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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 -
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