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Stochastic and substochastic solutions for infinite-state Markov chains with applications to matrix-analytic methods

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Winfried K. Grassmann  and
Javad Tavakoli
Winfried K. Grassmann*
Affiliation:
University of Saskatchewan
Javad Tavakoli*
Affiliation:
University of British Columbia Okanagan
*
∗ Postal address: Department of Computer Science, University of Saskatchewan, 101 Science Court, Saskatoon SK S7N 5C9, Canada. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Statistics and Physics, University of British Columbia Okanagan, 3333 University Way, Kelowna, Canada V1V 1V7. Email address: [email protected]
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Abstract

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This paper deals with censoring of infinite-state banded Markov chains. Censoring involves reducing the time spent in states outside a certain set of states to 0 without affecting the number of visits within this set. We show that, if all states are transient, there is, besides the standard censored Markov chain, a nonstandard censored Markov chain which is stochastic. Both the stochastic and the substochastic solutions are found by censoring a sequence of finite transition matrices. If all matrices in the sequence are stochastic, the stochastic solution arises in the limit, whereas the substochastic solution arises if the matrices in the sequence are substochastic. We also show that, if the Markov chain is recurrent, the only solution is the stochastic solution. Censoring is particularly fruitful when applied to quasi-birth-and-death (QBD) processes. It turns out that key matrices in such processes are not unique, a fact that has been observed by several authors. We note that the stochastic solution is important for the analysis of finite queues.

Keywords

Infinite-state Markov chainmatrix-analytic methodnonrecurrent Markov chainqueueing

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1157 - 1173
Copyright
Copyright © Applied Probability Trust 2008 

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