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On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Qi-Ming He  and
Marcel F. Neuts
Qi-Ming He*
Affiliation:
Dalhousie University
Marcel F. Neuts*
Affiliation:
University of Arizona
*
∗ Postal address: Department of Industrial Engineering, DalTech, Dalhousie University, Halifax, Nova Scotia, Canada B3J 2X4. Email address: [email protected]
∗∗ Postal address: Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA.
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Abstract

We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.

Keywords

matrix analytic methodsquasi-birth-and-death Markov chainnonnegative matrix

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 519 - 541
Copyright
Copyright © Applied Probability Trust 2001 

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