b-invariant measures for level-dependent quasi-birth-and-death processes

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

Abstract: We shall consider the problem of identifying the b-invariant measures and hence the quasistationary distributions of an absorbing level-dependent quasi-birth-and-death process (QBD), that is, an absorbing Markov chain with a block-tridiagonal transition matrix P. We examine successive lower truncations of P, obtained by removing rows and columns corresponding to levels.

The crucial factors in our technique are the Perron-Frobenius eigenvalue of a fundamental matrix and the sequence of convergence norms of the successive lower truncations: is the convergence norm of the transient class. We construct a -invariant measure for all . When , we show that a QBD admits one of two types of -invariant measure: which type depends on whether or . Together with a knowledge of whether or , this is sufficient to give the -classification of the process.

Keywords: Quasi-birth-and-death process; -invariant measures. Quasi-birth-and-death process.

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