The quasistationary distributions of homogeneous quasi-birth-and-death processes

Bean, N.G., Pollett, P.K. and Taylor, P.G.

Abstract: For evanescent Markov processes with a single transient communicating class, it is often of interest to examine the stationary probabilities that the process resides in the various transient states, conditional on absorption not having taken place. Such distributions are known as quasistationary distributions. In this paper we consider the determination of all the quasistationary distributions of a general level-independent quasi-birth-and-death process (QBD). These distributions are shown to have a form analogous to the quasistationary distributions exhibited by birth-and-death processes. We briefly discuss methods for the computation of these quasistationary distributions.

Keywords: Quasi-birth-and-death process; quasistationary distribution; limiting-conditional distribution.

Acknowledgement: This worked was funded by the Australian Research Council.

The authors:

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

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Last modified: 28th January 1997