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The joint distribution of sojourn times in finite Markov processes

Part of: Operations research and management science Markov processes

Published online by Cambridge University Press:  01 July 2016

Attila Csenki
Attila Csenki*
Affiliation:
Aston University
*
∗ Postal address: Department of Computer Science and Applied Mathematics, Aston University, Aston Triangle, Birmingham B4 7ET, UK.
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Abstract

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

Keywords

MARKOV CHAINMARKOV REWARD PROCESSSOJOURN TIMETRANSIENT ANALYSISRELIABILITY MODELLING

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 90B25: Reliability, availability, maintenance, inspection

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 141 - 160
Copyright
Copyright © Applied Probability Trust 1992 

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