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Poisson approximation for some point processes in reliability

Part of: Stochastic processes Markov processes Stochastic systems and control Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Jean-Bernard Gravereaux  and
James Ledoux
Jean-Bernard Gravereaux*
Affiliation:
INSA and IRMAR, Rennes
James Ledoux*
Affiliation:
INSA and IRMAR, Rennes
*
∗ Postal address: Centre de Mathématiques, INSA, 20 avenue des Buttes de Coësmes, 35043 Rennes Cedex, France
∗ Postal address: Centre de Mathématiques, INSA, 20 avenue des Buttes de Coësmes, 35043 Rennes Cedex, France
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Abstract

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

Keywords

Compensatorsoftware reliabilityMarkovian-arrival processdoubly stochastic process

MSC classification

Primary: 60G55: Point processes 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J75: Jump processes 90B25: Reliability, availability, maintenance, inspection 93E11: Filtering
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 455 - 470
Copyright
Copyright © Applied Probability Trust 2004 

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