Abstract
In this paper we consider semi-Markov reliability models of systems with discrete state space in a setup general enough to cover systems with maintenance and repair. The systems are assumed to consist of several components which can either be up or down in each state. In this framework we propose two different types of component importance measures which are based on transition rates and interval availability, respectively. For these importance measures we study both the time-dependent and the steady state situation, and express them in terms of quantities easily calculated from the building blocks of the semi-Markov process.
Similar content being viewed by others
References
Aven T, Nøkland TE (2010) On the use of uncertainty importance measures in reliability and risk analysis. Reliab Eng Syst Safety 95: 127–133
Barlow RE (1960) Applications of semi-Markov processes to counter and reliability problems. Stanford University, Thesis
Barlow RE, Proschan F (1965) Mathematical theory of reliability. Wiley, New York
Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Probability models. Holt, Rinehart and Winston, New York
Barlow RE, Proschan F (1975) Importance of system components and fault tree events. Stoch Proc Appl 3: 153–173
Birnbaum ZW, Esary JD, Saunders SC (1961) Multi-component systems and structures and their reliability. Technometrics 3: 55–77
Birnbaum ZW (1969) On the importance of different components in a multicomponent system. In: Krishnaiah PR (eds) Multivariate analysis II. Academic Press, New York
Boland PJ, El-Neweihi E (1995) Measures of component importance in reliability theory. Comput Oper Res 22: 455–463
Cheok MC, Parry GW, Sherry RR (1998) Use of importance measures in risk-informed regulatory applications. Reliab Eng Syst Safety 60: 213–226
Çinlar E (1969) Markov renewal theory. Adv Appl Prob 69: 123–187
Csenki A (1994) Dependability for systems with a partitioned state space: Markov and semi-Markov theory and computational implementation. Lecture Notes in Statistics, vol 90. Springer, New York
Csenki A (1995) An integral equation approach to the interval reliability of systems modelled by a finite semi-Markov process. Reliab Eng Syst Safety 47: 37–45
Do Van P, Barros A (2008) Reliability importance analysis of Markovian systems at steady state using perturbation analysis. Reliab Eng Syst Safety 93: 1605–1615
Do Van P, Barros A, Bérenguer C (2008) Importance measure on finite time horizon and application to Markovian multistate production systems. Proc Inst Mech Eng Part O J Risk Reliab 222: 449–461
Do Van P, Barros A, Bérenguer C (2010) From differential to difference importance measures for Markov reliability models. Eur J Oper Res 204: 513–521
Dutuit Y, Rauzy A (2001) Efficient algorithms to assess component and gate importance in fault tree analysis. Reliab Eng Syst Safety 72: 213–222
Gupta V, Dharmaraja S (2011) Semi-Markov modeling of dependability of VoIP network in the presence of resource degradation and security attacks. Reliab Eng Syst Safety 47: 1627–1636
Hong JS, Koo HY, Lie CH (2000) Computation of joint reliability importance of two gate events in a fault tree. Reliab Eng Syst Safety 68: 1–5
Huseby AB (2004) Importance measures for multicomponent binary systems. Statistical research report no. 11, Department of Mathematics, University of Oslo
Jung WS, Cho NZ (1995) Semi-Markov reliability analysis of three test/repair policies for standby safety systems in a nuclear power plant. Reliab Eng Syst Safety 31: 1–30
Korolyuk VS, Tomusyak AA (1965) Description of the functioning of reserve systems by means of semi-Markov processes. Kibernetika 5: 55–59
Korolyuk VS, Brodi SM, Turbin AF (1975) Semi-Markov processes and their applications. J Math Sci 4: 244–280
Limnios N (1997) Dependability analysis of semi-Markov systems. Reliab Eng Syst Safety 55: 203–207
Limnios N (2011) Reliability measures of semi-Markov systems with general state space. Methodol Comput Appl Probab. doi:10.1007/s11009-011-9211-5
Limnios N, Oprişan G (2001) Semi-Markov processes and reliability. Birkhäuser, Boston
Natvig B (1979) A suggestion of a new measure of importance of system components. Stoch Proc Appl 9: 319–330
Natvig B (1982) On the reduction of the remaining system lifetime due to the failure of a specific component. J Appl Prob 19:642–652 (Correction: J Appl Prob 20:713, 1983)
Natvig B (1985) New light on measures of importance of system components. J Scand Stat 12: 43–54
Natvig B, Gåsemyr J (2009) New results on the Barlow–Proschan and Natvig measures of component importance in nonrepairable and repairable systems. Methodol Comput Appl Probab 11: 603–620
Natvig B (2011) Measures of component importance in nonrepairable and repairable multistate systems. Methodol Comput Appl Probab 13: 523–547
Nollau V (1980) Semi-Markovsche Prozesse. Thun, Harri Deutsch (in German)
Ouhbi B, Limnios N (2002) The rate of occurrence of failures for semi-Markov processes and estimation. Stat Probab Lett 59: 245–255
Pyke R (1961) Markov renewal processes with finitely many states. Ann Math Stat 32: 1243–1259
Rubino G, Sericola B (1992) Interval availability analysis using operational periods. Perf Eval 14: 257–272
Störmer H (1970) Semi-Markoff-Prozesse mit endlich vielen Zuständen. Lecture Notes in Operations Research and Mathematical Systems, vol 34. Springer, Berlin (in German)
Vulpe A, Carausu A (2005) Markovian and semi-Markov models for availability evaluation of NPP subsystems and equipment. In: Proceedings of the 18th international conference on structural mechanics in reactor technology. SMiRT 18-M02-3, pp 3833–3842
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hellmich, M., Berg, HP. On the construction of component importance measures for semi-Markov systems. Math Meth Oper Res 77, 15–32 (2013). https://doi.org/10.1007/s00186-012-0413-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-012-0413-6