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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 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 -
Page 1 and 2: ISSN 1932-2321RELIABILITY:Theory &
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