Monotone Markov processes with respect to the reversed hazard rate ordering: An application to reliability. (English) Zbl 0981.60086

Bloch-Mercier, Sophie

J. Appl. Probab. 38, No. 1, 195-208 (2001).

First, the author derives some properties of reversed hazard rate monotone Markov processes with upper triangular generators. Then she considers a reliability system with a finite state space that evolves in time according to a Markov process up to its failure. Let \(1,2,\dots,m\) be the up-states of the system, and let \(m+ 1,m+ 2,\dots,m+ k\) be the down-states. Suppose that upon failure the system is repaired, and is brought from the present down-state into some (random) up-state, and then proceeds to evolve from there according to the Markov process as was described above. The repair duration is random. The author essentially shows that if the up-state distribution after a repair increases in the reversed hazard rate stochastic order, and if the Markov process is monotone in the reversed hazard rate stochastic order and has upper triangular generator, then its limiting availability decreases. It follows that under the above assumptions the limiting availability is maximized when the repair is perfect, namely, when the up-state distribution after a repair is degenerate at state 1.

Reviewer: Moshe Shaked (Tucson)

MSC:

60K20	Applications of Markov renewal processes (reliability, queueing networks, etc.)
60J35	Transition functions, generators and resolvents
60E15	Inequalities; stochastic orderings

Keywords:

reversed hazard rate monotone Markov processes; reliability system with a finite state space; limiting availability