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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: [email protected]
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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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