On the problem of establishing the existence of stationary distributions for continuous-time Markov chains

Phil Pollett and Peter Taylor

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 which 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 multi-dimensional Markov chains, namely networks of queues with batch movements, semi-reversible queues and partially balanced Markov processes.

AMS 1991 Subject Classification: 60J27; 60K30; 90B22.

Keywords: Regularity; positive recurrence.

Acknowledgement: This worked was funded by the Australian Research Council and the University of Queensland.

The authors:

• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au
• Peter Taylor, Department of Applied Mathematics, The University of Adelaide. ptaylor@maths.adelaide.edu.au

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Last modified: 26th December 1995