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On the Problem of Establishing the Existence of Stationary Distributions for Continuous-Time Markov Chains

Published online by Cambridge University Press:  27 July 2009

P. K. Pollett  and
P. G. Taylor
P. K. Pollett
Affiliation:
Department of Mathematics, The University of Queensland, Queensland 4072, Australia
P. G. Taylor
Affiliation:
Department of Applied Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide, South Australia 5001, Australia
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Abstract

We consider the problem of establishing the existence of stationary distributions for continuous-time Markov chains directly from the transition rates Q. Given an invariant probability distribution m for Q, we show that a necessary and sufficient condition for m to be a stationary distribution for the minimal process is that Q be regular. We provide sufficient conditions for the regularity of Q that are simple to verify in practice, thus allowing one to easily identify stationary distributions for a variety of models. To illustrate our results, we shall consider three classes of multidimensional Markov chains, namely, networks of queues with batch movements, semireversible queues, and partially balanced Markov processes.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 4 , October 1993 , pp. 529 - 543
Copyright
Copyright © Cambridge University Press 1993

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