoci, Szczyrk.[3] Guze, S. (2007). Numerical approach to reliabilityevaluation of two-state consecutive „k out of n: F”systems. Proc. Summer Safety and ReliabilitySeminars, SSARS 2007, Sopot.[4] Koowrocki, K. (2001). Asymptotyczne podejcie doanalizy niezawodnoci systemów. Monografia.Instytut Bada Systemowych PAN, Warszawa.[5] Koowrocki, K. (2004). Reliability of LargeSystems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney -Tokyo.[6] Kossow, A. & Preuss, W. (1995). Reliability oflinear consecutively connected systems withmultistate components. IEEE Transactions onReliability, 44, 3, 518-522.[7] Malinowski, J. & Preuss, W. (1995). A recursivealgorithm evaluating the exact reliability of aconsecutive k-out-of-n: F system. Microelectronicsand Reliability, 35, 12, 1461-1465.[8] Malinowski, J. (2005). Algorytmy wyznaczanianiezawodnoci systemów sieciowych o wybranychtypach struktur. Wydawnictwo WIT, ISBN 83-88311-80-8, Warszawa.[9] Shanthikumar, J. G. (1987). Reliability of sytemswith consecutive minimal cut-sets. IEEETransactions on Reliability, 26, 5, 546-550.[10] Xue, J. & Yang, K. (1995). Dynamic reliabilityanalysis of coherent multi-state systems. IEEETransactions on Reliability, 4, 44, 683–688.- 111 -

L.Knopik Some remarks on mean time between failures of repairable systems - RTA # 3-4, 2007, December - Special IssueKnopik LeszekUniversity of Technology and Agriculture, Bydgoszcz, PolandSome remarks on mean time between failures of repairable systemsKeywordsage replacement, increasing failure rate on average, mean time between failures, mean time between failures orrepairsAbstractThis paper describes a simple technique for approximating the mean time between failures (MTBF) of a systemthat has periodic maintenance at regular intervals. We propose an approximation of MTBF for vide range ofsystems than IFRA class of distributions.1. IntroductionOne important research area in reliability engineeringis studying various maintenance policies. Maintenancecan be classified by two major categories: correctiveand preventive. Corrective maintenance is anymaintenance that occurs when the system is failed.Some authors refer to corrective maintenance as repairand we will use this approach in this paper. Preventivemaintenance is any maintenance that occurs whensystem is not failed.A common measure used to describe the reliabilitycharacteristics of a repairable system is mean timebetween failures (MTBF). In most repairable systems,preventive maintenance used to reduce system failurefrequency and hence increase the MTBF. It is easy thatMTBF is the mean time to a repair service or an agereplacement.The MTBF for a system that has periodic maintenanceat a time t can be described by [1], [10]:t∫R(s )dsMTBF = 0 . (1)F( t )In paper [1], [10] an approximation to MTBF isprovided. Moreover in [10] is proved thattR( t )t≤ MTBF ≤ ,F( t ) F( t )and for increasing failure rate on average (IFRA) classof distributions in [1] the following relation byproposedtt− ≤ MTBF ≤ .ln( R( t ) F( t )In this paper, we propose an approximation to MTBFfor wide class than IFRA. The equality (1) can be abasis to introduction a new class of ageing distributions(see [7], [8]).Let us assumption the following notation1M (t) = MTBF.The case when M(t) is monotonic was considered byBarlow and Campo [2], Marshall and Proschan [9],Klefsjö [5] and Knopik [7], [8].Definition 1. The lifetime T belongs to the class (meantime to failure or replacement) MTFR, if the functionM (t) is non-decreasing for t ∈ { t : F( t ) > 0 } .It has been shown in Barlow [2] and Klefsjö [5] thatIFR ⊂ MTFR ⊂ NBUE,where IFR is increasing failure rate class, NBUE isnew better than used in expectation class.For absolutely continuous in [2] and for any randomvariable in [8] it has been proved, that- 112 -

L.Knopik Some remarks on mean time between failures of repairable systems - RTA # 3-4, 2007, December - Special IssueIFRA ⊂ MTFR,where IFRA is increasing failure rate on average classof distributions. Preservation of life distribution classesunder reliability operations has been studied in [7], [8].The class MTFR is closed under the operations:(a) formation of a parallel system for absolutelycontinuous random variables,(b) formation of series system with identicallydistributed and absolutely continuous randomvariables,(c) weak convergence of distributions,(d) convolution.In this paper we propose new approximation andbounds, applicable for a wide range of systems. It isMTFR class of distributions, such that MTBF is nonincreasing.The class MTFR contains distributions withunimodal failure rate function. We analyze special caseof distribution with MTBF non-increasing as mixtureof an exponential distribution and Rayleigh’sdistributions. This distribution has unimodal failurerate function. However, it is not easy to obtaindistribution from mixtures with unimodal failure ratefunction [12]. The mixtures of two increasing linearfailures rate functions were studied in [4]. In [4]showed that a mixture of two distributions withincreasing linear failure rate functions does not givedistributions with unimodal failure rate function.In section 2, we introduce the proposed model ofmixture and we estimate their parameters for anexample of lifetime data [11]. In section 3 we proposea simple approximation of to MTBF for lifetimedistributions with MTBF non-increasing.2. Mixture of distributionsWe consider a mixture of two lifetimes X 1 and X 2 withdensities f 1 (t), f 2 (t), reliability functions R 1 (t), R 2 (t),failure rate functions r 1 (t), r 2 (t) and weights p andq=1– p, where 0 < p < 1. The mixed density is thenwritten asf (t ) = pf1( t ) + ( 1 − p ) f 2(t)and the mixed reliability function isR (t) =pR 1 (t) + (1-p)R 2 (t).The failure rate function of the mixture can be writtenas the mixturer(t)=(t ) r 1 (t)+[1 – (t)] r 2 (t),where (t)=pR 1 (t)/R(t).Moreover, from [3], we have under some mildconditions, thatlim r( t ) = lim min{ r ( t ),r ( t )} .t→∞t→∞1In the following propositions, we give some propertiesfor the mixture failure rate function.Proposition 1. For the first derivative of (t), we have’(t) = (t)[1 – (t)][r 2 (t) – r 1 (t)].Proposition 2. For the first derivative of r(t), we haver’(t)=[1– (t)]( – (t)(r 2 (t) – r 1 (t)) 2 +r’ 2 (t)+ (t)r’ 1 (t).Proposition 3. If X 1 is exponentially distributed withparameter , thenr’(t)=[1 – (t)](– (t)(r 2 (t) – ) 2 +r’ 2 (t)).We suppose that r 2 (t) =t, where >0. Consequently,the reliability function of X 2 is the reliability functionof Rayleigh’s distribution of the formαR 2 (t) =exp { − t 2 }22for t0.Proposition 4. If 2 , then r (t) is unimodal.Proof. By Proposition 1, we conclude that (t) isdecreasing for t ∈ (0, t 1 ), where t 1 = and isincreasing for t ∈ (t 1 , ). Hence, if t < t 1 , then(t)(r 2 (t) – ) 2 is decreasing from p 2 < to 0, and if t> t 1 then (t)(r 2 (t) – ) 2 is increasing from 0 to .Thus the equation (t) (r 2 (t) – ) 2 = has only onesolution t 2 and r (t) is increasing for t < t 2 anddecreasing for t > t 2 .Proposition 5. If 2 > , then there exist t 3 , and t 4 , t 3< t 1 < t 4 , such that r (t) decreases in (0, t 3 ), increases in(t 3, t 4 ) and decreases in (t 4 , ).Proof. Let h(t)= – (t)(r 2 (t) – ) 2 . It is easy to findthat h(0)= – p 2 0, and is decreasing from h(t 1 )=>0 toh()=–.Thus, there exist t 3 and t 4 , 0

L.Knopik Some remarks on mean time between failures of repairable systems - RTA # 3-4, 2007, December - Special IssueBy maximizing the logarithm of likelihood function forgrouped data, we calculate p = 0.643316, = 0.001284, = 0.0288. For these values ofparameters, we prove Pearson’s test of fit and compute 2 = 0.68. By Proposition 4, we conclude that r (t) isunimodal.3. Bounds and ApproximationIn this section we cover some of the well knownbounds and approximations to the MTBF.By the inequalityt∫0R(s )ds≤ min{ t,ET } ,where ET is the mean value of T, we obtain the upperbound for MTBF:If f (t) is unimodal then exists t m and t 1 such that f’(t m )=0, t m 0.Proof. By Definition 1, if M (t) is non-decreasing, thenwe haveand[ M( t )]'=f ( t)t∫0R( s )ds − F( t )R( t )(t∫0R( s )ds )2≥ 0MTBF = 1U min{ t,ET }F( t ).MTBF ≥ MTBFL1=r( t )for t∈{t: r (t)>0}.In [1] for MTBF proposed is the following averageapproximationt + R( t )MTBF A = 1 .2 F( t )Proposition 8. If the lifetime T has unimodal failurerate function r (t), then T ∈ MTFR if and only ifr() ET – 1 0.Proof. LetProposition 6. If f (t) is unimodal, then these exist t 1such that MTBF A is a lower bound of MTBF for t ∈

L.Knopik Some remarks on mean time between failures of repairable systems - RTA # 3-4, 2007, December - Special IssueR(t) t MTBF MTBF U MTBF L MTBF A0.999990.99990.9990,990.90.80.70.60.50.40.30.20.10.000540.00540.053980.540315.4397110.926116.469622.161929.175134.800742.585452.817470.265553.9753.9753.9553.7651.6749.1546.6244.2141.9839.9438.1136.5035.2353.9753.9753.9854.0354.4054.6354.9055.4056.3558.0049.7343.5138.6853.9753.9753.9353.5549.2244.0539.1734.9034.8134.8134.8134.8134.8153.9753.9753.9553.7651.6849.1746.6644.3242.2640.6039.5439.6142.944. ConclusionIn this paper we show that, from a practical point view,the unimodal failure rate model can be obtained from amixture of two common IFR models. This model isflexibility. Practical relevance and applicability havebeen demonstrated using well known data. In thispaper a simple approximation of the MTBF of systemssubjected to periodic maintenance has been proposedas well.[8] Marschall, A. W. & Proschan, F. (1972). Classesof distributions applicable in replacement withrenewal theory implications, In: L. LeCam et aleds., Proc. of the Sixth Berkeley Symposium onMathematical Statistics and Probability. Vol. I,University of California Press, Berkeley, 395 –415.[9] Mondro, M. J. (2002). Approximation of meantime between failures when system has periodicmaintenance. IEEE Transaction on Reliability 2,51, 166 – 167.[10] Mudholkar, G. S., Srivastava, D. K. & Freiner,M. (1995). The Exponentiated Weibull Family:A Reanalysis of the Bus – Motor – Failure Data.Technometrics 4, 37, 436 – 445.[11] Wondmagegnehu, E. T., Navarro I. &Hernandez P. J. (2005). Bathtub Shaped FailureRates from a Mixtures: A Practical Point ofView. IEEE Transaction on Reliability 2, 54,270 – 275.References[1] Amari, S. V. (2006). Bounds on MTBF of SystemsSubjected to Periodic Maintenance. IEEETransaction on Reliability 3, 55, 469 – 475.[2] Barlow, R. E. & Campo, R. (1975). Total time on testprocesses and applications to failure data analysis.In: R. E. Barlow, J. Fussel & N. P. Singpurwalla, eds.Reliability and Fault Tree Analysis. SIAM,Philadelphia, 451 – 481.[3] Block, H. & Joe, H. (1997). Tail Behavior of theFailure Rate Functions. Lifetime Data Analysis 3, 209– 288.[4] Block, H.W., Savits, T.H. & Wondmagegnehu, E.T.(2003). E. T. Mixtures of distributions with linearfailure rates J. Appl. Probab. 40, 485 – 504.[5] Klefsjö, B. (1982). On ageing properties and totaltime on test transforms. Scandinavian Journal ofStatistics. Theory and Applications 9 (1), 37 – 41.[6] Klutke, G. A., Kiessler, R. C. & Wortman, M. A.(2003). A critical look at the bathtub curve. IEEETransactions on Reliability 1, 52, 125 – 129.[7] Knopik, L. (2006). Characterization of a class oflifetime distributions. Control and Cybernetics 35(2),407 – 411.- 115 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special IssueKoowrocki KrzysztofMaritime University, Gdynia, PolandReliability modelling of complex systems- Part 1Keywordsreliability, large system, asymptotic approach, limit reliability functionAbstractThe paper is concerned with the application of limit reliability functions to the reliability evaluation of largesystems. Two-state large non-repaired systems composed of independent components are considered. Theasymptotic approach to the system reliability investigation and the system limit reliability function are defined.Two-state homogeneous series, parallel and series-parallel systems are defined and their exact reliability functionsare determined. The classes of limit reliability functions of these systems are presented. The results of theinvestigation concerned with domains of attraction for the limit reliability functions of the consideredsystems and the investigation concerned with the reliability of large hierarchical systems as well arediscussed in the paper. The paper contains exemplary applications of the presented facts to the reliabilityevaluation of large technical systems.1. IntroductionMany technical systems belong to the class of complexsystems as a result of the large number of componentsthey are built of and their complicated operatingprocesses. As a rule these are series systems composedof large number of components. Sometimes the seriessystems have either components or subsystemsreserved and then they become parallel-series or seriesparallelreliability structures. We meet large seriessystems, for instance, in piping transportation of water,gas, oil and various chemical substances. Largesystems of these kinds are also used in electricalenergy distribution. A city bus transportation systemcomposed of a number of communication lines eachserviced by one bus may be a model series system, ifwe treat it as not failed, when all its lines are able totransport passengers. If the communication lines haveat their disposal several buses we may consider it aseither a parallel-series system or an “m out of n”system. The simplest example of a parallel system oran “m out of n” system may be an electrical cablecomposed of a number of wires, which are its basiccomponents, whereas the transmitting electricalnetwork may be either a parallel-series system or an“m out of n”-series system. Large systems of thesetypes are also used in telecommunication, in ropetransportation and in transport using belt conveyersand elevators. Rope transportation systems like portelevators and ship-rope elevators used in shipyardsduring ship docking are model examples of seriesparalleland parallel-series systems.In the case of large systems, the determination of theexact reliability functions of the systems leads us tocomplicated formulae that are often useless forreliability practitioners. One of the importanttechniques in this situation is the asymptotic approachto system reliability evaluation. In this approach,instead of the preliminary complex formula for thesystem reliability function, after assuming that thenumber of system components tends to infinity andfinding the limit reliability of the system, we obtain itssimplified form.The mathematical methods used in the asymptoticapproach to the system reliability analysis of largesystems are based on limit theorems on order statisticsdistributions, considered in very wide literature, forinstance in [4]-[5], [7], [12]. These theorems havegenerated the investigation concerned with limitreliability functions of the systems composed of twostatecomponents. The main and fundamental resultson this subject that determine the three-element classes- 116 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special Issueof limit reliability functions for homogeneous seriessystems and for homogeneous parallel systems havebeen established by Gniedenko in [6]. These results arealso presented, sometimes with different proofs, forinstance in subsequent works [1], [8]. Thegeneralizations of these results for homogeneous “mout of n” systems have been formulated and proved bySmirnow in [13], where the seven-element class ofpossible limit reliability functions for these systemshas been fixed. As it has been done for homogeneousseries and parallel systems classes of limit reliabilityfunctions have been fixed by Chernoff and Teicher in[2] for homogeneous series-parallel and parallel-seriessystems. Their results were concerned with so-called“quadratic” systems only. They have fixed limitreliability functions for the homogeneous seriesparallelsystems with the number of series subsystemsequal to the number of components in thesesubsystems, and for the homogeneous parallel-seriessystems with the number of parallel subsystems equalto the number of components in these subsystems.Kolowrocki has generalized their results for non-“quadratic” and non-homogeneous series-parallel andparallel-series systems in [8]. These all results mayalso be found for instance in [9].The results concerned with the asymptotic approachto system reliability analysis have become the basisfor the investigation concerned with domains ofattraction ([9], [11]) for the limit reliability functionsof the considered systems and the investigationconcerned with the reliability of large hierarchicalsystems as well ([3], [9]). Domains of attraction forlimit reliability functions of two-state systems areintroduced. They are understood as the conditionsthat the reliability functions of the particularcomponents of the system have to satisfy in orderthat the system limit reliability function is one of thelimit reliability functions from the previously fixedclass for this system. Exemplary theorems concernedwith domains of attraction for limit reliabilityfunctions of homogeneous series systems arepresented here and the application of one of them isillustrated. Hierarchical series-parallel and parallelseriessystems of any order are defined, theirreliability functions are determined and limittheorems on their reliability functions are applied toreliability evaluation of exemplary hierarchicalsystems of order two.All the results so far described have been obtainedunder the linear normalization of the system lifetimes.The paper contains the results described above andcomments on their newest generalizations recentlypresented in [9].2. Reliability of two-state systemsWe assume thatE i , i = 1,2,...,n, n ∈N,are two-state components of the system havingreliability functionsR i (t) = P(T i > t), t ∈( −∞,∞),whereT i , i = 1,2,...,n,are independent random variables representing thelifetimes of components E i with distribution functionsF i (t) = P(T i ≤ t), t ∈ ( −∞,∞).The simplest two-state reliability structures are seriesand parallel systems. We define these systems first.Definition 1. We call a two-state system series if itslifetime T is given byT = min{ T }.1≤i≤niThe scheme of a series system is given in Figure 1.Figure 1. The scheme of a series systemE 1 E 2 . . . E nDefinition 1 means that the series system is not failed ifand only if all its components are not failed, andtherefore its reliability function is given bynR n (t) = ∏i=1R i( t), t ∈( −∞,∞).(1)Definition 2. We call a two-state system parallel if itslifetime T is given byT = max{ T }.1≤i≤niThe scheme of a parallel system is given in Figure 2.Figure 2. The scheme of a parallel system- 117 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special IssueDefinition 2 means that the parallel system is failed ifand only if all its components are failed and thereforeits reliability function is given bynR n (t) = 1 − ∏ F i( t), t ∈( −∞,∞).(2)i=1Another basic, a bit more complex, two-state reliabilitystructure is a series-parallel system. To define it, weassume thatE ij , i = 1,2,...,k n , j = 1,2,...,l i , k n , l 1 , l 2 ,..., lk n∈N,are two-state components of the system havingreliability functionsR ij (t) = P(T ij > t), t ∈( −∞,∞),whereT ij , i = 1,2,...,k n , j = 1,2,...,l i ,are independent random variables representing thelifetimes of components E ij with distribution functionsF ij (t) = P(T ij ≤ t), t ∈ ( −∞,∞).Definition 3. We call a two-state system series-parallelif its lifetime T is given byT = max{min{T }.1ij1 ≤i≤kn≤ j≤liBy joining the formulae (1) and (2) for the reliabilityfunctions of two-state series and parallel systems it iseasy to conclude that the reliability function of thetwo-state series-parallel system is given byRk l , l ,...,( t)= 1 − ∏[1−∏Rij( t)], t ∈( −∞,∞),(3)n , 1 2 l k nE 1E 2...E nkn l ii=1j=1where k n is the number of series subsystems linked inparallel and l i are the numbers of components in theseries subsystems.Definition 4. We call a two-state series-parallel systemregular ifl 1 = l 2 = . . . =l k n= l n , l n ∈N,i.e. if the numbers of components in its seriessubsystems are equal.The scheme of a regular series-parallel system is givenin Figure 3.E 1 E 2E 21 E 2. . .1E1ln. . .2E2ln. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .E kn1E kn 2. . .E kn l nFigure 3. The scheme of a regular series-parallelsystemDefinition 5. We call a two-state system homogeneousif its component lifetimes have an identical distributionfunction F(t),i.e. if its components have the samereliability functionR( t)= 1−F(t),t ∈( −∞,∞).The above definition and equations (1)-(3) result in thesimplified formulae for the reliability functions of thehomogeneous systems stated in the followingcorollary.Corollary 1. The reliability function of thehomogeneous two-state system is given by- for a series systemR n (t) = [R(t)] n , t ∈ ( −∞,∞),(4)- for a parallel systemnR n (t) = 1− [ F(t)], t ∈( −∞,∞),(5)- for a regular series-parallel systemR k n, ( t)= 1− [1 −[R(t)]] , t ∈( −∞,∞).(6)n l nl n k- 118 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special Issue3. Asymptotic approach to system reliabilityThe asymptotic approach to the reliability of two-statesystems depends on the investigation of limitdistributions of a standardized random variable( T − b n) / an,where T is the lifetime of a system and a n > 0 andbn∈( −∞,∞)are suitably chosen numbers callednormalizing constants.SinceP (( T − bn) / an> t)= P(T > ant+ bn) = R n (a n t + b n ),where R n (t) is a reliability function of a systemcomposed of n components, then the followingdefinition becomes natural.Definition 6. We call a reliability function R(t) thelimit reliability function of a system having a reliabilityfunction R n (t) if there exist normalizing constants a n >0, b n ∈(-∞, ∞) such thatlim R n(a n t + b n ) = R(t) for t ∈C R ,n→∞where C R is the set of continuity points of R(t).Thus, if the asymptotic reliability function R(t) of asystem is known, then for sufficiently large n, theapproximate formulaR n (t) ≅ R( t − b ) /a n ), t ∈ ( −∞,∞).(7)(nmay be used instead of the system exact reliabilityfunction R n (t).3.1. Reliability of large two-state series systemsThe investigations of limit reliability functions ofhomogeneous two-state series systems are based on thefollowing auxiliary theorem.Lemma 1. If(i) R ( t)= exp[ −V ( t)]is a non-degeneratereliability function,(ii) R n (t)is the reliability function of a homogeneoustwo-state series system defined by (4),(iii) an> 0,b n∈ ( −∞,∞),thenlim R n (a n t + b n ) = R (t)for t ∈CRn→∞if and only iflim nF(a nt + b n ) = (t)n→∞V for t ∈CVProof. The proof may be found in [1], [6], [8].Lemma 1 is an essential tool in finding limit reliabilityfunctions of two-state series systems. It also is thebasis for fixing the class of all possible limit reliabilityfunctions of these systems. This class is determined bythe following theorem.Theorem 1. The only non-degenerate limit reliabilityfunctions of the homogeneous two-state series systemare:R ( ) = exp[ ( ) α− −t − ] for t < 0,1 tR ( ) = 0 for t ≥ 0, α > 0;1 tR ( ) = 1 for t < 0,2 tR ( ) = exp[ − tα ] for t ≥ 0, α > 0;2 tR ( ) = exp[ − exp[ t]]for t ∈(−∞,∞ ).3 tProof. The proof may be found in [1], [6], [8].3.2. Reliability of large two-state parallelsystemsThe class of limit reliability functions forhomogeneous two-state parallel systems may bedetermined on the basis of the following auxiliarytheorem.Lemma 2. If(i) R(t) = 1− exp[ −V ( t)]is a non-degeneratereliability function,(ii) R n (t) is the reliability function of a homogeneoustwo-state parallel system defined by (5),(iii) an> 0,b n∈ ( −∞,∞),thenlim R n(a n t + b n ) = R(t) for t ∈C R ,n→∞if and only iflim nR(a n t + b n ) = V(t) for t ∈C V .n→∞Proof. The proof may be found in [1], [6], [8].By applying Lemma 2 it is possible to fix the class oflimit reliability functions for homogeneous two-state- 119 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special Issueparallel systems. However, it is easier to obtain thisresult using the duality property of parallel and seriessystems expressed in the relationshipR n (t) = 1 − R ( −t)for t ∈( −∞,∞),nthat results in the following lemma, [1], [6], [8]-[9].Lemma 3. If R (t)is the limit reliability function of ahomogeneous two-state series system with reliabilityfunctions of particular components R (t),thenR(t) = 1 −R ( −t)for t ∈CRis the limit reliability function of a homogeneous twostateparallel system with reliability functions ofparticular componentsR( t)= 1−R(−t)for t ∈C R.At the same time, if a ) is a pair of normalizing( n, bnconstants in the first case, then ( an, − bn) is such a pairin the second case.The application of Lemma 3 and Theorem 1 yields thefollowing result.Theorem 2. The only non-degenerate limit reliabilityfunctions of the homogeneous parallel system are:R 1 (t) = 1 for t ≤ 0,R 1 (t) = 1 − exp[−t −α ] for t > 0, α > 0;R 2 (t) = 1 − exp[−(−t) α ] for t < 0,R 2 (t) = 0 for t ≥ 0, α > 0;R 3 (t) = 1 − exp[−exp[−t]] for t ∈(−∞,∞).Proof. The proof may be found in [1], [6], [8].3.3. Reliability evaluation of large two-stateseries-parallel systemsThe proofs of the theorems on limit reliabilityfunctions for homogeneous regular series-parallelsystems and methods of finding such functions forindividual systems are based on the following essentiallemmas.Lemma 4. If(i) k n → ∞,(ii) R(t) = 1 − exp[−V(t)] is a non-degeneratereliability function,(iii) Rk n , l n(t) is the reliability function of ahomogeneous regular two-state series-parallel systemdefined by (6),(iv) a n > 0, b n ∈(−∞,∞),thenlim R kn→∞n , lnif and only if(a n t + b n ) = R(t) for t ∈CRlim k lnn[R(a n t + b n )n→∞] = V(t) for t ∈C V .Proof. The proof may be found in [8].Lemma 5. If(i) k n → k, k > 0 , l n → ∞,(ii) R(t) is a non-degenerate reliability function,(iii) Rk n , l n(t) is the reliability function of ahomogeneous regular two-state series-parallel systemdefined by (6),(iv) a n > 0, b n ∈(−∞,∞),thenlim R kn→∞n , lnif and only iflim [R(a lnnt + b n )n→∞(a n t + b n ) = R(t) for t ∈C R ,] = R 0 (t) for t ∈C R 0,where R 0 (t) is a non-degenerate reliability function andmoreoverR(t) = 1 − [1 − R 0 (t)] kfor t ∈(−∞,∞).Proof. The proof may be found in [8].The types of limit reliability functions of a seriesparallelsystem depend on the system shape [7], i.e. onthe relationships between the number k n of its seriessubsystems linked in parallel and the number l n ofcomponents in its series subsystems. The results basedon Lemma 4 and Lemma 5 may be formulated in theform of the following theorem.Theorem 3. The only non-degenerate limit reliabilityfunctions of the homogeneous regular two-state seriesparallelsystem are:Case 1. k n = n, ⏐l n − c log n⏐ >> s, s > 0, c > 0.R 1 (t) = 1 for t ≤ 0,- 120 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special Issue−αR 1 (t) = 1 − exp[ − t ] for t > 0, α > 0;R 2 (t) = 1 − exp[− (−t) α ] for t < 0,R 2 (t) = 0 for t ≥ 0, α > 0;R 3 (t) = 1 − exp[−exp[−t]] for t ∈(−∞,∞);Case 2. k n = n, l n − c log n ≈ s, s ∈(−∞,∞), c > 0.R 4 (t) = 1 for t < 0,R 4 (t) = 1 - exp[−exp[−t α − s/c]] for t ≥ 0, α > 0;R 5 (t) = 1 − exp[−exp[(−t) α − s/c]] for t < 0,R 5 (t) = 0 for t ≥ 0, α > 0;R 6 (t) = 1 − exp[−exp[β(−t) α − s/c]] for t < 0,R 6 (t) = 1 − exp[−exp[−t α − s/c]] for t ≥ 0,α > 0, β > 0;R 7 (t) = 1 for t < t 1 ,R 7 (t) = 1 − exp[−exp[−s/c]] for t 1 ≤ t < t 2 ,R 7 (t) = 0 for t ≥ t 2 , t 1 < t 2 ;Case 3. k n → k, k > 0 , l n → ∞.−αR 8 (t) = 1 − [1 − exp[ − ( −t ) ]] k for t < 0,R 8 (t) = 0 for t ≥ 0, α > 0;R 9 (t) = 1 for t < 0,R 9 (t) = 1 − [1 − exp[−t α ]] k for t ≥ 0, α > 0;R 10 (t) = 1 − [1 − exp[−exp t]] k for t ∈(−∞,∞).Proof. The proof may be found in [8].Using the duality property of parallel-series and seriesparallelsystems similar to this given in Lemma 3 forparallel and series systems it is possible to prove thatthe only limit reliability functions of the homogeneousregular two-state parallel-series system areRi(t) =1 − R i (-t) for t ∈ C R, i = 1,2,..., 10.Applying Lemma 2, it is possible to prove thefollowing fact ([9]).iCorollary 2. If components of the homogeneous twostateparallel system have Weibull reliability functionsandR(t) = exp[−β t α ] for t ≥ 0, α > 0, β > 0a n = b n /(αlog n), b n = (log n/β) 1/α ,thenR 3 (t) = 1 − exp[−exp[−t]], t ∈(−∞,∞),is its limit reliability function.Example 1 (a steel rope, durability). Let us consider asteel rope composed of 36 strands used in ship ropeelevator and assume that it is not failed if at least oneof its strands is not broken. Under this assumption wemay consider the rope as a homogeneous parallelsystem composed of n = 36 basic components. Further,assuming that the strands have Weibull reliabilityfunctions with parametersα = 2, β = (7.07) –6 ,by (5), the rope’s exact reliability function takes theformR 36 (t) = 1 – [1 − exp[−(7.07) –6 t 2 ] 36 for t ≥ 0.Thus, according to Corollary 2, assuminga n = (7.07) 3 /(2 log 36 ), b n = (7.07) 3 log 36and applying (7), we arrive at the approximate formulafor the rope reliability function of the formR 36 (t) ≅ R 3 ((t − b n )/a n )for t ∈(−∞,∞).= 1 − exp[−exp[−0.01071t + 7.167]]The mean value of the rope lifetime T and its standarddeviation, in months, calculated on the basis of theabove approximate result and according to theformulaeE[T] = Ca n + b n , σ=πa n / 6,where C ≅ 0.5772 is Euler’s constant, respectively are:E[T] ≅ 723, σ ≅ 120.- 121 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special IssueThe values of the exact and approximate reliabilityfunctions of the rope are presented in Table 1 andgraphically in Figure 4. The differences between themare not large, which means that the mistakes inreplacing the exact rope reliability function by itsapproximate form are practically not significant.Table 1. The values of the exact and approximatereliability functions of the steel rope10.80.60.40.2t R 36 (t)t − bRn3 ( )an∆ = R 36 − R 30 1.000 1.000 0.000400 1.000 1.000 0.000500 0.995 0.988 −0.003550 0.965 0.972 −0.007600 0.874 0.877 −0.003650 0.712 0.707 0.005700 0.513 0.513 0.000750 0.330 0.344 −0.014800 0.193 0.218 −0.025900 0.053 0.081 −0.0281000 0.012 0.029 −0.0171100 0.002 0.010 −0.0081200 0.000 0.003 −0.003R 36(t), R 3((t − b n)/a n)0 200 400 600 800 1000 tFigure 4. The graphs of the exact and approximatereliability functions of the steel rope4. Domains of attraction for system limitreliability functionsThe problem of domains of attraction for the limitreliability functions of two-state systems solvedcompletely in [11] we will illustrate partly for twostateseries homogeneous systems only. FromTheorem 1 it follows that the class of limit reliabilityfunctions for a homogeneous series system iscomposed of three functions, R (t ) , i =1,2,3 . Nowwe will determine domains of attractionD R forthese fixed functions, i.e. we will determine theconditions which the reliability functions R(t) of theparticular components of the homogeneous seriessystem have to satisfy in order that the system limitreliability function is one of the reliability functionsR (t) , i = 1,2,3.Proposition 1. If R(t)is a reliability function of thehomogeneous series system components, thenR(t)∈D R 1if and only if1 − R(r)lim = tr → −∞1 − R(rt ) for t > 0.Proposition 2. If R(t)is a reliability function of thehomogeneous series system components, thenR(t)∈D R 2if and only if(i) ∃ y ∈ ( −∞,∞)R(y) = 1 and R(y + ε) < 1 for ε > 0,1 − R(rt + y)(ii) lim= t for t > 0.r→0+ 1 − R(r + y)Proposition 3. If R(t)is a reliability function of thehomogeneous series system components, thenR(t)∈D R3if and only iftlim n [1 − R(ant+ bn)] = e for t ∈( −∞,∞)n→∞with1b n= inf{ t : R(t + 0) ≤1−≤ R(t − 0)},na = inf{ t : R(t(1+ 0) + b )ne≤1 − ≤ R(t(1− 0) + bn)}.nnExample 2. If components of the homogeneous seriessystem have reliability functions- 122 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special Issue⎧1,⎪R(t)= ⎨1− t,⎪⎩0,thenR(t)∈D .R 2t < 00 ≤ t < 1t ≥ 1,The results of the analysis on domains of attraction forlimit reliability functions of two-state systems mayautomatically be transmitted to multi-state systems. Todo this, it is sufficient to apply theorems about twostatesystems such as the ones presented here to eachvector co-ordinate of the multi-state reliabilityfunctions ([9], [14]).5. Reliability of large hierarchical systemsPrior to defining the hierarchical systems of any orderwe once again consider a series-parallel system like asystem presented in Figure 3. This system here iscalled a series-parallel system of order 1.It is made up of components,i 1 j 1E i = 1,2,..., k , j = ,2,..., l ,1 n11i1with the lifetimes respectivelyT , i = 1,2,..., k , j = ,2,..., l .i 1 j 11 nIts lifetime is given byT = max{ min { Ti}}.1 j11≤i1≤kn1≤j1≤l i 111i1Now we assume that each component,i 1 j 1E i = 1,2,..., k , j = ,2,..., l ,1 n11i1(8)of the series-parallel system of order 1 is a subsystemcomposed of componentsE ,i1 j1 i 2 j2( i1j1)k ni2= 1,2,...,,j =2( i1j11,2,...,l ) i,2and has a series-parallel structure.This means that each subsystem lifetime T i 1 j is given1byT max i j= { min { Ti}},1 j1i(9)2 21 1( i1j1)( i11 1 j1 22) j≤i≤k n ≤ j ≤li2i = 1,2,..., k , j = ,2,..., l ,1 nwhere11i1T ,i1 j1i 2 j2( i1j1)k ni2= 1,2,...,,( i ) ,2,...,1 jj = ,211l i 2are the lifetimes of the subsystem components .E i 1 j1i2j2The system defined this way is called a hierarchicalseries-parallel system of order 2. Its lifetime, from (8)and (9), is given by the formulaT = max { min [ max ( min Ti)]},1 j1i2 j 21≤i1≤kn1≤j1≤li1 ( 1 1 ) ( i1j1)1 ≤ii j2 ≤k n 1≤j2≤li2where knis the number of series systems linked inparallel and composed of series-parallel subsystemsl are the numbers of series-parallel subsystemsE ,i 1 j 1 i1E in these series systems,i 1 j 1( i1 j1 )nk are the numbers ofseries systems in the series-parallel subsystems E i 1 j 1( i1j1)linked in parallel, and l are the numbers ofi2components in these series systems of the seriesparallelsubsystems E .i 1 j 1In an analogous way it is possible to define two-stateparallel-series systems of order 2.Generally, in order to define hierarchical series-paralleland parallel-series systems of any order r , r ≥1,weassume thatEi1 j1... i r j r,wherei = 11,2,...,k , nj1= 1,2,...,li,1and( i1j1)k ni2= 1,2,...,,( i1j1)( i1,2 l 1 j1...ir −1jr−1)i= 1,2,...,2rk n( i 1 j1...ir −1jr−1)= 1,2,...,j2= ,..., , ...,j rl i r( )i1 j1 ( i1j1)kn, l , k i 1 n,i2( i1 j1...ir −1jr−1)li∈ N ,rl , ...,i ,( i1 j1...ir −1j r −1)nk ,are two-state components having reliability functionsRt)= P(T ) , t ∈(–∞,∞),i(1 j1 ... i r j ri1j1... i r> tj rand random variablesTi1 j1... i r j r,wherei = 11,2,...,k , nj1= 1,2,...,li,1( i1j1)k ni2= 1,2,...,,- 123 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special Issue( i1j1)( i21,2,...,l 1 j1...ir −1jr−1)i= 1,2,...,2rk n( i 1 j1...ir −1jr−1)r= 1,2,...,l i rj = , ...,j ,i ,are independent random variables with distributionfunctionsFt)= P(T ≤ ) , t ∈(–∞,∞),i(1 j1 ... i r j ri1j1... i rtj rrepresenting the lifetimes of the components .E i 1 j1...i r jrDefinition 7. A two-state system is called a seriesparallelsystem of order r if its lifetime T is given byT = max{ min {1≤i1≤knmaxmin1≤j1≤li1 ( 1 1 ) ( i1j1)1 ≤ii j2 ≤k n 1≤j2≤li2... max ( min T( ) ( ) 1 1...1 1 1... 1 1 i1 1 ji j1... ir 1 jr 1ir jr≤i i j r ≤k nir−jr − ≤ jr ≤l− −ir( i1 j1 )n{( ))]...}}},i1 1i2j2...ir −1r −1where kn, k , …, k jjnare the numbersof suitable series systems of the system composed ofseries-parallel subsystems and linked in parallel, l i ,1l( i1j1)i2, …,( i1j 1i2j2 ... ir −2jr−2)ir−1l are the numbers of suitableseries-parallel subsystems in these series systems, and( i1 j1i2j2Kir−1jr−1)are the numbers of components in thel i rseries systems of the series-parallel subsystems.Definition 8. A two-state series-parallel system oforder r is called homogeneous if its componentlifetimes T i j ... i r j have an identical distribution1 1 rfunctionF t)= P(T ≤ t ) , t ∈( −∞,∞),1where(i j1... i r j ri = 11,2,...,k , nj1= 1,2,...,li,1( i1j1)k ni2= 1,2,...,,( i1j1)( i 1... 1 1 )j2= 1,2,...,l i,..., 1,2,...,1 j ir − jr−=2rk n( i 1 j1...ir −1jr−1)jr= 1,2,...,l i,ri.e. if its componentsi ,Ei j1... i r j rfunctionR(t) = 1 – F(t), t ∈(–∞,∞).1have the same reliabilityDefinition 9. A two-state series-parallel system oforder r is called regular if( i1j1) ( i1j1...ir −1jr −1)l = l = .. = l = li i.1 2irn( i1 j1) ( i j1...ir −1jr−1)n= ...= k1nk = kwhere knis the number of series systems in the seriesparallelsubsystems and l nare the numbers of seriesparallelsubsystems or respectively the numbers ofcomponents in these series systems.Using mathematical induction it is possible to provethat the reliability function of the homogeneous andregular two-state hierarchical series-parallel system oforder r is given byR ( t)= 1−[1[andRln knk , ,− Rkn lnk 1,( t )] ]kn , lnn−for k = 2,3,...,rln kn1,k( t ) 1 [1 [ R(t)]]n ,= − − , t ∈(–∞,∞),lnwhere k n and l n are defined in Definition 9.Corollary 3. If components of the homogeneous andregular two-state hierarchical series-parallel system oforder r have an exponential reliability functionR( t)= exp[ −λt]for t ≥ 0,λ > 0,then its reliability function is given byforR ( t)= 1 − [1 [Rln knk, ,− Rkn lnk 1,( t )] ]kn , lnk = 2,3,...,r and−for t ≥ 0kn1,kn,( t ) = 1−[1− exp[ −λllnnt]]for t ≥ 0.Theorem 4. If(i) R ( t)= 1−exp[ −V ( t)],t ∈( −∞,∞),is a nondegeneratereliability function,1−r−1ln(ii) liml k = 0 for r ≥1,(iii)nn→∞nr −1ln + ... + 1nlnlim k [ R(ant + bn)] = V ( t)for t ∈C V,n→∞r ≥1, t ∈( −∞,∞),thenlim R ( a t + b ) = R(t) for t ∈CR , r ≥1,n→∞t ∈( −∞,∞).r, kn , ln n nrand- 124 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special IssueProposition 4. If components of the homogeneous andregular two-state hierarchical series-parallel system oforder r have an exponential reliability functionR( t)= exp[ −λt]for t ≥ 0,λ > 0,1−r −1lnlim l k = 0 for r ≥1,nn→∞andn1 1 1 1 1an= , b ... )log ,r n= ( + + + k2 r nλln λ lnlnlnthenR 3 (t) = 1−exp[ − exp[ −t]]for t ∈( −∞,∞),(10)is its limit reliability function.Example 3. A hierarchical regular series-parallelhomogeneous system of order r = 2 is such thatkn= 200, ln= 3.The system components haveidentical exponential reliability functions with thefailure rate λ = 0.01.Under these assumptions its exact reliability function,according to Corollary 3, is given byR t)= 1 − [1 [ 1−[1− exp[ −0.01⋅3t]]2,200,3 ( −for t ≥ 0.Next applying Proposition 4 with normalisingconstants= a nb n=1= 11.1,0.01⋅91(0.011+31) log 2009= 235.5,200]3] 200we conclude that the system limit reliability function isgiven byR 3 (t) = 1−exp[ − exp[ −t]]for t ∈( −∞,∞),and from (7), the following approximate formula isvalidR2,200,3( t)≅ R 3 ( 0.09t− 21.2)= 1 − exp[ − exp[ −0.09t + 21.2]] for t ∈( −∞,∞).Definition 10. A two-state system is called a parallelseriessystem of order r if its lifetime T is given byT = min { max{1≤i1≤knminmax1≤j1≤li1 ( 1 1 ) ( i1j1)1 ≤ii j2 ≤kn1≤j2≤li2...[ min ( max T...1≤( i1j1...1 1 ) ( i1j1...1 1 )ir ≤knir − jr − 1≤lir − jr −jr ≤ir( i1 j1 )n{i1j1( )ir jr)]...}}},i1 1i2j2 ... ir −1r −1where kn, k , …, k jjnare the numbersof suitable parallel systems of the system composed ofparallel-series subsystems and linked in series, l ,i 1( i1j1)( i11i2j2 ... ir −2jr−2)l , ..., l jare the numbers of suitablei2ir−1parallel-series subsystems in these parallel systems,and ( i1 j 1i2j2 ... ir −1j r −1l ) are the numbers of components inirthe parallel systems of the parallel-series subsystems.Definition 11. A two-state parallel-series system oforder r is called homogeneous if its componentlifetimes Ti j ... i rhave an identical distribution1 1 j rfunctionF t)= P(T ≤ t ) ,1where(i j1... i r j r1 = k , ( i1j1)nj1= 1,2,...,li,1 21,2,...,k n( i1j1)21,2,...,l i,…,2( i 1 j1...ir −1jr−1)( i 1... 1 1 )r= 1,2,...,k 1 j ir − jr−nr= 1,2,...,l i ri 1,2,...,j =i ,i.e. if its componentsfunctionEi j1... i r j rR(t) = 1 – F(t), t ∈(–∞,∞).i = ,j ,1have the same reliabilityDefinition 12. A two-state parallel-series system oforder r is called regular if( i1j1) ( i1j1...ir −1jr −1)i= li= .. . = l1 2irl = land( i1 j1) ( i j1...ir −1jr−1)n= ...= k1nk = kwhere knis the number of parallel systems in theparallel-series subsystems and l nare the numbers ofparallel-series subsystems or, respectively, thenumbers of components in these parallel systems.nn- 125 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special IssueApplying mathematical induction it is possible toprove that the reliability function of the homogeneousand regular two-state hierarchical parallel-seriessystem of order r is given byRkandR, ,tkn ln1,,tkn ln( ) = 1 − [ 1−Rk( t )] ] for k = 2,3,...,rln kn[−1, kn , lnln kn( ) = [ 1−[F ( t)]] , t ∈(–∞,∞),where k n and l n are defined in Definition 12.Corollary 4. If components of the homogeneous andregular two-state hierarchical parallel-series system oforder r have an exponential reliability functionR( t)= exp[ −λt]for t ≥ 0,λ > 0,then its reliability function is given byRkandR, ,tkn ln1,,tkn lnln kn( ) = [ 1 − [ 1−R ( ) ] ] for k = 2,3,...,rk−1,,tkn lnln kn( ) = [ 1−[1− exp[ −λt ]] ] for t ≥ 0.Theorem 5. If(i) R (t)= exp[ −V ( t)],t ∈( −∞,∞),is a nondegeneratereliability function,1−r −1ln(ii) lim l k = 0 for r ≥1,(iii)nn→∞nr −1ln + ... + 1nlnlim k [ F ( ant + bn)] = V ( t)for t ∈C V,n→∞r ≥1, t ∈( −∞,∞),thenlimn→∞R a + ) = R (t)for t ∈ C , r ≥1,Rt ∈( −∞,∞).r , kn , l(n ntbnProposition 5. If components of the homogeneous andregular two-state hierarchical parallel-series system oforder r have an exponential reliability functionR( t)= exp[ −λt]for t ≥ 0,λ > 0,1−r −1lnlim l k = 0 for r ≥1,lim = l,nn→∞andnrl nn→∞l ∈ N,anthen1 1= , = 0,λk1 1+ ... +l n rlnnrb nl⎺R 2 (t) = exp[ −t ] for t ≥ 0(11)is its limit reliability function.Example 3. We consider a hierarchical regular parallelserieshomogeneous system of order r = 2 such thatkn= 200, ln= 3,whose components have identicalexponential reliability functions with the failure rateλ = 0.01.Its exact reliability function, according to Corollary 4,is given byR ( ) = [ 1 − [1 − [ 1−[1− exp[ −0.01t]]2,200,3tfor t ≥ 0.3] 200]3] 200Next applying Proposition 5 with normalisingconstants1 1an= ⋅ = 9.4912, b1 / 3+1 / 9n= 0,0.01 200we conclude thatR ( ) exp[ 9= −t ] for t ≥ 02 tis the system limit reliability function, and from (7),the following approximate formula is valid9R2,200,3( t)≅ R2(0.1054t) = exp[ −(0.1054t)]for t ≥ 0.6. ConclusionGeneralizations of the results on limit reliabilityfunctions of two-state homogeneous systems for theseand other systems in case they are non-homogeneous,are mostly given in [8] and [9]. These results allow usto evaluate reliability characteristics of homogeneousand non-homogeneous series-parallel and parallelseriessystems with regular reliability structures, i.e.systems composed of subsystems having the samenumbers of components. However, this fact does notrestrict the completeness of the performed analysis,since by conventional joining of a suitable number ofcomponents which do not fail, in series sub-systems ofthe non-regular series-parallel systems, leads us to theregular non-homogeneous series-parallel systems.- 126 -

K.Koowrocki Reliability modelling of complex systems-Part 1 - RTA # 3-4, 2007, December - Special IssueSimilarly, conventional joining of a suitable number offailed components in parallel subsystems of the nonregularparallel-series systems we get the regular nonhomogeneousparallel-series systems. Thus theproblem has been analyzed exhaustively.The results concerned with the asymptotic approach tosystem reliability analysis, in a natural way, have led toinvestigation of the speed of convergence of the systemreliability function sequences to their limit reliabilityfunctions ([9]). These results have also initiated theinvestigations of limit reliability functions of “m out ofn”-series, series-“m out of n” systems, theinvestigations on the problems of the system reliabilityimprovement and on the reliability of systems withvarying in time their structures and their componentsreliability described briefly in [9] and presented in Part2 ([10]) of this paper.More general and practically important complexsystems composed of multi-state and degrading in timecomponents are considered in wide literature, forinstance in [14]. An especially important role they playin the evaluation of technical systems reliability andsafety and their operating process effectiveness isdescribed in [9] for large multi-state systems withdegrading components. The most important resultsregarding generalizations of the results on limitreliability functions of two-state systems dependent ontransferring them to series, parallel, “m out of n”,series-parallel and parallel-series multi-state systemswith degrading components are given in [9]. Somepractical applications of the asymptotic approach to thereliability evaluation of various technical systems arecontained in [9] as well.The proposed method offers enough simplifiedformulae to allow significant simplifying of largesystems’ reliability evaluating and optimizingcalculations.[5] Frechet, M. (1927). Sur la loi de probabilite del’ecart maximum. Ann. de la Soc. Polonaise deMath. 6, 93–116.[6] Gniedenko, B. W. (1943). Sur la distribution limitedu terme maximum d’une serie aleatoire. Ann. ofMath. 44, 432–453.[7] Gumbel, E. J. (1962). Statistics of Extremes. NewYork.[8] Koowrocki, K. (1993). On a Class of LimitReliability Functions for Series-Parallel andParallel-Series Systems. Monograph. MaritimeUniversity Press, Gdynia.[9] Koowrocki, K. (2004). Reliability of LargeSystems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney -Tokyo.[10] Koowrocki, K. (2007). Reliability of complexsystems – Part 2. Proc. Summer Safety andReliability Seminars – SSARS 2007, Sopot.[11] Kurowicka, D. (2001). Techniques in RepresentingHigh Dimensional Distributions. PhD Thesis.Maritime University, Gdynia - Delft University.[12] Von Mises, R. (1936). La distribution de la plusgrande de n valeurs. Revue Mathematique del’Union Interbalkanique 1, 141–160.[13] Smirnow, N. W. (1949). Predielnyje ZakonyRaspredielenija dla Czlienow WariacjonnogoRiada. Trudy Matem. Inst. im. W. A. Stieklowa.[14] Xue, J. & Yang, K. (1995). Dynamic reliabilityanalysis of coherent multi-state systems. IEEETransactions on Reliability 4, 44, 683–688.References[1] Barlow, R. E. & Proschan, F. (1975). StatisticalTheory of Reliability and Life Testing. ProbabilityModels. Holt Rinehart and Winston, Inc., NewYork.[2] Chernoff, H. & Teicher, H. (1965). Limitdistributions of the minimax of independentidentically distributed random variables. Proc.Americ. Math. Soc. 116, 474–491.[3] Cichocki, A. (2003). Wyznaczanie granicznychfunkcji niezawodnoci systemów hierarchicznychprzy standaryzacji potgowej. PhD Thesis.Maritime University, Gdynia – Systems ResearchInstitute, Polish Academy of Sciences, Warszawa.[4] Fisher, R. A. & Tippett, L. H. C. (1928). Limitingforms of the frequency distribution of the largestand smallest member of a sample. Proc. Cambr.Phil. Soc. 24, 180–190.- 127 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special IssueKoowrocki KrzysztofMaritime University, Gdynia, PolandReliability modelling of complex systems- Part 2Keywordsreliability, large system, asymptotic approach, limit reliability functionAbstractSeries-“m out of n” systems and “m out of n”-series systems are defined and exemplary theorems on their limitreliability functions are presented and applied to the reliability evaluation of a piping transportation system and arope elevator. Applications of the asymptotic approach in large series systems reliability improvement are alsopresented. The paper is completed by showing the possibility of applying the asymptotic approach to the reliabilityanalysis of large systems placed in their variable operation processes. In this scope, the asymptotic approach toreliability evaluation for a port grain transportation system related to its operation process is performed.1. Reliability of large series-“m out of n”systemsDefinition 1. A two-state system is called a series-“mout of k n ” system if its lifetime T is given byT = T , m = 1,2,...,k( k n −m+1)n ,where T is the mth maximal order statistic in( k n −m+1)the set of random variablesT i = min { T } , i = 1,2,...,k n .1≤j≤liijThe above definition means that the series-“m out ofk n ” system is composed of k n series subsystems and itis not failed if and only if at least m out of its knseries subsystems are not failed.The series-“m out of k n ” system is a series-parallel form = 1 and it becomes a series system for m = k n .The reliability function of the two-state series-“m outof k n ” system is given either by),( t)= 1( mR k n l1,l2 ,..., lk n( )( m )R k , l1,l2 ,...,tn lk n=1∑k nr1,r2 ,..., rk = 0 ni=1r1+ r2+ ... + r k ≤mnl i∏[1− ∏ Rj=1r( t)]i[ ∏ Rfor t ∈ (–∞,∞), where m = k m.ijn −l ij=1ij( t)]1−r iDefinition 2. The series-“m out of k n ” system is calledregular ifl 1 = l 2 = . . . = l k n= l n , l n ∈ N.Definition.3. The series-“m out of k n ” system is calledhomogeneous if its component lifetimes T ij have anidentical distribution functionF(t) = P(T ij ≤ t), t ∈ (–∞,∞), i = 1,2,...,k n , j = 1,2,...,l i ,i.e. if its components E ij have the same reliabilityfunctionR(t) = 1 – F(t), t ∈ (–∞,∞).,−1∑k nr1,r2 ,..., rk = 0 ni=1r1+ r2+ ... + r k ≤m−1nl i∏[∏j=1rR ( t)]i[1 − ∏ijl ij=1R ( t)]ij1−r i,From the above definitions it follows that the reliabilityfunction of the homogeneous and regular series-“m outof k n ” system is given either byfor t ∈ (–∞,∞) or by- 128 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special IssueR( m)knnm−1k(n), l( t)= 1−∑ i [ Rn( t)][1 − Rn( t)]i=0lilkn−i=k( )−11 −m ∑ ii=0exp[ −itα][1 − exp[ −tα]]k −ifor t ≥ 0for t ∈ (–∞,∞) or byRm( m)( ) ( ) kn= ∑ [1 − Rtkn, lnii=0lni( t)][ Rln( t)]kn−ifor t ∈ (–∞,∞), m = k − nm,where k n is thenumber of series subsystems in the “m out of k n ”system and l n is the number of components of theseries subsystems.Corollary 1. If components of the homogeneous andregular two-state series-“m out of k n ” system haveWeibull reliability functionαR(t)= exp[ −βt] for t ≥ 0,α > 0,β > 0,then its reliability function is given either by( m)R kn , l nt=( )kn( )−11 −m ∑ ii=0for t ≥ 0 or byR( m)tkn, lnm= ∑i=0( )kn( )i[exp[ −il[1 − exp[ −lfor t ≥ 0,m = k m.n −nnβtα]][1 − exp[ −lnβtα]]kn −iα iαβ t ]] [exp[ −(k − i)l βt]] (2)Proposition 1. If components of the two-statehomogeneous and regular series-“m out of k n ” systemhave Weibull reliability functionandαR(t)= exp[ −βt] for t ≥ 0,α > 0,β > 0,k nn→∞lim = k , k > 0,0 < m ≤ k,lim l = ∞,1−αa ( βl) , = 0,thenn=n(2)R 9( t)b nnn→∞nnis its limit reliability function, i.e., for t ≥ 0,we have( m)(2)t − bnRk n , l n( t)≅R9( )am−1= 1 − ∑ii=0nkαα k −i( ) exp[ −il t ][1 − exp[ −βlt ]] .β (3)nExample 1. The piping transportation system is set up toreceive from ships, store and send by carriages or carsoil products such as petrol, driving oil and fuel oil.Three terminal parts A, B and C fulfil these purposes.They are linked by the piping transportation systems.The unloading of tankers is performed at the pier. Thepier is connected to terminal part A through thetransportation subsystem S 1 built of two piping lines. Inpart A there is a supporting station fortifying tankers’pumps and making possible further transport of oil bymeans of subsystem S 2 to terminal part B. Subsystem S 2is built of two piping lines. Terminal part B is connectedto terminal part C by subsystem S 3 . Subsystem S 3 is builtof three piping lines. Terminal part C is set up forloading the rail cisterns with oil products and for thewagon carrying these to the railway station.We will analyse the reliability of the subsystem S 3only. This subsystem consists of k n = 3 identical pipinglines, each composed of l n = 360 steel pipe segments.In each of lines there are pipe segments with Weibullreliability functionR(t) = exp[−0.0000000008t 4 ] for t ≥ 0.We suppose that the system is good if at least 2 of itspiping lines are not failed. Thus, according toDefinitions 2-3, it may be considered as ahomogeneous and regular series-“2 out of 3” system,and according to Proposition 1, assuming11a =, b = 0 ,n=αβ n11/ 4( l ) (0.000000288)and using (3), its reliability function is given by(2)(2) tR ( ) ≅ R ( )3,360t9a n3 41 ⎛ ⎞= ∑ ⎜ ⎟ exp[ −i⋅ 0.000000288ti=0⎝i ⎠⋅−i[ 1 − exp[ −0.000000288t4]] 3for .]nnt ≥ 0- 129 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issue2. Reliability of large “m out of n”-seriessystemsDefinition 4. A two-state system is called an “ m i out ofl ”-series system if its lifetime T is given byiT , m = ,2,..., l ,= min T(−m+ 1)1≤i≤li i kni1iwhere T is the m( l i −mi +1)i th maximal order statistic inthe set of random variablesT i1 , T i2 , ..., T , i = ,2,..., k .ili1 nThe above definition means that the “ m i out of l i ”-series system is composed of k n subsystems that are“ m i out of l i ” systems and it is not failed if all its “ m iout of l i ” subsystems are not failed.The “ m i out of l i ”-series system is a parallel-seriessystem if m 1 = m 2 = . . . = m = 1 and it becomes aseries system if m i = l i for all i = 1,2, …,k n .The reliability function of the two-state “ miout of li”-series system is given either byknDefinition 6. The “ m i out of l i ”-series system is calledregular ifandl 1 = l 2 = . . . = l k n= l nm 1 = m 2 = . . . = m = m, where l n , m∈ N, m ≤ l n .knThe reliability function of the two-state homogeneousand regular „ m out of l n ”-series system is given eitherbyR( m)knnm−1ln( )( t)= [1 − ∑ [ ( )] [1 − ( )]n], li R t R ti=0for t ∈ (–∞,∞) or by( m)tkn, lnm( ) = [ ∑( ) R nn ni=0liiil −ik[1 − R ( t)][ R(t)]]l −iknfor t ∈ (–∞,∞), m = l n− m where k n is thenumber of “m out of l n ” subsystems linked in seriesand l n is the number of components in the “m out of l n ”subsystems.( m1,m2,...,m ) knR k n , l1,l2,...,lknkn= ∏[1−i=1( t)1∑r1, r2,..., rl i= 0r1+ r2+ ... + r ≤mi−1lili[ ∏j=1ijrliR ( t)]i[1 − ∏j=1R ( t)]ij1−ri]Corollary 2. If the components of the two-statehomogeneous and regular “m out of l n ”-series systemhave Weibull reliability functionαR(t)= exp[ −βt] for t ≥ 0,α > 0,β > 0,for t ∈ (–∞,∞) or bythen its reliability function is given either by( m1,m2,..., m ) knR k n , l1, l2,..., lknkn= ∏[1∑( t)i=1 r1, r2,..., rl i= 0r1+ r2+ ... + r ≤mli ili[1 − ∏j=1ijrliR ( t)]i[ ∏ Rj=1ij( t)]1−rifor t ∈ (–∞,∞), where m = l − m , i = ,2,..., k .iii]1nDefinition 5. The two-state “ m i out of l i ”-seriessystem is called homogeneous if its componentlifetimes T ij have an identical distribution functionF(t) = P(T ij ≤ t), t ∈ (–∞,∞), i = 1,2,...,k n , j=1,2,...,l i ,i.e. if its components E ij have the same reliabilityfunctionR(t) = 1 – F(t), t ∈ (–∞,∞).( )( m)R kn , l ntm−1= [1 − ∑i = 0ln( )for t ≥ 0 or byiαα ln −iknexp[ −iβ t ][1 − exp[ −βt]] ] (4)Rm( )( m)ln( ) = [ ∑ [1 − exp[ −βttkn, lnii=0αi]] exp[ −(ln−i)βtfor t ≥ 0,m = l n− m.Proposition 2. If components of the two-statehomogeneous and regular “m out of l n ”-series systemhave Weibull reliability functionandαR(t)= exp[ −βt] for t ≥ 0,α > 0,β > 0,α]]kn- 130 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issuek nn→∞lim = k , k > 0,0 < m ≤ k,lim l = ∞,anthen1n→∞nbnlog n= , b [ ]αn = ,(5)α log n β(5)R ( ) =10,22t4[1 − ∑ (i=022ifor t ≥ 0.) exp[ −i0.05t2][1 − exp[ −0.05tNext, applying Proposition 2 with2]]22−i]10[ R(0)3(t)]km−1exp[−it]= [1 − exp[ − exp[ −t]]∑ ]i=0 i!k7.8626log 22a n = ≅ 1.2718, b [ ]2n = ≅ 7.8626,2 log 220.051for t ∈( −∞,∞),is its limit reliability function, i.e.( m)R kn , l nt( ) ≅R [(0)3t − bn( )]at − b= [1 − exp[ − exp[ −afor t ∈( −∞,∞),where anandnnnkt − bnexp[ −i]m−1an]] ∑]i=0 i!b nare defined by (5).Example 2. Let us consider the ship-rope transportationsystem (elevator). The elevator is used to dock andundock ships coming in to shipyards for repairs. Theelevator is composed of a steel platform-carriageplaced in its syncline (hutch). The platform is movedvertically with 10 rope hoisting winches fed byseparate electric motors. During ship docking theplatform, with the ship settled in special supportingcarriages on the platform, is raised to the wharf level(upper position). During undocking, the operation isreversed. While the ship is moving into or out of thesyncline and while stopped in the upper position theplatform is held on hooks and the loads in the ropes arerelieved.In our further analysis we will discuss the reliability ofthe rope system only. The system under consideration isin order if all its ropes do not fail. Thus we may assumethat it is a series system composed of 10 components(ropes). Each of the ropes is composed of 22 strands.Thus, considering the strands as basic components ofthe system and assuming that each of the ropes is notfailed if at least m = 5 out of its strands are not failed,according to Definitions 5-6, we conclude that the ropeelevator is the two-state homogeneous and regular„5 out of 22”-series system. It is composed of k n = 10series-linked “5 out of 22” subsystems (ropes) with l n =22 components (strands). Assuming additionally thatstrands have Weibull reliability functions withparameters α = 2,β = 0.05,i.e.2k(6)R(t)= exp[ −0.05t] for t ≥ 0,from (4), we conclude that the elevator reliabilityfunction is given byand (6) we get the following approximate formula forthe elevator reliability function(5),22R 10( t)≅(0)[ R3(0.7863t− 6.1821)]= [ 1−exp[ − exp[ −0.7863t+ 6.1821]]104 exp[ −0.7863it+ 6.1821 i]⋅ ∑]i=0 i!10,t ∈( −∞,∞).3. Asymptotic approach to systems reliabilityimprovementWe consider the homogeneous series system illustratedin Figure 1.Figure 1. The scheme of a series systemIt is composed of n components E i1,i =1,2,...,n,having lifetimes T i1,i = 1,2,...,n,and exponentialreliability functionsR( t)= exp[ −λt]for t ≥ 0,λ > 0.Its lifetime and its reliability function respectively aregiven byT(0)= min{Ti1},1≤i≤nnR ( t)= [ R(t)]= exp[ −λnt],t ≥ 0.nE 11 E 21 E n1. . .In order to improve of the reliability of this seriessystem the following exemplary methods can be used:– replacing the system components by the improvedcomponents with reduced failure rates by a factor ρ,0 < ρ < 1,– a warm duplication (a single reservation) of system- 131 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issuecomponents,– a cold duplication of system components,– a mixed duplication of system components,– a hot system duplication,– a cold system duplication.It is supposed here that the reserve components areidentical to the basic ones.The results of these methods of system reliabilityimprovement are briefly presented below, giving thesystem schemes, lifetimes and reliability functions.Case 1. Replacing the system components by the'improved components Ei1i = 1,2,...,n,with reducedfailure rates by a factor ρ, 0 < ρ < 1, having lifetimes'T , i = 1,2,...,n,and exponential reliability functionsi1. . .TR(3)(3)n= min{∑2 T1≤i≤nj=1ij},n( t ) = [1 − [ F(t)]∗[F(t)]]= [ 1 + λ ] exp[ −nλt],t ≥ 0.Case 4. A mixed reservation of the system componentsE 11E 12. E m1 . . .E m2E m+11E m+12Figure 5. The scheme of a series system withcomponents having mixed reservationt nE n1E n2E’ 11 E’ 21 E’ n1R( ρt)= exp[ −ρλt]for t ≥ 0,λ > 0.T(4)= min{min{ ∑T1≤i≤m2ijj=1},min {max{ T }}},m+1≤i≤n1≤j≤2ijFigure 2. The scheme of a series system with improvedcomponentsTR(1)(1)n= min{ T1≤i≤n'i1},n( t ) = [ R(ρt)]= exp[ −ρλnt],t ≥ 0.Case 2. A hot reservation of the system componentsR(4)n( t ) = [1 − [ F(t)]∗[F(t)]]mm2[1 − R ( t)]n−mn−m= [1 + λ t]exp[ −λnt][2− exp[ −λt]], t ≥ 0.Case 5. A hot system reservationE 11 E 21 . . . E n1Figure 3. The scheme of a series system withcomponents having hot reservationTRE 11 E 21 . E n-11E n1E 12 E 22(2)(2)n= min{max{ T1≤i≤n1≤j≤2ij2}},n2 n( t ) = [1 − [ F(t)]] = [1 −[1− exp[ −λt]]] , t ≥ 0.Case 3. A cold reservation of the system componentsE 11 E 21.E n-12E n2E 12 E 22E n-11E n1E n-12E n2Figure 4. The scheme of a series system withcomponents having cold reservationFigure 6. The scheme of a series system with hotreservationTR(5)(5)nE 12 E 22 . . . E n2= max{min{T1 ≤ j≤n1 ≤ j≤2ij}},n2( t)= 1−[1 − [ R(t)]] = 1 − [1 − exp[ −nλt]], t ≥ 0.Case 6. A cold system reservationE 11 E 21 . . . E n1E 12 E 22 . . . E n2Figure 7. The scheme of a series system with coldreservation2- 132 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special IssueT(6)2= ∑ min{ Tj= 1 1 ≤i≤nij},(R 6)nn( t)= 1 − [1 − [ R(t)]] ∗[1− [ R(t)]]n= [ 1 + nλt]exp[−nλt],t ≥ 0.The difficulty arises when selecting the right method ofimprovement of reliability for a large system. Thisproblem may be simplified and approximately solvedby the application of the asymptotic approach.Comparisons of the limit reliability functions of thesystems with different types of reserve and suchsystems with improved components allow us to findthe value of the components’ decreasing failure ratefactor ρ, which gives rise to an equivalent effect on thesystem reliability improvement. Similar results areobtained under comparison of the system lifetimemean values. As an example we will present theasymptotic approach to the above methods ofimproving reliability for homogeneous two-state seriessystems.Proposition 3.Case 1. Ifa n = 1/λρn, b n = 0,then(1)R ( t)= exp[–t] for t ≥ 0,(2)(2)2R n ( t)≅ R ( λ nt ) = exp[ −λ 2 nt ] for t ≥ 0andT(2)Case 3. Ifathen(2) 3 1= E [ T ] ≅ Γ() .2 λ nn=n2 / λ n,b = 0,R (3) (t) = exp[–t 2 ] for t ≥ 0,is the limit reliability function of the homogeneousexponential series system with cold reservation of itscomponents, i.e.RandT( nt 1 2R = exp[ − λ 2 nt ] for t ≥ 02 2(3)3)n( t)≅ ( λ )(3)Case 4. If(3) 3 1 2= E [ T ] ≅ Γ() .2 λ nis the limit reliability function of the homogeneousexponential series system with reduced failure rates ofits components, i.e.anthen1 2= , bnλ 2n− m= 0,(1)R n ( t)= R (1) ( λρ nt)= exp[ −λρnt]for t ≥ 0and(1) (1) 1T = E[T ] = .λρnCase 2. Ifathenn=n1 / λ n,b = 0,R (2) (t) = exp[–t 2 ] for t ≥ 0,is the limit reliability function of the homogeneousexponential series system with hot reservation of itscomponents, i.e.R (4) (t) = exp[–t 2 ] for t ≥ 0,is the limit reliability function of the homogeneousexponential series system with mixed reservation of itscomponents, i.e.RandT(4)( 4) n − mn( t)≅ R ( λ )(4)Case 5. If222n− m 2= exp[ − λ2 t ] for t ≥ 02(4) 3 1 2= E [ T ] ≅ Γ() .2 λ 2n − m- 133 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issueanthen1= , bnλn= 0,R (5) 2(t) = 1 − [1 − exp[ −t]]for t ≥ 0,is the limit reliability function of the homogeneousexponential series system with hot reservation, i.e.(5)n ( λnt)(5)R ( t)= R = 1− [1 − exp[ −λnt]]for t ≥ 0and(5) (5) 3T = E[T ] = .2λnCase 6. Ifanthen1= , bnλn= 0,R (6) (t) = [ 1+ t]exp[−t]for t ≥ 0,is the limit reliability function of the homogeneousexponential series system with cold reservation, i.e.and(6)n( λnt)(6)R ( t)= R = [ 1+ λnt]exp[−λnt]for t ≥ 0(6) (6) 2T = E[T ] = .λnCorollary 3. Comparison of the system reliabilityfunctionsR (i) (t) = R (1) (t), i = 2,3,…,6,results respectively in the following values of thefactor ρ :ρ = ρ( t)= λt for i = 2,1ρ = ρ( t)= λt for i = 3,22n− mρ = ρ(t)= λtfor i = 4,2n2ρ = ρ( t)= 1−log[2 − exp[ −λnt]]for i = 5,1ρ = ρ( t)= 1−log[1+λnt]for i = 6,λntwhile comparison of the system lifetimesT (i) (t) = T (1) (t), i = 2,3,...,6results respectively in the following values of thefactor ρ :1ρ = for i = 2,3Γ() n21ρ = for i = 3,3Γ() 2n21ρ =for i = 4,3 2Γ() n2 2n− m2ρ = for i = 5,31ρ = for i = 6.2Example 3. We consider a simplified bus servicecompany composed of 81 communication lines. Wesuppose that there is one bus operating on eachcommunication line and that all buses are of the sametype with the exponential reliability functionR( t)= exp[ −λt]for t ≥ 0,λ > 0.Additionally we assume that this communicationsystem is working when all its buses are not failed, i.e.it is failed when any of the buses are failed. The failurerate of the buses evaluated on statistical data comingfrom the operational process of bus service companytransportation system is assumed to be equal to 0.0049−1h.Under these assumptions the considered transportationsystem is a homogeneous series system made up ofcomponents with a reliability functionR( t)= exp[ −0.0049t]for t ≥ 0.Here we will use four sensible methods from thoseconsidered for system reliability improvement.- 134 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special IssueNamely, we apply the four previously consideredcases.Case 1. Replacing the system components by theimproved components with reduced failure rates by afactor ρ.Applying Proposition 3 with normalising constants1 1a 81 =,0.0049 ⋅ 81ρ =,0.397ρb 81 = 0we conclude thatR (1) (t) = exp[–t] for t ≥ 0,is the limit reliability function of the system, i.e.(1)R n ( t)= R (1) ( 0 .397 ρt) = exp[ −0.397ρt]for t ≥ 0andT(1)(1) 1= E[T ] = h.0.397ρCase 2. Improving the reliability of the system by asingle hot reservation of its components.This means that each of 81 communication lines has atits disposal two identical buses it can use and its task isperformed if at least one of the buses is not failed.Applying Proposition 3 with normalising constants1 1a 81 == , ,0.0049⋅81 0.0441b 81 = 0we conclude that(2)R ( t)= exp[ −t ], t ≥ 0.2is the limit reliability function of the system, i.e.traffic which are using two buses permanently (a hotreservation).Applying Proposition 3 with normalising constants1 2 1a n = , b0.0049 112 0.0367= nwe conclude thatR (4) (t) = exp[–t 2 ] for t ≥ 0,= 0,is the limit reliability function of the system, i.e.(4)R n ( t)≅ R (4) 2( 0.0367t ) = exp[ −0.00135t] for t ≥ 0and(4) (4) 3 1 2T = E[T ] ≅ Γ( )≅ 24.15 h.2 0.0049 112Case 5. Improving the reliability of the system by asingle hot reservation.This means that the transportation system is composedof two independent companies, each of them operatingon the same 81 communication lines and having attheir disposal one identical bus for use on each line.Applying Proposition 3 with normalising constants1 1a81= = , ,0.0049⋅810.397b 81 = 0we conclude thatR (5) 2(t) = 1− [1 − exp[ −t]]for t ≥ 0,is the limit reliability function of the system, i.e.(5)R n ( t)= R (5) ( 0.397t)(2)(2)2R81( t)≅ R (0.0441t ) ≅ exp[ −0.0019t], t ≥ 0,andand= 1− [1 − exp[ −0.397t]]for t ≥ 02(2) (2) 3 1T = E[T ] ≅ Γ( ) ≅ 20.10 h.2 0.0049 81Case 4. Improving the reliability of the system by asingle mixed reservation of its components.This means that each of 81 communication lines has atits disposal two identical buses. There are m = 50communication lines with small traffic which are usingone bus permanently and after its failure it is replacedby the second bus (a cold reservation) andn − m = 81 − 50 = 31 communication lines with large(5)(5)T = E[T ] =3≅ 3.78 h.2 ⋅ 0.0049⋅81Comparing the system reliability functions forconsidered cases of improvement, from Corollary 3,results in the following values of the factor ρ:ρ =ρ( t)= 0. 0049t for i = 2,ρ = 0.0340t for i = 4,- 135 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issueρ =ρ( t)= 1−log[2 − exp[ −0.397t]]for i = 5,while comparison of the system lifetimes resultsrespectively in:ρ = 0.1254 for i = 2,ρ = 0.1043 for i = 4,ρ = 0.6667 for i = 5.Methods of system reliability improvement presentedhere supply practitioners with simple mathematicaltools, which can be used in everyday practice. Themethods may be useful not only in the operationprocesses of real technical objects but also in designingnew operation processes and especially in optimisingthese processes. Only the case of series systems madeup of components having exponential reliabilityfunctions with single reservations of their componentsand subsystems is considered. It seems to be possibleto extend these results to systems that have morecomplicated reliability structures, and made up ofcomponents with different from the exponentialreliability functions.4. Reliability of large systems in their operationprocessesThis section proposes an approach to the solution ofthe practically very important problem of linkingsystems’ reliability and their operation processes. Toconnect the interactions between the systems’operation processes and their reliability structures thatare changing in time a semi-markov model ([1]) of thesystem operation processes is applied. This approachgives a tool that is practically important and notdifficult for everyday use for evaluating reliability ofsystems with changing reliability structures duringtheir operation processes. Application of the proposedmethods is illustrated here in the reliability evaluationof the port grain transportation system.We assume that the system during its operation processis taking different operation states. We denote by Z(t),t ∈< 0 , ∞ > , the system operation process that mayassume v different operation states from the setZ = { z1 , z2, . . ., zv}.In practice a convenient assumption is that Z(t) is asemi-markov process ([1]) with its conditional sojourntimes θblat the operation state zbwhen its nextoperation state is zl, b , l = 1,2,..., v,b ≠ l.In this casethis process may be described by:- the vector of probabilities of the initial operationstates [ p b (0)] 1xν,- the matrix of the probabilities of its transitionsbetween the states [ pbl]νxν,- the matrix of the conditional distribution functions[ H ( t)]of the sojourn times θ , b ≠ l,whereHandblνxν( t)= P(θ t)for b , l = 1,2,..., v,b ≠ l,bl bl

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issuehave an influence on the system components Eijreliability and on the system reliability structure aswell. Thus, we denote ([13]) the conditional reliabilityfunction of the system component E while thesystem is at the operational state zb, b = 1,2,...,v,by( i,j)( b)( b)ij[ R ( t)]= P ( T ≥ t / Z(t)= z ) ,for t ∈ t)k n , l ntν≅ ∑ p b[ R ( t)]b=1kn, ln( b)(12)for t ≥ 0 and T is the unconditional lifetime of theseries-parallel system.The mean values and variances of the series-parallelsystem lifetimes areνM ≅ ∑ p b M b,(13)whereMbb=1= ∞ ∫[R ( t)]dt,(14)0k n , ln( b)and( b)D[ T ] 2∫t[R ( t)]dt −[M b ] , (15)= ∞ 0for b = 1,2,...,ν .knln( b)Example 5. We analyse the reliability of one of thesubsystems of the port grain elevator. The consideredsystem is composed of four two-state nonhomogeneousseries-parallel transportation subsystemsassigned to handle and clearing of exported andimported grain. One of the basic elevator functions isloading railway trucks with grain.In loading the railway trucks with grain the followingelevator transportation subsystems take part: S 1 –horizontal conveyors of the first type, S 2 – verticalbucket elevators, S 3 – horizontal conveyors of thesecond type, S 4 – worm conveyors.We will analyze the reliability of the subsystem S 4only.Taking into account experts opinion in the operationprocess, Z(t),t ≥ 0 of the considered transportationsubsystem we distinguish the following as its threeoperation states:an operation state z 1– the system operation with thelargest efficiency when all components of thesubsystem S 4are used,an operation state z2– the system operation with lessefficiency system when the first and second conveyorsof subsystem S 4are used,an operation state z3– the system operation with leastefficiency when the first conveyor of subsystem S 4isused.On the basis of data coming from experts, theprobabilities of transitions between the subsystem S 4operation states are given by⎡ 0 0.357 0.643⎤[ p ]⎢0.8 0 0.2⎥bl=⎢⎥,⎢⎣0.385 0.615 0 ⎥⎦and their mean values, from (8), areM 1 = E[ θ 1 ] = 0.357⋅0.36 + 0.643⋅0.2 ≅ 0.257,M 2 = E[ θ 2 ] = 0.8 ⋅0.05+0.2 ⋅ 0.2 ≅ 0.08,M= E[ θ 3] = 0.385 ⋅ 0.08 + 0.615 ⋅ 0.053≅Since from the system of equations20.062.- 137 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issue⎧⎪[ π1, π⎨⎪⎪⎩π1 + πwe get2, π23+ π] = [ π31= 1, π2, π3⎡ 0]⎢⎢0.8⎢⎣0.3850.35700.6150.643⎤0.2⎥⎥,0 ⎥⎦p 11 = 2/242, p 12 = 160/242, p 13 = 80/242,p 21 = 2/242, p 22 = 240/242,and from (10)-(11) the reliability function of thissystem is given byπ 1 = 0.374, π 2 = 0.321,π 3 = 0.305,then the limit values of the transient probabilitiesp b(t) at the operation states zb, according to (9), aregiven byp 1 = 0.684, p 2 = 0.183,p 3 = 0.133.(16)The subsystem S 4 consists of three chain conveyors.Two of these are composed of 162 components and theremaining one is composed of 242 components. Thus itis a non-regular series-parallel system. In order tomake it a regular system we conventionally completetwo first conveyors having 162 components with 80components that do not fail. After this supplementsubsystem S 4 consists of k n = 3 conveyors, eachcomposed of l n = 242 components. In two of them thereare:- two driving wheels with reliability functionsR (1,1) (t) = exp[−0.0798t],- 160 links with reliability functionsR (1,2) (t) = exp[−0.124t],- 80 components with “reliability functions”R (1,3) (t) = exp[ −λ 1 (1)t], where λ 1 (1 ) = 0.The third conveyer is composed of:- two driving wheels with reliability functionsR (2,1) (t) = exp[−0.167t]- 240 links with reliability functionsR (2,2) (t) = exp[−0.208t].At the operation state z1the subsystem S 4becomes anon-homogeneous regular series-parallel system withparametersk n = 3, l n = 242, a = 2, q 1 = 2/3, q 2 = 1/3,e 1 = 3, e 2 = 2,[R3 ,242(t)](1)= 1 – [1 – exp[–19.9892t]] 2 [1 – exp[–50.2628t]]= 2 exp[ −19.9892t ] − 2 exp[ −70.252t]+ exp[ −50.2628t]+ exp[ −90.2412t]− exp[ −39.9784t]for t ≥ 0. (17)According to (14)-(15), the subsystem lifetime meanvalue and the standard deviation areM 1 ≅ 0.078, σ 1 ≅ 0.054.(18)At the operation state z2the subsystem S 4becomes anon-homogeneous regular series-parallel system withparametersk n = 2, l n = 162, a = 1, q 1 = 1, e 1 = 2,p 11 = 2/162, p 12 = 160/162.and from (10)-(11) the reliability function of thissystem is given by(2)[R2 ,162(t)]= 1 – [1 – exp[–20.007t]] 2= 2exp[−20.007t]− exp[ −40.014t]for t ≥ 0. (19)According to (14)-(15), the subsystem lifetime meanvalue and the standard deviation areM 2 ≅ 0.075, σ 2 ≅ 0.056.(20)At the operation state z3the subsystem S 4becomes anon-homogeneous regular series-parallel (series)system with parametersk n = 1, l n = 162, q 1 = 1, e 1 = 3,p 11 = 2/162, p 12 = 160/162,and from (10)-(11) the reliability function of thissystem is given by- 138 -

K.Koowrocki Reliability modelling of complex systems-Part 2 - RTA # 3-4, 2007, December - Special Issue[ R 1, 162(3)(t)]= exp[–19.999t] for t ≥ 0. (21)According to (14)-(15), the system lifetime mean valueand the standard deviation areM 3 ≅ 0.050, σ 3 ≅ 0.050.(22)Finally, considering (12), the subsystem S4unconditional reliability is given by(1)(2)3 ,242( t)]+ 0.183⋅[R2,162()]R( t)≅ 0.684⋅[Rt(3)1 ,162( )]+ 0.133⋅[R t , (23)(1)(2)(3)where [R3 ,242(t)], [R2 ,162(t)], [ R 1, 162 (t)], aregiven by (17), (19), (21).Hence, applying (16) and (18), (20), (22), we get themean values and standard deviations of the subsystemunconditional lifetimes given byM ≅ 0.684⋅ 0. 078 + 0.183⋅0.075+ 0.133⋅0.050 ≅ 0.074,(24)σ ( 1) ≅ 0.054.(25)References[1] Grabski, F. (2002). Semi-Markov Models of SystemsReliability and Operations. Warsaw: SystemsResearch Institute, Polish Academy of Sciences.[2] Koowrocki, K. (2004). Reliability of LargeSystems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney -Tokyo.[3] Koowrocki, K. & Soszyska, J. (2005). Reliabilityand availability of complex systems. Quality &Reliability Engineering International. 1, 22, J.Wiley & Sons Ltd., 79-99.[4] Koowrocki, K. (2006). Reliability and riskevaluation of complex systems in their operationprocesses. International Journal of Materials &Structural Reliability. Vol. 4, No 2.[5] Koowrocki, K. (2007). Reliability and safetymodelling of complex systems – Part 1. Proc.Safety and Reliability Seminars – SSARS 2007.Sopot.[6] Koowrocki, K. (2007). Reliability of LargeSystems, in Encyclopedia of Quantitative RiskAssessment. Everrit, R. & Melnick, E. (eds.), JohnWiley & Sons, Ltd., (to appear).[7] Kwiatuszewska-Sarnecka, B. (2003). Analizaniezawodnociowa efektywnoci rezerwowania wsystemach szeregowych. PhD Thesis. MaritimeAcademy, Gdynia – Systems Research Institute,Polish Academy of Sciences, Warsaw.[8] Kwiatuszewska-Sarnecka B. (2006). Reliabilityimprovement of large multi-state parallel-seriessystems. International Journal of Automation &Computing. 2, 157-164.[9] Kwiatuszewska-Sarnecka, B. (2006). Reliabilityimprovement of large parallel-series systems.International Journal of Materials & StructuralReliability. Vol. 4, No 2.[10] Milczek, B. (2004). Niezawodno duychsystemów szeregowo progowych k z n. PhD Thesis.Maritime Academy, Gdynia – Systems ResearchInstitute, Polish Academy of Sciences, Warsaw.[11] Soszyska, J. (2006). Reliability of large seriesparallelsystem in variable operation conditions.International Journal of Automation & Computing.2, 3, 199-206.[12] Soszyska, J. (2006). Reliability evaluation of aport oil transportation system in variable operationconditions. International Journal of PressureVessels and Piping. 4, 83, 304-310.[13] Soszyska, J. (2007). Analiza niezawodnociowasystemów w zmiennych warunkacheksploatacyjnych. PhD Thesis. Maritime Academy,Gdynia – Systems Research Institute, PolishAcademy of Sciences, Warsaw.- 139 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special IssueKudzys AntanasKTU Institute of Architecture and Construction, Kaunas, LithuaniaTransformed conditional probabilities in the analysis of stochastic sequencesKeywordssafety margin, stochastic sequence, conditional probability, system safetyAbstractThe need to use unsophisticated probability-based approaches and models in the structural safety analysis of thestructures subjected to annual extreme service, snow and wind actions is discussed. Statistical parameters of singleand coincident two extreme variable actions and their effects are analysed. Monotone and decreasing randomsequences of safety margins of not deteriorating and deteriorating members are treated, respectively, as ordinaryand generalized geometric distributions representing highly-correlated series systems. An analytical analysis of thefailure or survival probabilities of members and their systems is based on the concepts of transformed conditionalprobabilities of safety margin sequences whose statistically dependent cuts coincide with extreme loadingsituations of structures. The probability-based design of members exposed to coincident extreme actions isillustrated by a numerical example.1. IntroductionThe stochastic systems and their subsystems consist ofsome particular members representing the onlypossible failure mode. To particular members belongcross and oblique sections of tension, compression,flexural and torsional structures. The structuralmembers (beams, slabs, columns, walls) of buildingsconsist of two or three design particular members andmay be treated as auto systems representingmulticriteria failure modes. An overloading ofmembers during severe service and climate actionsmay provoke a failure of structures. Therefore, therequirements of design codes should be satisfied at allsections along structural members.Structural failures and collapses in buildings andconstruction works can be caused not only byirresponsibility and gross human errors of designers,builders or erectors but also by some conditionalities ofrecommendations and directions presented in designcodes and standards. A possibility to ensure objectivelythe safety degree of structures subjected to extremeservice loads, wind gust and snow pressures or wavesurfs is hardly translated into reality using the traditionaldeterministic design methods of partial safety factors inEurope or load and resistance factors in the USA.It is understandable that probabilistic designapproaches are inevitable for the calibration of partialfactors. However, it should be more expedient toanalyse the structural safety of particular members andtheir systems by probability-based methods.Regardless of efforts to improve and modifydeterministic design approaches, it is inconceivable tofix a real reliability index of structures a failure domainof which changes with time. The time-dependent safetyassessment and prediction of deteriorating membersand systems using unsophisticated methods is asignificant concern of researchers.Despite of fairly developed up-to-date concepts ofreliability, hazard and risk theories, including thegeneral principles on reliability for structures [6], [7],[15], it is difficult to apply probability-basedapproaches in structural safety analysis. Theseapproaches may be acceptable to designers andbuilding engineers only under the indispensablecondition that the safety performance of members andtheir systems may be considered in a simple and easyperceptible manner. In other words, probabilisticmethods may be implanted into structural designpractice only using unsophisticated mathematicalmodels helping us to assess all uncertainties due to thefeatures of resistances and action effects of structures.This paper deals with probability-based safety analysisof deteriorating and not deteriorating members andtheir systems under extreme gravity and lateral(horizontal) actions using unsophisticated but fairlyexact design models.- 140 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special Issue2. Time dependent safety marginAccording to probability-based approaches (designlevel III), the time-dependent safety margin as theperformance of deteriorating particular members maybe presented as follows:[ ,X(t)] = θ R(t)− θ S − θ S ( t −Z(t)= g)R− θ S ( t)− θ S ( t), (1)q2q2wwwhere is the vector of additional variablescharacterizing uncertainties of models which give thevalues of resistance R , permanent S , sustainedgggq1q1S q1and extraordinary S q 2service and extreme wind S waction effects of members (Figure 1, a). This vectormay represent also the uncertainties of probabilitydistributions of basic variables.According to Rosowsky and Ellingwood [11], theannual extreme sum of sustained and extraordinaryoccupancy live action effects S ( t)= S ( t)S ( t)q q +1can be modelled as an intermittent process anddescribed by a Type 1 (Gumbel) distribution with thecoefficient of variation S q = 0. 58 , characteristic Sqkand meanqmqkq2S = 0. 47Svalues. Latter on Ellingwoodand Tekie [4] recommended modelling extreme valuesof this sum during a 50 years period by a Type 1distribution with the coefficient of variation S q = 0.25 and mean value S qm = Sqk.It is proposed to model the annual extreme climate(wind and snow) action effects by Gumbel distributionlaw with the mean values equal toSwm= Swk( 1+ k0.98Sw ) and Ssm= Ssk( 1+ k0.98Ss )[3, 6, 7, 13, 15]. According to meteorological data, thestrong wind conditions are characterized by a smallwind extreme velocity variation, i.e. v ≈ 0. 1. On thecontrary, a large variation is characteristic of strongsnow loading. Therefore, the coefficients of variationof wind and snow loads depending on the feature of ageographical area are equal to w = 0.2− 0. 4 and s = 0.3− 0.7 .Probability distributions of material properties areclose to a Gaussian distribution [3], [6], [9], [12].Therefore, a normal distribution or a log-normaldistribution may be convenient in resistance analysismodels [5], [6], [7]. The permanent action effect Scan be described by a normal distribution law [4], [5],[6], [10], [12]. Thus, for the sake of designsimplifications, it is expedient to present the expression(1) in the form:Z( t)= Rc ( t)− S(t), (2)where the component processR ( t)= θ R(t)− θ S , (3)cRggmay be considered as the conventional resistance ofmembers which may be modelled by a normaldistribution;ga)S, Rtλ1 t λ2R(t)d1d2Sq1Sgtb)SgR(t)S1S2S 1+S2R c(t)0 1 1 2 2 kt nnP{N>n}tFigure 1. Real (a) and conventional (b) models for safety analysis of particular members (sections) ofdeteriorating structuresS( t)θ S ( t)+ θ S ( t)S( t)= θ S ( t)+ θ S ( t), (5)= [ ], (4) [ ]qqwwssww- 141 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special Issueare the joint processes of two annual extreme action effectswhen floor and roof structures, respectively, are underconsideration. The components in square brackets belongingto the wind action effect are used in design analysis of windresistantmembers and systems. The action effect S s (t)inEquation (5) is caused by extreme snow loads.3. Safety margin sequences with independent cutsThe data presented in Section 2 allow us to modelextreme service and climate action effects asintermittent rectangular pulse renewal processes. Thesetime-variant intermittent action effects belong topersistent design situations in spite of the short periodof extreme events being much shorter than the designworking life of structures. When variable action effectsmay be treated as rectangular pulse processes, thetime-dependent safety margin (2) may be expressed asthe finite rank random sequence and written as:ZkThereRSck= R − S , k = 1,2, 3, ..., n −1,n . (6)ckRkk= θ R − θ S , (7)gg= θ S + θ S , (8)k = θqSqk+ θwSwk or k s sk w wkSare the components of this non-stationary sequence;n = λt n is the number of sequence cuts as criticalevents (situations) during design working life t n ofmembers (Figure 1, b), where λ = 1 tλis a meanrenewal rate of these events per unit time when theirreturn period is t λ .Usually the components R ck and S k are stochasticallyindependent. The instantaneous survival probability ofa member at k-th extreme situation (assuming that itwas safe at the situations 1, 2, …, k −1) is:instantaneous survival probabilities of membersP f 1 < Pf2 < ... < Pfk< ... Pf, n−1< Pfncalculated byEquation (10).Ps1ZP1f1> P >…> P >…> P > PZ Z Z ZPs2 sk s, n-1 sn2 k n-1 nf2P P Pfk f, n-1 fnFigure 2. The scheme of series systemsFailure probabilities of structures should always bedefined for some reference period t n or as a number ofextreme events n during this period. The scheme ofseries systems representing the safety marginsequences is given in Figure 2. When the cuts of rankrandom sequences are statistically independent, thecumulative distribution function and similarly a failureprobability of members during their service life [ 0,t n ]with n extreme situations may be presented asfollows:Pf= FN= 1 − P... +nk −1( n)= P{ N ≤ n} =⎡∑ ( 1 − Psk) ∏ Pk = 1⎢⎣i=1+k −1( 1 − P ) P + ... + ( − P )∏s1 s2s11n−1n( 1 − ) P = 1 − P .k = 1si⎥ ⎦⎤ski=1P ∏ ∏(11)sni=1siWhen the resistance R (t)is a time-invariant functionand treated as a stationary process, the instantaneoussurvival probability P sk by (9) is characterized by thesame value for all cuts of the monotone sequence. Inthis case, Equation (11) becomes a cumulativedistribution function of an ordinary geometricdistribution as follows:skPsi∞sk = P ck k ∫0R ck S k){ R > S } = f ( x)F ( x dxP , (9)PfN{ N ≤ n} = 1−( − P ) n= F ( n)= P1 . (12)fkwhere f R ck(x)and FS k(x)are the density anddistribution functions of a conventional resistanceRckby (7) and an extreme action effect S k by (8). In thiscase, the instantaneous failure probability of membersmay be presented as:k( )∏ − 1 − Pski=1P =P . (10)fk1siThus, the random sequence of safety margins may betreated as a geometric distribution with rankedThe failure probability of members may beapproximated by Equations (11) and (12) only forsituations in which a variance of the action effect2 is much larger than the value Rcfor theirconventional resistance by (7).4. Safety margin sequences with dependent cutsIn design practice, only recurrent extreme actioneffects caused by extraordinary service and climateloads may be treated as stochastically independentvariables. Usually, random sequence cuts of the safetymargin (6) are dependent. The value of a coefficient of2S- 144 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special Issueautocorrelation ρ kl of sequence cuts depends onuncertainties of material properties and dimensions ofmembers. This coefficient may be defined as:kl( Z Z ) = Cov ( Z Z ) ( Z × Z)ρ = ρ , ,, (13)where ( )klkCov Z k , Z l and Z k , Zlare anautocovariance and standard deviations of the randomsafety margins Z k and Z l .The finite random sequence of member safety marginsmay be treated as a series stochastic system. Thesurvival probability of highly correlated series systemsconsisting of two dependent elements can be expressedas follows:P{ Z > I Z > 0} = Ps × P{ Z > 0 Z 0}1 0 21 2 1 >laPs1 × Ps2+ ρ ( P2− P1× P2), (14)=sa 12 is the bond index ofsurvival probabilities of second-order series systems.The data calculated by (14) and computed by thecomplex numerical integration method presented byAhammed and Melchers [1] are very close. Thus, awhere ≈ 4.5( 1 − 0.98ρ)conditional probability { Z > Z 0}kP may be2 0 1 >⎡⎤a ⎛ 1 ⎞transformed to a probability P s2⎢1+ ρ ⎜ ⎟−1⎥ .⎢⎣⎝ Ps1⎠⎥⎦Therefore, Equation (14) may be presented in the form:{ Z1 > 0 I Z2> 0} ≈ Ps1Ps2P ×⎡⎤a ⎛ 1 ⎞× ⎢1+ ρ12⎜⎟−1⎥⎢⎣⎝ Ps1⎠⎥⎦l(15)For not deteriorating structures, a member resistance isa time-invariant fixed random function the numericalvalues of which are random only at the beginning of aprocess. Therefore, the coefficient of correlation (13)of monotone sequence cuts may be expressed as:kl2 2( 1+ S R )ρ = 1 . (16)kcWhen the monotone rank sequence of safety marginsconsists of n dependent elements, a failure probabilityof members is:Pf= P{ N ≤ n}n⎧= P⎨UZ⎩k= 1kn⎫ ⎧≤ 0⎬= 1 − P⎨IZ⎭ ⎩k= 1k⎫> 0⎬⎭n−1⎡ 1 ⎤n a⎛ ⎞≈1− P ⎢1⎜ 1⎟sk+ ρkl− ⎥ .(17)⎢⎣⎝ Psk⎠⎥⎦2k R cWhen a ratio of variances S > 1, thecoefficient ρ a kl ≈ 0 and the failure probability (17)becomes P = − ( 1 − ) nf P fk21 as it is expressed byEquation (12).A long-term survival probability of not deterioratingmembers is:n−1⎡ 1 ⎤n a⎛ ⎞P 1 ⎢1⎜ 1⎟s= − Pf= Psk+ ρkl− ⎥ . (18)⎢⎣⎝ Psk⎠⎥⎦The decreasing rank sequence of safety margins ofdeteriorating members may be treated as a generalizedgeometric distribution. Similar to Equation (17), thefailure probability of these members as series systemsmay be calculated by the formula:Pf= P{ N ≤ n}⎡... × ⎢1+ ρ⎢⎣≈1− ∏ Pak...1⎡⎢1+ ρ⎢⎣⎛⎜1⎝ Ps,na...1= 1 sk nkn −1⎛⎜1⎝ Ps, k−1⎞⎤− 1⎟⎥⎠⎥⎦⎞⎤− 1⎟⎥⎠⎥⎦⎡⎤a ⎛ 1 ⎞... × ⎢1+ ρ21⎜⎟−1⎥ , (19)⎢⎣⎝ Ps1⎠⎥⎦where the transformed rank coefficient of correlation is( ρ + ρ + + ρ + ρ ) ( k 1)ρk ... 1 = k,k−1k,k−2... k 2 k1− (20)The long-term survival probability of deterioratingmembers Ps= 1 − Pf, where the probability P f isgiven in (19).The presented method of transformed conditionalprobabilities may also be successfully used in thereliability analysis of random systems consisting ofindividual components and characterizing differentfailure modes of structures. In this case, it is expedientto base the structural safety analysis of systems on theranked survival probabilities of their members as:P s1 > Ps2 > ... > Psk> ... > Ps,n−1> Psn(Figure 2). Arank correlation matrix of systems is constructedtaking into account this analysis rule.4. The system of safety margin sequences- 145 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special IssueDue to the complexity of mathematical models, it israther difficult to assess and predict a failure probabilityof structures subjected to two and more coincidentrecurrent and different by nature extraordinary actions.The methods based on the Markov-chain model andTurkstra’s rule [14] may be quite unacceptable in aprobabilistic analysis of not only deteriorating but alsonot deteriorating members and their systems. TheMarkov-chain model may be quite inaccurate forreliability analysis of members exposed to multiplecombination of action-effect processes [12]. TheTurkstra’s rule may be assumed only in the case when theprincipal extreme load is strongly dominant [10].Failure probabilities of members may be computed bymodified numerical integration methods. It is suggestedto use the theoretical expression of the cumulativedistribution function of the maximum intensity of twoload processes [10], the load overlap method [12] and theimproved upper bounding techniques [13]. It leads tosufficiently accurate values but it is hard to realize theserecommendations in engineering practice.The need to simplify a reliability analysis ofdeteriorating structures is especially urgent. In anyanalysis case, it must be taken into account that amember failure caused by two statistically independentextreme action effects may occur not only in the case oftheir coincidence but also when the value of one out oftwo effects is extreme. Therefore, three finite randomsequences of safety margins should be considered:= , k = 1,2, ..., n1, (21)M1kRck− S1k= , k = 1,2, ..., n2, (22)M 2kRck− S2kM 3kRck− S3k= , k = 1,2, ..., n3. (23)There S3 k = S1k+ S2kis the joint action effect, therecurrence number of which during the period of time[ 0 ,t n ] may be calculated by the equation:( d1+ 2 ) λ12n n , (24)3 = t d λwhere d 1 , d 2 and λ 1, λ 2 are durations and renewalrates of extreme actions [8].Mostly, the duration d of annual extreme gravityqservice loads is from 1 to 3 days. The durations of annualextreme snow and wind loads, respectively, are: ds= 14-28 days and dw= 8-12 hours. The renewal rates of theseactions are: λ q = λ s = λ w = 1/year. Therefore, for 50years reference period, the recurrence numbers ofextreme actions are: nqw= 0.2-0.5 and n = 2-4.swWhen probability distributions of random variables Xand Y obey a Gumbel distribution law, the bivariatedensity function of the random variable Z = X + Ymay be presented in the form:( z − y,X 0.45X)f ( z)∫ f −z= ∞ −∞xm× f ( y, Y − 0.45Y)dy , (25)ymwhere X m , Y m and X , Y are means and standarddeviations of these variables.Taking into account that Z = X + Y is thevariance of bivariate probability distribution, the jointdensity function may be expressed as:ZZ( z a )f ( z)≈ f , , (26)a0.60.50.4zf z(z)z22 2( X + Y) − 0.45( X + ) 1/ 2= X + Y − 0.19Y .mma)122 20.3110.2b)0.111 1 2 2205 6 7 8 9 10 11 12 13 14 15 16 z=x+yFigure 3. Bivariate density functions calculated byEquations (25) – 1 and (26) – 2: the coefficients ofcorrelation X= Y= 0.10 (a) and 0.224 (b)The probability density curves of joint extremevariable Z = X + Y are given in Figure 3. It is notdifficult to ascertain that the difference between thevalues computed by Equations (25) and (26) is fairlysmall. Besides, the upper tails of both density curvescoincides. Therefore, in design practice it is expedientto use the conventional bivariate distribution functionof two independent extreme action effects with themean = S S and the varianceS 3 k , m 1k,m + 2k,m2 2 2S3k S1k S2k = + .6. Numerical exampleThe knee-joints of not deteriorating concrete frames ofreliability class RC2 are under exposure of shear forcesduring 50 years period (Figure 4). The shear resistanceof knee-joints is expressed as: R = 0. 068bhf . The22c- 146 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special Issuecharacteristic, design and mean values of the concretecompressive strength and shear resistance of kneejointsare:f = 30 MPa, f = 20 MPa, f = 38 MPa;ckcdR = 306 kN, R = 204 kN, R = 387.6 kN.kdThe variance of shear resistance of knee-joints is:22( 0.128×387.6) = 2461. 4mcm R =(kN) 2 .2V g = 60.4 (kN) 2 ; = 0.6,VV s( 1+k V)= 15.21 kN,sm = Vsk0.98 s2V s = 83.25 (kN) 2 ; = 0.3,VV w( 1 + k V)= 21.86 kN;wm = Vwk0.98 w2V w = 43.0 (kN) 2 .The parameters of additional variables are:1 - 1V V Vg s wb = 0.3 m11θ Rm = 1.0, = 0.1;θ Rh = 0.5 mθ Vm = 1.0,Figure 4. The knee-joint of concrete framesThe characteristic and design values of shear forcescaused by permanent, snow and wind loads are:VgkVsk= 77.72 kN,V= wk= 38.86 kN;V = 77 .72 × 1.35 = 104.92 kN,gd θ = θ= = 0.1,gs = 0.15.θ swθ wThus, the variances of revised shear forces are:2 ( θ g V g )= 120.8 (kN) 2 ,2 ( θ )= 85.56 (kN) 2 ,s V sVsdVwd= 38.86 × 1.5 = 58.29 kN,= 38.86 × 0.7 × 1. 5 = 40.8 kN.2 ( θ )= 47.8 (kN) 2 ,w V w2 ( θ )= 157.17 (kN) 2 .sw V swThus, the joint design shear forceVd= Vgd+ Vsd+ Vwd= 204 kN = Rd.Therefore, according to deterministic calculation data,the frame knee-joints are reliable.The coefficients of variation, means and variances ofthese extreme shear forces are: = 0.1,VV ggm = V gk= 77.72 kN,The parameters of conventional shear resistance (3)are:Rcm2R c= 387.6 – 77.72 =309.9 kN, = 1.0 × 2461.4 + 387.6 2 × 0.01+ 120.8 = 4084.6 (kN) 2 .According to (16), the coefficients of autocorrelationof the safety margins Z w = Rc−Vw, Z s = Rc−VsandZ = R −V−Vof considered knee-joints are:swcρ w, kl = 0.9884,sw- 147 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special IssueDespite high-correlated cuts of the safety marginρ s, kl = 0.9795,sequences Z w , Z s and Z sw of knee-joints,considerable differences among their instantaneous andρ sw, kl = 0.9629.long-term survival probabilities are corroborated.The reliability verification of knee-joints of concreteframes by the deterministic partial factor method andThe recurrence number of joint action effectprobability-based approaches practically gave the sameV s + V w calculated by Equation (24) is:results.n3= 50 [21/365 + 12/(24 × 3.65)] 1 × 1 = 2.945. 7. ConclusionWhen the system may be subjected to annual extremeAccording to (9), the instantaneous survivalservice and climate actions, it is expedient to expressprobabilities of members are:its member performance processes by finite randomsequences of safety margins, the dependent cuts ofPsk,w= 0.99999617,which coincide with the extreme loading situations ofstructures. Therefore, the generalized geometricPsk,s= 0.99999728,distribution as the decreasing stochastic sequence maybe successfully used in failure or survival probabilityanalysis of highly correlated series systems. It leads toPsk,sw= 0.9999837.considerable perfections of probability-based analysisof deteriorating structures subjected to recurrent singleTherefore, according to (18), the partial long-term and coincident actions as intermittent rectangular pulsesurvival probabilities of analysed knee-joints are: renewal processes. A Gumbel distribution law may beused not only for joint sustained and extraordinaryPsw= 0.9999717,variable service loads but also for the sum of annualextreme action effects.Pss= 0.9999710,For the sake of simplifications of probabilistic timedependentsafety analysis of members, it isrecommended to use design models with theirPs,sw= 0.9999747.conventional resistances and correlated sequence cutsof safety margins representing a variety of loadAccording to (13), the coefficients of cross-correlation combinations. The presented unsophisticatedof safety margins are:probability-based approaches and models maystimulate engineers having minimum appropriate skillsρ sw = 0.9839,to use full probabilistic methods in their engineeringpractice more courageously and effectively. It shouldρ w, sw = 0.9871,be one more remedy in the struggle againstdeterministic approaches in the structural design.ρ s, sw = 0.9914.ReferencesFrom Equation (19), the total survival probability of [1] Ahammed, M. & Melchers, R. E. (2005). New boundsknee-joints is:for highly correlated structural series systems.ICOSSAR, G. Augusti, G. J. Schuëller, M. CiampoliP = 0.9999747 × 0.9999717 × 0.9999710(eds), Millpress, Rotterdam, 3556-3662.[2] Dey, A. & Mahadevan, S. (2000). Journal of⎡11.84⎛1 ⎞⎤× ⎢1+ 0.98767 ⎜ −1⎟⎥ Structural Engineering, ASCE 126 (5), 612-620.⎣⎝ 0.9999717 ⎠ ⎦ [3] Ellingwood, B. R. (1981). Wind and snow loadstatistics for probabilistic design, Journal of⎡11.73⎛1 ⎞⎤Structural Division, ASCE 107 (7), 1345-1349.× ⎢1+ 0.9871 ⎜ −1⎟⎥ = 0.9999635. [4] Ellingwood, B. R. & Tekie, P.B. (1999). Wind load⎣⎝ 0.9999747 ⎠ ⎦ statistics for probability-based structural design,Journal of Structural Engineering, ASCE 125 (4),It corresponds to the reliability indexβ = 3.97> β min ( = 3.8) [5]. 453-463.[5] EN 1990 (2002). Eurocode 1. Basis of design andactions on structure, CEN, Brussels.- 148 -

A. Kudzys Transformed conditional probabilities in the analysis of stochastic sequences - RTA # 3-4, 2007, December - Special Issue[6] ISO 2394 (1998). General principles on reliabilityfor structures, 2nd ed., Switzerland.[7] JCSS (2000). Probabilistic model code: Part 1 –Basis of design, 12 th draft.[8] Kudzys, A. (1992). Probability estimation ofreliability and durability of reinforced concretestructures. Technika, Vilnius.[9] Lundberg, J. E. & Galambos, T. V. (1996). Loadand resistance factor design of composite columns,Structural Safety, 18 (2, 3), 169-177.[10] Mori, V., Kato, T. & Murai, K. (2003).Probabilistic models of combinations of stochasticloads for limit state design, Structural Safety, 25(1), 69-97.[11] Rosowsky, D. & Ellingwood, B. (1992). Reliabilityof wood systems subjected to stochastic live loads,Wood and Fiber Science, 24 (1), 47-59.[12] Raizer, V. P. (1998). Theory of Reliability inStructural Design. ACB Publishing House, Moscow.[13] Sykora, M. (2005). Load combination model based onintermittent rectangular wave renewal processes.ICOSSAR, G. Augusti, G. J. Schuëller, M. Ciampoli(eds), Millpress, Rotterdam, 2517-2524.[14] Tukstra, C. J. (1970). Theory of structural designdecision. Study N o 2. Solid Mechanics Division,Waterloo.- 149 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special IssueKwiatuszewska-Sarnecka BoenaGdynia Maritime University, PolandOn asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy:series and parallel systemsKeywordsreliability improvement, limit reliability functionsAbstractThe paper is composed of two parts, in this part after introducing the multi-state and the asymptotic approaches tosystem reliability evaluation the multi-state homogeneous series and parallel systems with reserve components aredefined and their multi-state limit reliability functions are determined. In order to improve of the reliability ofthese systems the following methods are used: (i) a warm duplication of components, (ii) a cold duplication ofcomponents, (iii) a mixed duplication of components, (iv) improving the reliability of components by reducingtheir failure rate. Next, the effects of the systems’ reliability different improvements are compared.1. IntroductionMost real systems are very complex and it is difficultto analyze and to improve their reliability. Largenumbers of components and subsystems and theiroperating complexity cause that the evaluation of theirreliability is complicated. As a rule these are seriessystems, parallel systems or “m out of n” systemscomposed of a large number of components. One ofthe important techniques for reliability evaluation oflarge systems is the asymptotic approach. Themathematical methods are based on the limit theoremsof order statistics distributions considered in a wideliterature. These theorems generated investigations onlimit reliability functions for systems with two-statecomponents. Next, more general systems with multistatecomponents began to be considered. Theasymptotic approach is also very useful in reliabilityimprovement of large multi-state systems because ofsimplifying the calculation.2. Multi-state and asymptotic approachIn multi-state reliability analysis presented in thispaper it is supposed that:- E i , i = 1,2,...,n, are components of a system,- all components and a system under considerationhave the state set {0,1,...,z},- the state indices are ordered, the state 0 is the worstand the state z is the best,- T i (u), i = 1,2,...,n, are independent random variablesrepresenting the lifetimes of the components E i in thestate subset {u,u+1,...,z} while they were in the statez at the moment t = 0,- T(u) is a random variable representing the lifetime ofa system in the state subset {u,u+1,...,z} while it wasin the state z at the moment t = 0,- the system state degrades with time t without repair,- e i (t) is a component E i state at the time t, t > 0,- s(t) is a system state at the moment t, t > 0.Definition 2.1. A vectorRi ( t,⋅ ) = [ Ri( t,0),Ri( t,1),...,Ri( t,z)],t ∈(-∞,∞), i = 1,2,...,n,whereR ( t,u)= P(e ( t)≥ u e (0) = z)= P(T ( u)t),i iii >t ∈(-∞,∞), u = 0,1,...,z,is the probability that the component E i is in the statesubset {u,u+1,...,z} at the time t, t ∈(-∞,∞) while it wasin the state z at the moment t = 0, is called the multistatereliability function of a component E i .Definition 2.2. A vector- 150 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special IssueR ( t,⋅ ) = [ R ( t,0),R ( t,1),...,R ( t,z)],nt ∈(-∞,∞),wherennR ( t,u)= P(s(t)≥ u s(0)= z)= P(T ( u)t),n >t ∈(-∞,∞), u = 0,1,...,z,is the probability that the system is in the state subset{u,u+1,...,z} at the moment t, t ∈(-∞,∞) while it was inthe state z at the moment t = 0, is called the multi-statereliability function of a system.In the asymptotic approach to system reliabilityanalysis we are interested in limit distributions of astandardized random variable(T(u) - b n (u))/a n (u), u = 1,2,...,z,where T(u) is the lifetime of the system in the statesubset {u,u+1,...,z} and a n( u)> 0,b n( u)∈ ( −∞,∞),u =1,2,...,z, are some suitably chosen numbers, callednormalizing constants.SinceP((T ( u)− bn= P(T(u)> a= nwhere( u))/ ann( u)t + b( u))n( u)> t)R ( a ( u)t b ( u),u),u = 1,2,...,z,n+nnR ( t,⋅ ) = [ R ( t,0),R ( t,1),...,R ( t,z)],t ∈(-∞,∞),nnnis the multi-state reliability function of the system, thenwe assume the following definition.Definition 2.3. A vectorR ( t,⋅)= [1, R ( t,1),...,R ( t,z)],t ∈(-∞,∞),is called the limit multi-state reliability function of thesystem if there exist normalizing constants a n (u) > 0,b n (u) ∈(-∞,∞) such thatlim R ( a ( u)t + b ( u),u)= R ( t,u),n→∞nnt ∈C R(u) , u = 1,2,...,z,nwhere C R(u) is the set of continuity points of R(t,u).nThe knowledge of the system limit reliability functionallow us, for sufficiently large n, to apply the followingapproximate formulaRn ( t,⋅)= R (( t − bn( u)) / an( u),⋅),t ∈(-∞,∞). (1)3. System reliability improvement3.1. Reliability improvement of a multi-stateseries systemDefinition 3.1. A multi-state system is called a seriessystem if its lifetime T(u) in the state subset{u,u+1,...,z} is given byT ( u)= min{ T ( u)},u = 1,2,...,z.1≤i≤niFigure 1. The scheme of a homogeneous series systemDefinition 3.2. A multi-state series system is calledhomogeneous if its component lifetimes T i (u) in thestate subsets {u,u+1,...,z} have an identical distributionfunctionF i (t,u) = F(t,u), u = 1,2,...,z, t ∈(-∞,∞), i = 1,2,...,n,The reliability function of the homogeneous multi-stateseries system is given byR ( t,⋅ ) = [1, R ( t,1),...,R ( t,z)],nwherennnR ( t , u)= [ R(t,u)], t ∈(-∞,∞), u = 1,2,...,z.nDefinition 3.3. A multi-state series system is called asystem with a hot reserve of its components if itslifetime T (1) (u) in the state subset {u,u+1,...,z} is givenbyT(1)E 1 E 2 E n-1 E n( u)= min{max{ T ( u)}},u = 1,2,...,z,1≤i≤n1≤j≤2ijwhere T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multistateseries system with a hot reserve of itscomponents is given by(1)I R n (t , ⋅ ) = [1,(1)I R n (t,1),...,(1)I R n (t,z)],- 151 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special Issuewhere( 1)2 nIR( t,u)= [1 − ( F(t,u))] , t ∈(-∞,∞). (2)nLemma 3.1. If(1)(i) Ι R ( t,u)= exp[- V ( t,u)], u = 1,2,...,z, is nondegeneratereliability function,(1)(ii) I R n(t,u), t ∈ (-∞,∞), u = 1,2,...,z, is thereliability function of non-degenerate multistateseries system whit a hot reserve of itscomponents defined by (2),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,then(1)lim ( ( ) ((1)I R nan u t + bnu))= Ι R ( t,u), t ∈C R,n→∞u = 1,2,...,z,if and only iflim n[F(an→∞nu = 1,2,...,z.( u)t + bn( u))]2=⎺V(t,u), t ∈C V,Proposition 3.1. If components of the homogeneousmulti-state series system with a hot reserve of itscomponents have multi-state exponential reliabilityfunctions1and a n (u) = , b n (u) =0, u = 1,2,...,z,λ(u)then(1)Ι R (t,u) = 1, t < 0,Ι R(1)(t,u) = exp[-t 2 ], t ≥ 0, u = 1,2,...,z,is its limit reliability function.The proof of Proposition 3.1 is given in [9].Corollary 3.1. The reliability function of exponentialseries system whit a hot reserve of its components isgiven by(1)I ( t,u)= 1 , t < 0,R n(1)n2 2I R ( t,u)≅ exp[ −λ( u)nt ] , t ≥ 0 , u = 1,2,...,z. (3)Definition 3.4. A multi-state series system is called asystem with a cold reserve of its components if itslifetime T (2) (u) in the state subset {u,u+1,...,z} is givenbyT(2)2( u)= min{ ∑T1≤i≤nijj=1( u)}, u = 1,2,...,z,where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multi-stateseries system with cold reserve of its components isgiven by(2)I R n (t , ⋅ ) = [1,where(2)I (t,1),..., I ( t,z)],(2)R n(2)nI R n( t , u)= [1 − F(t,u)∗ F(t,u)], t ∈ (-∞,∞), (4)u = 1,2,...,z.Lemma 3.2. If(2)(i) Ι R ( t,u)= exp[- V ( t,u)] u = 1,2,...,z, is nondegeneratereliability function,(2)(ii) I R n(t,u), t ∈ (-∞,∞), u = 1,2,...,z, is thereliability function of non-degenerate multistateseries system whit a cold reserve of itscomponents defined by (4),(iii) a n (u) > 0, b n (u)∈ (-∞,∞),then(2)lim ( ( ) ((2)I R nan u t + bnu))= Ι R ( t,u), t ∈C ΙR,n→∞u = 1,2,...,z,if and only ifn→∞R nlim n[F(a ( u)t + b ( u))∗ F(a ( u)t b ( u ))]=⎺V(t,u), t ∈CVn nn+, u = 1,2,...,z.Proposition 3.2. If components of the homogeneousmulti-state series system with a cold reserve of itscomponents have multi-state exponential reliabilityfunctions2and an( u)= , bn( u)= 0,u = 1,2,...,z,λ(u)nthen(2)Ι R ( t,u)= 1, t < 0,Ι R(2)( t,u)= exp[-t 2 ], t ≥ 0, u = 1,2,...,z,is its limit reliability function.The proof of Proposition 3.2 is given in [9].n- 152 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special IssueCorollary 3.2. The reliability function of exponentialseries system whit a cold reserve of its components isgiven by(2)I ( t,u)= 1, t < 0,R n(2)2 2I Rn( t,u)≅ exp[ −λ( u)nt / 2] , t ≥ 0, (5)u =1,2,...,z.Definition 3.5. A multi-state series system is called asystem with a mixed reserve of its components if itslifetime T (3) (u) in the state subset {u,u+1,...,z} is givenbyT(3)( u)= min{ min {max{ T1≤i≤s1n 1≤j≤2u = 1,2,...,z,ij( u)}},mins1n+ 1≤i≤n2{ ∑Tijj=1( u)}},where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve componentsand s 1 , s 2 , where s 1 + s 2 = 1 are fractions of thecomponents with hot and cold reserve, respectively.The reliability function of the homogeneous multi-stateseries system with a mixed reserve of its components isgiven by(3)I R n (t , ⋅ ) = [1,where(3)IR n(3)R n(3)I (t,1),..., I ( t,z)],2 s 1ns2n( t,u)= [1 − ( F(t,u))] [1 − F(t,u)∗ F(t,u)], (6)t ∈ (-∞,∞), u = 1,2,...,z.Lemma 3.3. If(3)(i) Ι R ( t,u)= exp[ − V ( t,u)], u = 1,2,...,z, isnon-degenerate reliability function,(3)(ii) I R n(t,u), t ∈(-∞,∞), u = 1,2,...,z, is thereliability function of non-degenerate multistateseries system whit a mixed reserve of itscomponents defined by (6),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,then(3)lim ( ( ) ((3)I R nan u t + bnu))= Ι R ( t,u), t ∈C ΙR,n→∞u = 1,2,...,z,if and only iflim 2 ns1 [ F ( a ( u ) t b ( u))]n→∞n+nR n+ ns [ F(a ( u)t + b ( u))∗ F(a ( u)t b ( ))] =⎺V(t,u),2 n nn+nut ∈CV, u = 1,2,...,z.Proposition 3.3. If components of the homogeneousmulti-state series system with a mixed reserve of itscomponents have multi-state exponential reliabilityfunctions1and an( u)= , bn( u)= 0,u = 1,2,...,z,λ(u)nthenΙ R ( t,u)= 1 for t < 0,Ι R ( t,u)= exp[-(2s 1 +s 2 ) t 2 /2] , t ≥ 0, u = 1,2,…,z,is its limit reliability function.Corollary 3.3. The reliability function of exponentialseries system whit mixed reserve of its components isgiven by(3)I ( t,u)= 1 , t < 0,R n(3)22I R n( t , u)≅ exp[ −λ ( u)n(2s1+ s2) t / 2] , t ≥ 0, (7)u = 1,2,...,z.Proposition 3.4. If components of the homogeneousmulti-state series system have improved componentreliability functions i.e. its components failure ratesλ(u) is reduced by a factor ρ(u), ρ(u)∈ ,u = 1,2,...,z, i.e.R ~ (t,u) = 1, t < 0,R ~ (t,u) = exp[- λ ( u ) ρ(u)t ], t ≥ 0, λ (u)> 0,u = 1,2,…,z,and a n (u) =then(4)Ι R ( t,u)= 1, t < 0,1, b n (u) = 0, u = 1,2,...,z,λ(u ) ρ(u)n(4)Ι R ( t,u)= exp[-t], t ≥ 0, u = 1,2,...,z,is its limit reliability function.Corollary 3.4. The reliability function of exponentialseries system whit improved reliability functions of itscomponents is given by- 153 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special Issue(4)I ( t,u)= 1, t < 0,R n(4)I ( t,u)= exp[ −λ(u)nρ(u)t], t ≥ 0, u = 1,2,...,z. (8)R n3.2. Reliability improvement of a multi-stateparallel systemDefinition 3.6. A multi-state system is called a parallelsystem if its lifetime T(u) in the state subset{u,u+1,...,z} is given byT ( u)= max{ T ( u)},u = 1,2,...,z.1≤i≤ni(1)IR n (t , ⋅ ) = [1,where(1)IR n (t,1),...,(1)IR n (t,z)],( 1)2 nIRn( t,u)= 1−[(F(t,u))] , t ∈(-∞,∞), (9)u = 1,2,...,z.Lemma 3.4. If(1)(i) ΙR (t,u) = exp[- V ( t,u)], u = 1,2,...,z, is nondegeneratereliability function,(1)(ii) IRn(t,u), t ∈ (-∞,∞), u = 1,2,...,z, is thereliability function of non-degenerate multistateparallel system whit a hot reserve of itscomponents defined by (9),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,then(1)lim IR ( ( ) (nanu t + bnu))= ΙR ( t,u), t ∈C ΙR,n→∞u = 1,2,...,z,Figure 2. The scheme of a homogeneous parallelsystemDefinition 3.7. A multi-state parallel system is calledhomogeneous if its component lifetimes T i (u) in thestate subsets {u,u+1,...,z} have an identical distributionfunctionF i (t,u) = F(t,u), u = 1,2,...,z, t ∈(-∞,∞), i = 1,2,...,n.The reliability function of the homogeneous multi-stateparallel system is given byR t,⋅ ) = [1, R ( t,1),...,R ( t,z)],wheren(nnnR ( t , u)= 1−[F(t,u)], t ∈(-∞,∞), u = 1,2,...,z.nif and only iflim 2n[R(an→∞u = 1,2,...,z.n( u)t + bn( u))]=V(t,u), t ∈ C V ,Proposition 3.5. If components of the homogeneousmulti-state parallel system with a hot reserve of itscomponents have multi-state exponential reliabilityfunctions1 log 2nand a n (u) = , b n (u) = , u = 1,2,...,z,λ(u)λ(u)thenIR (1) (t) = 1 - exp[-exp[-t]], t ∈ (-∞,∞), u = 1,2,...,z,is its limit reliability function.Proof: Since for all fixed u, we haveDefinition 3.8. A multi-state parallel system is called asystem with a hot reserve of its components if itslifetime T (1) (u) in the state subset {u,u+1,...,z} is givenbyt + log 2nan( u)t + bn( u)= → ∞λ(u)for t ∈ ( −∞,∞),u = 1,2,...,z.asn → ∞T(1)( u)= max{max{ T ( u)}},u = 1,2,...,z,1≤i≤n1≤j≤2ijwhere T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multi-stateparallel system with a hot reserve of its components isgiven byThereforeV ( t,u)= lim 2nR(a ( u)t + b ( u))n→∞= lim 2nexp[ −λ(u)(an→∞n= lim 2nexp[ −t− log 2n]n→∞nn( u)t + bn( u))]- 154 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special Issue= exp[ −t], t ∈( −∞,∞),which by Lemma 3.4 completes the proof.Corollary 3.5. The reliability function of exponentialparallel system whit a hot reserve of its components isgiven by(1)IRn( t , u)≅ 1−exp[ − exp[ −λ(u)t + log 2n], (10)t ∈ ( −∞,∞),u = 1,2,...,z.Definition 3.9. A multi-state parallel system is called asystem with a cold reserve of its components if itslifetime T (2) (u) in the state subset {u,u+1,...,z} is givenbyT(2)2( u)= max{ ∑T1≤i≤nijj=1( u)}, u = 1,2,...,z,where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multi-stateparallel system with a cold reserve of its components isgiven by(2)IR n (t , ⋅ ) = [1,where(2)IR (t,1),..., IR ( t,z)],(2)n(2)nIRn( t , u)= 1−[F(t,u)∗ F(t,u)], (11)t ∈ (-∞,∞), u = 1,2,...,z.Lemma 3.5. If(2)(i) ΙR (t,u) = exp[-V(t,u)], u = 1,2,...,z, is nondegeneratereliability function,(2)(ii) IRn(t,u), t ∈ (-∞,∞), u = 1,2,...,z, is thereliability function of non-degenerate multistateparallel system whit a cold reserve of itscomponents defined by (11),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,then(2)lim ( ( ) ((2)IR a u t b u))= ΙR ( t,u),n→∞t ∈ CRif and only ifn→∞n n+, u = 1,2,...,z,nlim n[1 − F(a ( u)t + b ( u))∗ F(a ( u)t b ( u ))]n nn+= V(t,u), t ∈ C V , u = 1,2,...,z.nnProposition 3.6. If components of the homogeneousmulti-state parallel system with a cold reserve of itscomponents have multi-state exponential reliabilityfunctions1and a ( u)= exp[ λbn( u)]n ,= nλ(u)λ(u)b ( u), u = 1,2,...,z,nthenIR (2) (t,u) = 1 - exp[-exp[-t]], t ∈ (-∞,∞),u = 1,2,...,z,is its limit reliability function.Proof: Since for all fixed u, we havean ( u)t + bn( u)→ ∞ as n → ∞ , t ∈( −∞,∞),u =1,2,...,z.ThereforeV(t,u)= lim n[1 − F(a ( u)t + b ( u))∗ F(a ( u)t b ( u ))]n→∞= lim n[(1+ λ(u)(an→∞exp[ −λ(u)(ann nn+n( u)t + b( u)t + bn( u))]n( u)))⋅= lim n[(1+ t + λ ( u)b ( u)) exp[ −(t + λ(u)bn→∞1+t= lim n[n→∞exp[ λ(u)b= exp[ −t]for t ∈( −∞,∞),nnnn( u))]λ(u)bn( u)+]exp[ −t]( u)]exp[ λ(u)b ( u)]which by Lemma 3.5 completes the proof.Corollary 3.6. The reliability function of exponentialparallel system whit a cold reserve of its components isgiven by(2)IRn( t,u)≅ 1−exp[ − exp[ −λ(u)t + λ(u)bn( u)], (12)t ∈( −∞,∞),u = 1,2,...,z.Definition 3.10. A multi-state parallel system is calleda system with a mixed reserve of its components if itslifetime T (3) (u) in the state subset {u,u+1,...,z} is givenbyn- 155 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special IssueT(3)( u)= max{max{max{ Tu = 1,2,...,z,2max { ∑Ts1n+ 1≤i≤n1≤i≤s1n 1≤j≤2ijj=1ij( u)}},( u)}},where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve componentsand s 1 , s 2 , where s 1 + s 2 = 1 are fractions of thecomponents with hot and cold reserve, respectively.The reliability function of the homogeneous multi-stateparallel system with mixed reserve of its components isgiven by(3)IR n (t , ⋅ ) = [1,where(3)IR (t,1),..., IR ( t,z)],(3)n(3)2 ns nIR s 12n( t,u)= 1−[(F(t,u))] [ F(t,u)∗ F(t,u)], (13)t ∈ (-∞,∞), u = 1,2,...,z.Lemma 3.6. If(3)(i) ΙR (t,u) = exp[ − V ( t,u)], u = 1,2,...,z, is nondegeneratereliability function,(3)(ii) IRn(t,u), t ∈(-∞,∞), u = 1,2,...,z, is the reliabilityfunction of non-degenerate multi-state parallelsystem with a mixed reserve of its componentsdefined by (13),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,then(3)lim ( ( ) ((3)IRnanu t + bnu))= ΙR ( t,u),t∈ CR,n→∞u = 1,...,z,if and only iflim 2 ns1 [ R ( a ( u ) t b ( u))]n→∞n+n+ ns2[1− F(an ( u)t + bn( u))∗ F(an( u)t + bn( u))]=⎺V(t,u), t ∈ C V , u = 1,2,...,z.Proposition 3.7. If components of the homogeneousmulti-state parallel system with a mixed reserve of itscomponents have multi-state exponential reliabilityfunctions anda n (u) =then1 exp[ λ(u)bn( u)], = s2n,λ(u)b ( u)−1λ(u)b ( u)nnnIR (3) (t,u) =1 - exp[-exp[-t]], t∈ (-∞,∞), u = 1,…,z,is its limit reliability function.Proof: Since for all fixed u, we havean ( u)t + bn( u)→ ∞ as n → ∞ , t ∈( −∞,∞),and1→ 0 as n → ∞ , t ∈( −∞,∞).λ(u)b ( u)−1nThereforeV ( t,u)= lim n[2s1 exp[ −λ(u)(a ( u)t + b ( u))]n→∞+ s2[(1+ λ(u)(an ( u)t + bn( u)))exp[ −λ(u)(an ( u)t + b= lim n exp[ −λ(u)(an→∞s2λ(u)(an ( u)t + bn( u))[1 +s2s2+ 2sλ(u)(an ( u)t + b= lim exp[ −λ(u)an→∞n1nnn( u))]]]( u)t + bn( u)t]]( u))nn( u))]exp[ −λ ( u)b ( u)+ log ns2λ(u)b ( u)[ 1+ +o (1)]][1 + o(1)]= exp[ −t],t ∈( −∞,∞),u = 1,2,…,z,which by Lemma 3.6 completes the proof.nCorollary 3.7. The reliability function of parallelsystem whit a mixed reserve of its components is givenbyIR n(3)( t,u)λ(u)bn( u)−1≅ 1−exp[ −exp[−t + ( λ(u)bn( u)−1)],bn( u)t ∈( −∞,∞),u = 1,2,...,z. (14)Proposition 3.8. If components of the homogeneousmulti-state parallel system have improved componentreliability functions i.e. its components failure ratesλ(u) is reduced by a factor ρ(u), ρ(u)∈,u = 1,2,...,z, andn- 156 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special Issuea n (u) =1log n, b n (u) = , u = 1,2,...,z,λ(u)ρ ( u)λ(u)ρ(u)then(4)ΙR ( t,u)= 1−exp[ −exp[−t]],t ∈ (-∞,∞),u = 1,2,...,z,is its limit reliability function.Corollary 3.8. The reliability function of exponentialparallel system whit improved reliability functions ofits components is given by(4)IRn( t , u)≅ 1−exp[ − exp[ −λ(u)ρ(u)t + log n], (15)t ∈ ( −∞,∞),u = 1,2,...,z.4. Comparison of reliability improvementeffectsThe comparisons of the limit reliability functions of thesystems with different kinds of reserve and suchsystems with improved components allow us to findthe value of the components decreasing failure ratefactor ρ(u), which warrants an equivalent effect of thesystem reliability improvement.4.1 Series systemThe comparisons of the system reliability improvementeffects in the case of the reservation to the effects inthe case its components reliability improvement maybe obtained by solving with respect to the factorρ ( u)= ρ(t,u)the following equationsIR(4)((t −b( u))/ a ( u))n( k)= I ((t − b ( u))/ a ( u))nRn n, u = 1,2,...,z, (16)k = 1,2,3.The factors ρ ( u)= ρ(t,u)decreasing componentsfailure rates of the homogeneous exponential multistateseries system equivalent with the effects of hot,cold and mixed reserve of its components as a solutionof the comparisons (16) are respectively given byk = 1k = 2k = 3ρ ( u ) = ρ ( t,u)= λ(u)t , u = 1,2,...,z,λ(u)tρ ( u)= ρ(t,u)= , u = 1,2,...,z,22s1+ s2ρ(u ) = ρ(t,u)= λ(u)t , u = 1,2,...,z.24.2. Parallel systemThe comparisons of the system reliability improvementeffects in the case of the reservation to the effects inthe case its components reliability improvement maybe obtained by solving with respect to the factorρ ( u)= ρ(t,u)the following equationIR(4)((t − b ( u))/ a ( u))n( k )= R ((t − b ( u))/ a ( u))nIn n, u = 1,2,...,z, (17)k = 1,2,3.The factors ρ ( u)= ρ(t,u)decreasing componentsfailure rates of the homogeneous exponential multistateparallel system equivalent with the effects of hot,cold and mixed reserve of its components as a solutionof the comparisons (17) are respectively given byk = 1log 2ρ(u)= ρ(t,u)= 1−, u = 1,2,...,z,λ(u)tλ(u)bn( u)− log nk = 2 ρ(u)= ρ(t,u)= 1−,λ(u)tu = 1,2,...,z,k =3λ(u)bn( u)−1ρ(u)= ρ(t,u)=λ(u)λ(u)b−u = 1,2,...,z.5. Conclusionn( u)−1− log n,λ(u)tProposed in the paper application of the limit multistatereliability functions for reliability of largesystems evaluation and improvement simplifiescalculations. The methods may be useful not only inthe technical objects operation processes but also intheir new processes designing, especially in theiroptimization. The case of series, parallel and ”m out ofn” (in part 2) systems composed of components havingexponential reliability functions with double reserve oftheir components is considered only. It seems to bepossible to extend the results to the systems havingother much complicated reliability structures andcomponents with different from the exponentialreliability function. Further, it seems to be reasonableto elaborate a computer programs supportingcalculations and accelerating decision making,addressed to reliability practitioners.References- 157 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: series and parallel systems - RTA # 3-4, 2007, December - Special Issue[1] Barlow, R. E. & Proschan, F. (1975). StatisticalTheory of Reliability and Life Testing. ProbabilityModels. Holt Rinehart and Winston, Inc., NewYork.[2] Chernoff, H. & Teicher, H. (1965). Limitdistributions of the minimax of independentidentically distributed random variables. Proc.Americ. Math. Soc. 116, 474–491.[3] Kolowrocki, K. (2000). Asymptotic Approach toSystem Reliability Analysis. System ResearchInstitute, Polish Academy of Sciences, Warsaw, (inPolish).[4] Kolowrocki, K. (2001). On limit reliabilityfunctions of large multi-state systems with ageingcomponents, Applied Mathematics andComputation, 121, 313-361.[5] Koowrocki, K. (2004). Reliability of LargeSystems. Elsevier.[6] Kolowrocki, K., Kwiatuszewska-Sarnecka, B. at al.(2003). Asymptotic Approach to ReliabilityAnalysis and Optimisation of ComplexTransportation Systems. Part 2, Project founded byPolish Committee for Scientific Research, Gdynia:Maritime University, (in Polish).[7] Kwiatuszewska-Sarnecka, B. (2002). Analyse ofReserve Efficiency in Series Systems. PhD thesis,Gdynia Maritime University, (in Polish).[8] Kwiatuszewska-Sarnecka, B. (2006). Reliabilityimprovement of large multi-state series-parallelsystems. International Journal of Automation andComputing 2, 157-164.[9] Kwiatuszewska-Sarnecka, B. On reliabilityimprovement of large series systems withcomponents in hot reserve. Applied Mathematicsand Computation, (in review).[10] Sarhan, A. (2000). Reliability equivalence ofindependent and non-identical components seriessystems. Reliability Engineering and System Safety67, 293-300.[11] Xue, J. (1985). On multi-state system analysis,IEEE Transactions on Reliability, 34, 329-337.- 158 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special IssueKwiatuszewska-Sarnecka BoenaGdynia Maritime University, PolandOn asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy:„m out of n” systemsKeywordsreliability improvement, limit reliability functionsAbstractThe paper is composed of two parts, in this part the multi-state homogeneous „m out of n” systems with reservecomponents are defined and their multi-state limit reliability functions are determined. In order to improve of thereliability of these systems the following methods are used: (i) a warm duplication of components, (ii) a coldduplication of components, (iii) a mixed duplication of components, (iv) improving the reliability of componentsby reducing their failure rate. Next, the effects of the systems’ reliability different improvements are compared.1. IntroductionPresented paper is continuation of a work aboutreliability improvement of large system. In the firstpart of this work are defined the component’s andsystem’s multi-state reliability functions and next theasymptotic approach are brought forward. There arepresented results concerned with improvement of largeseries and parallel systems, their multi-state limitreliability functions in case when the systems havereserve components and in case when the reliability ofcomponents is improved by reducing their failure rate.As the main result are found the forms of reducingtheir failure rate factor for both kinds of large systems.2. Reliability improvement of a multi-state „mout of n” systemDefinition 2.1. A multi-state system is called an „m outof n” system if its lifetime in the state subset{u,u+1,...,z} is given byT(u) = T ( ), m = 1,2,...,n, u = 1,2,...,z,( n −m+1) uwhere T( n − m+1)( u)is the m th maximal order statistics inthe sequence of the component lifetimesT1(u), T2(u),..., Tn(u).E 1E 2EE 3. .E nFigure 1. The scheme of a homogeneous „m out of n”systemThe above definition means that the multi-state „m outof n” system is in the state subset {u,u+1,...,z} if andonly if at least m out of n its components is in this statesubset and it is a multi-state parallel system if m = 1and it is a multi-state series system if m = n.Definition 2.2. A multi-state „m out of n” system iscalled homogeneous if its component lifetimes T i (u) inthe state subsets have an identical distribution functionF i (t,u) = F(t,u), u = 1,2,...,z, t ∈(-∞,∞), i = 1,2,...,n.The reliability function of the homogeneous multi-state„m out of n” system is given either byR (mn) (t , ⋅ ) = [1,R (mn ) (t,1),...,R (m)(t,z)],nwhere- 159 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issuem−1( m)nR ( , u)= 1 − ∑ ( )nn−it [ R(t,u)][ F(t,u)],ii=0t ∈(-∞,∞), u = 1,2,...,z,or by( m)( m)( m)R ( t,⋅)= [1, R ( t,1),..., R ( t,z)],nwhereR( m )n( t,u)=nmn∑ ( )ii=0m = n − m, u = 1,2,...,z.i[ F(t,u)][ R(t,u)]nin−i, t ∈(-∞,∞),Definition 2.3. A multi-state system is called an „m outof n” system with a hot reserve of its components if itslifetime T (1) (u) in the state subset {u,u+1,...,z} is givenbyT (1) (u) = T ( ), m = 1,2,...,n, u = 1,2,...,z,( n −m+1) uwhere T( n − m+1)( u)is the m-th maximal order statistics inthe sequence of the component lifetimesT i (u) = max{ ( u)},i = 1,2,..,n, u = 1,2,...,z,T ij1≤ j≤2where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multi-state„m out of n” system with a hot reserve of itscomponents is given either byIRwhere( 1) ( m)n(1) ( m)( m)( m)(1)(1)(t , ⋅ ) = [1, IR n (1,z),..., IR n ( t,z)],m−1n( )2 i2( n−i)IR n ( t,u)= 1−∑ [1 − ( F(t,u))] [ F(t,u)], (1)ii=0t ∈(-∞,∞), u = 1,2,...,z,or by(1) ( m )(1) ( m )( m )(1)I R n ( t,⋅)= [1, I R n ( t,1),..., I R n ( t,z)],whereIR= ∑(1) ( m )nmii=0( t,u)n2i2 ( n−i)( )[(( t,u))][1 − ( F(t,u))] ,F (2)m = n − m, t ∈(-∞,∞), u = 1,2,...,z.Lemma 2.1.case 1: If(1)(i) ( m)ΙR ( t,u)= − ∑ − 1[( , )]1 m iV t uexp[ −V( t,u)],i=0 i!u = 1,2,...,z, is non-degenerate reliabilityfunction,(1) ( m)(ii) IR n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a hot reserve of its components defined by(16),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m = constant ( m / n → 0,as n → ∞ ),thenlim IR n ( a ( u ) t + b ( u))= ΙR ( t,u),n→∞t ∈CΙR(1) ( m)if and only ifn, u = 1,2,...,z,limn→∞n[1-F2 ( a ( u)t bn( u))u = 1,2,...z,case 2: Ifn(1) ( m)n+ ] = V(t,u), t ∈ C V ,(1) ( µ )−v(t,u)1−(i) ΙR ( t,u)= 1−∫ e2dx ,2π−∞u = 1,2,...,z, is non-degenerate reliabilityfunction,(1) ( m)(ii) IR n ( t,u)is the reliability function ofnon-degenerate multi-state „m out of n”system with a hot reserve of its componentsdefined by (16),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m / n → µ , 0 < µ < 1, as n → ∞ ,thenn→∞(1) ( m)x2lim IR ( ( ) ((1)n a u t + b u))= ΙR ( t,u),t ∈ C IR , u = 1,2,...,z,if and only ifn( n + 1)[1 − F ( an( u)t + bnlimn→∞m(n − m + 1)n + 1u = 1,2,...,z.2n( µ )( u))]− m=ν ( t,u),case 3: If(1) ( m )mi[ V ( t,u)](i) Ι R ( t,u)= ∑ exp[ −V( t,u)],i=0 i!- 160 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issuem = n − m, u = 1,2,...,z, is non-degeneratereliability function,( m )(ii)(1)I R n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a hot reserve of its components definedby (17),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) n - m = m = constant ( m / n →1as n → ∞),then(1)lim ( ( ) ((1)I R n a u t + b u))= Ι R ( t,u),n→∞t ∈CIR( m )if and only iflim n[F(an→∞n, u = 1,2,...,z,nu = 1,2,...,z.( u)t + bnn( u)]2( m )= V ( t,u),t ∈C V,Proposition 2.1. If components of the homogeneousmulti-state „m out of n” system with a hot reserve of itscomponents have multi-state exponential reliabilityfunctionsandcase 1 m = constant,a n (u) =then11, b n (u) = log 2n, u = 1,2,...,z,λ(u)λ(u)( 1) ( )IRm m−1exp[−it](t,u) = 1- ∑ exp[-exp[-t]],i=0 i!t ∈ (-∞,∞), u = 1,2,...,z,case 2 m / n → µ , 0 < µ < 1, n → ∞ ,µ1a n (u) =, b n( u)= 1− µ ,λ( u)2n + 1 λ(u)u = 1,2,...,z,thenx2−2(1) ( µ )1tIR ( t,u)= 1 - ∫ e dx ,2π−∞t ∈ (-∞,∞), u = 1,2,...,z,a n (u) =then(1) ( m )1nλ(u)IR ( t,u)= 1, t < 0,(1) ( m )IR ( t,u)= ∑ −i=, b n (u) = 0 , u = 1,2,...,z,nm02iis its limit reliability function.Proof:case 1: Since for all fixed u, we havean ( u)t + bn( u)→ ∞ as n → ∞ .ThereforeV ( t,u)= lim n[1− Fn→∞ti! exp[-t2 ], t ≥ 0 , u = 1,2,...,z,2( an( u)t + b= lim n[2 exp[ −λ(u)(an→∞− exp[ −2λ(u)(an ( u)t + b= lim 2nexp[ −λ(u)(an→∞nnn( u))]( u)t + bn( u))]]( u)t + bnn( u))]( u))]1[ 1−exp[ −λ(u)(an ( u)t + bn( u))]]2= lim exp[ −t][2nexp[ −λ(u)bn→∞− n exp[ −t]exp[−2λ(u)bnn( u)]]( u)]1 1= lim exp[ −t][2n− n exp[ −t]]n→∞2n n= exp[ −t], t ∈( −∞,∞),u = 1,2,...,z,which by case 1 in Lemma 2.1 completes the proof.case 2: Since for all fixed u, we havean ( u)t + bn( u)→ ∞ as n → ∞ ,moreovercase 3n − m = m = constant, ( m / n →1,n → ∞ ),1−F2( an ( u)t + bn( u))- 161 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issue= 2 exp[ −λ(u)(a ( u)t b ( u))]n+− exp[ −2λ(u)(an ( u)t + bnn( u))]= 2[1− λ(u)(a ( u)t b ( u))+ λ2n +1 2( u)(a ( u)t b ( u))2n+n−[ 1−2λ(u)(a ( u)t b ( u))+ 4λ2n +1 2( u)(a ( u)t + b2n nnn]1( u))] + o()( n + 1)= [ 1 − exp[ −λ(u)(a ( u)t b ( u))]]n+t 2= [1 − exp[ − ]] , t ≥ 0.nThereforeV ( t,u)= lim n[F(aandn→∞u = 1,2,...,z,nn( u)t + bn( u))]V ( t,u)= lim n[F(a ( u)t + b ( u))]n→∞nn22 =20, t < 0,22 1= 1−λ ( u)(an ( u)t + bn( u))+ o(),( n + 1)= lim n[1− exp[ −n→∞t]]n2next( n + 1)[1 − Fν ( t,u)= limn→∞2( an( u)t + bm(n − m + 1)n + 1n( u))]]− mµ (1 −µ ) 1( n + 1)[ − t + µ − o()] − mn + 1n + 1= limn→∞m(n − m + 1)n + 1− µ (1 − µ ) t + o(1)= lim = −t, t ∈( −∞,∞),n→ ∞µ (1 − µ )u = 1,2,...,z,which by case 2 in Lemma 2.1 completes the proof.case 3: Since for all fixed u, we havet 1 2= lim n[+ o()] = tn→∞n nu = 1,2,...,z,2, t ≥ 0,which by case 3 in Lemma 2.1 completes the proof.Corollary 2.1. The reliability function of exponential„m out of n” system with a hot reserve of itscomponents is given bycase 1IR(1) ( m)nm−1exp[−i(λ(u)t − log 2n)]( t,u)≅ 1−∑i=0 i!exp[ −exp[−λ ( u ) t + log 2n], (3)t ∈( −∞,∞),u = 1,2,...,z.case 2tan( u)t + bn( u)= < 0λ(u)nandtan( u)t + bn( u)= ≥ 0λ(u)nthenFandF22( an ( u)t + bn( u))= 0 , t < 0( an ( u)t + bn( u))for t < 0for t ≥ 0 ,x2− 2I(1) ( µ )1IR n ( t,u)≅ 1 − ∫ e dx(4)2π−∞where2λ(u)n + 1 2 n + 1 1−µI =t −, t ∈( −∞,∞),(5)µµu = 1,2,...,z.case 3(1) ( m )IR n ( t,u)= 1, t < 0,- 162 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issue(1) ( m ) n m [ λ(u)nt]2 2IR n ( t,u)≅ ∑− exp[ −λ( u)nt ] , (6)i=0 i!t ≥ 0 , u = 1,2,...,z.Definition 2.4. A multi-state system is called an „m outof n” system with a cold reserve of its components ifits lifetime T (2) (u) in the state subset {u,u+1,...,z} isgiven byT (2) (u) = T ( ), m = 1,2,...,n, u = 1,2,...,z,( n −m+1)uwhere T( n − m+1) ( u)is the m-th maximal order statistics inthe sequence of the component lifetimes22iT i (u) = ∑T( u), i = 1,2,..,n, u = 1,2,...,z,ijj = 1where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multi-state„m out of n” system with a cold reserve of itscomponents is given either byIRwhereIR=( 2) ( m)n(2) ( m)n(t , ⋅ ) = [1,( t,u)n( )−11 −m ∑ ii=0( 2) ( m)nIR (t,1),...,[1 − F(t,u)∗ F(t,u)]= )]t ∈ (-∞,∞), u = 1,2,...,z,or byi( 2) ( m)nIR (t,z)],n−i⋅[ F(t,u)∗ F(,u(7)(2) ( m)(2) ( m)(2) ( m)I R n ( t,⋅)= [1, IR n ( t,1),..., I R n ( t,z)],where(2)R( m )I ( t,u)mii=0nnin−i( )[F(t,u)∗ F(t,u)][1 − F(t,u)∗ F(t,u= ∑)]t ∈ (-∞,∞), m = n − m,u = 1,2,...,z.Lemma 2.2.case 1: If, (8)(2) ( m)(i) ΙR ( t,u)= − ∑ − 1[( , )]1 m V t uexp[ −V( t,u)],i=0 i!u = 1,2,...,z, is non-degenerate reliabilityfunction,(2) ( m)(ii) IR n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a cold reserve of its components definedby (24),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m = constant ( m / n → 0 , as n → ∞ ),then(2)lim IR n ( a ( u ) t + b ( u))= ΙR ( t,u), t ∈n→∞C IR ,( m)if and only ifn→∞nni(1) ( m)lim n [ F(a ( u)t + b ( u))∗ F(a ( u)t b ( u ))]]n nn+= V(t,u), t ∈ C V , u = 1,2,...z,case 2: If(2)(i) ( µ )−v(t,u)1−ΙR ( t,u)= 1−∫ e2dx ,2π−∞u = 1,2,...,z, is non-degenerate reliabilityfunction,( m)(i)(2)IR n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a cold reserve of its components definedby (24),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m / n → µ , 0 < µ < 1, as n → ∞ ,thenlimn→∞IR(2) ( m)( an( u)t bn( u))t ∈ C IR ,if and only if( n + 1)[1 − F(alimn→∞nx2(2) ( µ )n+ = ΙR ( t,u),n( u)t + b= ν ( t,u), u = 1,2,...,z.n( u))∗ F(am(n − m + 1)n + 1n( u)t + bn( u))]− mcase 3: If(2)(i) ( m )mi[ V ( t,u)]Ι R ( t,u)= ∑ exp[ −V( t,u)],i=0 i!- 163 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issuem = n − m, u = 1,2,...,z, is non-degeneratereliability function,(2) ( m )(ii) I R n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a cold reserve of its components definedby (25),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) n - m = m = constant ( m / n →1as n → ∞),thenlim I R n ( a ( u ) t + b ( u))= Ι R ( t,u),n→∞t ∈C ΙR,(2) ( m )if and only iflim n[F(an→∞nn( u)t + bt ∈C ⎺V , u = 1,2,...,z.nn( u)∗ F(an(2) ( m )( u)t + bn( u)]= V ( t,u),Proposition 3.2. If components of the homogeneousmulti-state „m out of n” system with a cold reserve ofits components have multi-state exponential reliabilityfunctionsandcase 1 m = constant,a n (u) =then1 exp[ λ(u)bn( u)], = n , u = 1,2,...,z,λ(u)λ(u)b ( u)(2) ( m)m−1exp[−it]ΙR ( t,u)= 1- ∑ exp[-exp[-t]],i=0 i!t ∈ (-∞,∞), u = 1,2,...,z,case 2 m / n → µµa n (u) = ,λ( u)n + 1thenn0 < µ < 1, n → ∞ ,b n1− µ( u)= , u = 1,2,...,z,λ(u)x2− 2(2) ( µ )1tΙR ( t,u)=1 - ∫ e dx , t ∈ (-∞,∞),2π−∞u = 1,2,...,z,then(2) ( m )IR ( t,u)= 1, t < 0,(2) ( m )I R ( t,u)= ∑ −i=nm02iis its limit reliability function.Proof:case 1: Since for all fixed u, we haveti! exp[-t2 ] , t ≥ 0 , u = 1,2,...,z,an ( u)t + bn( u)→ ∞ as n → ∞ , t ∈( −∞,∞).ThereforeV ( t,u)= lim n[1+ λ(u)(a ( u)t + b ( u))]n→∞exp[ −λ(u)(a1 + t= lim n[n→∞exp[ λ(u)bn( u)tn+( u)]λ(u)bn( u)+]exp[ −t]exp[ λ(u)b ( u)]nnnb ( u))]= exp[ −t], t ∈( −∞,∞),u = 1,2,...z,which by case 1 in Lemma 2.2 completes the proof.case 2: Since for all fixed u, we haveandµan ( u)t + bn( u)= tλ( u)n + 1n1→ 1−µλ(u)1+ 1− µλ(u)> 0 asn → ∞1 – F( a ( u)t + b ( u))∗ F(a ( u)t b ( u))= [1 + λ(u)(aexp[ −λ(u)(an nn +nn( u)t + bn( u)t + bn( u))]( u))]ncase 3 n - m = m = constant ( m / n → 1,n → ∞ ),= [1 + λ(u)(an( u)t + bn( u))]a n (u) =2nλ(u), b n (u) = 0 , u = 1,2,...,z,[1 − λ(u)(an( u)t + bn( u))+- 164 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issue1 22 1λ ( u)(an ( u)t + bn( u))− o()]2n + 11 2= 1−λ ( u)(an ( u)t + bn( u))21− o ( )], t ∈( −∞,∞).n + 12t 2 t= 1−(1 + ) exp[ −n2t 1= + o() , t ≥ 0.n nTherefore2]nThereforev( t,u)( n + 1)[1 − F(a= limn→∞n( u)t + bn( u))∗ F(am(n − m + 1)n + 1n( u)t + bn( u))]− mv(t,u)= lim n[F(a∗ F(ann→∞( u)t + bt= lim n[1− (1 +n→∞nn( u))]( u)t + b2) exp[nnt−( u))2]]n( n + 1)[ −= limn→∞µ (1 − µ ) 1t + µ − o()] − mn + 1 n + 1m(n − m + 1)n + 12t= lim n[n→∞n+ o1( )]n= t2, t ≥ 0,u = 1,2,...z,which by case 3 in Lemma 2.2 completes the proof.−= limµ (1 − µ ) t + o(1)= −t, t ∈( −∞,∞),µ (1 − µn→ ∞)which by case 2 in Lemma 2.2 completes the proof.case 3: Since for all fixed u, we haveCorollary 2.2. The reliability function of exponential„m out of n” system whit a cold reserve of itscomponents is given bycase 1IR(2) ( m)n( t,u)≅−1exp[−iλ(u)(t − b− ∑= 0 i!1 m in( u))]2tan( u)t + bn( u)= < 0λ(u)nfor t < 0t ∈( −∞,∞),exp[ − exp[ −λ ( u)t + λ(u)bn( u)], (9)u = 1,2,...,z.andcase 2t 2an( u)t + bn( u)= ≥ 0λ(u)nthenfor t ≥ 0,x2− 2I(2) ( µ )1IR n ( t,u)≅ 1 − ∫ e dx(10)2π−∞whereF( an ( u)t + bn( u))= 0 for t < 0andF( a ( u)t + b ( u))∗ F(a ( u)t b ( u))n nn+= [1 − [1 + λ(u)(aexp[ −λ(u)(ann( u)t + b( u)t + bnn( u))]]( u))]nλ( u)n + 1 n + 1 1−µI = t −µµcase 3(2) ( m )IR n ( t,u)= 1, t < 0,IR(2) ( m )n( t,u)], t ∈( −∞,∞).(11)- 165 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issue2in m [ λ(u)nt / 2]2 2≅ ∑− exp[ −λ( u)nti=0 i!t ≥ 0 , u = 1,2,...,z./ 2], (12)Definition 2.5. A multi-state series system is called an„m out of n” system with a mixed reserve of itscomponents if its lifetime T (3) (u) in the state subset{u,u+1,...,z} is given byT (3) (u) = T ( ), m = 1,2,...,n, u = 1,2,...,z,( n −m+1) uwhere T( n − m+1) ( u)is the m-th maximal order statistics inthe sequence of the component lifetimesT i (u) = { max {max{ T ( u)}},max { ∑T( u)}},1≤i≤s1n1≤j≤2i = 1,2,..,n., u = 1,2,...,z,ijs1n+ 1≤i≤n2ijj=1where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve componentsand s 1 , s 2 , where s 1 + s 2 = 1 are fractions of thecomponents with hot and cold reserve, respectively.The reliability function of the homogeneous multi-state„m out of n” system with a mixed reserve of itscomponents is given either byIRwhereIR( 3) ( m)n(3) ( m)n(t , ⋅ ) = [1,( t)= 1 −m−1∑( 3) ( m)nIR (t,1),...,n( )ii=0( 3) ( m)nIR (t,z)],2 s1i[1 − ( F(t,u))][1 − F(t,u)∗ F(t,u)]( n−i)s2s2 i))] 2( n−is1[ F(t,u[ F(t,u)∗ F(t,u)], (13)t ∈ (-∞,∞), u = 1,2,...,z,or by(3) ( m )(3) ( m )( m )(3)I R n ( t,⋅)= [1, I R n ( t,1),..., I R n ( t,z)],wherem(3) ( m )nI R ( t)= ∑ ( )nii=0[ F(t,u)]2s1i[ F(t,u)∗ F(t,u)]s2icase 1: If(3) ( m)(i) ΙR ( t,u)= − ∑ − 1[( , )]1 m iV t uexp[ −V( t,u)],i=0 i!u = 1,2,...,z, is non-degenerate reliabilityfunction,(3) ( m)(ii) IR n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwhit a mixed reserve of its componentsdefined by (30),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m = constant ( m / n → 0 , as n → ∞ ),then(3)lim IR ( ( ) ((3)n a u t + b u))= ΙR ( t,u),n→∞( m)t ∈ C IR , u = 1,2,...,z,if and only ifnlim n[s1[1−[F(an( u)t + bn( u))]n→∞n2]( m)+ s2[1− F(an ( u)t + bn( u))∗ F(an( u)t + bn( u))]]=V(t,u), t ∈ C V , u = 1,2,...z,case 2: If(3) ( µ )−v(t,u)1−(i) ΙR ( t,u)= 1−∫ e2dx , u =2π−∞1,2,...,z, is non-degenerate reliability function,(3) ( m)(ii) IR n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwhit a mixed reserve of its componentsdefined by (30),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m / n → µ , 0 < µ < 1, przy n → ∞ ,thenlimn→∞IR(3) ( m)( an( u)t bn( u))( µ )x2(3)n+ = ΙR ( t,u),t ∈ C IR , u = 1,2,...,z,if and only if( n + 1)[ s1[1−[F(an( u)t + bn( u))]]lim→∞m(n − m + 1)n + 1n22 ( n−i)s1( n−i)s2[1 − ( F(t,u))] [1 − F(t,u)∗ F(t,u)], (14)t ∈ (-∞,∞), m = n − m,u = 1,2,...,z.Lemma 2.3.- 166 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issues2[1− F(a( u)t + b= ν ( t,u),u = 1,2,..., z.nn( u))∗ F(anm(n − m + 1)n + 1( u)t + bn( u))]]− mcase 3: If(3) ( m )mi[ V ( t,u)](i) Ι R ( t,u)= ∑ exp[ −V( t,u)],i=0 i!m = n − m, u = 1,2,...,z, is non-degeneratreliability function,( m )(ii)(3)I R n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a mixed reserve of its componentsdefined by (31),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) n - m = m = constant ( m / n →1as n → ∞),thenlim ( ( ) ((3)I R n a u t + b u))= Ι R ( t,u),n→∞t ∈CIR(3) ( m )if and only ifn, u = 1,2,...,z,lim n[s1 [ F(an( u)t + bn( u))]n→∞+ s2[F(a= V ( t,u)t ∈ C ,Vn( u)t + bnu = 1,2,...,z.n2( u))∗ F(an( m )( u)t + bn( u))]]Proposition 2.3. If components of the homogeneousmulti-state „m out of n” system with a mixed reserve ofits components have multi-state exponential reliabilityfunctionsandcase 1 m = constant,a nthen1 exp[ λ(u)bn( u)]( u)= , = n,u = 1,2,...,z,λ(u)2s+ s λ(u)b ( u)1(3) ( m)m−1exp[−it]ΙR ( t,u)= 1- ∑ exp[-exp[-t]],i=0 i!t ∈ (-∞,∞), u = 1,2,...,z,2n,case 2 m / n → µa n( u)=λ(u)0 < µ < 1, n → ∞ ,µ / 2,(2s1 + s2)( n + 1)1 2(1 − µ )b n( u)=, u = 1,2,...,z,λ(u)2s 1+ s 2thenx2−2(3) ( µ )1tΙR ( t,u)= 1 - ∫ e dx ,2π−∞t ∈ (-∞,∞), u = 1,2,...,z,case 3 n - m = m = constant ( m / n → 1,n → ∞ ),2a n ( u)= , b n( u)= 0,u = 1,2,...,z,λ ( u)(2ss ) nthen(3) ( m )1 +Ι R ( t,u)= 1, t < 0,(3) ( m )Ι R ( t,u)= ∑ −i=nm022iis its limit reliability function.Proof:case 1: Since for all fixed u, we haveandti! exp[-t2 ], t ≥ 0 , u = 1,2,...,z,an ( u)t + bn( u)→ ∞ as n → ∞ for t ∈ ( −∞,∞),1 – [ F(a ( u)t b ( u))]n+= 2 exp[ −λ(u)(a ( u)t b ( u))]nn+− exp[ −2λ ( u)(a ( u)t b ( u))], t ∈ ( −∞,∞),n+1 – F( a ( u)t + b ( u))∗ F(a ( u)t b ( u))= [1 + λ(u)(aexp[ −λ(u)(aThereforen nn +nn( u)t + b( u)t + bnn2nn( u))]( u))],t ∈(−∞,∞).n- 167 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special IssueV ( t,u)= lim n[s1[1−[F(an( u)t + bn( u))]n→∞+ s2[1− F(a∗ F(ann( u)t + b( u)t + bnn( u))]]( u))= lim[ns 1 [2exp[ −λ(u)(a ( u)t + b ( u))]n→∞− exp[ −2λ(u)(an ( u)t + bnn( u))]]+ ns2[1+ λ(u)(an ( u)t + bn( u))]exp[ −λ(u)(an ( u)t + bn( u))]]= lim[ns 1 2exp[ −λ(u)(a ( u)t + b ( u))]n→∞[ 1−1/2 exp[ −λ(u)(a ( u)t b ( u))]]+ ns λ(u)b2exp[ −λ(u)(an ( u)t + bnnn+1+λ(u)an( u)t( u)[1+]λ(u)b ( u)nn( u))]]= lim exp[ −t][ns12 exp[ −λ(u)b ( u)]n→∞1[ 1−exp[ −t]exp[−λ(u)bn( u)]]2+ ns λ(u)b2nexp[ −λ(u)b1+t( u)[1+ ]λ(u)b ( u)n( u)]]= lim exp[ −t][ns1 2exp[ −λ(u)b ( u)]n→∞− ns1 exp[ −t]exp[−2λ(u)bn( u)]+ ns2λ(u)bn( u) exp[ −λ(u)bn( u)]+ ns2[1+ t]exp[−λ(u)bn( u)]]= lim exp[ −t]n→∞nnnn2nn]s2+(1 + t)]( 2s1+ λ(u)bn( u)s2)= exp[ −t], t ∈( −∞,∞),u = 1,2,...,z,which by case 1 in Lemma 2.3 completes the proof.case 2: Since for all fixed u, we have1 2(1 − µ )an ( u)t + bn( u)→> 0 asλ(u)2s+ st ∈( −∞,∞).and21 – [ F(a ( u)t b ( u))]n += 2 exp[ −λ(u)(a ( u)t b ( u))]nn+− exp[ −2λ(u)(an ( u)t + bnn1( u))]= 2[1− λ(u)(a ( u)t b ( u))+ λ2n +1 2( u)(a ( u)t b ( u))2n+n−[ 1−2λ(u)(a ( u)t b ( u))+ 4λ2n +1 2( u)(a ( u)t + b ( u))2n nnn]21] + o()n + 122 1= 1−λ ( u)(an ( u)t + bn( u))+ o ( ) ,n + 1t ∈( −∞,∞),1 – F( a ( u)t + b ( u))∗ F(a ( u)t b ( u))n nn += [ 1+λ(u)(a ( u)t b ( u))]n +[ 1− λ(u)(a ( u)t b ( u))n+1 22 1+ λ ( u)(an ( u)t + bn( u))− o()]2n + 1n1 22 1= 1−λ ( u)(an ( u)t + bn( u))− o ( )],2n + 1t ∈( −∞,∞).nnn → ∞ ,[1 −n(2s1s1+ λ(u)bn( u)s2)2exp[ −t]Therefores1[1−[F(an ( u)t + bn( u))]2]- 168 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issue+ s2[F(an ( u)t + bn( u))∗ F(an( u)t + bn( u))]2= s1[1− λ ( u)(an ( u)t + bn( u))1 22 1+ s2[1− λ ( u)(an ( u)t + bn( u))] + o()2n + 12s1+ s222 1= 1−λ ( u)(an ( u)t + bn( u))+ o()2n + 12]and for t ≥ 0[ F(a ( u)t b ( u))]n+n2= [ 1−exp[ −λ ( u)(an ( u)t + bn( u))]],F( a ( u)t + b ( u))∗ F(a ( u)t b ( u))n nn+= [ 1−[1+ λ(u)(a ( u)t b ( u))]2n+nn= −µ (1 − µ ) 1t + µ − o(),n + 1 n + 1t ∈ ( −∞,∞),exp[ −λ ( u)(a ( u)t b ( u))]].Thereforen+nnext( n + 1)[ s1[1−[F(an( u)t + bn( u))]]ν ( t,u)= limn→∞m(n − m + 1)n + 1s2[1− F(a+= limn→∞n( n + 1)[ −( u)t + bn( u))∗ F(anm(n − m + 1)n + 1m(n − m + 1)n + 1( u)t + bn2( u))]]− mµ (1 − µ ) 1t + µ − o()] − mn + 1 n + 1− µ (1 − µ ) t + o(1)= lim= −t, t ∈( −∞,∞),n→ ∞µ (1 − µ )u = 1,2,..., z,which by case 2 in Lemma 2.3 completes the proof.case 3: Since for all fixed u, we haveanda ( u)t + bnna ( u)t + bnthenn( u)=λ(u)( u)=λ(u)2t(2st1(2s21+ s+ sF( an ( u)t + bn( u))= 0 , t < 0 ,22) n) n< 0≥ 0, t < 0, t ≥ 0,V ( t,u)= lim n[s1 [ F(an( u)t + bn( u))]n→∞+ s2[F(an ( u)t + bn( u))∗ F(an( u)t + bn( u))]]= lim[ns1 [1 − exp[ −λ(u)(an( u)t + bn( u))]]n→∞+ ns2[1−[1+ λ(u)(an ( u)t + bn( u))]exp[ −λ(u)(a( u)tn+b ( u))]]]= lim[ns1 [ λ(u)(an( u)t + bn( u))]n→∞+ ns λ− λ2222[ ( u)(an ( u)t + bn( u))1 2( u)(a ( u)t b ( u))2n+n2s= lim n[n→∞21+ s22n( λ(u)an]]( u)t)t= lim n[] = t 2 , t ≥ 0,t ∈ ( −∞,∞),u = 1,2,...,z,n→∞nwhich by case 3 in Lemma 2.3 completes the proof.Corollary 2.3. The reliability function of exponential„m out of n” system with a mixed reserve of itscomponents is given bycase 1IR(3) ( m)nm−1exp[−iλ(u)(t − b( t,u)≅ 1 − ∑i=0 i!2]22n2( u))]exp[ − exp[ −λ ( u)t + λ(u)b ( u)], (15)n- 169 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issuet ∈( −∞,∞),u = 1,2,...,z.case 2x2− 2I(3) ( µ )1IR n ( t,u)≅ 1 − ∫ e dx , (16)2π−∞case 3a n (u) =thenn − m = m constant ( m / n →1,n → ∞ ),1nλ(u)ρ(u), b n (u) = 0 , u = 1,2,...,z,whereλ( u)(2s+ s2)( n + 1)2 2I =t −µt ∈( −∞,∞),u = 1,2,...,z.case 3(3) ( m )IR n ( t,u)= 1, t < 0,1 n+ 1µ1−µ, (17)(4) ( m )IR ( t,u)= 1, t < 0,(4)I( m )nR ( t,u)= ∑ − m ti=0 i! exp[-t], t ≥ 0 ,is its limit reliability function.iCorollary 2.3. The reliability function of exponential„m out of n” system with improved reliability functionsof its components is given bycase 1IR(3) ( m )n( t,u)n≅ ∑− mi=0[ λ(u)(2s1+ si!2) nt /2]2iIR(4) ( m)nm−1exp[−iλ(u)ρ(u)t − log n)]( t,u)≅ 1 − ∑i=0i!t ≥ 022exp[ −λ ( u )(2s1+ s2) nt / 2] , (18)u = 1,2,...,z.t ∈( −∞,∞),exp[ −exp[−λ ( u ) ρ(u)t + log n], (19)u = 1,2,...,z.Proposition 2.3. If components of the homogeneousmulti-state „m out of n” system have improvedcomponent reliability functions i.e. its componentsfailure rates λ(u) is reduced by a factor ρ(u),ρ(u)∈, u = 1,2,...,z,andcase 1 m = constant,a n (u) =then1, b n (u) =λ(u)ρ(u)log n, u = 1,2,...,z,λ(u)ρ(u)(4) ( m)m−1exp[ −it]ΙR ( t,u)= 1- ∑ exp[-exp[-t]],i=0 i!t ∈ (-∞,∞),case 2 m / n → µ , 0 < µ < 1, n → ∞ ,1 1−µ log(1/ µ )a n (u) = , b n (u) = ,λ(u)ρ(u)n + 1 µ λ(u)ρ(u)u = 1,2,...,z,thenx2− 2(4) ( µ )1tΙR ( t,u)= 1 - ∫ e dx , t ∈ (-∞,∞),2π−∞case 2(4) ( µ )1I −IR ( t,u)1 e2n ≅ − ∫ dx,(20)2π−∞whereλ(u)ρ(u)( n + 1) µI =t −1−µt ∈( −∞,∞),u = 1,2,...,z.case 3(4) ( m )IR n ( t,u)= 1 , t < 0,IR(4) ( m )n( t,u)n m [ λ(u)ρ(u)nt]≅ ∑−i=0 i!t ≥ 0 , u = 1,2,...,z.ix2n + 1 µlog(1/ µ ) , (21)1−µexp[ −λ(u)ρ(u)nt], (22)3. Comparison of reliability improvementeffects- 170 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special IssueThe comparisons of the limit reliability functions of thesystems with different kinds of reserve and suchsystems with improved components allow us to findthe value of the components decreasing failure ratefactor ρ(u), which warrants an equivalent effect of thesystem reliability improvement.3.1. The “m out of n” systemThe comparison of the system reliability improvementeffects in the case of the reservation to the effects inthe case its components reliability improvement maybe obtained by solving with respect to the factorρ ( u)= ρ(t,u)the following equationsIR(4) ( m)( t − b ( u))/ a ( u))( k) = ( m)R ((t − b ( u)) / a ( u))nnnI , (23)u = 1,2,...,z, k = 1,2,3.nThe factors ρ ( u)= ρ(t,u)decreasing componentsfailure rates of the homogeneous exponential multistate„m out of n” system equivalent with the effects ofhot, cold and mixed reserve of its components as asolution of the comparisons (23) are respectively givenbyk = 1case 1ln 2ρ(u)= ρ(t,u)= 1−, u = 1,2,...,z,λ(u)t2 1−µ 2(1 − µ ) + µ log µcase 2 ρ(u)= −,µ λ(u)µ tu = 1,2,...,z,case 3k = 2case 1case 2ρ ( u ) = ρ(t,u)= λ(u)t , u = 1,2,...,z,λ(u)bn( u)− log nρ(u)= 1−λ(u)tlog λ(u)b n ( u)= 1−, u = 1,2,...,z,λ(u)t1 − µ 1 − µ + µ log µρ(u)= −, u = 1,2,...,z,µ λ(u)µ tcase 3λ(u)tρ ( u)= ρ(t,u)= , u = 1,2,...,z,2k = 3λ(u)bn( u)− log ncase 1 ρ(u)= 1−λ(u)tcase 2log(2s1+ s2λ(u)bn( u))= 1−, u = 1,2,...,z,λ(u)t2(2s1 + s2) 1−µ 2(1 − µ ) + µ log µρ(u)= −,µλ(u)µ tu = 1,2,...,z,case 3(2s1 + s2) λ(u)tρ(u)= ρ(t,u)=, u = 1,2,...,z.24. ConclusionProposed in the paper application of the limit multistatereliability functions for reliability of largesystems evaluation and improvement simplifiescalculations. The methods may be useful not only inthe technical objects operation processes but also intheir new processes designing, especially in theiroptimization. The case of series, parallel (in part 1) and„m out of n” systems composed of components havingexponential reliability functions with double reserve oftheir components is considered only. It seems to bepossible to extend the results to the systems havingother much complicated reliability structures andcomponents with different from the exponentialreliability function. Further, it seems to be reasonableto elaborate a computer programs supportingcalculations and accelerating decision making,addressed to reliability practitioners.References[1] Barlow, R. E. & Proschan, F. (1975). StatisticalTheory of Reliability and Life Testing. ProbabilityModels. Holt Rinehart and Winston, Inc., NewYork.[2] Koowrocki, K. (2004). Reliability of LargeSystems. Elsevier.- 171 -

B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issue[3] Kwiatuszewska-Sarnecka, B. (2002). Analyse ofReserve Efficiency in Series Systems. PhD thesis,Gdynia Maritime University, (in Polish).[4] Kolowrocki, K., Kwiatuszewska-Sarnecka, B. at al.(2003). Asymptotic Approach to Reliability Analysisand Optimisation of Complex TransportationSystems. Part 2, Project founded by PolishCommittee for Scientific Research, Gdynia:Matitime University, (in Polish).[5] Kwiatuszewska-Sarnecka, B. (2006). Reliabilityimprovement of large multi-state series-parallelsystems. International Journal of Automation andComputing 2, 157-164.[6] Xue, J. (1985). On multi-state system analysis,IEEE Transactions on Reliability, 34, 329-337.- 172 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special IssueRakowsky Uwe KayUniversity of Wuppertal, Germany, and Vossloh Kiepe, Düsseldorf, GermanyFundamentals of the Dempster-Shafer theory and its applications tosystem safety and reliability modellingKeywordsDempster-Shafer Theory, evidence, uncertainty, expert assessment, Event Tress Analysis, Fault Tree AnalysisAbstractThe Dempster-Shafter Theory is well-known for its usefulness to express uncertain judgments of experts. Thiscontribution shows how to apply the calculus to safety and reliability modelling, especially to expert judgement;Failure Modes, Effects, and Criticality Analysis; Event Tree Analysis, Fault Tree Analysis, and Reliability CentredMaintenance. Including a tutorial introduction to the Dempster-Shafer Theory, the differences between theProbability and the Dempster-Shafer Theory are discussed widely.1. IntroductionIn the middle of the 1960s Arthur P. Dempster developeda theory [1], [2], [3] that includes a kind of “upperand lower probabilities”. Later, it turned out thatthis approach is very useful to express uncertainjudgments of experts.About ten years later the work of Dempster was extended,refined, recast, and published by Glenn Shafer[18] as a “Mathematical Theory of Evidence”. Shafere.g. rebuilt the mathematical theory around the Dempsterconcept and introduced degrees of belief insteadof lower probabilities. The Theory of Evidence wasalso denoted as the Dempster-Shafer Theory (DST) orthe Dempster-Shafer Evidential Theory.The more the Dempster-Shafer Theory was furtherdeveloped, the more the evidence measures of DSTdeparted from being probabilities, e.g. Klir & Folger[11] revised DST in that sense. As stated by [4] and[20], the advantage of DST is that it allows copingwith absence of preference, due to limitations of theavailable information, which results in indeterminacy.2. FundamentalsThe DST became known to the safety and reliabilitycommunity in the early 1990s, refer e.g. Guth [7]. Thereliability-oriented approach to DST as presented hereis based on a scenario that contains the system withall hypotheses, pieces of evidence and data sources.The hypotheses represent all the possible states (e.g.faults) of the system considered. It is required that allhypotheses are elements (singletons) of the frame ofdiscernment, which is given by the finite universal setΩ. The set of all subsets of Ω is its power set 2 Ω . Asubset of those 2 Ω sets may consist of a single hypothesisor of a conjunction of hypotheses. Moreover,it is required that all hypotheses are unique, not overlappingand mutually exclusive.In this context pieces of evidence are symptoms orevents (e.g. failures) that occurred or may occurwithin a system. One piece of evidence is related to asingle hypothesis or a set of hypotheses. It is not allowedthat different pieces of evidence lead to thesame hypothesis or set of hypotheses.The qualitative relation between a piece of evidenceand a hypothesis corresponds to a cause-consequencechain: A piece of evidence implies a hypothesis or aset of hypotheses, respectively. The strength of anevidence-hypothesis assignment, and thereby thestrength of this implication, is quantified by a statementof a data source.Data sources are persons, organisations, or any otherentities that provide information for a scenario. Insafety and reliability engineering, data sources areusually the results of empirical studies or they are experts,who give subjective quantifiable statements. Asrequired by O'Neill [13], data sources have to be representative(e.g. studies) or as free from bias as possible(e.g. experts).Some misunderstandings in interpretations concerningthe plot of hypotheses have to be cleared up. Froman objective point of view, which might e.g. be lo-- 173 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issuecated outside the system (e.g. observer), exactly onesingle hypothesis is true; from a subjective point ofview of a data source (e.g. expert or operator), itmight be uncertain which hypothesis fits best to reality.Therefore, DST makes it possible to model several• single pieces of evidence within single hypothesisrelations or• single pieces of evidence within multi hypothesesrelationsas uncertain assessments of a system in which exactlyone hypothesis is objectively true. Both points ofview, the objective and the subjective, have to be distinguishedclearly. The DST calculus describes thesubjective viewpoint as an assessment for an unknownobjective fact.By means of a data source, a mappingm: 2 Ω → [0, 1] (1)assigns an evidential weight to a set A ⊆ Ω, whichcontains a single hypothesis or a set of hypotheses.This is the most significant difference to the ProbabilityTheory: The DST mapping distinguishes clearlybetween the evidence measures and probabilities withmapping Ω → [0, 1]. Each A that holds m(A) > 0 iscalled a focal element. The function m is called a basicassignment and fulfils∑A ⊆ Ω m ( A)= 1 . (2)This equation means that all statements of a singledata source have to be normalised, just to ensure thatthe evidence presented by each data source is equal inweight, e.g. no data source is more important than anotherone. – For the “sake of simplicity” (Klir & Folger[11]), it is assumed thatm(∅) = 0 ; (3)however, this property requires an appropriate choiceof the universal set Ω. That means, the set Ω has to becomplete and contain all possible hypotheses of thescenario considered.In some publications m is called the basic probabilityassignment, refer Shafer [18], which misleads tothe assumption m(A) might be a probability. Furtherdenotations are the basic belief assignment [19], thebelief structure [21], [4] or the mass assignment function[14].A clear distinction has to be made between probabilitiesand basic belief assignment: probability distributionfunctions are defined on Ω and basic assignmentfunctions on the power set 2 Ω . In addition, m has threefurther properties, which distinguishes it from being aprobability function, refer Klir & Folger [11]: It is notrequired• that m(Ω) = 1,• that m(A) ≤ m(B) if A ⊂ B, or• that there is a relationship between m(A) andm(¬A).Therefore, it seems to be useful to avoid the termsprobability and belief (which is defined next) in thedenotation of m.By applying the basic assignment function, severalevidential functions can be created. A belief measureis given by the function bel: 2 Ω → [0,1]. There isbel ( A ) ∑ ; m(B). (4)= B⊆A B≠φThe counterpart of bel is the plausibility measure pl:2 Ω → [0,1] withpl ( A ) ∑ m(B). (5)= B∩A≠φThe measure pl(A) shall not be understood as thecomplement of bel(A). Only{ A Ω | m(A)> 0} ≠ ∅ → bel(A)≤ pl(A)⊆ (6)has to be fulfilled. In addition to bel and pl, a thirdevidential function can be defined. Shafer [18] introducedthe commonality measure with cmn: 2 Ω → [0,1]andcmn ( ) = ∑B⊇ Am(B)A . (7)Figure 1 shows a graphical representation of theabove-defined measures belief and plausibility. Thedifference pl(A) – bel(A) describes the evidential intervalrange, which represents the uncertainty concerningthe set A.Beliefbel(A)Plausibilitypl(A)0 1UncertaintyDoubt1 - bel(A)Disbelief1 - pl(A)Figure 1. Measures of belief and plausibility and itscomplements for a given bel(A) < pl(A). The evidentialor the uncertainty interval, respectively, is shadedgrey.The complements to the measures belief and plausibilityare doubt and disbelief, respectively. AlthoughShafer ([18], page 43) defines doubt as a complement- 174 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issueto the measure plausibility, it seems to make moresense to distinguish between doubt and disbelief in theway given above because DST does not require acausal relationship between a hypothesis and its negation.As Flack [6] emphasises, lack of belief does notimply disbelief. The disbelief of set A is the belief inthe complement. There isbel(¬A) = 1 – pl(A) , pl(¬A) = 1 – bel(A) (8)withbel(¬A) ≤ pl(¬A) . (9)The difference pl(A) – bel(A) describes the uncertaintyconcerning the hypothesis A represented by theevidential interval, see Figure 1.2.1. Interpretations on Evidence MeasuresSome helpful and interesting interpretations of theevidence measures are given in the literature and citedhere.Basic Assignment• The measure m(A) assigns an evidential weight tothe set A, refer Flack [6].• The measure m(A) is the degree of evidence thatthe element in question belongs exactly to the setA, refer Klir & Folger [11].• The measure m(A) is the degree of evidence supportingthe claim that a specific element of Ω belongsto the set A, but not to any special subset ofA, refer Klir & Folger [11].• The quantity m(A) is the degree of belief that theabove specified claim is warranted, refer Klir &Folger [11].Belief• The measure bel(A) is the degree of evidence thatthe element in question belongs to the set A as wellas to the various special subsets of A, refer Klir &Folger [11].• The measure bel(A) can be interpreted as the totalamount of justified support given to A, refer Denoeux[4].• The measure bel(A) is the degree of evidence supportingthe claim that a specific element of Ω belongsto the set A, but not to any special subset ofA, refer Klir & Folger [11].Plausibility• The quantity pl(A) is the degree of evidence thatthe element in question belongs to the set A or toany of its subsets [or to any set that overlaps withA], refer Klir & Folger [11].• The quantity pl(A) can be interpreted as the maximumamount of specific support that could begiven to A, if justified by additional information,refer Smets [19].2.2. Bayesian Statistical Modelling versusDempster-Shafer TheoryFlack [6] describes the differences between the Bayesianstatistical modelling and the Dempster-ShaferTheory as a difference in concepts. A Bayesian modeldescribes a Boolean type of phenomena, which eitherexist or do not exist. Little belief in the existence of aphenomenon implies a strong belief in its nonexistence.This implication does not necessarily holdfor DST. Here, no causal relationship is required betweenboth, belief in existence and belief in nonexistence.For example, a statement concerning thefailure probability of an item also implies a statementabout its counterpart, the reliability of the same item.DST does not require this sub-proposition; and thatadds new aspects and possibilities to reliability modelling.As stated by Ferson et al. [5], the Dempster-ShaferTheory has been widely studied in computer scienceand artificial intelligence, but has never achievedcomplete acceptance among probabilists and traditionalstatisticians. (By this, the question arises if anyother theory than the Probability Theory would everbe accepted by probabilists or traditional statisticians.)However, there are still some disadvantages of theProbability Theory for the DST, which should also bestated here. There are three undesired main propertiesas listed by [4]:• Lack of introspection or assessment strategies: Themain criticisms of the Bayesian statistical modellingis its unreasonable requirement for precision.But the necessity to assign precise numbers in DSTapplications to each subset A ⊆ Ω by the basic assignmentm is constraining in the same way. Precisedegrees of the desired measures may exist, butit is perhaps too difficult to determine them withthe necessary precision.• Instability: Underlying beliefs may be unstable.Estimated beliefs may be influenced by the conditionsof its estimation.• Ambiguity: Ambiguous or imprecise judgementcould not be expressed by the evidence measures.Additional statements to disadvantages of the Dempster-ShaferTheory are:• DST lists all the hypotheses into the frame of discernmentΩ, which resembles the fault space in theBayesian Theory. However, given k hypotheses, itcan consist of up to 2 k elements, representing allpossible subsets of Ω. This leads to a similar problemencountered in the Bayesian Theory, exceptthat it is worse because human experts have to es-- 175 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issuetimate a larger number of belief values than afterthe Bayesian theory [12].• Another caveat of the applicability of DST is that itdoes not offer a procedure for implementation of adiagnostic system [10].2.3. Dempster-Shafer Rule of CombinationDempster [2], [3] followed by Shafer [18] suggested arule of combination which allows that the basic assignmentsare combined. There is∑ m(A)⋅ m(B)A∩B=Z ≠φm ( Z)=, (10)1−∑m(A)⋅m(B)A∩B=φwith A, B, Z ⊆ Ω. Verbally: the numerator representsthe accumulated evidence for the sets A and B, whichsupports the hypothesis Z, and the denominator sumquantifies the amount of conflict between the two sets.Depending on the application, the denominator of∑ m(A)⋅m(B)A∩B=Z ≠φm ( Z)=∑ m(A)⋅ m(B), (11)A∩B≠φThe faults can be determined to at most three preciselydefined hypotheses represented by the set ΩwithΩ = {h 1 , h 2 , h 3 } . (12)With that, the frame of discernment of this context isgiven. It should be noted that Ω is postulated by theoperators based on their subjective points of view,assuming that Ω is complete. The correspondingpower set of Ω is2 Ω = {∅, {h 1 }, {h 2 }, {h 3 }, {h 1 , h 2 }, {h 1 , h 3 },{h 2 , h 3 }, Ω} . (13)The first operator mainly states that the faults h 1 or h 2are the reason for the problems. For example, the failureev 3 might have occurred and resulted in the consequencesh 1 or h 2 . The assignments of the second operatorare slightly different. Here, focus is on thefaults h 1 or h 3 . Both operators give their statements tothe four pieces of evidence found. The complete surveyof the qualitative failure-fault(s) assignments isgiven in Table 1.Table 1. Qualitative failure-fault(s) assignments givenby the operators involvedis easier to apply.FailureFault(s)3. IllustrationThe following illustration is strictly step-wise structuredand gives an easy-to-understand introduction tothe calculus of the Dempster-Shafer Theory.The given scenario discusses a typical situation in apower plant. The operators at the control panel detectserious changes of the system properties. Some failuresare detectable; however, their consequences orthe system fault, respectively, can neither be determinedexactly nor interpreted certainly. (This situationis widely discussed e.g. in the ATHEANA Report [14]and by Hollnagel [8].) To avoid an error forcing context,pieces of evidence are collected, hypotheses arepostulated, and conclusions are made on this basis.Therefore, a Dempster-Shafer approach is applied tosupport the operators in decision making.Step – Creating the ScenarioThe scenario consists of a power plant (the systemconsidered), two operators (data sources, denoted bythe index l), the failures detected (pieces of evidence),and the system fault states (set of hypotheses). As describedabove, the pieces of evidence correspond tofailures or causes and the hypotheses to faults or consequences.Operatorh 1 st ev 1ev 4 h 1 , h 2 , h 3ev 2ev 31h 2h 1 , h 2h 2 nd ev 1ev 4 h 1 , h 2 , h 3ev 2ev 31h 3h 1 , h 3Please note that contrary to the Fault Tree Analysis(FTA), DST does not allow that more than one failurelead to the same fault (hypothesis). However, differentfailures may have different set of consequences,which may contain the same hypotheses as elements,e.g. ev 1 , ev 3 , and ev 4 in Table 1 lead to hypotheses,which all contain h 1 as a fault. Again, DST allowsmodelling several• single-failure-single-fault relations and• single-failure-multi-fault relationsas an uncertain assessment of a system, which cantake exactly one state at a time. Generally, DST emphasizesmore on the hypotheses (faults) than on thepieces of evidence (failures), which are of minor interestin the next steps.As described in Section 0, the system is exactly inone state of Ω. In other words, exactly one hypothesis- 176 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issueof {h 1 , h 2 , h 3 } is true for the given scenario and situationif the system would be observed from an objective{h 1 ∪ h 2 }, { h 1 ∪ h 2 ∪ h 3 } ⊇ {h 1 ∪ h 2 } , (17)There is for A 4point of view. Subjectively, the operators are notsure, in which state the system actually is.cmn(A 4 ) = m(A 4 ) + m(A 7 ) = 0.7 . (18)Step – Quantification of StatementsThe plausibility includes basic assignments of allAt this step, both operators quantify their statementsstatements which have got at least one hypothesis withas given in Table 2. The set of hypotheses A is assignedto the first operator, B to the second operator.4 , there isthose of the discussed statement in common. ConcerningAFor example, the second operator claims that the consequencesh 1 or h 3 may have occurred with a basicassignment of 0.4. (Formulating this sentence verbally,it is rather difficult to avoid that the tongue{h 1 }, {h 2 }, {h 1 ∪ h 2 }, {h 1 ∪ h 3 }, {h 2 ∪ h 3 },{h 1 ∪ h 2 ∪ h 3 } ∩ {h 1 ∪ h 2 } ≠ ∅ , (19)mentions “probability”. Again, basic assignments arenot probabilities, see equation (1).) Non-specifiedstatements are assigned by 0 and are not focal elements.The subjective quantifications of the operators arebased on their system experiences and mostly on theirwhich results in the plausibilitypl(A 4 ) = m(A 1 ) + m(A 2 ) + m(A 4 ) + m(A 5 )+ m(A 6 ) + m(A 7 ) = 1 (20)“engineering feelings”. Certainly, these quantificationsare imprecise.Table 2. Quantitative statements given by the operatorsand finally in no disbelief at all1 – pl(A 4 ) = 0 . (21)involved (outer columns). The inner column con-tains all subsets of the power set 2 .Table 3 shows the results for belief and plausibility ofall statements (k = 1,…, 7).1 st operator 2 Ω 2 nd operatorTable 3. This table corresponds to Table 2 and showsm(A 1 ) = 0.2 {h 1 } m(B 1 ) = 0.2the values of basic assignments (bold typing), belief,m(A 2 ) = 0.1 {h 2 } m(B 2 ) = 0and plausibility for each statement and operator (firstm(A 3 ) = 0{h 3 } m(B 3 ) = 0.2m(A 4 ) = 0.6left side, second right side).{h 1 ∪ h 2 } m(B 4 ) = 0m(A 5 ) = 0 {h 1 ∪ h 3 } m(B 5 ) = 0.4m(A 6 ) = 0 {h 2 ∪ h 3 } m(B 6 ) = 0m(A k) bel(A k) pl(A k) 2 Ω m(B k) bel(B k) pl(B k)m(A 7 ) = 0.1 {h 1 ∪ h 2 ∪ h 3 } m(B 7 ) = 0.20.2 0.2 0.9 {h 1 } 0.2 0.2 0.80.1 0.1 0.8 {h 2 } 0 0 0.2Based on the basic assignments given by both operators,the belief and doubt, commonality, plausibility 0.6 0.9 1 {h 1 ∪ h 2 } 0 0.2 0.80 0 0.1 {h 3 } 0.2 0.2 0.80 0.2 0.9and disbelief measures can be calculated. For example,the belief in the set of hypotheses {h 1 ∪ h 2 } is the{h 1 ∪ h 3 } 0.4 0.8 10 0.1 0.8 {h 2 ∪ h 3 } 0 0.2 0.80.1 1 1sum of its own basic assignment with those of all ofΩ0.2 1 1its subsetsStep – Combining Hypotheses{h 1 }, { h 2 }, {h 1 ∪ h 2 } ⊆ {h 1 ∪ h 2 } , (14) The third step combines each hypothesis or set of hypotheses,respectively, from one data source (operator)with one from the other source and builds the cutsee equation (4). For the fourth statement of the firstoperator there isset of both, see Table 4. Depending on quantificationsbel(A 4 ) = m(A 1 ) + m(A 2 ) + m(A 4 ) = 0.9, (15)given at Step, some of the cut sets may not be focalelements before or thereafter. Actually, this step waswith the corresponding doubt measure1 – bel(A 4 ) = 0.1 . (16)fit in for illustrative purpose and can be combinedwith Step.Table 4. The Combination Table contains the full plotThe commonality takes every statement into account,which includes the discussed statement completely.of hypotheses cut sets of A and B- 177 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issue∩ A 1 A 2 A 3 A 4 A 5 A 6 A 7B 1 h 1B 2 ∅B 3 ∅B 4 h 1B 5 h 1B 6 ∅B 7h 1∅h 2∅h 2∅h 2h 2∅∅h 3∅h 3h 3h 3h 1h 2∅h 1 ∪h 2h 1h 2h 1 ∪h 2h 1∅h 3h 1h 1 ∪h 3h 3h 1 ∪h 3Step – Reducing the Combination Table∅ h 1h 2h 3h 2h 3h 2 ∪h 3h 2 ∪h 3h 2h 3h 1 ∪h 2h 1 ∪h 3h 2 ∪h 3ΩTo avoid mathematical effort, those columns and rowsof the Combination Table 4 were dropped, which arerelated to non-focal elements (non-specified statementswith m(A k ) = 0, m(B k ) = 0). In this context, columnsA 3 , A 5 , A 6 and rows B 2 , B 4 , B 6 are not applicable.Table 5 shows the reduced plot containing thecombinations of focal elements exclusively.Table 5. The reduced Combination Table∩ A 1 A 2 A 4 A 7B 1 h 1B 3 ∅B 5 h 1B 7h 1∅∅∅h 2h 1∅h 1h 1 ∪h 2h 1h 3h 1 ∪h 3ΩStep – Calculating Products and Sums of CombinedBasic AssignmentsAt this step, products of the related basic assignmentsare calculated from the non-empty sets. Products ofbasic assignments corresponding to the same cut sethave to be added. For {h 1 } yieldsZ 1 = A 1 ∩ B 1 = {h 1 }⇒ m(Z 1 ) = m(A 1 ) ⋅ m(B 1 ) = 0.04 , (22)Z 2 = A 1 ∩ B 5 = {h 1 }⇒ m(Z 2 ) = m(A 1 ) ⋅ m(B 5 ) = 0.08 , (23)Z 3 = A 1 ∩ B 7 = {h 1 }⇒ m(Z 3 ) = m(A 1 ) ⋅ m(B 7 ) = 0.04 , (24)Z 4 = A 4 ∩ B 1 = {h 1 }⇒ m(Z 4 ) = m(A 4 ) ⋅ m(B 1 ) = 0.12 , (25)Z 5 = A 4 ∩ B 5 = {h 1 }⇒ m(Z 5 ) = m(A 4 ) ⋅ m(B 5 ) = 0.24 , (26)Z 6 = A 7 ∩ B 1 = {h 1 }⇒ m(Z 6 ) = m(A 7 ) ⋅ m(B 1 ) = 0.02 (27)with the sum6∑k = 1m(Z ) = 0.54 . (28)kHypotheses h 2 or h 3 are supported byZ 7 = A 2 ∩ B 7 = {h 2 }⇒ m(Z 7 ) = m(A 2 ) ⋅ m(B 7 ) = 0.02 , (29)Z 8 = A 7 ∩ B 3 = {h 3 }⇒ m(Z 8 ) = m(A 7 ) ⋅ m(B 3 ) = 0.02 . (30)There is for the sets {h 1 ∪ h 2 }, {h 1 ∪ h 3 }, and {h 1 ∪ h 2∪ h 3 }:Z 9 = A 4 ∩ B 7 = {h 1 ∪ h 2 }⇒ m(Z 9 ) = m(A 4 ) ⋅ m(B 7 ) = 0.12 , (31)Z 10 = A 7 ∩ B 5 = {h 1 ∪ h 3 }⇒ m(Z 10 ) = m(A 7 ) ⋅ m(B 5 ) = 0.04 , (32)Z 11 = A 7 ∩ B 7 = {h 1 ∪ h 2 ∪ h 3 }⇒ m(Z 11 ) = m(A 7 ) ⋅ m(B 7 ) = 0.02 . (33)To illustrate this formal procedure, the results aregiven in Table 6.Table 6. This table corresponds directly to Table 5 andrepresents the products (•) of basic assignments;ßmeans no focal element• A 1 A 2 A 4 A 7B 1 0.04B 3 ßB 5 0.08B 7 0.04ßßß0.020.12ß0.240.120.020.020.040.02Step – Combining Basic AssignmentsThe sum over all combinations calculated in Step,11∑k = 1m(Z ) = 0.76 . (34)kis identical with the denominator of equation (11).With that, the basic assignment of every hypothesiscan be calculated. There is- 178 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issue6∑ m(Zk)m({h }) 11 = k = ≈ 0.710511∑ m(Zk)k = 1(35)Starting at the same low value, h 3 takes roughly halfthe range of uncertainty that h 2 takes. However, bothhypotheses alone should not be considered further dueto the low values of belief and plausibility. With about~0.24, the single hypothesis h 1 is assigned with a widerange of uncertainty. The combination of h 1 and h 3covers a smaller range (~0.18) than h 1 alone, and ithas a higher plausibility. Finally, a combination of h 1and h 2 shows the smallest range of uncertainty (~0.08)with the same (highest) plausibility as in case of thecombination of h 1 and h 3 .The conclusion is that a combination of h 1 and h 2may be responsible for the serious changes of the systemproperties. Please note that a probabilistic approachwould have blamed h 1 alone for being responsible.(And please consider the consequences.) Thisresult clearly shows the differences between boththeories, based on their different mappings Ω → [0, 1]versus 2 Ω → [0, 1], see Section 0.for the hypothesis h 1 andm(Z )m({h }) 72 = ≈ 0.026311∑ m(Zk)k = 1m(Z )m({h }) 83 = ≈ 0.026311∑ m(Zk)k = 1m(Z )m({h }) 91 ∪ h2= ≈ 0.157911∑ m(Zk)k = 1m(Z )m({h }) 101 ∪ h3= ≈ 0.052611∑ m(Zk)k= 1m(Z )m({h }) 111 ∪ h2∪ h3= ≈ 0.026311∑ m(Zk)k = 1for all relevant sets of hypotheses.(36)(37)(38)(39)(40)Step – Evidence Measures of Combined HypothesesThe evidence measures of combined hypotheses arecalculated according to Step, Table 7. The set {h 1 ∪h 2 } does not occur because it vanished in Table 5.Step – InterpretationTable 7. The basic assignments (bold) and the resultingevidence measures belief, commonality, and plausibilityare given; hypotheses are ranked by their beliefmeasures, all values are rounded.2 Ω m bel cmn plΩ{h 1 ∪ h 2 }{h 1 ∪ h 3 }{h 1 }{h 2 }{h 3 }0.02630.15790.05260.71050.02630.026310.89470.78950.71050.02630.02630.02630.18420.07890.94740.21050.105310.97370.97370.94710.21050.10534. Applications to System Safety and ReliabilityModellingThe introductory descriptions of the Failure Modes,Effects, and Criticality Analysis, the Event TreeAnalysis, and the Fault Tree Analysis are taken fromthe “System Safety Analysis Handbook” written byStephens & Talso [29] and published by the SystemSafety Society. The descriptions are shortened andslightly revised. The introductory description of theReliability Centred Maintenance is taken from [23].Additionally, the IEC standards are recommended forthe application of all listed methods, refer to Section0.4.1. Failure Modes, Effects, and CriticalityAnalysisAs described by Stephens & Talso [29], the FailureModes, Effects, and Criticality Analysis (FMECA)tabulates a list of items in a process along with all thepossible failure modes for each item. The effect ofeach failure is evaluated and ranked according to aseverity classification. An FMECA includes the followingsteps:• Define the worksheet formats and ground rules.• Give analysis assumptions.• Identify the lowest indenture level of analysis.• Code the system description.• Give failure definitions and evaluations.The usefulness of the FMECA as a design tool and inthe decision making process depends on the effectivenesswith which problem information is communicatedfor early design attention. Probably the mostsevere criticism of the FMECA has been its limiteduse for improvement of designs, as Stephens & Talso[29] claim. The main causes for this have been the- 179 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issueuntimeliness and the isolated performance of theFMECA without adequate inputs to the design process.While the objective of an FMECA is to identifyall modes of failure within a system design, its firstpurpose is the early identification of all critical failureprobabilities so that they can be eliminated or minimisedthrough design correction at the earliest possibletime.The Dempster-Shafer calculus, as described by theillustration in Section 0, can easily be applied to theFMECA. If more than one expert is involved in thequantitative assessments of the criticality and the occurrenceof an item failure. The results are narrow orwide ranges of uncertainties, which require an interpretationsimilar to Step Section 0.However, it is an interesting question if institutions,which conduct system homologations, would acceptthese results.4.2. Event Tree AnalysisAs summarised by Stephens & Talso [29], the EventTree Analysis (ETA) is an analytical tool that can beused to organise, characterise, and quantify potentialfailures in a methodical manner. An event tree modelsthe sequence of events that results from a single initiatingevent. The ETA is a bottom-up analysis versusthe top-down approach for the Fault Tree Analysis,see Section 0.Conducting an ETA starts with selection of the initiatingevents, both the desired events and the ones notdesired. Thereafter, their consequences are developedthrough consideration of component, module, and systemfailure-and-success alternatives, respectively. Theidentification of initiating events may be based on reviewof the system design and operation, the results ofanother safety analysis, or personal operating experienceacquired with a similar system. Then the successand failure of the mitigating systems are postulatedand continued through all alternate paths, consideringeach consequence as a new initiating event. The basicsteps for construction of an event tree include the following:• List all possible initiating events.• Identify functional system responses.• Identify support system responses.• Group initiating events with all responses.• Define failure sequences.• Assign probabilities to each step in the event treeto arrive at total probability of occurrence for eachfailure sequence.The method is universally applicable to all kinds ofsystems, with the limitation that all events must beanticipated to produce meaningful analytical results.Among the methods presented in the “System SafetyAnalysis Handbook” by Stephens & Talso [29], theEvent Tree Analysis is definitely one of the most exhaustive,if it is applied properly. Axiomatically, theiruse also consumes large quantities of resources. Theiruse, therefore, is well reserved for systems in whichrisks are regarded as high and well concealed.As described by Stephens & Talso [29], probabilitiesare assigned to each step in the event tree. To applyevidence measures instead of probabilities, the followingsteps are conducted, which lead to the Dempster-Shafer Event Tree Analysis DS-ETA.It is assumed that the considered event tree consistsof bifurcations only; i.e., any symbol within an eventtree has one input and two outputs (the event “failure”or the event “no failure”), see Figure 2.InitialeventA 2A 1A 1A 2A 2A 1 r 1Figure 2. Event tree with three bifurcations resultingin four sequences, A 1 denotes “failure” and A 2 “nofailure”Every bifurcation i = 1, …, n of an event tree is consideredseparately and independently from the n – 1other bifurcations. The assigned set of hypothesescontains Ω i = {A 1 , A 2 , A 3 } ≡ {“failure”, “no failure”,“uncertain”}. Hence, any expert (data source) l has togive three values for the basic assignments m l,i (A k ), k= 1, 2, 3 of any bifurcation i, representing his/her degreeof belief that A k may occur. Obviously, the uncertaintyof an expert concerning an event A 3 does notappear in the graphical representation. There is for theExpert Assessment Matrix⎡ml,1(A1)⎢M⎢Ml= ⎢ml,i ( A1)⎢⎢M⎢⎣ml,n(A1)ml,1(A2)Mml,i ( A2)Mml,n(A2)ml,1(A3) ⎤M⎥⎥ml,i ( A3) ⎥⎥M⎥ml,n(A3) ⎥⎦r 2r 3r 4(41)with m l,i (A 1 ) + m l,i (A 2 ) + m l,i (A 3 ) = 1.The Combination Matrix C i combines each hypothesisor set of hypotheses from two experts and buildsthe cut set of both. Depending on the quantificationsgiven, some of the cut sets (here: the cut sets of “failure”and “no failure”) may not be focal elements beforeor afterwards. To avoid mathematical effort,those columns and rows were dropped, which are relatedto empty cut sets or non-specified statements(non-focal elements with a 0% basic assignment). In- 180 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issuecase of two experts and three possible answers, as presentedhere, the matrix⎡m1,i(A1) ⋅ m2,i(A1)⎢Ci= ⎢ 0⎢⎣m1,i ( A1) ⋅m2,i(A3)0m1,i ( A2) ⋅m2,i(A2)m1,i ( A2) ⋅m2,i ( A3)m1,i(A3) ⋅ m2,i ( A1) ⎤⎥m1,i(A3) ⋅m2,i ( A2) ⎥m1,i(A3) ⋅ m2,i ( A3) ⎥⎦(42)represents the combinations of focal elements exclusively assigned to the i-th event within the tree. The focal sumσ(i) is the sum of all matrix elements C i . In this case, it is given by( m ( A ) ⋅ m ( A ) + m ( A ) ⋅ m ( ))σ ( i)= 1−1,i 1 2, i 2 1, i 2 2, i A1, (43)considering the fact that “failure” (A 1 ) and “no failure” (A 2 ) are mutually exclusive. The combined basic assignmentsm i (A k ) for a “failure”, “no failure”, and “uncertain” arem1,i ( A1) ⋅m2,i ( A1) + m1,i ( A3) ⋅ m2,i ( A1) + m1,i ( A1) ⋅m2,i ( A3)mi( A1) = ,(44)σ ( i)m1,i ( A2) ⋅m2,i ( A2) + m1,i(A3) ⋅ m2,i(A2) + m1,i ( A2) ⋅ m2,i ( A3)mi( A2) = ,(45)σ ( i)m1,i ( A3) ⋅m2,i ( A3)mi( A3) = . (46)σ ( i)With m i (A k ) as given above, the evidence measures for a “failure” and a “no failure” decision can be calculated bybel i ( A1 ) = mi( A1) ,pl i ( A1 ) = mi( A1) + mi( A3) ; (47)bel i ( A2 ) = mi( A2) ,pl i ( A2 ) = mi( A2) + mi( A3) . (48)The next step of modelling applies the basic operationsof interval arithmetic to the given event tree andassigns the evidential measures as input variables. Theaddition and multiplication operations are commutative,associative and sub-distributive. There is for i ≠ii,[bel i (A k ), pl i (A k )] + [bel ii (A k ), pl ii (A k )]= [bel i (A k ) + bel ii (A k ), pl i (A k ) + pl ii (A k )] , (49)[bel i (A k ), pl i (A k )] ⋅ [bel ii (A k ), pl ii (A k )]= [min[bel i (A k ) ⋅ bel ii (A k ), bel i (A k ) ⋅ pl ii (A k ),pl i (A k ) ⋅ bel ii (A k ), pl i (A k ) ⋅ pl ii (A k )],max[bel i (A k ) ⋅ bel ii (A k ), bel i (A k ) ⋅ pl ii (A k ),pl i (A k ) ⋅ bel ii (A k ), pl i (A k ) ⋅ pl ii (A k )],] . (50)Instead of subtraction, the complements of the evidencemeasures are applied; this yieldsbel i (A 1 ) = 1 – pl i (A 2 ) , (51)pl i (A 1 ) = 1 – bel i (A 2 ) . (52)With the inputs and operations given, the evidence ofa sequence r j , see Figure 2.,can be calculated easily.As described above, the basic operations of intervalarithmetic are applied within the DS-ETA. Fortunately,the structure avoids the trouble that the subdistributivityof subtraction operations may cause, e.g.as known from the Fuzzy Fault Tree Analysis (f-FTA),see [17].4.3. Fault Tree AnalysisThe Fault Tree Analysis (FTA) can model the failureof a single event or multiple failures which lead to asingle system failure denoted as the top event, refer toStephens & Talso [29]. However, the FTA is a topdownanalysis versus the bottom-up approach for theEvent Tree Analysis; i.e., the method identifies anundesirable top event and the contributing elements(down to the so-called basic events) that would pre-- 181 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issuecipitate it. The contributors are interconnected withthe top event, using network paths through Booleanlogic gates. The following basic steps are used to conducta fault tree analysis:• Define the top event of interest.• Define the physical and analytical boundaries.• Define the tree-top structure.• Develop the path of failures for each branch to thelogical initiating failure, represented by the basicevent.Figure 3 shows a typical fault tree with meshed basicevents 1, 4, and 7.≥1≥118 20 && 19 21≥1&≥1≥1≥1Once the fault tree has been developed to the desireddegree of detail, the various paths can be evaluated toarrive at a probability of occurrence. Cut sets arecombinations of components failure causing the topevent. Minimal cut sets are the smallest combinationscausing the top event. The method is universally applicableto systems of all kinds, with the followingground rules:• The top events which are to be analysed, and theircontributors, must be foreseen.• Each of those top events must be analysed individually.• The contributing factors have been adequatelyidentified and explored in sufficient depth.The FTA has got several strengths. The proceduresare well defined and focus on failures. The top-downapproach requires analysis completeness at each levelbefore proceeding. It cannot guarantee identificationof all failures, but the systematic approach enhancesthe likelihood of completeness. The FTA addresseseffects of multiple failures by identifying interrelationshipsbetween components and identifying minimalfailure combinations that cause the system to fail(minimal cut sets). The method addresses the effectsof design, operation, and maintenance. The FTA canhandle complex systems. It provides a graphical representationthat helps to understand these complexoperations and interrelationships between modulesand components. Finally, FTA provides both qualitativeand quantitative information.The method is capable of producing numericalstatements of the probability of occurrence of undesirableevents, given probabilities of contributing factors.As Stephens & Talso ([29], page 136) claim, this capabilityleads to a common abuse: much effort can beexpended in producing refined numerical statementsof probability, based on contributing factors whoseindividual probabilities are hardly known and towhich broad confidence limits should be attached.Applying the Dempster-Shafer Theory to FTA canhelp modelling uncertainties with less effort as shownby Guth [7].Guth discusses Ω = {h 1 , h 2 , h 3 } ≡ {“event occurs”,“uncertain”, “event does not occur”}. In the following,two events A and B are considered as inputs andZ as output of an And or Or gate, respectively. Thereis&&m(A 1 ) = bel(A) , (53)2 3 8 14 912 15 1 4 7 10 13 16 11 17 5 6Figure 3. Typical fault tree with basic events, gates,and top event; the example represents one fault of abraking module as applied in railway vehiclesm(A 2 ) = pl(A) – bel(A) , (54)m(A 3 ) = 1 – pl(A) , (55)m(A 1 ) + m(A 2 ) + m(A 3 ) = 1 ; (56)the same holds for B. Please note that the Guth approachto the Dempster-Shafer Fault Tree Analysis(DS-FTA) considers events where the DS-ETA considersbifurcations. Table 8 shows the (underlying)combination of hypotheses with the different resultsgiven for the And and Or gate.Table 8. Combination TableAnd A 1 A 2 A 3 Or A 1 A 2 A 3B 1 h 1 h 2 h 3 B 1 h 1 h 1 h 1B 3 h 3 h 3 h 3 B 3 h 1 h 2B 2 h 2 h 2 h 3 B 2 h 1 h 2 h 2Following Step in Section 0, the combined basicassignments are calculated. An And gate yieldsm(Z 1 ) = m(A 1 ) m(B 1 ) , (57)m(Z 2 ) = m(A 1 ) m(B 2 ) + m(A 2 ) m(B 1 )+ m(A 2 ) m(B 2 ) , (58)m(Z 3 ) = m(A 1 ) m(B 3 ) + m(A 2 ) m(B 3 )+ m(A 3 ) m(B 1 )+ m(A 3 ) m(B 2 ) + m(A 3 ) m(B 3 )h 3- 182 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special IssueFor an or gate= m(A 1 ) m(B 3 ) + m(A 2 ) m(B 3 ) + m(A 3 ) . (59)m(Z 1 ) = m(A 1 ) m(B 1 ) + m(A 1 ) m(B 2 ) + m(A 1 ) m(B 3 )+ m(A 2 ) m(B 1 ) + m(A 3 ) m(B 1 )= m(A 1 ) + m(A 2 ) m(B 1 ) + m(A 3 ) m(B 1 ) , (60)m(Z 2 ) = m(A 2 ) m(B 2 ) + m(A 2 ) m(B 3 )+ m(A 3 ) m(B 2 ) , (61)m(Z 3 ) = m(A 3 ) m(B 3 ) (62)holds similarly. With that, both evidence measuresbel(Z) and pl(Z) can now be calculated recursively tothe equations (53) to (55).Cheng [22] claims that a calculus based on intervalarithmetic is more concise and efficient in operationthan the calculus proposed by Guth [7] and presentedabove. However, contrary to an event tree structure, afault tree structure may cause trouble with the subdistributivityproperty of subtraction operations asknown from the Fuzzy Fault Tree Analysis, see [17].This applies especially if events are meshed within afault tree, see Figure 3.Some authors apply an m: Ω×Ω → [0, 1] mappinginstead of the well-known m: 2 Ω → [0, 1] mappingwhich mainly characterises the calculus of the Dempster-ShaferTheory. However, an Ω×Ω rather representsoperations in interval arithmetic, where thelower bound and the upper bound are just labelled asbelief and plausibility, respectively.4.4. Reliability Centred MaintenanceReliability Centred Maintenance (RCM), which wasfirst introduced in the aircraft industry, has been usedwith considerable success in the last decades in manyindustrial branches. As described in [23], the RCManalysis starts with establishing an expert group andinitiating the collection of important component andsystem data based on the system documentation. Thenthe system functions are broken down to the desiredcomponent level. All relevant component informationshould be collected in the form of a modifiedFMECA. The main modification of the FMECA consistsof the inclusion of information facilitating thechoice of the optimum maintenance strategy. This isgenerally performed by an RCM decision diagram.Many different decision diagrams are proposed to applyin an RCM analysis. To illustrate the approach, adiagram as given in Figure 4 is discussed, see [28].Note that the given diagram is not necessarily completeor applicable in every context given.Criticality 1st line maint Detectability Maint strategySignificant yes First line no Detectability no Testability yesconsequences maintof a failure of failurenoyesOther reasonsfor prev maintyesnoFirst linemaint, alone?yesCond basedmaint effectivenonoIncreasingfailure rateyesyesnoyesnoPeriodicaltestsScheduledmaintenanceFind abetter designCond basedmaintenanceFirst linemaintenanceCorrectivemaintenanceFigure 4. Example of an RCM decision diagram [23]Supported by the diagram and the integrated questions,a choice for the best fitting maintenance strategyshould be made. Finally, the maintenance programshould be implemented, and feedback from operationexperience and new data should be used toimprove the program regularly.The choice of the best maintenance strategy is theobjective of applying RCM; however, reasoning maybe difficult, due to questions without a definite answer.For example, the question whether or not acomponent is critical could not easily be answeredwith either “yes” or “no”. A Dempster-Shafer basedalternative makes the avoidance of crisp “yes” or “no”decisions possible and leads to weighted recommendationson which maintenance strategy to choose.The application of the DST to RCM (denoted as DS-RCM [28]) follows the procedure of the DS-ETA asdescribed in Section 0. The RCM decision diagramcorresponds to the event tree, where the decisions arethe counterparts of the events. Consequently, the decisions“yes” and “no” correspond to the events “failure”and “no failure”. The Expert Assessment MatrixM l , the Combination Matrix C i , and the focal sum σ(i)are defined and applied analogously. Additionally, anRCM Decision Diagram Matrix D with⎡m1(A1) m1( A2) m1(A3) ⎤D =⎢M M M⎥(63)⎢⎥⎢⎣m8(A1) m8( A2) m8(A3) ⎥⎦is defined, which collects the combined basic assignmentvalues (results) of each decision within the RCMdiagram. Finally, the weighted recommendations onall maintenance strategies are listed by the RecommendationMatrix- 183 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issue⎡bel(r1) pl(r1) ⎤R =⎢M M⎥, (64)⎢⎥⎢⎣bel(r6) pl(r6) ⎥⎦which collects the values of the evidence measuresbelief and plausibility for every strategy.The main advantages of the DS-RCM as againstthe qualitative RCM can be summarised as follows:• Experts feel more comfortable giving degrees ofbelief instead of taking “yes” or “no” decisions. Itmight therefore be easier to obtain relevant data forthe RCM analysis.• The DS-RCM approach results in a profile of allpossible maintenance strategies. Decision makingbased on this profile helps preventing “weak decisions”and may in any case be more comprehensivethan relying on a single strategy.• In some RCM studies it may be desirable to analysemodules and not separate components. Inthese cases DS-RCM is especially useful, since itdoes not force a single strategy.• As shown in the discussion of the example case,the DS-RCM approach helps to reveal possible designproblems and their causes.A possible disadvantage of this approach is that someexperts may find the evidential numbers more complicatedthan a simple “yes” or “no” decision. A shortdiscussion about the nature of these numbers shouldtherefore be given as an introduction to an RCM session.5. ConclusionsThe Dempster-Shafter Theory is well-known for itsusefulness to express uncertain judgments of experts.It is shown in this contribution, how to apply the calculusto safety and reliability modelling. Approachesto expert judgement; Failure Modes, Effects, andCriticality Analysis; Event Tree Analysis; Fault TreeAnalysis, and Reliability Centred Maintenance arediscussed.Generally, the Dempster-Shafter Theory adds a newflavour to safety and reliability modelling comparedto probabilistic approaches. The illustration (Section0) clearly shows the differences between the Probabilityand the Dempster-Shafer Theory, based on theirdifferent mappings Ω → [0, 1] versus 2 Ω → [0, 1], seeSection 0. Probability theory would identify a singlehypothesis and DST a combination of two to be responsiblefor the serious changes of the system considered.6. References6.1. Dempster-Shafer Theory[1] Dempster, A.P. (1966). New Methods for Reasoningtowards Posterior Distributions based onSample Data. The Annals of Mathematical Statistics37: 355-374.[2] Dempster, A.P. (1967). Upper and Lower ProbabilitiesInduced by a Multivalued Mapping. TheAnnals of Mathematical Statistics 38: 325-339.[3] Dempster, A.P. (1968). A Generalization ofBayesian Inference. Journal of the Royal StatisticalSociety, Series B (methodological) 30: 205-247.[4] Denoeux, T. (1999). Reasoning with imprecisebelief structures. International Journal of ApproximateReasoning 20 (1): 79-111.[5] Ferson, S., Kreinovich, V., Ginzburg, L., Myers,D.S. & Sentz, K. (2003). Constructing ProbabilityBoxes and Dempster-Shafer Structures. SandiaReport SAND2002-4015.[6] Flack, J. (1996). On the Interpretation of RemotelySensed Data Using Guided Techniquesfor Land Cover Analysis. Unpublished PhD thesis,Department of Geographic Information Science,Curtin. University, Perth, Australia.[7] Guth, M. A. S. (1991). A Probabilistic Foundationfor Vagueness and Imprecision in Fault-TreeAnalysis. IEEE Transactions on Reliability 40(5), 563-571.[8] Hollnagel, E. (1998). Cognitive Reliability andError Analysis Method – CREAM. Amsterdam:Elsevier.[9] International Electrotechnical Commission IEC,ed. (2002). International Electrotechnical Vocabulary– Chapter 191: Dependability and Qualityof Service. IEC 60050-191: 2002-05, 2nd Edition.[10] Keung-Chi, N. & Abramson, B. (1990). UncertaintyManagement in Expert Systems. IEEE Expert5 (2), 29-48.[11] Klir, G. J. & Folger, T. A. (1988). Fuzzy Sets,Uncertainty and Information. Englewood Cliffs.Prentice-Hall.[12] Leang, S. (1995). A Control and DiagnosticSystem For The Photolithography Process Sequence,Ph.D. Dissertation.[13] O’Neil, A. (1999). The Dempster-Shafer Engine.http://www.quiver.freeserve.co.uk/Dse.htm[14] Nuclear Regulatory Commission (1999). TechnicalBasis and Implementation Guidelines for aTechnique for Human Event Analysis (AT-HEANA), NUREG-1624.[15] Parsons, S. (1994). Some Qualitative Approachesto Applying the Dempster-Shafer Theory. Informationand Decision Technologies 19, 321-337.- 184 -

U. Rakowsky Fundamentals of the Dempster-Shafer theory and its applications to system safetyand reliability modelling - RTA # 3-4, 2007, December - Special Issue[16] Rakowsky, U. K. (2005). Some Notes on Probabilitiesand Non-Probabilistic Reliability Measures.Proceedings of the ESREL 2005, Tri-City/Poland. Leiden: Balkema, 1645-1654.[17] Rakowsky, U. K. (2002). System Reliability (inGerman), Hagen/ Germany: LiLoLe Publishing.[18] Shafer, G. (1976). A Mathematical Theory ofEvidence. Princeton: Princeton University Press.[19] Smets, P. & Kennes R. (1994). The TransferableBelief Model. Artificial Intelligence 66, 191-243.[20] Walley, P. (1991). Statistical Reasoning withImprecise Probabilities. London: Chapman andHall.[21] Yager, R. R. (1982). Generalized Probabilities ofFuzzy Events from Fuzzy Belief Structures. InformationSciences 28: 45-62.6.2. Safety and Reliability Methodology[22] Cheng, Y.-L. (2000). Uncertainties in Fault TreeAnalysis. Tamkang Journal of Science and Engineering,Vol. 3, No. 1, 23-29.[23] Eisinger, S. & Rakowsky, U. K. (2001). Modelingof Uncertain-ties in Reliability CenteredMaintenance – A Probabilistic Approach. ReliabilityEngineering and System Safety, 71 (2),159-164.[24] International Electrotechnical Commission IEC(2007). Analysis techniques for dependability –Event Tree Analysis. IEC 62502, working document.[25] International Electrotechnical Commission IEC(2006-a). Analysis techniques for system reliability– Procedure for failure mode and effectsanalysis (FMEA). IEC 60812:2006-01, 2 nd Edition.[26] International Electrotechnical Commission IEC(2006-b). Fault tree analysis (FTA). IEC61025:2006-12, 2 nd Edition.[27] International Electrotechnical Commission IEC(1999). Dependability Management – Part 3-11:Application guide – Reliability centred maintenance.IEC 60300-3-11:1999-03, 1 st Edition.[28] Rakowsky, U. K. & Gocht, U. (appears 2007).Modelling of uncertainties in Reliability CentredMaintenance – a Dempster-Shafer approach.Proceedings of the European Conference onSafety and Reliability – ESREL 2007, Stavanger/Norway.Approved as full paper.[29] Stephens, R. A. & Talso, W. (1999-08). SystemSafety Analysis Handbook – A Source Book forSafety Practitioners. Unionville/Virginia, U.S.A.:System Safety Society, 2nd Edition.7. SymbolsA, B set of hypothesesbel belief measureC Combination Matrixcmn commonality measureD RCM Decision Diagram Matrixh single hypothesisi, n index, resp. number of an elementj sequence index (ETA) or statement index(RCM)k set or statement indexl data source indexM Expert Assessment Matrixm basic assignmentspl plausibility measureR Recommendation Matrixr j sequence evidence (ETA) or evaluation value ofthe maintenance strategies (RCM)σ(i) focal sumZ combined set of hypotheses- 185 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special IssueSoszyska JoannaMaritime University, Gdynia, PolandSystems reliability analysis in variable operation conditionsKeywordsreliability function, semi-markov process, large multi-state systemAbstractThe semi-markov model of the system operation process is proposed and its selected parameters are defined.There are found reliability and risk characteristics of the multi-state series- “m out of k ” system. Next, the jointmodel of the semi-markov system operation process and the considered multi-state system reliability and risk isconstructed. The asymptotic approach to reliability and risk evaluation of this system in its operation process isproposed as well.1. IntroductionMany technical systems belong to the class of complexsystems as a result of the large number of componentsthey are built of and complicated operating processes.This complexity very often causes evaluation ofsystems reliability to become difficult. As a rule theseare series systems composed of large number ofcomponents. Sometimes the series systems have eithercomponents or subsystems reserved and then theybecome parallel-series or series-parallel reliabilitystructures. We meet these systems, for instance, inpiping transportation of water, gas, oil and variouschemical substances or in transport using beltconveyers and elevators.Taking into account the importance of safety andoperating process effectiveness of such systems itseems reasonable to expand the two-state approach tomulti-state approach in their reliability analysis. Theassumption that the systems are composed of multistatecomponents with reliability state degrading intime without repair gives the possibility for moreprecise analysis of their reliability, safety andoperational processes’ effectiveness. This assumptionallows us to distinguish a system reliability criticalstate to exceed which is either dangerous for theenvironment or does not assure the necessary level ofits operational process effectiveness. Then, animportant system reliability characteristic is the time tothe moment of exceeding the system reliability criticalstate and its distribution, which is called the systemrisk function. This distribution is strictly related to thesystem multi-state reliability function that is a basiccharacteristic of the multi-state system.The complexity of the systems’ operation processesand their influence on changing in time the systems’structures and their components’ reliabilitycharacteristics is often very difficult to fix and toanalyse. A convenient tool for solving this problem issemi-markov modelling of the systems operationprocesses which is proposed in the paper. In thismodel, the variability of system components reliabilitycharacteristics is pointed by introducing thecomponents’ conditional reliability functionsdetermined by the system operation states. Therefore,the common usage of the multi-state system’s limitreliability functions in their reliability evaluation andthe semi-markov model for system’s operation processmodelling in order to construct the joint general systemreliability model related to its operation process isproposed. On the basis of that joint model, in the case,when components have exponential reliabilityfunctions, unconditional multi-state limit reliabilityfunctions of the series- m out k n system are determined.2. System operation processWe assume that the system during its operation processhas v different operation states. Thus, we can defineZ (t), t ∈< 0 , +∞ > , as the process with discrete statesfrom the setZ = { z1 , z 2 , ..., z v }.- 186 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special IssueIn practice a convenient assumption is that Z(t) is asemi-markov process [1] with its conditional sojourntimes θ blat the operation state z bwhen its nextoperation state is z l , b , l = 1,2,..., v,b ≠ l.In this casethis process may be described by:- the vector of probabilities of the initial operationstates [ p b (0)] 1xν,- the matrix of the probabilities of its transitionsbetween the states [ p bl ] νxν,- the matrix of the conditional distribution functions[ H ( t)]of the sojourn times θ , b ≠ l.blνxνIf the sojourn times θ , b, l = 1,2,...,v,b ≠ l,haveWeibull distributions with parameters αbl,for b , l = 1,2,..., v,b ≠ l,blblβblH bl (t) = P( θbl< t) = 1 − exp[ −α t β blbl], t > 0,then their mean values are determined by−βbl1Mbl= E[θbl] = αbl(1+ ),β blb , l = 1,2,..., v,b ≠ l.1, i.e., if(1)The unconditional distribution functions of the processZ (t) sojourn times θ bat the operation states z b ,b = 1,2,...,v,are given by(t) H bvp bll=1= ∑b = 1,2,...,v,v[ 1 - exp[-αblbl t βt]],= 1 − ∑ p −blblexp[ α t ], t > 0,(2)l=1and, considering (1), their mean values areMb= E[ θb] = ∑ p bl M blvl = 11blβν −βbl1= ∑ pblαbl (1 + ) , b = 1,2,...,v, (3)l=1βand variances areblE[(θb)2]= ∞02∫tdH ( t)v= ∑ pl=1b∞2bl ∫ t0αblβblexp[ −αv −bl2= ∑ p bl α β bl (1 +l=β21 bl),βtblblLimit values of the transient probabilitiesp b (t) = P Z(t)= z ), t ≥ 0,b = 1,2,...,v,(bat the operation statesbzbare given by] tβbl−1dtb = 1,2,...,v.p = lim ( t)= π M / ∑πM , b = 1,2,...,v,(5)wherep bt→∞of the vectorbbvl=1Mbare given by (3) and the probabilities[ π ] satisfy the system of equations⎧[πb] = [ π⎪⎨ v⎪∑ πl= 1.⎩l=1bb 1xν][ pbl]3. Multi-state series- “m out of kn” systemIn the multi-state reliability analysis to define systemswith degrading components we assume that allcomponents and a system under consideration have thereliability state set {0,1,...,z}, z ≥ 1,the reliabilitystates are ordered, the state 0 is the worst and the statez is the best and the component and the systemreliability states degrade with time t without repair.The above assumptions mean that the states of thesystem with degrading components may be changed intime only from better to worse ones. The way in whichthe components and system states change is illustratedin Figure 1.ltransitions0 1 . . . u-1 u . . . z-1 z.lπ bDb22= D[ θb] = E[(θ ) ] − ( ) ,(4)bM bworst statebest statewhere, according to (2),Figure 1. Illustration of states changing in system withageing components- 187 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special IssueOne of basic multi-state reliability structures withcomponents degrading in time are series- “m out ofkn” systems.To define them, we additionally assume that E ij , i =1,2,...,k n , j = 1,2,...,l i , k n , l 1 , l 2 ,..., lk n, n ∈ N, arecomponents of a system, T ij (u), i = 1,2,...,k n , j =1,2,...,l i , k n , l 1 , l 2 ,..., lk n, n ∈N, are independent randomvariables representing the lifetimes of components E ijin the state subset { u , u + 1,..., z},while they were inthe state z at the moment t = 0, e ij (t) are components E ijstates at the moment t, t ∈< 0,∞),T(u) is a randomvariable representing the lifetime of a system in thereliability state subset {u,u+1,...,z} while it was in thereliability state z at the moment t = 0 and s(t) is thesystem reliability state at the moment t, t ∈< 0,∞).Definition 1. A vectorR ij (t , ⋅ ) = [R ij (t,0), R ij (t,1),..., R ij (t,z)], t ∈< 0,∞),whereR ij (t,u) = P(e ij (t) ≥ u | e ij (0) = z) = P(T ij (u) > t)for t ∈< 0,∞),u = 0,1,...,z, i = 1,2,...,k n , j = 1,2,...,l i , isthe probability that the component E ij is in thereliability state subset { u , u + 1,..., z}at the moment t,t ∈< 0,∞),while it was in the reliability state z at themoment t = 0, is called the multi-state reliabilityfunction of a component E ij .Definition 2. A vectorR (m)k nlnwhereR (m)nln(t , ⋅ ) = [1, R )k(t,0), R )k(t,1),..., R ) (t,z)],(m nln(m nln(m k nlnk(t,u) = P(s(t) ≥ u | s(0) = z) = P(T(u) > t)for t ∈< 0,∞), u = 0,1,...,z, is the probability that thesystem is in the reliability state subset { u , u + 1,..., z}atthe moment t, t ∈< 0,∞),while it was in the reliabilitystate z at the moment t = 0, is called the multi-statereliability function of a system.It is clear that from Definition 1 and Definition 2, foru = 0, we have R ij (t,0) = 1 and R )k(t,0) = 1.(m nlnDefinition 3. A multi-state system is called series- “mout of k ” if its lifetime T(u) in the state subsetn{ u , u + 1,..., z}is given byT u)= T ( ( u), u = 1,2,...,z,( k n −m+1)where T( k m 1) ( u)n − + is m-th maximal statistics in therandom variables setT ( u)= min { T ( u)}, i =1,2,...,k n , u =1,2,...,z.i1≤j≤liijDefinition 4. A multi-state series- “m out ofsystem is called regular if l 1 = l 2 = . . . =lk nkn”= l n , l n ∈N.Definition 5. A multi-state series- “m out of kn”system is called homogeneous if its componentlifetimes T (u) have an identical distribution function,i.e.ijF( t,u)= P(Tij ( u)≤ t),t ∈< 0,∞),u =1,2,...,z,i = ,2,..., k , j = ,2,..., l ,1ni.e. if its componentsfunction, i.e.1iE ij have the same reliabilityR(t,u) = 1 − F(t,u), t ∈< 0,∞),u =1,2,...,z.From the above definitions it follows that the reliabilityfunction of the homogeneous and regular series- “mout of k ” system is given by [3]( m)knlnn( m)knln( m)knln( m)knlnR ( t,⋅ ) = [1, R ( t,1),R ( t,2),..., R ( t,z)],(6)whereR( m)kn , ln= 1 −( t,u)m−1∑(kn)ii=0[ Rlni( t,u)][1 − Rfor t ∈< 0,∞),u = 1,2,...,z,or by( m )nln( m )knlnln( m )knln( t,u)]kn −i( m )knln(7)R k ( t,⋅ ) = [1, R ( t,1),R ( t,2),..., R ( t,z)],(8)where( m )n lnm(kn)lnkn−iR k , ( t,u)= ∑ [1 − R ( t,u)][ R ( t,u)](9)ii=0for t ∈< 0,∞),u = 1,2,...,z, m = k m,ilnn −- 188 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issuewhere knis the number of series subsystems in the “mout of kn” system and l nis the number of componentsof the series subsystems.(m)Under these definitions, if Rk n l n(t,u) = 1 for t ≤ 0,u = 1,2,...,z,or R (t,u) = 1 for t ≤ 0, u = 1,2,...,z,thenor(m )k n l n∞( m)M(u) = ∫ R ( t,u)dt,u = 1,2,..., z, (10)0knln∞( m)M(u) = ∫ R ( t,u)dt,u = 1,2,..., z, (11)0knlnis the mean lifetime of the multi-state nonhomogeneousregular series “m out of kn” system inthe reliability state subset { u , u + 1,..., z},and thevariance is given byD[ T ( u)]2∫tR ( t,u)dt − E [ T(u)],(12)or by= ∞ 0( m)knlnD[ T ( u)]2∫tR ( t,u)dt − E [ T(u)].(13)= ∞ 0( m )knlnThe mean lifetime M (u),u = 1,2,...,z,of this systemin the particular states can be determined from thefollowing relationshipsM ( u)= M ( u)− M ( u + 1), u = 1,2,...,z −1,M ( z)= M ( z).(14)Definition 6. A probabilityr(t) = P(s(t) < r | s(0) = z) = P(T(r) ≤ t), t ∈< 0,∞),that the system is in the subset of states worse than thecritical state r, r ∈{1,...,z} while it was in the reliabilitystate z at the moment t = 0 is called a risk function ofthe multi-state homogeneous regular series “m out ofk ” system.nConsidering Definition 6 and Definition 2, we haver(t) =1−(m)k n l n22R (t,r), t ∈< 0,∞),(15)and if τ is the moment when the system risk functionexceeds a permitted level δ, thenτ = r −1 ( δ ),(16)where r −1 ( t ) , if it exists, is the inverse function of therisk function r(t).4. Multi-state series- “m out of kn” system in itsoperation processWe assume that the changes of the process Z(t) stateshave an influence on the system components Ereliability and the system reliability structure as well.Thus, we denote the conditional reliability function ofthe system component E while the system is at theoperational state z b , b = 1,2,...,v,by[( i,j)( b)( i,j)( b)R ( t,⋅ )] = [1, [ ( t,1)] ,ij( i,j)( b)R ..., [ R ( t,z)]],where for t ∈< 0,∞),u = 1,2,...,z,b = 1,2,...,v,( i,j)( b)( b)ij[ R ( t,u)]= P(T ( u)> t Z(t)= zand the conditional reliability function of the systemwhile the system is at the operational state z b ,b = 1,2,...,v,by( m)knl n( b)( m)kn, l n( b)[ R,( t,⋅)]= [1, [ R ( t,1)], ..., [ Rfor t ∈< 0,∞),u = 1,2,...,z,b = 1,2,...,ν ,where according to (7), we have( m)knl n( b)( b)[ R,( t,u)]= P ( T ( u)> t Z(t)= zb)=−−1∑ (kn)1 m ii=0u = 1,2,...,z,b = 1,2,...,ν ,or by( m)knl n( b)[[ R(t,u)]]b)( b)lni( b)ln kn −i⋅ [1 − [[ R(t,u)]] ]( m)kn, l n( b)[ R,( t,⋅)]= [1, [ R ( t,1)], ..., [ Rfor t ∈< 0,∞),u = 1,2,...,z,b = 1,2,...,ν ,where according to (9), we have( m)knl n( b)( b)[ R,( t,u)]= P ( T ( u)> t Z(t)= zb)( m)knl n,( t,z)]ij( b)for t ∈< 0,∞),( m)knl n,( t,z)]( b)- 189 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issuem= ∑(kn)ii=0u = 1,2,...,z,b =1,2,...,ν .[1 − [[ R(t,u)]( b)ln kn −i⋅ [[[ R(t,u)]] ]( i,j)( b)]lnifor t ∈< 0,∞),( b)The reliability function [ R ( t,u)]is theconditional probability that the component Eijlifetime( )T bij ( u)in the reliability state subset { u , u + 1,..., z}isnot less than t, while the process Z(t) is at the operationstate z . Similarly, the reliability function( m)knl nb( b)[ R,( t,u)]or[ R( m)knl n,( t,u)]( b)is the conditional( )probability that the system lifetime T b ( u)in thereliability state subset { u , u + 1,..., z}is not less than t,while the process Z(t) is at the operation state z b . Inthe case when the system operation time θ is largeenough, the unconditional reliability function of thesystem( m)k n , l n⋅( m)k n , l nt( m)k n , l nzR ( t,) = [1, R ( ,1),..., R ( t,) ],whereor( m),ukn lnR ( t,) = P ( T ( u)> t)for u =1,2,...,z,( m)k n , l n⋅( m)k n , l nt( m)k n , l nzR ( t,) = [1, R ( ,1),,..., R ( t,) ],where( m)k n , l nuR ( t,) = P ( T ( u)> t)for u =1,2,...,z,and T (u)is the unconditional lifetime of the system inthe reliability state subset { u , u + 1,..., z},is given byorRR( m)kn, ln( m )knlnν( t,u)≅ ∑ pb=1νb=1b[ R( m)kn , ln( m )kn , ln( t,u)],( t,u)≅ ∑ pb[R ( t,u)]( b)( b),(17)(18)for t ≥ 0 and the mean values and variances of thesystem lifetimes in the reliability state subset{ u , u + 1,..., z}areνM ( u)≅ ∑ p M ( u)for u = 1,2,...,z,(19)whereorMMandorbbb=1bb( u)= ∞ ∫[R ] ( t,u)dt,(20)0( m)kn,ln( b)( u)= ∞ ∫[R ] ( t,u)dt,(21)( b)0( m )kn,ln( b)D[ T ( u)]2∫t[R ( t,u)]dt − E [ T ( u)],(22)( b)= ∞ 0( m)knln( b)2( b)D[ T ( u)]2∫t[R ( t,u)]dt − E [ T ( u)](23)= ∞ 0( m)knlnfor b = 1,2,...,ν , t ≥ 0,and pbare given by (4).The mean values of the system lifetimes in theparticular reliability states u , by (14), are( b)2( b)M ( u)= M ( u)− M ( u + 1) , u = 1,2,...,z −1,M ( z)= M ( z).(24)5. Large multi-state series- “m out of kn”system in its operation processDefinition 7. A reliability functionR ( t,⋅)= [1, R ( t,1),...,R ( t,z)],t ∈ ( −∞,∞),whereR ( t,u)= ∑ p Rvb=1b( b)( t,u),is called a limit reliability function of a multi-statehomogeneous regular series- “m out of kn” system inits operation process with reliability functionor(m)knln( m)( m)R (t, ⋅ ) = [1, R ( t,1),..., R ( t,z)],knlnknln- 190 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issue(m)knln( m )( m )R (t, ⋅ ) = [1, R ( t,1),..., R ( t,z)],knlnknln( m)( m)where R ( t,u),R ( t,u),u = 1,2,...,z,are givenknlnknlnby (17) and (18) if there exist normalising constants( )a b( )b bnb = 1,2,...,v,n( u)> 0, ( u)∈ ( −∞,∞),u = 1,2,...,z,such that for t ∈ C , u = 1,2,...,z,b = 1,2,...,v,orlim[ Rn→∞lim[ Rn→∞R( b)( u)( m)( b)( b)( b)( b),( an( u)t + bn( u),u)]= R ( t,ukn ln( m)( b)( b)( b)( b),( an( u)t + bn( u),u)]= R ( t,ukn lnHence, the following approximate formulae are validor( b)v( m)b ( b)t − bnRk n l( t,u)≅ ∑ p R (n( b)b=1 an, u),(25)u = 1,2,...,z,( b)v( m )b ( b)t − bnRk n l( t,u)≅ ∑ p R (n( b)b=1 an, u),(26)u =1,2,...,z.The following auxiliary theorem is proved in [7].Lemma 1. If(i) lim k = ∞ , m = constantn→∞n( m → 0 and kn→ ∞ ),kn( m)(ii) R ( t,u)= 1 − ∑ p[ Vv m−1( b)( b)b ∑ exp[ −V( t,u)]b=1 i=0i!is a non-degenerate reliability function,( t,u)]( m)( m)( m)Rk n , l=,,znkn lnkn ln(iii) ( t,⋅)[1, R ( t,1),...,R ( t,)],t ∈( −∞,∞),whereRv( m)m ( b)k ,( t)≅ ∑ pb[R,( t)]n lnkn lnb=1is the reliability function of a homogeneous regularmulti-state series- “m out of k ” system, whereni),).[ R=( m)knln,( t,u)](kn)−11 −m ∑ ii=0t∈(-∞,∞),( b)( b)lni( b)kn −i[ R ( t,u)][1 − [ R ( t,u)]] ,u = 1,2,..., z ,is its reliability function at the operational statethen( m)R ( t,⋅)= [1, Rt ∈( −∞,∞),( m)( t,1),...,R( m)( t,z)],lnzb,is the multi-state limit reliability function of thatsystem if and only if [7]n→∞n( b)( b)n( b)nlim k [ R ( a ( u)t + b ( u),u)]( )= V b ( t,u),t ∈ C ,(27)V( b)( u)u = 1,2,...,z,b =1,2,...,v.Proposition 1. If components of the multi-statehomogeneous, regular series- “m out of kn” system atthe operational state z b(i) have exponential reliability functions,( )R b ( t,u)= 1for t < 0,( b)( b)R ( t,u)= exp[ −λ ( u)t]for t ≥ 0,(28)u = 1,2,...,z,b =1,2,...,v,(ii) m = constant , kn= n, ln> 0,( b)1 ( b)1(iii) an( u)= , blog n,( )n=b( b)λ ( u)lnλ ( u)lnu = 1,2,...,z,b =1,2,...,v,then( m)( m)( m)R3( t,⋅ ) = [1, R3( t,1),...,R3( t,z)],(29)t ∈( −∞,∞),Wherev m−1( m)exp[ −it]R3( t,u)= 1−∑ pb∑ exp[ − exp( −t)](30)b=1 i=0i!for t ∈ ( −∞,∞), u = 1,2,...,z,is the multi-state limitreliability function of that system , i.e. for n largeenough we haveln- 191 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special IssueR(m)k nv m−1( b)t − bn( u), l n( t,u)≅ 1 − ∑ pb∑ exp[ − exp( − )]( b)b=1 i=0a ( u)t − bexp[ −i( ban⋅i!v≅ 1 − ∑ pb=1exp[ −iλ⋅m−1( b)n)( u)]( u)∑ exp[ − exp( −λbi=0( b)( u)li!for t ∈ ( −∞,∞), u = 1,2,...,z.nt − i log n]( b)n( u)lnt − log n)](31)mk−η = o(1n k n) ,( b)2(ii)~ 1 v −v( uxt upt ,( η )R ( , ) = 1 −)∑ b ∫ exp[ − ] dx ,2πb=1 −∞ 2is a non-degenerate reliability function, where( )v b ( t,u)is a non-increasing function( m)( m)( m)Rk n , l=,,znkn lnkn ln(iii) ( t,⋅)[1, R ( t,1),...,R ( t,)],t ∈( −∞,∞),wherev( m)( m)( b)k ,( t,u)≅ ∑ pb[R,( t,u)]n lnkn lnb=1R , t ∈( −∞,∞),is the reliability function of a homogeneous regularmulti-state series- “m out of k ” system, wherenProof. For n large enough we have( b)( b)t + log nan( u)t + bn( u)=( b)λ ( u)lnu = 1,2,...,z,b =1,2,...,v.≥ 0for t ∈ ( −∞,∞)[ R=( m)knln,( t,u)](kn)−11 −m ∑ ii=0t ∈(-∞,∞),( b)( b)lni( b)kn −i[ R ( t,u)][1 − [ R ( t,u)]] ,u = 1,2,..., z , b =1,2,...,v,lnTherefore, according to (28) for n large enough, weobtainR( b)( a( b)n= exp[ −λ( u)t + b( b)( u)(a( b)n( b)n( u),u)( u)t + b( b)n( u))]− t − log n= exp[ ] for t ∈ ( −∞,∞), u =1,2,...,z,l nb =1,2,...,v.Hence, considering (27), it appears that( b)( b)( b)( b)V ( t,u)]= lim klnn[R ( an( u)t + bn( u))]n→∞[= lim n exp[ ln→∞n− t − log n]ln= exp[ −t]for t ∈( −∞,∞),u = 1,2,...,z,b = 1,2,...,v,which means that according to Lemma 1 the limitreliability function of that system is given by (29)-(30).The next auxiliary theorem is proved in [7].Lemma 2. Ifm(i) →η, 0 < η < 1k nforn → ∞ ,is its reliability function at the operational statethen~ ( η)~ ( η)~ ( ηR ( t,⋅)= [1, R ( t,1),...,R) ( t,z)],t ∈ ( −∞,∞)zb,is the multi-state limit reliability function of thatsystem if and only if [7]kn+ 1[ R ( an( u)t + blimn→∞η 1( b)l n( b)( b)( b)n( u),u)]( −η)−η]= v ( t,u)for t ∈C, u = 1,2,...,z,(32)b =1,2,...,v.v( b)( u)Proposition 2. If components of the multi-statehomogeneous, regular series- “m out of kn” system atthe operational state z b(i) have exponential reliability functions,( )R b ( t,u)= 1for t < 0,( b)( b)R ( t,u)= exp[ −λ ( u)t]for t ≥ 0,(33)u = 1,2,...,v,b = 1,2,...,v,(ii)m→ η , 0 < η < 1 for n → ∞ , k n, l > 0,k n(iii) a( b)( b)n( u), bbn( u)=( )( b)λ ( u)lnηnλn=n1−η− logη= ,( u)ln- 192 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issuethenu = 1,2,...,z,b = 1,2,...,v,( b)( b)( b)lk + 1[[ ( ( ) + ( ), )]nnR anu t bnu u= limn→∞η( 1 −η)~ ( η)~ ( η)~ ( ηR ( , ) [1, ( ,1),...,)7t ⋅ = R7t R7( t,z)],(34)1−ηlogηt ∈( −∞,∞),n + 1(exp[ −ln( t − )] −η)lnηnln= limn→∞η( 1−η)tx2~v −( η)1R t u = − ∑ pb∫ e27 ( , ) 1dx (35)1−η2πb=1n + 1(exp[ − t + logη]−η)−∞ηnfor t ∈( −∞,∞),u = 1,2,...,z,= limn→∞η( 1−η)1−ηn + 1( η(exp[− t]−1))( b)t−bn( u)ηn( b)an u x2= limv( ) −n→∞(m)1η( 1 −η)R,( t,u)≅ 1 − ∑ p e2dxkn lnb ∫2πb=1 −∞1−η1−ηn + 1( η(1− t + o(t)−1))1≅ 1 −ηnηn= lim2πn→∞η( 1−η)ηn( λ( b)( u)lnt+log nx2v1−η−η(1−η)1−η⋅ ∑ pb∫ e2dx (36)n + 1( − t + η ⋅ o(t))b=1−∞nηn= limt ∈ ( −∞,∞ u = 1,2,...,z.n→∞η( 1−η)= − t for t ∈ ( −∞,∞), b =1,2,...,v,( b)( b)1 1 −ηan( u)t + bn( u)= ( t − logη)> 0( b)λ ( u)lnηnfor t ∈ ( −∞,∞),u = 1,2,...,z,b =1,2,...,v,The next auxiliary theorem is proved in [7].( b)( b)( b)R ( an( u)t + bn( u),u),( b)( b)( b)= exp[ −λ( u)(an( u)t + bn( u))]Lemma 3. If1 1 −η= exp[ − ( t − logη)]for t ∈ ( −∞,∞),ml n ηn(i) kn→ ∞ , →1, kk n− m)→ mnu = 1,2,...,z,b =1,2,...,v.for n → ∞ ,( b)v m( m)( b)[ V ( t(ii) R ( t,u)= ∑ pb∑ exp[ −V( t,u)]b=1 i=0i!( )v b ( t,u)is a non-degenerate reliability function,( m )( m )( m )(iii) Rk n , l( t,⋅)= [1, R,( t,1),...,R,( t,z)],nkn lnkn lnwhereis the multi-state limit reliability function of thatsystem , i.e. for n large enough we havefor ),Proof. Since, for sufficiently large n, we havethen according to (33) for sufficiently large n, weobtainHence, considering (32), it appears that−η]which means that according Lemma 2 the limitreliability function of that system is given by (34)-(35).( = constanti, u)]- 193 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issuet ∈ ( −∞,∞),where( m )R kn , ln ( t,uv) ≅ ∑ p [ Rb=1b( m )kn , ln( t,u)]( b), t ∈ ( −∞,∞),is the multi-state limit reliability function of thatsystem , i.e. for n large enough we haveis the reliability function of a homogeneous regularmulti-state series- “m out of k ” system, where( m )n ln[ R k , ( t)]kn= − m∑i=0t∈(-∞,∞),( b)(kn( b))ln i ( b)ln ( kn −i)[1 − [ R ( t)]] [ R ( t)],iu = 1,2,..., z , b =1,2,...,v,is its reliability function at the operational statethen( m )R ( t,⋅)= [1, Rt ∈ ( −∞,∞),( m )n( t,1),...,R( m )( t,z)],zb,is the multi-state limit reliability function of thatsystem if and only if [7]lim k ln→∞n nF( b)( a( b)n( u)t + b( b)n( u),u)( b)= V ( t,u)for t ∈CV( b)(37)( u)u = 1,2,...,z,b =1,2,...,v.Proposition 3. If components of the multi-statehomogeneous, regular series- “m out of k ” system atz bthe operational state(i) have exponential reliability functions,( )R b ( t,u)= 1for t < 0,( b)( b)R ( t,u)= exp[ −λ ( u)t]for t ≥ 0,(38)u = 1,2,...,z,b = 1,2,...,v,(ii) k n → ∞ , lim − m = m = constant ,k nn→∞( b)1( b)(iii) an( u)= , b ( u)= 0( )n,bλ ( u)lnknu = 1,2,...,z,b =1,2,...,v,then( m )( m )( m )R2( t,⋅ ) = [1, R2( t,1),...,R2( t,z)],(39)t ∈ ( −∞,∞),where⎧ 1,t < 0,( m ) ⎪2 ( t,u)= ⎨miR vt(40)⎪∑pb∑ exp[ −t], t ≥ 0,⎩b=1 i=0 i!n(m)R kn l n,( t,u)⎧1,t < 0,⎪v m( b)⎪t − bn( u)∑ pb∑ exp[ − ]⎪( b)b= 1 i=0 an( u)⎪≅ ⎨⎪( b)⎪ t − bn( u)i[ ]⎪ ( b)an⎪⋅, t ≥ 0,⎩ i!⎧1,t < 0,⎪ v m( b)⎪∑p ∑⎪≅ ⎨= b exp[ tλ( u)l= nknb 1 i 0⎪⎪ ( b)i[ tλ( u)lnkn]⎪⋅, t ≥ 0.⎩ i!Proof. Since( b)( b)tan( u)t + bn( u)=( b)λ ( u)lnku = 1,2,...,z,b = 1,2,...,v,and( b)( b)tan( u)t + bn( u)=( b)λ ( u)lnku = 1,2,...,z,b = 1,2,...,v,nn< 0≥ 0therefore, according to (38), we obtain( b)( b)( b)F ( an( u)t + bn( u),u)= 0u = 1,2,...,z,b = 1,2,...,v,andF( b)( a( b)n( u)t + b( b)n( u),u)for t < 0,for t < 0,for t ≥ 0,t= 1 − exp[ − ] for t ≥ 0,u = 1,2,...,z,k n l nb = 1,2,...,v.Hence, considering (37), it appears that( )V b( t,u)(41)- 194 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issue( b)( b)( b)= lim k l F ( a ( u)t + b ( u),u)= 0n→∞n nu = 1,2,...,z,b = 1,2,...,v,andnnfor t < 0,( )V b ( b)( b)( b)( t,u)= lim k l F ( a ( u)t + b ( u),u)n→∞n n= lim k l ( 1−exp[ − ])n→∞n nntk ln n= lim k l ( 1−1+− o())n→∞n ntk ln nntk ln n= t for t ≥ 0,u = 1,2,...,z,b = 1,2,...,v,which means that according Lemma 3 the limitreliability function of that system is given by (39)-(40).The next auxiliary theorem is proved in [7].Lemma 4. If(i) lim k = k,k > 0, 0 < m ≤ k,lim l = ∞ ,n→∞nvnn→∞(ii) R ( t,u)= ∑ p R ( t,u)is a non-degenerateb=1b( b)reliability function,( m)( m)( m)(iii) ( t,⋅)[1, R ( t,1),...,R ( t,)],RRk n , l=,,znkn lnkn lnt ∈ ( −∞,∞),where( m)knlnv,( t)≅ ∑ pb[R ( t)]b=1( m)knln( b)is the reliability function of a homogeneous regularmulti-state series- “m out of k ” system, where[ R=( m)knln,( t,u)](kn)−11 −m ∑ ii=0( b)[ R( b)( t,u)]lnin[1 − [ R( b)t∈(-∞,∞), u = 1,2,...,z,b = 1,2,...,v,( t,u)]ln]kn −iis its reliability function at the operational statethenR ( t,⋅)= [1, R ( t,1),...,R ( t,z)],t ∈ ( −∞,∞),zb,lim[ Rn→∞( b)( a( b)n( u)t + b( b)n( u),u)]ln= R( b)0for t ∈ C , u = 1,2,...,z,b =1,2,...,v,( b)R ( u)0( b)( t,u)(42)where R0( t,u), u = 1,2,...,z,is a non-degeneratereliability function andR ( t,u)v m−1⎛k ⎞ ( b)i ( b)k −i= 1 − ∑ pb∑ ⎜ ⎟[R0( t,u)][1 − R0( t,u)](43)b=1 i=0⎝i ⎠for t ∈(-∞,∞), u = 1,2,...,z.Proposition 4. If components of the multi-statehomogeneous, regular series- “m out of kn” system atthe operational state z b(i) have exponential reliability functions,( )R b ( t,u)= 1for t < 0,( b)( b)R ( t,u)= exp[ −λ ( u)t]for t ≥ 0,(44)u = 1,2,...,z,b =1,2,...,v,(ii) k n → k, k > 0 , l n → ∞, m = const ,( b)1 ( b)(iii) an( u)= , b ( u)= 0( )n,bλ ( u)lnu = 1,2,...,z,b = 1,2,...,v,then( m)( m)( m)R9( t,⋅ ) = [1, R9( t,1),...,R9( t,z)],(45)t ∈(-∞,∞),where( m)R9( t,u)⎧1,⎪v m−1⎪ ⎛k⎞1 − ∑ pb∑ ⎜ ⎟ [exp[ −t]]≅ ⎨ b=1 i=0⎝i ⎠⎪⎪⎪k −i⎩⋅[1 − exp[ −t]],it < 0,t ≥ 0,(46)is the multi-state limit reliability function of thatsystem , i.e. for n large enough we haveR( m)9( t,u)is the multi-state limit reliability function of thatsystem if and only if [7]- 195 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issue⎧1,t < 0,⎪v m−1( b)⎪ ⎛k⎞ t − bn( u)i1 − ∑ pb∑ −⎪⎜⎟[exp[]]( b)b=1 i=0⎝i ⎠ an( u)≅ ⎨⎪⎪( b)t − bn( u)k −i⎪⋅[1 − exp[ − ]] , t ≥ 0,( b)⎪⎩an( u)⎧1,⎪v m−1⎪ ⎛k⎞( b)1 − ∑ pb∑ −⎪⎜⎟[exp[tλ( u)ln]]b=1 i=0⎝i ⎠⎪≅ ⎨⎪( b)k −i⎪⋅[1 − exp[ −tλ( u)ln]] ,⎪⎪⎩Proof . Since( b)( b)tan( u)t + bn( u)=( b)λ ( u)lu = 1,2,...,z,b = 1,2,...,v,and( b)( b)tan( u)t + bn( u)=( b)λ ( u)lu = 1,2,...,z,b = 1,2,...,v,nn< 0≥ 0therefore, according to (44), we obtainit < 0,t ≥ 0.for t < 0,for t ≥ 0,(47)n→∞( b)( b)n( b)n= lim[ R ( a ( u)t + b ( u),u]= limn→∞t [exp[ − ]]lnln= exp[ − t]for t ≥ 0 , u = 1,2,...,z,b = 1,2,...,v.which, by Lemma 4, completes the proof.6. ConclusionThe purpose of this paper is to give the method ofreliability analysis of selected multi-state systems invariable operation conditions. As an example a multistateseries-“m out of k” systems are analyzed. Theirexact and limit reliability functions, in constant and invarying operation conditions, are determined. Thepaper proposes an approach to the solution ofpractically very important problem of linking thesystems’ reliability and their operation processes. Toinvolve the interactions between the systems’ operationprocesses and their varying in time reliabilitystructures a semi-markov model of the systems’operation processes and the multi-state systemreliability functions are applied. This approach givespractically important in everyday usage tool forreliability evaluation of the large systems withchanging their reliability structures and componentsreliability characteristic during their operationprocesses. The results can be applied to the reliabilityevaluation of real technical systems.ln( b)( b)( b)l[ R ( a ( ) + ( ), )]nnu t bnu u = 1 for t < 0 ,u = 1,2,...,z,b = 1,2,...,v,and( b)( b)( b)t[ R ( an( u)t + bn( u),u)]= exp[ − ] for t ≥ 0 ,lnu = 1,2,...,z,b = 1,2,...,v.Hence, according (42)-(43), it appears that( b)( b)( b)( b)lnR 0 ( t , u)= lim[ R ( an( u)t + bn( u),u]= 1n→∞for t < 0 , u = 1,2,...,z,b = 1,2,...,v,and( b)R0( t,u)References[1] Grabski, F. (2002). Semi-Markov Models of SystemsReliability and Operations. Systems ResearchInstitute, Polish Academy of Sciences, Warsaw.[2] Hudson, J. & Kapur, K. (1985). Reliability boundsfor multi-state systems with multi-statecomponents. Operations Research 33, 735- 744.[3] Kolowrocki, K. (2004). Reliability of LargeSystems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney –Tokyo.[4] Kolowrocki, K. & Soszynska, J. (2005). Reliabilityand Availability Analysis of Complex PortTrnsportation Systems. Quality and ReliabilityEngineering International 21, 1-21.[5] Lisnianski, A. & Levitin, G. (2003). Multi-stateSystem Reliability. Assessment, Optimisation andApplications. World Scientific Publishing Co., NewJersey, London, Singapore , Hong Kong.- 196 -

J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issue[6] Meng, F. (1993). Component- relevancy andcharacterisation in multi-state systems. IEEETransactions on reliability 42, 478-483.[7] Soszyska, J. (2002). Asymptotic approach toreliability evaluation of non-renewal multi-statesystems in variable operation conditions. Chapter20 (in Polish). Gdynia Maritime University. Projectfounded by the Polish Committee for ScientificResearch, Gdynia.[8] Xue, J. & Yang, K. (1995). Dynamic reliabilityanalysis of coherent multi-state systems. IEEETransactions on Reliability 4, 44, 683–688.- 197 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special IssueVali DavidUniversity of Defence, Brno, Czech RepublicReliability of complex system with one shot itemsKeywordsreliability, complex system, one shot itemAbstractThis article deals with modelling and analysis of the reliability of complex systems that use one-shot items duringtheir operation. It includes an analysis of the impact of the reliability of used one-shot items on the resultingreliability of the system as a whole. Practical application of theoretical knowledge is demonstrated on an exampleof a model of reliability of an aircraft gun that was used for optimization of the gun’s design during itsdevelopment and design. The analysed gun uses two types of one-shot items – rounds intended for conducting offire and special pyrotechnic cartridges designed for re-charging a gun after a possible failure of the round.1. IntroductionThis contribution is supposed to contribute to asolution of dependability qualities of the complex (inthis case) weapon system as an observed object. Iwould like to show one of the ways how to specify avalue of single dependability measures of a set. Theaim of our paper is to verify the suggested solution inrelation to some functional elements which influencefulfillment of a required function in a very significantmanner. [1], [3]A weapon set is a complex mechatronics system whichis designed and constructed for military purposes. Weare talking about a barrel shooting gun – a fastshooting two-barrel cannon. It is going to beimplemented in military air force in particular.Generally speaking the set consists of mechanicalparts, electric, power and manipulation parts,electronic parts and ammunition. For the purpose ofuse in our paper we are going to deal with isolatedfunctional blocks and ammunition only. In this case weview the ammunition as recommended standardisedrounds and pyrotechnic cartridges.Single parts of the set can be described with qualitativeand most importantly quantitative indices whichpresent their quality. In my paper I am dealingespecially with quality in terms of dependabilitycharacteristics. We are working first and foremost withprobability values which characterize single indices,and which describe functional range and requiredfunctional abilities of the set. We focus on the parthandling rounds and pyrotechnic cartridges which arecrucial for this case. In order to continue our work it isnecessary to define all terms and specify everyfunction.2. Essential terms and definitionsWe are always talking about an object in terms ofreliability analyses. The definition for object is thesame as the used in IEC 60500 (191/50). Consequentlywe need to describe the basic object’s measures [2].Object’s function:The main function: The main function of the object isputting into effect a fire from a gun using standardammunition.The step function: Manipulation with ammunition, itscharging, initiation, detection and indication ofammunition failure during initiation, initiation ofbackup system used for re-charging of a failedcartridge.It is expected that the object will be able to work underdifferent operating conditions especially in differenttemperature spectra, under the influence of variedstatic, kinetic and dynamic effects, in various zones ofatmospheric and weather conditions.In this case we will not take into account any of theoperating conditions mentioned above. However, theirinfluence might be important while consideringsuccessful mission completion.One of the main terms we are going to develop is:- 198 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special IssueMission: It is an ability to complete a regarded missionby an object in specified time, under given conditionsand in a required quality.In our contribution it is a case of cannon ability to putinto effect a fire in a required amount – in a number ofshot ammunition at a target in required time, and undergiven operating and environmental conditions.As it follows from the definition of a mission it is acase of a set of various conditions which have to befulfilled all at once in a way to satisfy us completely.Our object is supposed to be able to shoot a requiredamount of ammunition which has to hit the target withrequired accuracy (probability). We will not take intoconsideration circumstances relating to evaluation ofshooting results, weapon aiming, internal and externalballistics, weather conditions and others. We will focusonly on an ability of an object to shoot. [4]As we have stated above we will deal with isolatedfunction blocks only. We are presuming that theseblocks act according to required and determinedboundary conditions. In order to understand functionallinks fully we introduce our way of dividing an object.We are talking about the following block:- manipulation with ammunition, its charging,initiation, failure detection and indicationduring initiation, initiation of a backup systemin order to recharge a failed cartridge, allmechanical parts, all electric and electronicparts, interface elements with a carrying device- Block A;- ammunition – Block B;- pyrotechnic cartridges – Block C.3. Description of a processThe process as a whole can be described this way:From a mathematical and technical point of view it is afulfilling of requirements´ quee which gradually comesinto the service place of a chamber. The requirements´quee is a countable rounds´ chain where the roundswait for their turn and are transported from the linewhere they wait in to a service place (fulfilment of arequirement) of a chamber and there they are initiated.After the initiation the requirement is fulfilled. Anempty shell (one of the essential parts of a round)leaves a chamber taking a different way than acomplete round. When the requirement is fulfilled,another system which is an integral part of a set detectsprocess of fulfilling the requirement. The process isdetected and indicated on the basis of interconnectedreaction processes. In this case fulfilling therequirement is understood as a movement of a barrelbreech going backwards. Both fulfilling therequirement and its detection are functionallyconnected with transport of another round waiting in aline to go into a chamber.Let’s presume that rounds are placed in an ammunitionfeed belt of an exactly defined length. A maximumnumber of rounds which could be placed in a belt islimited by the length then. The length is given eitherby construction limitations or by tactical and technicalrequirements for a weapon set. Let’s presume thatdespite different lengths of an ammunition belt, thiswill be always filled with rounds from the beginning tothe end. Let’s also assume that the rounds are not nonstandardand are designed for the set.The process of fulfilling the requirement is monitoredall the time by another system which is able todifferentiate if it is fulfilled or not. The fulfilment itselfmeans that a round is transported into a chamber, it isinitiated, shot, and finally an empty shell leaves achamber according to a required principle. If theprocess is completed in a required sequence, thesystem detects it as a right one.Because of unreliability of rounds the whole system isdesigned in the way to be able to detect situations inwhich the requirement is not fulfilled in a demandedsequence and that is why it is detected as faulty.Although a round is transported into a chamber and isinitiated, it is not fired. A function which is essentialfor a round to leave a chamber is not provided either,and therefore another round waiting in line cannot betransported into a chamber. That is the reason whyfulfilling of the requirement is not detected.The system is designed and constructed in such a waythat it is able to detect an event like this and takesappropriate countermeasures. A redundant systemwhich has been partly described above is initiated.After a round is initiated and the other steps don’t carryout (non-fire, non-movement of a barrel breechbackwards, non-detection of fulfilling the requirement,non-leaving of a chamber by an empty shell, and nontransportof another round into a chamber) a system ofpyrotechnic cartridges is initiated. It is functionallyconnected with all the system providing missioncompletion. A pyrotechnic cartridge is initiated andowing to this a failed round is supposed to leave achamber. A failed functional link is established andanother round waiting in line is transported into achamber.In order to restore the main function we use a certainnumber of backup pyrotechnic cartridges. Our task isto find out a minimum number which is essential forcompleting the mission successfully.4. Mathematical modelTo meet the needs of our requirements we are going touse a mathematical way which helps us to expresssuccessful completing the mission. We know that thenumber of rounds n in an ammunition belt is final. Wealso know that an event-failure of a round B(ammunition block – B) can occur with a probabilityp n . All the requirements and specifications mentionedabove will be used in further steps.- 199 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special IssueBecause it is about a stream of rounds of a number nwhich wait in line to meet the requirement, and each ofthem has a potential quality p n, a number of failedrounds has a binomial distribution (Bi) of a an eventoccurrence. The distribution is specified by theparameters n and p n : Bi(n,p n ). A number of occurrencesX n of an event B follows the distribution inBernoulli’s row n of independent experiments, andprobability of event occurrence P( B ) = p n . A numberp n is the same in every experiment. [5]; [6]Because there is an occurrence of a number of eventsin an observed file we are talking about a countingdistribution of an observed random variable. A randomvariable is in this case a number of failed rounds. Aprobability function of a binomial distribution can beput that way:⎛n⎞x n−xP( X n = x)=⎜ pn( − pn)x⎟ 1 ; x∈{0,1,2,...,n}. (1)⎝ ⎠Qualities of binomial distribution like a mean valueE(X n ) and dispersion D(X n ) are obtained by calculatingthe formula:E(X n ) = n . p n , (2)D(X n ) = n . p n . (1- p n ). (3)A number of failed rounds follows a binomialdistribution with parameters n – a number of roundsand p n – failure occurrence probability of a round.In order to specify a mean number of possible failuresin an ammunition belt of a given length (there is acertain amount of rounds) we quantify the formula (2)and replace n by a real number of rounds in anammunition belt.On the basis of construction, technical and technicalrequirements we can have ammunition belts ofdifferent length at a given moment, and consequentlywe have a different number of rounds. Only amaximum number of rounds in an ammunition belt isconsidered in another calculation. The ammunition beltis supposed to be of a maximum length which is ableto fit a loading deviceIn case a round fails initiation of a backup system forfunction restoration occurs according to a mechanismdescribed above. It is a case of successive initiation ofpyrotechnic cartridges (in a system of pyrotechniccartridges) which are supposed to guarantee restoringof a required broken chain of function. A number ofpyrotechnic cartridges in a backup system is m.Pyrotechnic cartridges have also a probability p m of afailure occurrence which unable their initiation.Pyrotechnic cartridges too are placed in line waitingfor meeting the requirement which results from theirfunction. In case of a failure of the first pyrotechniccartridge the next one is initiated up to the momentwhen either a function is restored or all pyrotechniccartridges are used up.On the basis of the facts mentioned above it is obviousthat the process of fulfilling the requirements followsgeometrical distribution (Ge). It means that the processof fulfilling the requirements repeats so often until itmeets them in terms of reversion of all the process toan operational state. It is a case of an observed discreetrandom variable. Pyrotechnic cartridges also havefailure rate p m (failure probability) and there is alimited number of them. It means that a failure canoccur up to m-times. A geometrical distribution Ge(p m )generally follows this outline.We are going to assess the succession of independentattempts, and probability of an observed eventoccurrence equals the same number p m in each attempt.The quantity X m is a serial number of the first successwhich means that a required event occurs. The eventhere means a function of a block C, and a probabilityp m means an event occurrence C . Characteristics ofthe process are as follow. A probability function:P(X m =x) = p m x-1 (1-p m );x∈{1,2,3,…,m}. (4)It is a special case of a geometrical distribution when aprobability of an event occurrence (a pyrotechniccartridge failure) does not depend on a number ofprevious unsuccessful attempts of a value 0.Characteristics of a geometrical distributions, forexample mean value E(X m ) (a mean number ofpyrotechnic cartridges necessary for removing onefailed round) and dispersion D(X m ) are obtained by acalculation of a formula:E(Xm)( X = x)= ∑ ∞ x.Prm =x= 011− pm. (5)While completing the mission during either training ora real deployment a few scenarios can occur, and thecourse of them depends on single functional blocks. Tocomplete the mission M successfully single blocks areexpected to be failure free as stated above. Thefunction of the blocks mentioned above are designatedas A, B, C, the opposite is A ; B;C . The relation canbe expressed by using events this way:M = A ∩(B∪ C). (6)Using probability expression we talk about probabilityof mission completion M. We can put it that way:P(M) = P(A) . [P(B) + P(C) - P(B ∩C)]; (7)- 200 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special Issue5. Description of scenariosDescription of the scenarios which can occur duringcompleting or defaulting the mission relate only to anammunition block and to a redundant mechatronicssystem with pyrotechnic cartridges.The mission is completed. In the first case there can bea situation when all the ammunition of a certainamount which is placed in an ammunition belt is usedup and a round failure occurs or it is used up and around failure does not occur. In this case a backupsystem of pyrotechnic cartridges is able to reverse asystem into an operational state. Using up can besingle, successive in small bursts with breaks betweendifferent bursts, or it might be mass using one burst.Shooting is failure free or there is a round failureoccurrence n. In case a round failure occurs, a systemwhich restores a function of pyrotechnic cartridges isinitiated. There are two scenarios too – a systemrestoring a pyrotechnic cartridges function is failurefree, or a pyrotechnic cartridge fails. If a function ofpyrotechnic cartridges is applied, it can remove afailure m-times. So a number of restorations of thefunction is the same as the number of availablepyrotechnic cartridges. In order to complete themission successfully we need a higher amount ofpyrotechnic cartridges m, or in the worst case thenumber of pyrotechnic cartridges should be equal to anumber of failures. Another alternative is the situationthat a round fails and in this case a pyrotechniccartridge fails too. A different pyrotechnic cartridge isinitiated and it restores the function. This must satisfythe requirements that an amount of all round failures nis lower or at least equal to a number of operational(undamaged) pyrotechnic cartridges m. The mission iscompleted in all the cases mentioned above and whenfollowing a required level of readiness of a block A.The mission is not completed. In the second case theshooting is carried out one at a time, in small bursts orin one burst, and during the shooting there will be nround failures. At the time the failure occurs a backupsystem for restoring the function will be initiated.Unlike the previous situation there will be mpyrotechnic cartridges´ failures and a total number ofpyrotechnic cartridges´ failures equals at least anumber of round failures, and is equal to a number ofimplemented pyrotechnic cartridges M at the most. Itmight happen in this case that restoring of the functiondoes not take place and the mission is not completed atthe same time because there are not enoughimplemented pyrotechnic cartridges.The relation of transition among the states can beexpressed by the theory of Markov chains.Figure 1. Description of transitions among the statesCharacteristics of the states:0 state: An initial state of an object until a roundfailure occurs with a probability function of around P(B). It is also a state an object can get with apyrotechnic cartridge probability P(C) in case a roundfailure occursorP( B) − P( B)P(C| B ) == 1 ,P(C ∩ B).P(B)m 1 …m m state: A state an object can get whilecompleting the mission. Either a round failure occursP B = 1 − P B , or there is a pyrotechnicin probability ( ) ( )cartridge failure in probability P( C) − P( C)= 1 .1 state: A state an object can get while completing themission. It is so called an absorption state. Transitionto the state is described as probability P( C) = 1−P( C)of a failure of last pyrotechnic cartridge as long as anobject was in a state „k n “ before this state, or it can bedescribed as probability of a round failure occurrenceP( B) = 1 − P( B)as long as an object was in a state 0before this state and all pyrotechnic cartridges areeliminated from the possibility to be used.Transitions among different states as well as absoluteprobability might be put in the following formulae:( 0) P( B) + P( C ) + P( C ) + ... P( C )P = + , (8)PP(B)1. An alternative of a function when the missionis completed.1 - P(B)1 - P(B)1 - P(C)0 m 1 m 2m 1mP(C)P(C)( m ) = 1−P( B)1 - P(B)P(C)1 - P(B)1 - P(C) 1 - P(C)…k10 k20k1n01, (9)( m ) = ( − P( B)) + ( 1 P( C))2. An alternativeof a functionwhen the missionis not completed.P m1 − , (10)1P ( 1 ) = 1 . (11)- 201 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special IssueWe suggest the subsequent steps for all the scenariosmentioned above. Following the mathematical formula(1) it is possible to find out probability of a number ofround failures´ occurrences in an ammunition belt of alength n. Following the equation (2) we can specify anexpected mean value of a mean number of roundfailures in an ammunition belt of a given length.The mean value result is recommended to be used for amaximum length of an ammunition belt (a maximumnumber of rounds) which could be implemented into aweapon set concerning construction as well as tacticaland technical views. The result informs us of aminimum number of pyrotechnic cartridges which areto be applied for a successful completing the mission.In this case there is a threat of a pyrotechnic cartridgefailure which could cause a system failure (as far as anumber of round failures is higher than a number ofavailable pyrotechnic cartridges). In this case wewould not complete the mission.In order to assess dependability of a shooting functionit is necessary to know a number of pyrotechniccartridges and, depending on this, probability ofcompleting the mission. To fulfil the requirements Isuggest three steps:1) To determine a required number ofpyrotechnic cartridges;2) To quantify generally probabilities ofcompleting the mission;3) To quantify exactly probabilities of completingthe missionFollowing the steps mentioned above we suggest thismethod.Ad 1) To determine a required number of pyrotechniccartridgesWhen we calculate a mean number of failed roundsE(X n ) which is determined from a maximum number ofrounds n in a ammunition belt (see above) andprobability of a round failure occurrence p n , see theformula (2), we get a minimum recommended numberof pyrotechnic cartridges which are supposed toguarantee completing the mission in case a round fails.The calculation would be successful in case apyrotechnic cartridge failure does not occur. However,even a system of pyrotechnic cartridges concerning afailure occurrence depends on counting distribution ofa discreet random variable which is specified in ourcase by a geometrical distribution. (Because the systemis activated so long until the observed and requiredevent occurs – in terms of repairing the failure.) Wesuggest calculating a mean number of pyrotechniccartridges´ failures following the formula (5). For thecalculation we will need only pyrotechnic cartridgefailure probability p m . On the basis of this calculationwe get an average number of pyrotechnic cartridgesrequired to repair a failure of one round.In order to complete the mission a number of available(operational) pyrotechnic cartridges should be at leastthe same as a number of failed rounds. When wemultiply the mean values we obtain a total number ofpyrotechnic cartridges M which will guaranteecompleting the mission (even in the situation whenbesides failed rounds there are failed pyrotechniccartridges too)n.pnM = E(X n ) . E(X m )=1− pm. (12)Logically a number of pyrotechnic cartridges whichare essential for completing the mission successfully iscontinually proportioned to a number of rounds n andto probability of their failure p n , and inverselyproportioned to probability of pyrotechnic cartridge“success” 1-p m . The Figure 2 shows a typical course ofdependability M (p n ;p m ), it means a invariant M whichdepends on variables p n a p m . This way might be thefirst of the alternatives how to solve the problem. Itsuggests a total number of pyrotechnic cartridgeswhich are essential for completing the mission but itdoes not show the way how to quantify probability ofmission completion.While recording distribution parameters we are goingto use an equivalent m standing for a value M.600400M20000 0.2 0.40.6p n 0.81 00.210.80.60.4p mFigure 2. Course of dependability of a number ofpyrotechnic cartridges M on variables p n and p mAd 2) To quantify generally probability of completingthe missionIn this case we follow the solution which has beenstated in the part Ad 1. We take into account that thereis a number of pyrotechnic cartridges required forcompleting the mission. So, we determine an α fractilewhich provides an upper limit of a number of roundswhich fail in probability α. After we specify β fractile- 202 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special Issuewhich provides an upper limit of pyrotechniccartridges which fail in probability β.While working with fractiles we follow the generalinformation. 100% fractile of a random variable X is anumber x p , and a probability p where 0

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special Issueround failure occurrence p n , a number of pyrotechniccartridges m, and probability of a pyrotechnic cartridgefailure occurrence p m . A general function of missioncompletion probability and its variables is put thatway:p mis (n,p n ,m,p m ). (22)Further steps follow well known assumptions. Thefunction of a rounds´ failure takes form of a binomialdistribution with parameters n and p n – Bi(n,p n ), andthe rounds which may fail can be marked with k wherek ∈ {0;1;2;…..;n}. Moreover, we introduce functionsof a pyrotechnic cartridges´ failure Y k where k∈{0;1;2;…..;m}. They show us possibility of apyrotechnic cartridge failure while shooting as soon asit is necessary to remove a failed round. Let us assumethat a sum of functions of a pyrotechnic cartridges´failure will be lower than a number of availablepyrotechnic cartridges used for removing a failedround. We put it in the following formulam∑k = 0Y k ≤ m ≈ Y0 + Y1+ ...... Yk ≤ m . (23)Following the assumption mentioned above weconsider the case that the first available pyrotechniccartridge follows geometrical distribution of a functionof its activity Ge(p m ) during the failure of the k-thround Y k . The function p m means probability ofpyrotechnic cartridge failure occurrence. It can bedescribed asY k ~ Ge(p m ). (24)The equation showing probability of completing themission is put that wayP( n,p , m,p )n= ∑ Pk = 0nm( X = k ).P( Y + Y + + Y ≤ m)0 1 ....k(25)where in case k=0 (it reflects a situation where there isno round failure) a function would be specifiedadditionally provided that P(Y 1 +….Y k ≤ m)=1. And inorder to solve a probability value of completing themission we would use so called completing theformula taking advantage of forming functions. From amathematical point of view this is much moredemanding but it offers a very exact value expressingprobability of completing the mission p mis while usinga variation of function factors. On its basis it is easy toprove a dependability of a total number of usedpyrotechnic cartridges on a level of missioncompletion probability p mis .An example of a possible solution:Given:p n = 0,000 1 - round failure probability;n = 200 - maximum rounds´ number during oneprocess;p m = 0,01 - pyrotechnic cartridge failure probability;p mis = 0,99 - probability of mission success.Solution according to “Ad 1)”: We are looking for asufficient number of pyrotechnic cartridges used forremoving a possible failurenα . 200.0,0001M = =≅ 0,02 .1−β 1−0,01The formula shows us that having at least onepyrotechnic cartridge is enough to complete themission successfully. However, we cannot quantifyprobability for completing the mission.Solution according to “Ad 2)”: We are looking for alevel of mission completion probability p mis as well asa required number of pyrotechnic cartridges. Wefollow the values described above. The solution is putin the table.Table 1. Results of examplex αp misβ =αm0,991 1 0,998991 20,992 1 0,997984 20,993 1 0,996979 20,994 1 0,995976 20,995 1 0,994975 20,996 1 0,993976 20,997 1 0,992979 20,998 1 0,991984 20,999 1 0,990991 2If we take into account this solution and startingmarginal conditions, two pyrotechnic cartridges will beenough to complete the mission successfully in 0,99probability.6. ConclusionThis contribution is supposed to serve as one of thealternatives solving the problems connected withproviding a function of an object whose function isredundant (backed up) because its failure is importantto complete the mission. In order to solve the problemwe chose the methods which are supposed to be themost suitable for it. Other ways are also likely to beused in order to reach the aim but it is not the intentionof this contribution.- 204 -

D. Vali Reliability of complex system with one shot items - RTA # 3-4, 2007, December - Special IssueReferences[1] Birolini, A. (2004). Reliability Engineering –Theory and Practice. New York, Springer-VerlagBerlin Heidelberg.[2] Blischke, W. R. & Murthy, D. N. P. (2000).Reliability Modeling, Prediction and Optimization,New York, Reliability Modeling, Prediction andOptimization, New York, John Wiley & Sons.[3] DEF STAN 00-42 (1997). (Part1)/Issue 1.Reliability and Maintainability Assurance Guides.Part 1: One-shot Devices/Systems, Glasgow, UKMinistry of Defence - Directorate ofStandardization.[4] Dodson, B. & Nolan, D. (1999). ReliabilityEngineering Handbook. New York, Marcel Dekker.[5] Ebeling, C. E. (1997). An introduction to Reliabilityand Maintainability Engineering. New York,McGraw-Hill.[6] Hoang, P. (2003). Handbook of ReliabilityEngineerin. London, Springer-Verlag.- 205 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special IssZio EnricoDepartment of Nuclear Engineering, Polytechnic of Milan, ItalySoft computing methods applied to condition monitoring and faultdiagnosis for maintenanceKeywordssoft computing, artificial neural networks, fuzzy logic, genetic algorithms, condition monitoring, fault diagnosis,maintenanceAbstractMalfunctions in equipment and components are often sources of reduced productivity and increased maintenancecosts in various industrial applications. For this reason, machine condition monitoring is being pursued torecognize incipient faults in the strive towards optimising maintenance and productivity. In this respect, thefollowing lecture notes provide the basic concepts underlying some methodologies of soft computing, namelyneural networks, fuzzy logic systems and genetic algorithms, which offer great potential for application tocondition monitoring and fault diagnosis for maintenance optimisation. The exposition is purposely kept on asomewhat intuitive basis: the interested reader can refer to the copious literature for further technical details.1. IntroductionManaging an industrial plant entails evaluating andtrading off the conflicting objectives of economicservice and safe operation. The first scientificapproaches to this management problem date back tothe 1950’s and 1960’s and can be found in the reviewpaper by McCall [1] and in the book from Barlow andProschan [3]. As a result, various so-called periodicmaintenance optimisation models were introduced inwhich both costs and benefits of periodic maintenancewere quantified and an optimum compromise betweenthe two was sought. Well-known models originatingfrom this period are the so-called age and blockreplacement models.From the practical point of view, at that time,preventive maintenance was strongly advocated as ameans to reduce failures, for safety reasons, andunplanned downtime, for economic reasons. In manycompanies, large time-based preventive maintenanceprograms were set up.Nowadays, modern production plants are expected torun continuously for extended hours. In this situation,unexpected downtime due to components andequipment failures has become more costly than everbefore. The faults can degrade the quality of a productline or even cause the entire plant to functionincorrectly, possibly resulting in downtime of theproduction system with consequent economic loss.On the other hand, proper monitoring of theconditions of the components and systems can behighly cost effective in minimizing maintenancedowntime by providing advanced warning and leadtime to prepare the appropriate corrective actionsupon an adequate fault diagnosis.For this reason, condition monitoring has become apopular approach for predicting component failuresusing physical information on the actual state of theequipment. The possibility of monitoring the systemstate, continuously for operating systems or by testsand inspections for stand-by safety systems, allows amore dynamic preventive maintenance practice, calledcondition-based maintenance (CBM), in which thedecision of maintaining the system is taken on thebasis of the observed condition of the system. This, inprinciple, allows saving resources by preventivelymaintaining the system only when necessary. In manypractical instances, this approach has proved moreeffective than the previous large preventivemaintenance programs.Analytical results for single-component deterioratingsystems have been established under simplifyingassumptions. Markov and semi-Markov models havebeen the preferred approach in modelling CBM [4],[12], [16]- [17], [21], [27] but other approaches, likecounting processes [1], have also been proposed. Themajority of the models appeared in the literature- 206 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Issassume that the system’s degradation level can onlybe known through periodic inspection as typical insafety systems such as those employed in nuclearplants [2]-[3], [26]-[27]; Kopnov [16] considers thecase in which the system is continuously monitoredand Lam [17] considers both cases. Another commonassumption is to consider that repairs/replacementsalways restore the system to a ‘good-as-new’condition, which, in practice, may not be veryrealistic; Kopnov [16] has allowed also for partialrecovery.The dynamic CBM policies for single-componentsystems whose condition can only be known throughinspection, developed in [10], [12] and [19], are allbased on control-limit rules which define when torepair/replace a component and when to schedule thenext inspection.For the continuously inspected systems investigatedby Kopnov [16], the two-level policies from theInventory Theory have been adapted to the CBMproblem of degrading systems. Semi-Markovprocesses are also considered; a death process isproposed for a unit subject to corrosion and a Markovchain is used for modelling fatigue crack growth.A common feature of the models discussed is that thestate of the system is described as a state of a Markovprocess and then the analysis proceeds to findinganalytically the probabilities of the various states.However, if the system is made of several multi-statecomponents the analysis becomes excessivelycomplicated. Simulation tools are hence needed whentreating more complex systems. Bérenguer et al. [[4]have extended the work of Grall et al. [10] byinvestigating two-component deteriorating systemsusing simulation. Their maintenance model takes intoconsideration economic dependence betweencomponents and again the state of the system is onlyknown through periodic inspections. Barata et al. [2]developed a stochastic degradation model forrepairable multi-component systems and embedded itssimulation within a maintenance optimisation scheme.The condition of each component is knowncontinuously. The novelty of the model stems fromthe fact that the component’s failures can occur notonly because of excessive degradation which leads toa critical state of the system, but also because ofrandom shocks which suddenly fail the system andwhose occurrence probability is degradationdependent.While in some cases the systemdegradation level depends on the combination ofmany mechanisms and can only be known throughinspection [4], [12], [12], [19], [26] there are othermechanisms such as fatigue and corrosion ofstructures in which deterministic laws are known andthe uncertainty is on the value of the parameters thatgovern those laws.Regarding the deterioration models themselves,Hontelez et al. [12] give several examples, all ofdeterministic nature, from the civil engineering field.Grall et al. [10] use a model in which the degradationlevel increases randomly according to an exponentialdistribution. Degradation models describing fatigueand corrosion of metal structures are described byGuedes Soares and Garbatov in [23], [24].The success of condition monitoring and conditionbasedmaintenance strongly relies on the capability ofmodelling the degradation processes and thecorresponding plant dynamic responses underdifferent configurations and conditions. However, thecomplexity and non-linearities of the involvedprocesses are such that analytical modelling becomesburdensome, if at all feasible without resorting tounrealistic simplifying assumptions. For this reason,empirical modelling is becoming very popular since itdoes not require a detailed physical understanding ofthe processes nor knowledge of the materialproperties, geometry and other characteristics of theplant and its components and it does not resort tosimplifying assumptions: the underlying dynamicmodel is identified by fitting plant operational data,with a procedure often referred to as ‘learning’ or‘training’.Among the various techniques of empirical modelling,the so-called soft computing methods offer powerfulalgorithms for constructing non-linear models fromoperational data. As a fact, they are being used withincreasing frequency as an alternative to traditionalmodels in a variety of engineering applicationsincluding monitoring, prediction, diagnostics, controland safety.The main soft computing methodologies are NeuralNetworks (NNs), Fuzzy Logic Systems (FLSs) andGenetic Algorithms (GAs). These methodologies areinspired by biology and natural behaviour and providepotentially powerful tools for effectively tacklingdifficult multivariate, non-linear problems, whichoften cannot be solved with ease by means oftraditional analytical or numerical methods.In the present lecture notes, we shall try to give a briefdescription of the concepts underlying the differentmethodologies and point out their main advantagesand limitations. With this objective in mind, we shallrefer our discussion to a multidimensional non-linearinput/output mapping, for NNs and FLSs, or searchingspace, for GAs optimisation.NNs and FLSs are capable of establishing the existingnon-linear input/output relationships, which map theinputs of a system to its outputs. They reconstruct thecomplex non-linear relations by combining multiplesimple functions. More precisely, through an analogywith the functioning of the human brain, NNs formthe shape of the mapping of interest by appropriatelycombining a large number of sigmoid, radial or other- 207 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Isssimple parameterised functions, which are adjusted(enlarged, shrunk, shifted, etc.) by means ofappropriate parameters and synaptic weights [20],[21]. The great power of this technique lies in the factthat the adjustments can be made ‘automatically’through a training phase based on availableinput/output data: this training phase allows to adjustthe NN-model parameters so as to obtain the bestinterpolation of the multivariate, non-linear functionalrelation between input and output.FLSs, on the contrary, partition the input/outputspaces into several typically overlapping areas, whoseshapes are established by assigned membershipfunctions and whose mapping relationships aregoverned by distinct, simple IF-THEN rules [28],[13]. The great advantage of this method lies in theinherent capability of handling imprecise data and inthe physical transparency and interpretability offeredby this particular way of representing the underlyingmodel relations.Finally, if the input/output multidimensional space isseen as a searching space in which the inputs are thedecision variables and the outputs are the performanceindicators of the search problem, the GAs offer apowerful method for evaluating a best input solutionwith respect to the optimisation (minimization ormaximization) of the performance indicators ofinterest [9], [11]. The main advantages of the methodare that the search is performed by manipulation of apopulation of points, contrary to classical methodswhich proceed from a single solution point to another,and that the search is solely based on the evaluation ofthe performance indicators, with no need of otherinformation, e.g. of derivative nature.2. Artificial Neural NetworksArtificial neural networks (ANNs) are informationprocessing systems composed of simple processingelements (nodes) linked by weighted connections.Their functioning is inspired by the biological neuralnetworks.A biological neuron consists of dendrites, a cell bodyand axons (Figure 1a). The connections between adendrite and the axons of other neurons are calledsynapses. In correspondence of each synapse, electricpulses from other neurons are transformed intochemical information which is input to the cell body:if the sum of the inputs received by the neuronthrough all its synapses exceeds a given threshold,then it fires an electric pulse which activates theneuron function. The network of all these neuronsmakes up the most essential part of the human brainand its operation enables the incredible variety ofhuman activities. In synthesis, the function of abiological neuron is ‘simply’ to output pulses, withthe characteristics of a quasi-step switching function,according to a weighed combination of the multiplesignals received from the other connected neurons. Asecond important function of the neuron is toappropriately modify the rate of transition through thedifferent synapses to optimise the whole network.Figure 1. A biological neuron (a) and an artificialneuron model (b) [25]An artificial neuron (node) aims at simulating theoperation of a biological neuron: thus, it acceptsmultiple inputs x1, x2,..., xm, it weighs them by meansof adaptive synaptic weights, w0, w1,..., wm, and itsimulates the switching function characteristic of theinput/output relation to provide the output (Figure1b). Connecting several artificial neurons together oneobtains an artificial neural network which, byconstruction, constitutes an information processingsystem composed of simple processing elements(nodes) linked by weighted synaptic connections [21].The adaptation of the synaptic weights is realizedthrough a training phase during which properlydevised learning algorithms are used to change thesynaptic weights of the network in an effort tooptimise its mapping performance [20].Here, we limit ourselves to briefly describing the mostcommonly used multi-layered feed-forward neuralnetwork which, in its simplest form, consists of threelayers of processing elements: the input, the hiddenand the output layers, with ni, nhand nonodes,respectively (Figure 2). The signal is processedforward from the input to the output layer. Each nodecollects the output values, weighted by the connectionweights, from all the nodes of the preceding layer,processes this information through a sigmoid functionf( x) = 1+−x−( e ) 1and then delivers the result towards all the nodes ofthe successive layer. Typically, both input and hiddenlayers are provided with an additional bias node,which serves as a threshold in the argument of theactivation function and whose output always equalsunity.As for the determination of the connection weights,i.e. the model parameters, the learning technique mostcommonly employed is the so-called error backpropagationalgorithm, which follows from the- 208 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Issgeneral gradient-descent method [20]. In short,starting from random values of the synaptic weightsthe back-propagation algorithm performs the steepestdescent in the weight space on a surface whose heightat any point is equal to the error function; in practice,it consists of an iterative gradient algorithm designedto minimize the mean square error between the actualnetwork output and the true value. A number n ofsets (patterns) of input and associated outputs arerepeatedly presented to the network and the values ofthe connection weights are modified so as to minimizethe average squared output deviation error function, orEnergy function, defined as:1E =2nnpnp no∑∑(y€o n= 1l= 1nl− ynl)2p(1)where ynland y € nlare the true output value of the n-th pattern and the corresponding network-computedoutput value at the l-th node, l = 1, 2,..., no. Throughthis training procedure, the network is able to build aninternal representation of the input/output mapping ofthe problem under investigation. The success of thetraining strongly depends on the normalization of thedata and on the choice of the training parameters.Typically, each signal is transformed by an affinedmapping in an interval such as (0.2, 0.8) or similarand the connection weights are initialised randomlywithin an interval such as (-0.3, 0.3) or similar.After the training is completed, the final connectionweights are kept fixed. New input patterns arepresented to the network, which is capable of recallingthe information stored in the connection weightsduring training to produce the corresponding output,coherent with the internal representation of theinput/output mapping. Notice that the non-linearity ofthe sigmoid function of the processing elementsallows the neural network to learn arbitrary non-linearmappings [6], [15]. Moreover, each node actsindependently of all the others and its functioningrelies only on the local information provided throughthe adjoining connections. In other words, thefunctioning of one node does not depend on the statesof those other nodes to which it is not connected. Thisallows for efficient distributed representation andparallel processing, and for an intrinsic fault-toleranceand generalization capability.These attributes render the artificial neural networks apowerful tool for signal processing, non-linearmappings and near-optimal solution to combinatorialoptimisation problems.Figure 2. Scheme of a three-layered, feedforwardneural network2.1. Feedforward artificial neural networks forregressionIn order to understand further the way of functioningof neural networks, let us consider an artificialfeedforward neural network to be trained forperforming the task of non-linear regression, i.e.estimating the underlying non-linear relationshipexisting between a multi-dimensional vector of inputvariables x and an output target y, assumed monodimensionalfor simplicity of illustration ( n0= 1, inesq. (1)), based on a finite set of input/output dataexamples (the above mentioned patterns),{( n, n), 1, 2,...,p}D ≡ x y n= n .It is assumed that the target y is related to the inputvector x by an unknown deterministic function µ y( x)corrupted by a white noise ε , viz.2y = µy( x) + ε( x); ε( x) N(0, σ ε( x))(2)The objective of the regression task is to estimateµ ( x)by means of a regression function f( xw, ; € )ydependent on a set of parameters €w to be properlydetermined on the basis of an available set ofinput/output patterns D.A feedforward neural network provides a non-linearform of the function f( xw ; € ) for the regression task.As above explained, the parameters €w are callednetwork weights and are usually determined by atraining procedure which aims at minimizing thequadratic error function1E =2nn p∑ ( y€− y )p n=1nn2(3)where for simplicity of notation the output nodesubscript l has been dropped since the case consideredconcerns a single output.- 209 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special IssThe network output corresponding to input xnis afunction of the weight values, y€ ( ; €n= f xw). If thenetwork architecture and training parameters aresuitably chosen and the minimization done todetermine the weights values is successful, theobtained function f( xw ; € ) gives a good estimate ofthe unknown, true regression function µ ( x). Indeed,it is possible to show that in the ideal case of aninfinite training data set and perfect minimizationalgorithm, a neural network trained to minimize theerror function in (3) provides a function f whichperforms a mapping from the input x into the expectedvalue of the target y conditioned on x, i.e. the truedeterministic function E[ y x] = µ ( x)[5]. In otherwords, the network averages over the noise on thedata and discovers the underlying deterministicgenerator. Unfortunately, all the training sets are finiteand there is no guarantee that the selectedminimization algorithm can achieve the globalminimum.The quadratic error function in (3) can be motivatedfrom the principle of maximum likelihood applied tothe set of available training patternsD ≡ x , y , n= 1, 2,..., n . The likelihood of the{( n n)p}observed data set D is defined asL( w€D)n p= ∏ p(x , yn pw€)= ∏ p(yn nn= 1 n=1nnnyx , w€)p(x w€)y(4)where it is assumed that each pattern ( xn, yn)isdrawn independently from the same distributionP( xn, yn). The unknown weights €w of the neuralmodel f( xw ; € ) are determined by maximization ofthe likelihood L( wD € ) of observing the training setD [5]. Instead of maximizing the likelihood, it iscomputationally more convenient to minimize itsnegative logarithm,( € ) ln ( € )*L wD L wDnp∑=− =np∑=− ln p( y x , w€ ) − ln p( x w€)n n nn= 1 n=1(5)The distribution of the input values xn, i.e. the secondterm in the rhs of (5), is independent of the parameters€w of the neural model f( xw; ; € ) thus, the parameters€w can be found by minimization of the first termonly, i.e. the following error functionn pE = − ∑ln p(y x , w€)(6)n=1nnDifferent forms of the conditional distributionp ( y x,w€)lead to different error functions. Inparticular, the assumption of a Gaussian distributionfor the target as in (2) leads to a quadratic errorfunction of the kind in (3) [20].Indeed, from eq. (2) we have( 0, σ ( ))2 xε(x)= y −µ y ( x)~ N εand using the regression function y € = f ( x;w€) toestimate µ ( x), we getyp(y x,w€)=1e2πσ ε( x)1 ( y−y€)2−2 σ2( x)εand the error function in (6) becomes⎡np⎢E = − ∑ ln⎢n=1⎢⎣1e2πσ ε ( xn)1−2(7)( y n − y€)2 n ⎤σ2ε ( x n ) ⎥⎥ . (8)⎥⎦When the noise variance is independent of the input x,2 2i.e. σ ( x)= σ ,ε1E =2σ+ nplnσ2εn=1εn p∑ε( y − y€)nn2n p+ ln 2π,2(9)and the error function reduces to the form (3) since theother terms do not depend on the weights €w of theneural model.Obviously, the quadratic error function in (3) can beused also for regression on targets, which are notGaussian-distributed: in this case, the resultingregression function f( xw ; € ) cannot distinguishbetween the true distribution and any other with samemean and variance.Finally, notice that the value of the error function (3)at the minimum gives a measure of the variance of thetarget data, averaged over the input.2.2. Neural network uncertaintyIn practical regression problems, there are two typesof prediction that one may want to obtain incorrespondence of a given input x: an estimate- 210 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Issf( xw ; € ) of the underlying deterministic functionµ ( x)and an estimate of the target value y itself, asygiven by eq.(2), with their corresponding measures ofconfidence. This requires that the various sources ofuncertainty affecting the determination of the weights€w be properly accounted for [7].For what concerns the estimate f( xw ; € ) of µ ( x), itmust be considered that, from a probabilistic point ofD ≡ x , y , n=1, 2,..., nview, the data set ( ){ n n p}used for training the network is only one of an infinitenumber of possible data sets. This variability in thetraining data set is due to the variability in thesampling of the input vectors x , n= 1,2,..., n and inthe random fluctuation of the corresponding targetoutput y . Each possible training set D can give risento a different set of network weights €w .Correspondingly, there is a distribution of regressionfunctions f( xw ; € ) with variance (with respect to thetraining set D):{[ f ( x,w€)E[ f ( x,w€)] }2E − (10)Since in practice a neural network structure is not aperfect algorithm, it systematically under/overestimates the correct result, i.e. the expected valueE f( xw , € ) is not equal to the true underlying[ ]deterministic function, µ ( x), their difference beingythe so-called bias. Of course, the bias would be zeroin the case of a perfect neural network.To quantify the confidence in the estimate f( xw ; € ) ofthe true deterministic function µ ( x), it is customaryto refer to the confidence intervals of the errorf ( x,w€)− µ ( x)whose variance with respect to allypossible training data sets is:E{[f ( x,w€)⎧⎡f ( x⎪= E ⎢⎨⎢⎪⎩⎢⎣+ E=⎧⎪⎪⎪⎪⎨⎪+⎪⎪⎪⎩− µ y( x)]2}2, w€)− E[ f ( x,w€)] + ⎤ ⎫⎥ ⎪⎥ ⎬[ f ( x,w€)] − ( x)⎪⎭µ y[ f ( x,w€)− E[ f ( x,w€)]+⋅[ E[ f ( x,w€)] − µ ( x)]2[ f ( x,w€)− E[ f ( x,w€)]⎥⎦⎫[ E[ f ( x,w€)] − µ ( x)] ⎪⎪⎪⎪ y ⎭y22n⎪⎪⎪⎪⎬ypy={[ f ( x,w€)− E[ f ( x,w€)] }2{ E[ f ( x,w€)] − µ ( x)} 2+ (11)ywhere the first term is the variance of the distributionof regression function values f( xw ; € ) and measuresthe extent to which the network regression functionf( xw ; € ) differs from the ensemble average overdifferent training data sets, whereas the second term isthe square of the bias which measures the extent towhich the average (over all possible training data sets)of the network regression function f( xw ; € ) differsfrom the true underlying deterministic function,µ ( x).y2.3. Sources of uncertaintyA first source of uncertainty comes from a wrongchoice of the network architecture. Indeed, in case ofa network with too few nodes, i.e. too few parameters,a large bias occurs since the regression functionf( xw ; € ) has insufficient flexibility to model the dataadequately, which results in poor generalizationproperties of the network. On the other side,excessively increasing the flexibility of the model byintroducing too many parameters increases thevariance term because the network regression functiontends to over-fit the training data. Thus, in both cases,the network performs poorly when fed with new inputpattern in the generalization phase. A trade-off is,then, necessary. This trade-off is typically achieved bycontrolling the model complexity (i.e., the number ofparameters) and the training procedure (by adding aregularization term in the error function or by earlystopping of the training [5]) so as to achieve a good fitof the training data but with a reasonably smoothregression function which is not over-fit to the data.An additional source of uncertainty in the networkperformance, due to uncertainty in the weights €w ,arises from the minimization algorithm itself, whichmay get stuck in a local minimum of the errorfunction. Furthermore, the training may be stoppedprematurely, before reaching the minimum.Besides the above uncertainties in the regressionfunction f( xw ; € ) due to uncertainty in the weights€w , in practice there is also uncertainty in the inputvalues x due to noise and uncertainty in the modelstructure f () ⋅ itself.For what concerns the prediction of the target value, y,it is clear that even in the ideal case of a regressionfunction f( xw ; € ) equal to the true deterministicfunction, µ ( x), the target y could not be predictedy- 211 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Isswith certainty due to the presence of the noise termε ( x)in (eq. 2) which accounts for the intrinsicrandom fluctuations. To quantify the accuracy of theestimate of y, it is customary to refer to the predictionintervals of the deviation[ µ ( x)− f ( x,€ ] + εy − f ( x,w€)= y w).The variance of such deviation is:E{[ f ( x,w€)− y]}2{[ f ( x,w€)− µ ( x)] } {[ ] }2 E ε2= E y +(12)2{[ f x,w€)− µ ( )]} 2 +σ= Ex( yεNote that the first term is the variance of thedistribution of the error f ( x,w€)− µ ( x)(eq. 11), sothat the prediction intervals include the confidenceintervals.From the above said, it appears that artificial neuralnetworks are unstable predictors: small changes in thetraining data may produce very different regressionmodels and consequently different generalizationperformances on new, unseen data. For this reason,the generalization performance of a single artificialneural network, particularly if trained on small datasets, should be tested by means of a k-fold crossvalidation, where the available set of n input/outputpatterns is divided into k subsets of (approximately)equal size and the network is trained k times on atraining set in which each time one of the k subsets isleft out and used to verify the network generalizationperformance. If k equals the sample size, theprocedure is called leave-one-out cross-validation[14].3. Fuzzy logic systemFuzzy logic systems are founded on the theory offuzzy sets, which, in general, deals with vagueinformation, where vagueness is defined as theuncertainty associated with linguistic or intuitiveinformation. For example, the quality of an imagemay be judged as bad, medium or good. From thisexample, it appears that vagueness is related toimmeasurable issues and involves situations in whichthe transitions among linguistic statements occuracross boundaries, which are not sharp.A few words on the concepts underlying fuzzy settheory seem then in order [28]. Let us consider avariable x, for example a measured output of a plant.Mutating from classical logic, the set U whichcontains all the possible values of x is usually calledypthe universe of discourse (UOD) of x or the universalset. Suppose that the UOD U has been subdivided in asequence of subsets X ⊂ U . In classical set theory,ithe Xi’s are mutually exclusives so that a given valueof x may belong to only one of them. These sets arecalled crisp and the membership of a crisp value of xto a set X is specified by the (rectangular)characteristic functioniχX i, which is equal to unity orzero according to whether the value of x belongs ornot to Xi.In fuzzy set theory, the situation is quite different: thesubsets Xiof the universe of discourse U of alinguistic variable x are not necessarily exclusive, sothat a given crisp value of x∈ U , maysimultaneously belong to more than one of them withdifferent degrees of membership. This feature clearlydistinguishes fuzzy set theory from probability theory,which operates on crisp events. In fuzzy set theory,then, the subsets are not identified by sharpboundaries but by linguistic labels (words). Forexample we may consider the linguistic variabletemperature defined in the universe of discourseU = (0,40) ° ° subdivided in the subsetsX1= (0 ° ,20 ° ) , X2= (10,30) ° ° , X3= (20,40) ° ° ,labelled by the words cold, warm and hot,respectively. Clearly a given temperature value maybelong to more than one set, e.g. 15° belongs to X1and X2.Fuzzy set theory aims at quantifying the meanings ofthe words attached by the analyst to the subsets Xi(such as cold, warm or hot in the above example)within the framework of set theory. To this aim, toeach set X the analyst associates, for all values ofix∈ U , the membership function µ (x), whichrepresents the degree to which he postulates that xbelongs to Xi. As opposed to the characteristicfunctions of classical set theory, which are rectangularin shape and disjoint, the membership functionsassociated to fuzzy sets have subjective shapes andmay overlap to describe a continuous transition fromone set to another, thus providing for the possibilitythat a given value of x∈ U simultaneously belongsto several sets with different degrees of membership.In summary, in the fuzzy context we deal withlinguistic variables (e.g. temperature) whosearguments are words, also called fuzzy values (e.g.negative, approximately zero, low, positive, high).Each of these words refers to a subset of the universeof discourse and the degree of membership of thecrisp values within the subset is analytically specifiedby the associated membership function.X i- 212 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special IssFuzzy logic systems build on the theory of fuzzy setsto realize a complex non-linear input/output relationas a synthesis of multiple simple input/outputrelations. This idea is similar to that of NNs. Thedifference is that in FLSs each simple input/outputrelation is embedded in a different IF-THEN rule with‘fuzzy’, and not sharp, boundaries so that going fromone rule to the other the system output graduallychanges [28].In general, the generic j-th fuzzy rule is made up of anumber of antecedent and consequent linguisticstatements, suitably related by fuzzy connections:IF ( x 1isX1 jTHEN ( y1is) AND (…) AND ( xmisY1 j) AND (…) AND (Xmj)ykisYkj)The linguistic variables x , p = 1,2,..., m, are theantecedents, represented in terms of the fuzzy setsXpjof the universe of discourse (range) Xp, withmembership functions µ ( x ). The linguisticpXpjvariables y , q= 1, 2,..., k , are the consequents,qrepresented by the fuzzy sets Yqjof the universe ofdiscourse Y , with membership functions µ ( y ).qThe connective operator AND links two fuzzyconcepts and it is generally implemented by means ofa t-norm, typically the minimum operator ∧ or thealgebraic product. Since one of the key features ofFLS lies in allowing the overlapping of the rules, agiven input vector (FACT) will typically activatemore than one rule.Another feature of FLSs is the ability to separate logicand fuzziness [25]. Conventional binary logic systemsare unable to do so and thus their governing rules haveto be modified when either the system logic or thevariables fuzziness needs to be changed. On thecontrary, FLSs modify their rules when the logicneeds to be changed whereas they modify thesupporting membership functions when fuzzinessshould be changed. To clarify this, consider theperformance of an inverted pendulum controller [25].'Define as ω and ω the angle that the pole forms onthe right side with the vertical line and the associatedangular velocity, respectively. Let two correct logicrules for the control of the pendulum be:1) IF ω is positive big AND ω ' is big, THEN movethe car to the right quickly2) IF ω is negative small AND ω ' is small, THENmove the car to the left slowlyIf the performance of the controller is unsatisfactory,one needs not change the fuzzy rules themselves,pYqjqwhich are logically correct, but rather must onlymodify appropriately the definition of fuzziness in thelinguistic terms big, small, quickly, slowly.On the other hand, binary logic rules such as−1 ' −13)IF 40°< ω < 60° AND 50° s < ω < 80° s ,THEN move the car at 0.5 m/s−1 ' −14)IF − 20°< ω < − 10° AND 10° s < ω < 20° s ,THEN move the car at −0.1 m/smust be modified whenever the logic of the system orthe quantitative definitions of angle, angular velocityand car speed are changed.To understand FLSs, we address on an intuitive basisthe problem of controlling a plant [25]. Themathematical-based approach of classical and moderncontrol theory stems on the observation of the system,the construction of its mathematical model and thedesign of a model-based controller. The focus isplaced on the behaviour of the target system and on itsmathematical representation.On the contrary, fuzzy-logic control does not utilizethe target system for modelling but it is based, inprinciple, on the linguistic control rules used byexperienced and skilled operators. Although mostskilled operators do not know the mathematicalbehaviour of the systems they are required to control,they can still perform successfully. For example, askilled driver most likely ignores the mathematicalequations underlying the physical behaviour of the carwhen turning to the right while driving up an unpavedhill and, yet, he or she can still handle the car safelyand successfully. In this view, a fuzzy logic controlleraims at reproducing the knowledge and experiencesupporting the control actions of skilled humanoperators using IF-THEN fuzzy rules.Clearly the set of IF-THEN fuzzy rules constitutes theheart of the input/output mapping model provided bythe FLS. When the experience of the skilled humanoperators is unavailable or insufficient, because of thecomplexity of the system, input/output data can beused to generate a set of fuzzy rules representative ofthe mapping from the input space into the output one.This phase of rule construction during which both thesystem input and output are known is often referred towith the term ‘training’, in analogy to the procedurefor determining the weights of a neural network modelillustrated in Section 2.3.1. Establishing the antecedent part of a ruleThe IF part of a rule is called antecedent. Establishingthe antecedent parts of the rules of the FLSs is relatedto the partitioning of the multivariate input space. Forsimplicity, let us consider a two-dimensional inputspace ( x1, x2)and a one-dimensional output space y.- 213 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special IssMost of the times it is possible to assume that all inputvariables are independent and, thus, partitionseparately the input space in each direction (Figure 3).This assumption makes it easy not only to partitionthe input space but also to interpret the partitionedareas in linguistic terms. For example, the rule IFtemperature is A1AND humidity is A2, THEN …, iseasy to understand because the variables oftemperature and humidity are separated. Thedifference between ‘crisp’ and ‘fuzzy’ rule-basedsystems lies in the way the input space is partitioned(Figure 3). The idea behind FLSs is that in the realanalog world, changes are not sudden and sharp butgradual in nature so that overlapping of rules domainsshould be allowed. The degree of overlapping isdefined in terms of membership functions and theintrinsic gradual property allows for smooth control.Figure 3. Rule partition of an input space (a) partitionfor crisp rules and (b) partition for fuzzy rules [25]3.2. Establishing the consequent part of a ruleThe THEN part of a rule is called consequent. In thecontrol case, establishing the consequent parts of therules of the FLSs must eventually lead to defining thecontrol action value corresponding to each rule. In thisrespect, fuzzy models are classified into three typesaccording to the form used for the consequent y:Model ConsequentCharacteristictype expressionMamdani Y is Y Y is a fuzzy setTakagiai’s are constantSugenoy = cKang0 +Σ i a i xand the xii’s arethe input(TSK)variablesSimplifiedy = c c is constantfuzzyIn the Mamdani type FLSs the consequent is a fuzzyvariable defined by a membership function. Thesesystems are more difficult to compute than thosewhose consequents are numerically defined but theybetter describe the qualitative knowledge related tothe consequent.The consequents of the TSK models are expressed asa weighed linear combination of the input variables. Itis also possible to use non-linear combination forbetter performance, but at the expense of thetransparency of the rules.The simplified fuzzy model has fuzzy rules whoseconsequents are constant values. Thus, it is a specialcase of both Mamdani and TSK types. Even if theoutput of each rule is a constant, the overall FLSoutput is non-linear because it contains thecharacteristics of the underlying model membershipfunctions.3.3. Inferring the output corresponding to agiven input: fuzzy reasoning and aggregationNow that the IF and THEN parts of the rules havebeen designed, the next step is to infer the output ofthe FLS resulting from a given m-dimensional crisp(*)input vector x , also called FACT. This is done intwo steps: 1) determination of the rules strengths; 2)aggregation of each rule output into the final FLSnumerical output.As mentioned, the connective operator AND links twofuzzy concepts in a rule and it is generallyimplemented by means of a t-norm applied to themembership functions of the rule’s antecedentsevaluated in correspondence of the crisp input vectors(*)x constituting the FACT. Typically, the minimumoperator ∧ or the algebraic product is employed. Thisgives the strength of the rule for the given FACT: ameasure of how active that rule is for the given FACTor, in other words, how much the FACT is describedby the antecedents of the rule. Considering a generic(*)rule l, the strength sl( x ) is given, in the case of thealgebraic product, by the product of its antecedentmembership values in correspondence of the crispinputs:m(*) (*)l( ) µX( x )pl pp=1s x= ∏ (13)where Xpldenotes the fuzzy set characterizing the p-th antecedent of the l-th rule.In the case of the min operator:s x= µ x(14)(*) (*)l( ) min(X(p))µpl(*)Obviously, if the l-th rule is not activated by x thenits strength is zero. This occurs if at least one of theelements of the crisp input vector, say x , does notbelong toXpl(*)p, the corresponding fuzzy set in the rule.Let us now consider the aggregation step and denoteby R the number of rules, which are activated (fired)f- 214 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Iss(*)lby the input x and by y the consequent in the l-thactivated rule, l = 1, 2,..., Rf. The output y can bedetermined as a normalized weighed sum of thelconsequents y of the R active rules, the weightsbeing the strengths of the rules,R f r∑ sl( xl = 1y =R f r∑ s ( xll=1(*)) y(*))lf(15)yη2∫ *η1= yCOA=η2∫η1y ⋅ µµY ′Y ′( y)dy( y)dy(17)*The crisp number y thereby obtained can be taken asthe output resulting from the given input vector(*)(FACT) x .Figure 4 shows an example of a simple TSK-typeFLS with four fuzzy rules [25]. The first rule forexample could be:IF x 1is small AND x 2is small,THEN y = 3x1+ 2x2−4Corresponding to the input vector ( x1, x2) = (10,0.5) ,the membership functions of the fuzzy setsconstituting the antecedents of the first rule are readilyevaluated as 0.8 and 0.3.If the algebra product is used as the t-norm operatorfor the AND connection, then the rule strength iss1(10,0.5) = 0.8⋅ 0.3 = 0.24 . Similarly, the strengthsof the second, third and fourth rules ares 2(10,0.5) = 0.8 , s 3(10,0.5) = 1.0 , s 4(10, 0.5) = 0.3 ,respectively. The output of each rule corresponding to12the given input vector is y = 27 , y = 23.5 ,3y =− 9 ,4y =−system output is20.5, respectively. Then, the final0.24 ⋅ 27 + 0.8 ⋅ 23.5 + 1⋅( −9)+ 0.3 ⋅ ( −20.5)y =0.24 + 0.8 + 1.0 + 0.3≈ 4.33(16)In the Mamdani type model, with fuzzy consequents,the output of the fuzzy inference engine consists of afuzzy set Y′ ⊆ Y with compact support ( η1, η2),whose membership function is µY ′ ( y). However,eventually we are interested in finding a crisp number*y that represents the information encoded in the'output fuzzy set Y . This conversion, calleddefuzzification, may be done in several ways, the mostcommonly used being the Centre of Area (COA)method:Figure 4. Example aggregation of TSK model [25]3.4. Interpretation of fuzzy rulesThe major difference between fuzzy systems andother non-linear approximates, such as neuralnetworks, is the possibility of interpretation of therules underlying the fuzzy model. This, however, doesnot automatically follow from the existence of a rulestructure. Therefore, it is relevant to discuss thecircumstances under which a fuzzy system is reallyinterpretable and this depends on the application. Yet,some general guidelines can be given. The followingfactors may influence the interpretability of a fuzzylogic system [25]:• Number of rules. If the number of rules is toolarge, the fuzzy system can be hardly interpreted.Especially for systems with many inputs, thenumber of rules often becomes overwhelminglylarge if all antecedent combinations are realized.• Number of antecedents in the rule premise. Ruleswith premises that have many, say more thanthree or four, antecedents are hard to interpret. Inthe human language, most rules include only veryfew antecedents even if the total number of inputsrelevant for the problem is large.• Dimension of input fuzzy sets. One-way to avoid,or at least, reduce the difficulties with highdimensionalinput spaces and to decrease thenumber of rules is to resort to multi-dimensionalinput fuzzy sets. However, multidimensional inputfuzzy sets with more than three inputs arecertainly beyond human imagination and it isprecisely the conjunction of one-dimensional- 215 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Issinput fuzzy sets that make a fuzzy logic modelinterpretable.• Arrangement of fuzzy sets. Fuzzy sets should bearranged in proper order over the universe ofdiscourse so that, for example, very small isfollowed by small, followed by medium, large andso on. If the fuzzy model is developed on the basisof expert knowledge such ordering comes natural.However, if rule construction by training oninput/output data is used to optimise the FLS, theordering of the fuzzy sets might be lost if noprecautions are taken. This typically leads toconstraint optimisations in which for example theordering of the input membership functions isconstrained. This not only leads to an easierinterpretation of the fuzzy system but, often, alsoprovides an improved performance.4. Genetic algorithmsSearch or optimisation algorithms inspired on thebiological laws of genetics are called evolutionarycomputing algorithms [8]. The main features of thesealgorithms are that the search is conducted i) using apopulation of multiple solution points or candidates,ii) using operations inspired by the evolution ofspecies, such as breeding and genetic mutation, iii)based on probabilistic operations, iv) using onlyinformation on the objective or search function andnot on its derivatives. Typical paradigms belonging tothe class of evolutionary computing are geneticalgorithms (GAs), evolution strategies (ESs),evolutionary programming (EP) and geneticprogramming (GP). In the following we shall focus onthe more popular GAs.As a first definition, it may be said that geneticalgorithms are numerical search tools aiming atfinding the global maximum (or minimum) of a givenreal objective function of one or more real variables,possibly subject to various linear or non linearconstraints [11]. Genetic algorithms have proven to bevery powerful search and optimisation tools especiallywhen only little about the underlying structure in thedata is known. They employ operations similar tothose of natural genetics to guide their path throughthe search space. Essentially, they embed a survival ofthe fittest optimisation strategy within a structured, yetrandomised, information exchange [9].Since the GAs owe their name to the fact that theirfunctioning is inspired by the rules of the naturalselection, the adopted language contains many termsborrowed from biology, which need to be suitablyredefined to fit the algorithmic context. Thus, whenwe say that the GA operates on a set of (artificial)chromosomes, these must be understood as strings ofnumbers, generally sequences of binary digits 0 and 1.If the objective function has many arguments, eachstring is partitioned in as many substrings of assignedlengths, one for each argument and, correspondingly,we say that each chromosome is analogouslypartitioned in (artificial) genes. The genes constitutethe so-called genotype of the chromosome and thesubstrings, when decoded in real numbers calledcontrol factors, constitute its phenotype. When theobjective function is evaluated in correspondence ofthe values of the control factors of a chromosome, itsvalue is called the fitness of that chromosome. Thuseach chromosome gives rise to a trial solution to theproblem.The GA search is performed by constructing asequence of populations of chromosomes, theindividuals of each population being the children ofthose of the previous population and the parents ofthose of the successive population. The initialpopulation is generated by randomly sampling the bitsof all the strings. At each step, the new population isthen obtained by manipulating the strings of the oldpopulation in order to arrive at a new populationhopefully characterized by an increased mean fitness.This sequence continues until a termination criterionis reached. As for the natural selection, the stringmanipulation consists in selecting and mating pairs ofchromosomes in order to groom chromosomes of thenext population. This is done by repeatedlyperforming on the strings the four fundamentaloperations of reproduction, crossover, replacementand mutation, all based on random sampling. Theseoperations will be detailed below [18].Finally, it is by now acknowledged that GAs take amore global view of the search space than many otheroptimisation methods. The main advantages are i) fastconvergence to near global optimum, ii) superiorglobal searching capability in complicated searchspaces, iii) applicability even when gradientinformation is not readily achievable. The first twoadvantages are related to the population-basedsearching property (Figure 5). Indeed, while thegradient method determines the next searching pointusing the gradient information at the current searchingpoint, the GA determines the next set of multiplesearch points using the evaluation of the objectivefunction at the current multiple searching points.When only gradient information is used, the nextsearching point is strongly influenced by the localgeometric information of the current searching pointso that the search may remain trapped in a localminimum. On the contrary, the GA determines thenext multiple searching points using the fitness valuesof the current searching points, which are spreadthroughout the searching space, and it can also resortto the additional mutation to escape from localminima.The key disadvantage of a GA is that its convergencespeed becomes slow near the global optimum.- 216 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Issgenotype and the external environment, i.e. thephenotype, constituted by three control factors,x1, x2,x3, one for each gene. The passage from thegenotype to the phenotype and vice versa is ruled bythe phenotyping parameters of all genes, whichperform the coding/decoding actions. Each individualis characterized by fitness, defined as the value of theobjective function calculated in correspondence of thecontrol factors pertaining to that individual. Thus apopulation is a collection of points in the solutionspace, i.e. in the space of f.Figure 6. Components of an individual (achromosome) and its fitnessFigure 5. GA search and gradient-based search [25]4.1. DefinitionsIt is important to be acquainted with the technicalterms of GAs.Individuals and Population: An individual is achromosome, constituted by n ≥ 1 genes and apopulation is a collection of individuals. Tocode/decode the i-th gene in a control factor, that is inan argument of the objective function, the user:• defines the range ( ai, bi)of the correspondingargument in the objective function;• assigns the resolution of that independent variableniby dividing the range ( a , b ) in 2 intervals. Anumber n i of bits is then assigned to the substringrepresentative of the gene and the relationbetween a real value x ∈ a i, b ) and its binarycounterpart β isxb − aii(ii i= ai+ β (18)n i2The values ai, bi, niare called the phenotypingparameters of the gene.Figure 6 shows the constituents of a chromosomemade up of three genes and the relation between theAn important feature of a population is its geneticdiversity: if the population is too small, the scarcity ofgenetic diversity may result in a population dominatedby almost equal chromosomes and then, afterdecoding the genes and evaluating the objectivefunction, in the quick convergence towards anoptimum which may well be a local one. At the otherextreme, in too large populations, the overabundanceof genetic diversity can lead to clustering ofindividuals around different local optima: then themating of individuals belonging to different clusterscan produce children (newborn strings) lacking thegood genetic part of either of the parents. In addition,the manipulation of large populations may beexcessively expensive in terms of computer time.In most computer codes the population size is keptfixed at a value set by the user so as to suit therequirements of the model at hand. The individualsare left unordered, but an index is sorted according totheir fitnesses. During the search, the fitnesses of thenewborn individuals are computed and the fitnessindex is continuously updated.4.2. Creation of the initial populationAs said above, the initial population is generated byrandom sampling the bits of all the strings. Thisprocedure corresponds to uniformly sampling eachcontrol factor within its range. The chromosomecreation, while quite simple in principle, presents- 217 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Isssome subtleties worth to mention: indeed it mayhappen that the admissible hypervolume of the controlfactors is only a small portion of that resulting fromthe Cartesian product of the ranges of the singlevariables, so that one must try to reduce the searchspace by resorting to some additional condition interms of suitable physical criteria to be satisfied. Thisremark also applies to the chromosome replacement,below described.4.3. The traditional breeding algorithmThe breeding algorithm is the way in which the( n + 1)-th population is generated from the n-thprevious one.The first step of the breeding procedure is thegeneration of a temporary new population. Assumethat the user has chosen a population of size N(generally an even number). The populationreproduction is performed by resorting to the StandardRoulette Selection rule: to find the new population,the cumulative sum of the fitnesses of the individualsin the old population is computed and normalized tosum to unity. The new population is generated byrandom sampling individuals, one at a time withreplacement, from this cumulative sum, which thenplays the role of a cumulative distribution function(cdf) of a discrete random variable (the position of anindividual in the population). By so doing, on theaverage, the individuals in the new population arepresent in proportion to their relative fitness in the oldpopulation. Since individuals with relatively largerfitness have more chance to be sampled, mostprobably the mean fitness of the new population islarger.The second step of the breeding procedure, i.e. thecrossover, is performed as indicated in Figure 7: afterhaving generated the new (temporary) population asabove said, N/2 pairs of individuals, the parents, aresampled at random without replacement andirrespectively of their fitness, which has already beentaken into account in the first step. In each pair, thecorresponding genes are divided into two portions byinserting at random a separator in the same position inboth genes (one-site crossover): finally, the firstportions of the genes are exchanged. The twochromosomes so produced, the children, are thus acombination of the genetic features of their parents. Avariation of this procedure consists in performing thecrossover with an assigned probability p (generallyrather high, say pc≥ 0.6 ): a random number R isuniformly sampled in (0,1] and the crossover isperformed only if R< pc. Vice versa, if R≥pc, thetwo children are copies of the parents.cFigure 7. Crossover in a population withchromosomes constituted by three genesThe third step of the breeding procedure, performedafter each generation of a pair of children, concernsthe replacement in the new population of two amongthe four involved individuals. The simplest recipe,again inspired by natural selection, just consists in thechildren replacing the parents: children live, parentsdie. In this case, each individual breeds only once.The fourth and last step of the breeding procedureeventually gives rise to the final ( n + 1)-th populationby applying the mutation procedure to the (up to thistime temporary) population obtained in the course ofthe preceding steps. The procedure concerns themutation of some bits in the population, i.e. thechange of some bits from their actual values to theopposite one (0 → 1) and vice versa. The mutation isperformed on the basis of an assigned mutationprobability for a single bit (generally quite small, say310 − ). The product of this probability by the totalnumber of bits in the population gives the meannumber µ of mutations. If µ< 1 a single bit ismutated with probability µ . Those bits to be actuallymutated are then located by randomly sampling theirpositions within the entire bit population.The sequence of successive population generations isusually stopped according to one of the followingcriteria:1. when the mean fitness of the individuals in thepopulation increases above an assignedconvergence value;2. when the median fitness of the individuals in thepopulation increases above an assignedconvergence value;3. when the fitness of the best individual in thepopulation increases above an assigned- 218 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special Issconvergence value. This criterion guarantees thatat least one individual is good enough;4. when the fitness of the weakest individual in thepopulation drops below an assigned convergencevalue. This criterion guarantees that the wholepopulation is good enough;5. when the assigned number of populationgenerations is reached.More sophisticated techniques of reproduction,crossover and replacement can be employed for amore effective search.Furthermore, in general, the initial population sampledcontains a majority of second-rate individuals togetherwith few chromosomes, which, by chance, havemoderately good fitnesses. Then, the selection rulesare such that, in a few generations, almost all themoderately good chromosomes, which are actuallymediocre individuals, are selected as parents andgenerate children of similar fitnesses; thus, almost allthe second-rate individuals disappear, and most of thepopulation gathers in a small region of the searchspace around one of the mediocre individuals. In thiscase, the genetic diversity is drastically reduced andthe algorithm may achieve a premature convergenceof the population fitness to a local maximum. In thecourse of the successive generations, the crossoverprocedure generates mediocre individuals, which aresubstituted in place of other mediocre individuals, sothat the genetic selection may be seen as a randomwalk among mediocres. To obviate to this unpleasantpremature convergence to mediocrity, a pre-treatmentof the fitness function is often welcome. This istypically done by means of an affined transform of thefitnesses. Instead of applying the selection rules to thefitness f( x ) one works with its affined transformf' ( x ), viz.,f '(x)= a f ( x)+ b(19)where a and b are chosen so as to favour, at thebeginning, the less fit individuals, thus maintaininggenetic diversity and avoiding premature convergence[18].5. ConclusionThese lecture notes have briefly sketched some of theconcepts underlying the modern computationalparadigms of neural networks, fuzzy logic systemsand genetic algorithms, which are becoming ofsignificant interest for application to conditionmonitoring and fault diagnostics for maintenance. Dueto the limitation in the number of pages, only anintuitive and non-exhaustive treatment has beenprovided. Whereas some examples of practicalapplication will be illustrated during the lecture, theinterested reader is invited to refer to the specializedliterature for further, in-depth details on the differenttechniques.References[1] Aven, T. (1996). Condition based replacementpolicies – a counting process approach. ReliabilityEngineering and System Safety, Vol. 51, 275-282.[2] Barata, J., Guedes Soares, C., Marseguerra, M. &Zio, E. (2001). Monte Carlo simulation ofdeteriorating systems. Proceedings ESREL 2001,879-886.[3] Barlow, R. E. & Proschan, F. (1965).Mathematical Theory of Reliability. John Wiley,New York.[4] Bérenguer, C., Grall, A. & Castanier, B. (2000).Simulation and evaluation of condition-basedmaintenance policies for multi-componentcontinuous-state deteriorating systems. Foresightand Precaution. Cottam, Harvey, Pape & Tait(eds), Rotterdam, Balkema, 275-282.[5] Bishop, C. M. (1995). Neural Networks for PatternRecognition. Oxford University Press.[6] Cybenko, G. (1989). Approximation bySuperpositions of a Sigmoidal Function,Mathematics of Control, Signals and Systems,Vol.2, 303-314.[7] Dybowski, R. & Roberts, S. J. (2000). Confidenceand Prediction Intervals for Feed-Forward NeuralNetworks, in Clinical Applications of ArtificialNeural Networks, Eds. R. Dybowski and V. Gant,Cambridge University Press.[8] Fonseca, C. M. & Fleming, P. J. (1995). AnOverview of Evolutionary Algorithms inMultiobjective Optimisation. EvolutionaryComputation, 3(1):1-16, Spring.[9] Goldberg, D.E. (1989). Genetic Algorithms inSearch, Optimisation, and Machine Learning.Addison-Wesley Publishing Company.[10] Grall, A., Bérenguer, C. & Chu, C. (1998).Optimal dynamic inspection/replacement planningin condition - based maintenance. Safety andReliability. S. Lydersen, G. Hansen & H. Sandtorv(eds), Trondheim, Balkema, 381-388.[11] Holland, J. H. (1975). Adaptation in natural andartificial system. Ann Arbor, MI: University ofMichigan Press.[12] Hontelez, J. A. M., Burger, H. H. & Wijnmalen, D.J. D. (1996). Optimum condition-basedmaintenance policies for deteriorating systemswith partial information. Reliability Engineeringand System Safety, Vol.51, 267-274.[13] Klir, G. J. & Bo, Yuan (1995). Fuzzy Sets andfuzzy logic: Theory and Application. Prentice Hall.[14] Kohavi, R. (1995). A Study of Cross-Validationand Bootstrap for Accuracy Estimation and Model- 219 -

E. Zio Soft computing methods applied to condition monitoring and fault diagnosis for maintenance - RTA # 3-4, 2007, December - Special IssSelection, in C. S. Mellish Ed. Proceedings of the14th International Joint Conference on ArtificialIntelligence. Morgan Kaufmann Publishers.[15] Kolmogorov, A. N. (1957). On the Representationof Continuous Functions of Many Variables bySuperposition of Continuous Functions of OneVariable and Addition. Doklady Akademii NaukSSR, Vol. 114, 953-956.[16] Kopnov, V. A. (1999). Optimal degradationprocess control by two-level policies. ReliabilityEngineering and System Safety. Vol. 66, 1-11.[17] Lam, C. & Yeh, R. (1994). Optimal maintenancepolicies for deteriorating systems under variousmaintenance strategies. IEEE Transactions onReliability. Vol. 43, 423-430.[18] Marseguerra, M. & Zio, E. (2001). GeneticAlgorithms: Theory and Applications in the SafetyDomain, In “The Abdus Salam InternationalCentre for Theoretical Physics: Nuclear ReactionData and Nuclear Reactors”, N. Paver, M. Hermanand A. Gandini Eds., World Scientific Publisher,655-695.[19] McCall, J. J. (1965). Maintenance policies forstochastically failing equipment: a survey,Management Sci., 11, 493-524.[20] Muller, B. & Reinhardt, J. (1991). NeuralNetworks - An introduction. Springer-Verlag, NewYork.[21] Rumelhart, D. E. & McClelland, J. L. (1986).Parallel distributed processin.. Vol. 1, MIT Press,Cambridge, MA.[22] Samanta, P. K., Vesely, W. E., Hsu, F. & Subudly,M. (1991). Degradation Modelling WithApplication to Ageing and MaintenanceEffectiveness Evaluations. NUREG/CR-5612, U.S.Nuclear Regulatory Commission.[23] Soares, G. C. & Garbatov, Y. (1996). Fatiguereliability of the ship hull girder accounting forinspection and repair. Reliability Engineering andSystem Safety. Vol. 51, No. 2, 341-351.[24] Soares, G. C. & Garbatov, Y. (1999). Reliability ofmaintained, corrosion protected plates subjected tonon-linear corrosion and compressive loads.Marine Structures. Vol. 12, No. 6, 425-446.[25] Takagi, H. (1997). Introduction to Fuzzy Systems,Neural Networks, and Genetic Algorithms, inIntelligent Hybrid Systems. Da Ruan Ed., KluwerAcademic Publishers.[26] Wang, W., Christer, A. H. & Jia, X. (1998).Determining the optimal condition monitoring andPM intervals on the basis of vibration analysis - Acase study. Safety and Reliability. S. Lydersen, G.Hansen & H. Sandtorv (eds), Trondheim, Balkema,241-246.[27] Yeh, R. H. (1997). State-age-dependentmaintenance policies for deteriorating systemswith Erlang sojourn time distributions. ReliabilityEngineering and System Safety. Vol. 58, 55-60.[28] Zadeh, L. A. (1965). Fuzzy Sets, Information andControl, Vol. 8.- 220 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special IssueZio EnricoBaraldi PieroPopescu Irina CrengutaDepartment of Nuclear Engineering, Polytechnic of Milan, Milan, ItalyOptimising a fuzzy fault classification tree by a single-objective geneticalgorithmKeywordsfault classification, decision tree, fuzzy logic, genetic algorithmAbstractIn this paper a single-objective Genetic Algorithm is exploited to optimise a Fuzzy Decision Tree for faultclassification. The optimisation procedure is presented with respect to an ancillary classification problem builtwith artificial data. Work is in progress for the application of the proposed approach to a real fault classificationproblem.1. IntroductionIn recent years, many efforts have been devoted to thedevelopment of automatic diagnostic techniques basedon statistical or geometric methods, neural networks,expert systems, fuzzy and neuro-fuzzy approaches[22], [11], [15], [5], [7]. These techniques have provento be very effective but often remain “black boxes” asto the interpretation of the physical relationshipsunderpinning the fault classification.In an effort to overcome this limitation, a systematicapproach to fault classification has been introduced bythe authors leading to a Fuzzy Decision Tree (FDT)[25], [26]. The main advantages of the proposedapproach from the operator point of view are thetransparency of the resulting classification model andits visualization in the form of a DT [14], [20].The construction of the Decision Tree (DT) is pursuedstarting from the fuzzy rules of a Fuzzy Rule Base(FRB) derived from a clustering algorithm tailored tofault classification [25]. To do this, every Fuzzy Set(FS) representing a deviation of the monitored signalsin the respective ranges of variability (Universes ofDiscourse, UODs, in Fuzzy Logic terminology) isassociated to a symptom of a fault class and the FRBof the model is translated into a Symptom Table inwhich the relationships between fault classes andsymptoms are explicitly laid out.In practice, however, it is often difficult to attribute thedetected symptoms to a given fault class, given thatone fault may cause several symptoms and dually asymptom may describe more than one possible fault.To solve this problem, the relationships between faultclasses and symptoms contained in the SymptomsTable are systematically represented in a DT, which isthen quantified by applying the rules of Fuzzy Logic.The design of the DT entails the successiveconsideration of the symptoms. These can beconsidered in different orders, leading to differentstructures of the DT and thus different classificationperformances. Hence, a combinatorial optimisationproblem arises with regards to the DT design.In this paper, a single-objective genetic algorithmsearch is devised to find the sequence of symptomsleading to the optimal configuration of the DT, i.e. thatwhich achieves the maximum classificationperformance.The paper is organized as follows. Section 2 illustratesthe procedure adopted for the construction of the DTand its fuzzy quantification. In Section 3, the resultsrelated to its application to an artificial case studyregarding the classification of data randomly extractedfrom six different Gaussian distributions are reported.Section 4 presents the optimisation of the FDT by asingle-objective genetic algorithm maximizing thepercentage of correct classifications. A synthetic- 221 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special Issuediscussion of the findings of the work is provided inthe last Section.2. From a Fuzzy Classification Model to aFuzzy decision TreeLet us consider an industrial system or plant whose“state of health” is monitored by a set of sensors,which collect the relevant parameters data at a givenfrequency. These data (also called “signals”) provide apicture of the health state of the plant. A particularpicture corresponds to the plant functioning in nominalconditions, with all the signals within their designenvelope. Deviations from the nominal states are dueto faults of different types (classes), which may occurto the components of the plant, leading to different“pictures” of the monitored signals.When a generic fault of class Γ , j = 1,..., c, occurs inthe plant, corresponding representative symptoms areobserved by the monitoring system, in terms ofvariations in the signal values. A symptom associatedto the fault of class Γ is a deviation of a monitoredjsignal from its reference value, outside of the alloweddesign envelope. The objective of fault identification isto build a system capable of recognizing the fault as ofclass Γ on the basis of the measured symptoms, i.e.jthe monitored signals.In this work, we assume that the classification of thefault is performed by applying a previously built FRB(for example, a possible method for building an FRBfrom available pre-classified data is presented in [25]).Such FRB has one fuzzy rule for each fault class: thegeneric rule j associates the symptoms of themonitored signals (input data) to the fault class Γj. Inthe fuzzy rules, each one of the FSs of the antecedentsdescribes a deviation of a monitored signal, i.e. asymptom, except those FSs representing the stillnominal conditions of those monitored signals whichare unaffected by the particular fault. Correspondingly,the generic FS X associated to the p -th antecedentpjin rule j , p= 1,..., n, j = 1,..., c, represents asymptom for the class of faults Γj.Notice that the relations between fault classes andsymptoms (signals deviations) are not univocal: thefaults of a given class may initiate several symptomsand in turn one symptom may be a legitimaterepresentative of several possible fault classes.On the other hand, an adequately designed monitoringsystem should be capable of associating to each faultclass a unique set of symptoms (signal deviations).This leads to a Symptom Table such as the onereported in Table 1, where Sr, r = 1,..., s, denotes thegeneric symptom.jThe binary vector σj= [ Ij1, Ij 2,..., Ijs] represents thereference symptoms vector for fault class Γj,j = 1,..., c. Each element Ijris a binary value thatcorresponds to the presence or absence of symptom rwhen a fault of class Γ has occurred, r = 1,...,s ,j = 1,...,c.jTable 1. Symptom Table: Reference relations betweenfault classes and symptoms [13]FaultClassS1S2Symptom Type…S …rS sΓ1I11I …12I …1rI 1sΓ2I21I …22I …2rI 2s......ΓjIj1j 2......ΓcIc1 c2.........I … I …jrI js.........I … I …crI csDuring operation, the monitored signals could then betranslated into an observation vector' ' ' 'σ = ( I1, I2,..., I s) , which carries the information onthe presence or absence of the symptoms. As explainedearlier, a symptom is present in the system if itsrepresentative measured signal has deviated from itsnominal value beyond the design envelope. Forexample, a patient has the symptom “fever” if his orher monitored temperature rises to a “high” value, i.e.above 37°C.However, in practice often the presence or absence of asymptom remains uncertain and ambiguous due to thecomplexity of the non-linear signal behavioursassociated to the various faults, to the measurementerrors of the monitoring sensors and to the impreciseand ambiguous definition of the signal deviationranges and the associated linguistic labels [25]. Toreflect this uncertainty, a fuzzy observation vector' ' ' 'σ = ( µ , µ ,..., µ ) is associated to a pattern off1 2s............- 222 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special Issuedeviations of the monitored signals measured in'correspondence of a given fault, where µr, r = 1,..., s,is the value of the membership of the FS correspondingto the symptom S rand gives the degree with which itis present in the monitored situation being examined.The fault identification problem is then to identifywhich fault class is occurring in the plant on the basisof the fuzzy observation vector σ . To tackle thisproblem a systematic procedure for constructing a DTis presented below [26].2.1. Decision TreeBased on the Symptom Table 1, a complete DT can bediagrammed by examining all s symptoms one by one[7]. Taking into consideration all possiblecombinations of symptoms, the DT will have 2 sbranches given that each of the s symptoms can beeither present or absent. On the other hand, only onecombination of symptoms corresponds to a given fault:thus, only c of the 2 s tree branches correspond to aclass while the remaining 2 s − c combinations ofsymptoms cannot be associated to a class.For building a smaller, more transparent and easier tointerpret DT, two main hypotheses are assumed [26],[13]:- if a symptom is indicated as present in the measured'observation vector σ , it is certainly present in thesystem;- the presence of a single symptom characteristic of afault suffices to conclude that the measured pattern ofsignals belongs to that fault class.In this context, defining an “unwanted” symptom as asymptom that, although not present in the system,somehow is present by mistake in the observationvector and a “missing” symptom as a symptom that isnot observed although it is present in the system [13],the first hypothesis can be called of “impossibility ofunwanted symptoms” and the second of “possibility ofmissing symptoms”.The procedure for building the DT proceeds asfollows:1. A root node is placed at the top of the tree.This node refers to all possible fault classes identifiedfor the system under analysis.2. A symptom from the Symptom Table isassociated to this node.3. The root node is split into two branches: theleft corresponding to the presence of the symptom, theright to the absence of the symptom.4. The fault classes for which the symptom ispresent are associated to a node under the left branch.'fIf only one fault class is found to contain the symptom,then the associated node is a terminal leaf of the branchand its identification is guaranteed by the fact that ithas been assumed that a symptom that is absent in thesystem cannot be indicated as present (impossibility ofunwanted symptoms hypothesis). The fault classassociated to the identified leaf may be also associatedto other leaves, at the end of other branches in the tree.This accounts for the possibility that a symptom is notindicated as present by the monitoring system althoughit actually is (possibility of missing symptomshypothesis). If more than one fault class are associatedto the node characterized by the identified symptom, anew symptom is searched in the Symptom Table andassociated to the node in order to differentiate betweenthe identified fault classes. To select the newdifferentiating symptom, the previous procedure isapplied, starting from step 2.5. The right branch from the root node is furtherdeveloped by first adding a node associated to allpossible fault classes. This node is then treated as alocal root node to which the branching procedure isapplied starting from step 2.6. The tree development terminates when allsymptoms have been considered and their associatedbranches developed down to the distinguishing leavesof the individual fault classes.A path through the branches of the tree, from the root'node to a leaf, identifies a crisp observation vector σof symptoms representative of the fault classassociated to the corresponding leaf. As pointed outabove, different paths may lead to different leavesassociated to the same fault class, due to the possibilityof missing symptoms.In operation, the DT gives the correct diagnosis whenthe measured symptom vector matches completelywith the reference symptom vector of a fault class; onthe contrary, the diagnosis is conservative in case of amissing symptom, i.e. it is not necessary to have all thesymptoms to diagnose the fault.Finally, in case of unwanted symptoms, theclassification is driven by the structure of the tree andthe classification will be wrong if the first symptomconsidered is an unwanted symptom.From the above it appears that an issue of adequate DTdesign arises with respect to the order with which thesuccessive symptoms are considered for optimalclassification performance.2.2. Classification by the FDTIn the realistic case of ambiguity in the actual presenceor absence of a symptom, in correspondence of a givenpattern of signal deviations the degree of activation of- 223 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special IssueFigure 1. Propagation of fuzzy information through the DTeach symptom S r, r = 1,..., s, is computed from theMF of the corresponding FS. The DT then becomes aFDT and the possibility classification of a givenpattern of measured signal deviations is performed byproceeding through all the branches of the tree andcomputing the MFs to each fault class, at the treeleaves.The symptoms degrees of activation are thenpropagated through the DT according to the rules of FStheory. In particular, the logic operator of negation of asymptom− in the rightS ris implemented by ( 1 µ S )rbranch corresponding to the absence of the symptomwhereas its complement µ is propagated along theleft branch associated to its presence (Figure 1). Theconnection between two nodes of the tree represent alogic operator of intersection (and), implemented asthe algebraic product of the membership values in thiswork.Finally, since more than one terminal leaf can indicatethe same class, the final membership to a given class iscomputed through the logic operation of union (or) ofall the leaves associated to that class. The logicoperator or is here implemented as the MFs sumlimited to 1, accordingly to the rules of FS arithmetic.As mentioned at the end of Section 2.1, differentsequences of symptoms lead to different DTs and,implicitly, to different classification performances. Forrealistic problems, the number of possible sequences ofsymptoms for building the DT is combinatorial, so thata trial and error process for finding the optimalstructure of the tree, i.e. that which allows obtainingthe maximum classification performance, would not bepractical. For example, the number of possiblesequences of a group of 15 symptoms, would be11approximately 10 and to each sequence correspondsa different DT whose classification performance mustbe evaluated.S r3. Application of the FDT to an artificial casestudyThe classification approach has been applied to theartificial four-dimensional, six-classes data set ofFigure 2. The data have been obtained by randomsampling from 6 different Gaussian distributions andcan be assumed to represent the system responsesignals resulting from 6 different types of systemfaults.The previously illustrated procedure for classifying thedata into the six classes (Section 2) consists of twomain steps: the first one is the building of the FRB, i.e.a set of transparent and accurate fuzzy rules and thesecond one is the construction and quantification of thecorresponding FDT.A fuzzy clustering – based method has been used forobtaining the FRB from available pre-classified data[25]. The resulting FRB is composed of c = 6 rules,one for each class (Table 2).To build the associated DT, first each antecedent of therules in the FRB is associated to a symptom, resultingin 15 possible symptoms, indicated as Si,i = 1, 2,...,15 , in Table 2. This allows the translationof the FRB in the Symptom Table 3.Then, by applying the steps 1.– 6. of the procedure forbuilding the DT (Section 2.1) on the sequence ofsymptoms Σ0= [ S1; S2; ... ; S15], one obtains the DTreported in Figure 3.The possibility quantification of the degree ofmembership to different classes can be performed asdescribed in Section 2.2, i.e. propagating through thebranches of the tree the degree of activation of eachsymptom according to the rules of Fuzzy Logic. Thefinal assignment of an incoming pattern of signals to aclass is conservatively realized as follows:- 224 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special IssueFigure 2. Four-dimensional data set comprised of six classesTable 2. The Table of rules of the FRBRulex1x x23x4Γ1Γ Γ2 3 Γ Γ4 5Γ61 Low S 1Low S 4Low S 9Medium S 12 Yes No No No No No2 High S2Medium S 5Medium S 10High S13No Yes No No No No3 IF High S THEN2High S6Medium S 10High S13No No Yes No No No4 High S2Low S 4Medium S 10Low S 14No No No Yes No No5 High S2Higher S7Medium S 10Medium S 12 No No No No Yes No6 Higher S3Highest S8High S Higher S1115No No No No No Yes- 225 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special IssueTable 3. Symptom Tablex1 lowx1 highx1 higherx2 lowx2 mediumx2 highx2 higherx2 highestx3 lowx3 mediumx3 highx4 mediumx4 highx4 lowx4 higherS1S2S3S4S5S6S7S8S9S10S11S12S13S14S15Γ11 0 0 1 0 0 0 0 1 0 0 1 0 0 0Γ20 1 0 0 1 0 0 0 0 1 0 0 1 0 0Γ30 1 0 0 0 1 0 0 0 1 0 0 1 0 0Γ 0 1 0 1 0 0 0 0 0 1 0 0 0 1 04Γ50 1 0 0 0 0 1 0 0 1 0 1 0 0 0Γ60 0 1 0 0 0 0 1 0 0 1 0 0 0 1- the pattern is declared assigned to a class (inpossibility terms), if the membership grade to therespective class is larger than a confidencethreshold γ (here chosen equal to 0.6);- the pattern is declared ‘atypical’, if none of themembership grades is larger than γ ;- the pattern is declared ‘ambiguous’, if more thanone membership grade is larger than γ .A test on a set of 600 data has resulted in only 40.67%correct classifications to the six fault classes, while10.5% of the data are considered as atypical, 2.33% asambiguous and 46.5% are assigned to the wrong class.The obtained performance is obviously unacceptableand motivates the search for an optimal or near-optimalsequence of symptoms upon which to build the DT.The objective of the optimisation algorithm is to findthe sequence of symptoms that leads to the DT with thebest classification performance in terms of percentageof correct classifications. The number of possiblesequences of symptoms is 15! (~ 10 11 ), in Section 4,this combinatorial optimisation problem is tackled by asingle-objective genetic algorithm.- 226 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special IssueFigure 3. DT for classification, built with the ordered sequence of symptoms Σ04. Genetic algorithm optimisation of thedecision tree design, a procedure based on a single-objective genetic Inthis Section algorithm (Appendix A) is carried out fordetermining the sequence of symptoms to whichcorresponds the FDT with the maximum classificationperformance. The genetic algorithm can be seen asperforming a wrapper search [12] around theclassification algorithm (Figure 4): the symptomssequence selected during the search is evaluated usingas criterion (fitness) the percentage of correct classifieddata achieved by the FDT itself.The data and rules of the genetic algorithm search aregiven in Table 4. These parameters have beenestablished through a systematic procedure ofexperimentation. The objective (fitness) function to bemaximized is the percentage of correct dataclassifications; the decision variable is the symptomssequence.Table 4. GA run parametersNumber of chromosomes in thepopulationNumber of generations (terminationcriterion)SelectionReplacement10050Mutation probability 0.01Crossover probability (one-site) 1StandardRouletteChildren -ParentsFigure 4. Single-objective genetic algorithm “wrapper” searchEach chromosome is made up by 15 genes, one genefor each symptom. The single gene can assume anyinteger value in [0,15] that encodes the “swap”position of the symptom along the sequence. An- 227 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special Issueexample of a chromosome coding a particularsequence is given in Figure 5. To decode thechromosome in its corresponding symptom sequence, a15 – steps procedure is performed, one for each gene.At the generic step i = 1, 2,...,15 , the orderedsequence Σi−1and the value k contained in the i -thgene are considered: the symptom in the i -th positionof Σi−1is swapped with the symptom in the k -thposition of the sequence. For example in the first stepof Figure 5, the value 7 in gene 1 means that thesymptom S 1 is placed in position 7 of the sequenceand simultaneously the symptom that occupiedposition 7 is swapped to position 1. This operation iscarried out until the 15 th gene of the chromosome isworked out, leading to the final sequence:Σ = [ S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ]15 3 11 5 12 6 8 7 2 1 10 13 9 14 4 15Note that this original chromosome random designleads to a coherent symptom sequence, i.e. withoutrepetition of symptoms, thus avoiding computationallyburdensome chromosome coherence checking aposterior.The optimal sequence found at convergence of thegenetic algorithm is:Σ = [ S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ; S ]1 4 6 7 3 12 10 13 15 5 1 14 11 8 9 2The FDT built following this sequence ends into 46leaves and achieves a classification performance of91.34%, while 5.33% of the data are considered asatypical, 0.33% as ambiguous and only 3% areassigned to the wrong class.Figure 5. Example of a chromosome and the corresponding sequence- 228 -

E.Zio, P.Baraldi, I.Popescu Optimizing a fuzzy fault classification treeby a single-objective genetic algorithm - RTA # 3-4, 2007, December - Special Issue5. ConclusionIn realistic applications, fault classification is usuallybased on ambiguous information, which can beeffectively handled within a fuzzy logic framework. AFuzzy Decision Tree can then be built to logicallystructure the uncertain information available. To eachfault class corresponds a classification rule, withmono-dimensional Fuzzy Sets representing thecharacteristic symptoms for the corresponding faultclass.The classification performance by the resulting FDT isdependent on the order in which the symptoms areconsidered in the building procedure of the DT. Thisleads to an optimisation problem with respect to theconstruction of the tree. In this work, this problem hasbeen tackled by means of a single-objective geneticalgorithm in which the different sequences ofsymptoms are coded into the chromosomes of thegenetic population by an original procedure, whichguarantees coherence, i.e. no repetition of symptoms inthe sequence.The genetic algorithm-based optimisation proceduredeveloped has been successfully applied to a test caseregarding the development of Fuzzy Decision Trees forthe classification of artificial data. Undergoingresearch concerns the application of the developedclassification procedure to a real diagnostic problem.Appendix A: A brief recall of GeneticAlgorithmsIn the following, only a concise snapshot is providedon the basics of genetic algorithms optimisation. Foran extensive and detailed presentation of thiscomputational paradigm, the interested reader isinvited to consult the available copious literature [18],[17], [3], [6], [9], [4], [2].Genetic Algorithms (GAs) are optimisation methodsaiming at finding the global optimum of a set of realobjective functions, F ≡{ f (⋅) }, of one or more decisionvariables, U ≡ { u}, possibly subject to various linearor non linear constraints. Their main properties are thatthe search is conducted i) using a population ofmultiple solution points or candidates, ii) usingoperations inspired by the evolution of species, such asbreeding and genetic mutation, iii) using probabilisticoperations, iv) using only information on the objectiveor search function and not on its derivatives [16].GAs owe their name to their operational similaritieswith the biological and behavioural phenomena ofliving beings. After the pioneering theoretical work byJohn Holland [10], in the last decade a flourishingliterature has been devoted to their application to realproblems. The basics of the method may be found inGoldberg [8]; some applications in various contexts areincluded in Chambers [1].The terminology adopted in GAs contains many termsborrowed from biology, suitably redefined to fit thealgorithmic context. Thus, GAs operate on a set of(artificial) chromosomes, which are strings of numbers,generally sequences of binary digits 0 and 1. If theobjective function of the optimisation has manyarguments (typically called control factors or decisionvariables), each string is partitioned in as manysubstrings of assigned lengths, one for each argumentand, correspondingly, we say that each chromosome ispartitioned in (artificial) genes. The genes constitutethe so-called genotype of the chromosome and thesubstrings, when decoded in real numbers, constituteits phenotype. When the objective functions areevaluated in correspondence of a set of values of thecontrol factors of a chromosome, its values are calledthe fitness of that chromosome. Thus, eachchromosome gives rise to a trial solution to theproblem at hand in terms of a set of values of itscontrol factors.The GA search is performed by constructing asequence of populations of chromosomes, theindividuals of each population being the children ofthose of the previous population and the parents ofthose of the successive population. The initialpopulation is generated by randomly sampling the bitsof all the strings. At each step, the new population isthen obtained by manipulating the strings of the oldpopulation in order to arrive at a new populationhopefully characterized by increased mean fitness.This sequence continues until a termination criterion isreached. As for the natural selection, the stringmanipulation consists in selecting and mating pairs ofchromosomes in order to groom chromosomes of thenext population. This is done by repeatedly performingon the strings the four fundamental operations ofreproduction, crossover, replacement and mutation, allbased on random sampling: the parents’ selection stepdetermines the individuals which participate in thereproduction phase; reproduction itself allows theexchange of already existing genes whereas mutationintroduces new genetic material; the substitutiondefines the individuals for the next population. Thisway of proceeding enables to efficiently arrive atoptimal or near-optimal solutions.With regards to their performance, it is acknowledgedthat GAs takes a more global view of the search spacethan many other optimisation methods. The mainadvantages are i) fast convergence to near globaloptimum, ii) superior global searching capability incomplicated search spaces, iii) applicability even whengradient information is not readily achievable.- 229 -

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