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Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability

Part of: Special processes Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Sophie Bloch-Mercier
Sophie Bloch-Mercier*
Affiliation:
Université de Marne-la-Vallée
*
∗ Postal address: Equipe d'Analyse et de Mathématiques Appliquées, Cité Descartes, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex, France. Email address: merciers@univ-mlv.fr
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Abstract

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

Keywords

reliabilityreversed hazard rate orderingmonotone Markov processesstationary availability

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 60J35: Transition functions, generators and resolvents
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 195 - 208
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Anderson, W. J. (1991) Continuous-Time Markov Chains: an Applications-Oriented Approach. Springer, New York.Google Scholar
Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing (International Series in Decision Processes). Holt, Rinehart and Winston, Inc., New York.Google Scholar
Block, H. W., Savits, T. H., and Singh, H. (1998). The reversed hazard rate function. Prob. Eng. Inf. Sci. 12, 69–90.Google Scholar
Brown, M., and Chaganty, N. R. (1983). On the first passage times distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.CrossRefGoogle Scholar
Cheng, D. W., and Righter, R. (1995). On the order of tandem queues. Queueing Systems 21, 143–160.CrossRefGoogle Scholar
Ҫinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Cocozza-Thivent, C. (1997). Processus Stochastiques et Fiabilité des Systèmes (Mathématiques et Applications 28). Springer, Berlin.Google Scholar
Eeckhoudt, L., and Gollier, C. (1995). Demand for risky assets and the monotone probability ratio order. J. Risk Uncertainty 11, 113–122.Google Scholar
Joag-dev, K., Kochar, S., and Proschan, F. (1995). A general composition theorem and its applications to certain partial orderings of distributions. Statist. Prob. Lett. 22, 111–119.CrossRefGoogle Scholar
Karasu, I. and Özekici, S. (1989). NBUE and NWUE properties of increasing Markov processes. J. Appl. Prob. 26, 827–834.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press.Google Scholar
Keilson, J., and Kester, A. (1977). Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.CrossRefGoogle Scholar
Keilson, J., and Sumita, U. (1982). Uniform stochastic ordering and related inequalities. Can. J. Statist. 10, 181–198.CrossRefGoogle Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modelling. Chapman and Hall, London.Google Scholar
Kijima, M. (1998). Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains. J. Appl. Prob. 35, 545–556.CrossRefGoogle Scholar
Li, H., and Shaked, M. (1995). On the first passage times for Markov processes with monotone convex transition kernels. Stoch. Proc. Appl. 58, 205–216.Google Scholar
Li, H., and Shaked, M. (1997). Ageing first-passage times of Markov processes: a matrix approach. J. Appl. Prob. 34, 1–13.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1987). IFRA properties of some Markov processes with general state space. Math. Operat. Res. 12, 562–568.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1988). On the first-passage times of pure jump processes. J. Appl. Prob. 25, 501–509.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications. Academic Press, Boston.Google Scholar
Shanthikumar, J. G. (1988). DFR property of first-passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.Google Scholar
Shanthikumar, J. G., Yamazaki, G., and Sakasegawa, H. (1991). Characterization of optimal order of services in a tandem queue with blocking. Operat. Res. Lett. 10, 17–22.CrossRefGoogle Scholar
Stoyan, D. (1983). Comparison Methods for Queues and other Stochastic Models. John Wiley, Chichester.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.Google Scholar