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Perturbation analysis for denumerable Markov chains with application to queueing models

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Eitan Altman ,
Konstantin E. Avrachenkov  and
Rudesindo Núñez-Queija
Eitan Altman*
Affiliation:
INRIA
Konstantin E. Avrachenkov*
Affiliation:
CWI and Eindhoven University of Technology
Rudesindo Núñez-Queija*
Affiliation:
CWI and Eindhoven University of Technology
*
∗ Postal address: INRIA Sophia Antipolis, 2004 Route des Lucioles, BP 93, Sophia Antipolis Cedex 06902, France.
∗ Postal address: INRIA Sophia Antipolis, 2004 Route des Lucioles, BP 93, Sophia Antipolis Cedex 06902, France.
∗∗∗∗ CWI, PO Box 94079, Amsterdam, 1090 GB, The Netherlands. Email address: sindo@cwi.nl
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Abstract

We study the parametric perturbation of Markov chains with denumerable state spaces. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, with several ergodic classes, are perturbed such that (rare) transitions among the different ergodic classes of the unperturbed chain are allowed. Singularly perturbed Markov chains have been studied in the literature under more restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. We relax these conditions so that our results can be applied to queueing models (where the conditions mentioned above typically fail to hold). Assuming ν-geometric ergodicity, we are able to explicitly express the steady-state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our results to quasi-birth-and-death processes and queueing models.

Keywords

Denumerable Markov chainperturbation analysisgeometric ergodicityquasi-birth-and-death processqueueing model

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J22: Computational methods in Markov chains 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 839 - 853
Copyright
Copyright © Applied Probability Trust 2004 

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