Journal of Applied Probability

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

Sophie Bloch-Mercier

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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.

Article information

J. Appl. Probab. Volume 38, Number 1 (2001), 195-208.

First available in Project Euclid: 5 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60E15: Inequalities; stochastic orderings

reliability reversed hazard rate ordering monotone Markov processes stationary availability


Bloch-Mercier, Sophie. Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability. J. Appl. Probab. 38 (2001), no. 1, 195--208. doi:10.1239/jap/996986653.

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