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Simple formulae for counting processes in reliability models

Part of: Operations research and management science Markov processes

Published online by Cambridge University Press:  01 July 2016

James Ledoux  and
Gerardo Rubino
James Ledoux*
Affiliation:
INSA
Gerardo Rubino*
Affiliation:
ENST
*
*Postal address: INSA, Campus de Beaulieu 35043 Rennes Cédex, France. Email: ledoux@{univrennes1}{irisa}.fr
**Postal address: ENST, rue de la Châtaigneraie, 35512 Cesson-Sevigné Cédex, France. Email: rubino@{rennes.enst-bretagne}{irisa}.fr
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Abstract

Dependability evaluation is a basic component in the assessment of the quality of repairable systems. We develop a model taking simultaneously into account the occurrence of failures and repairs, together with the observation of user-defined success events. The model is built from a Markovian description of the behavior of the system. We obtain the distribution function of the joint number of observed failures and of delivered services on a fixed mission period of the system. In particular, the marginal distribution of the number of failures can be directly related to the distribution of the Markovian arrival process extensively used in queueing theory. We give both the analytical expressions of the considered distributions and the algorithmic solutions for their evaluation. An asymptotic analysis is also provided.

Keywords

COUNTING PROCESSESMARKOV CHAINSUNIFORMIZATION

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90B25: Reliability, availability, maintenance, inspection
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 1018 - 1038
Copyright
Copyright © Probability Trust 1997 

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