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Computing Stationary Expectations in Level-Dependent QBD Processes

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Hendrik Baumann  and
Werner Sandmann
Hendrik Baumann*
Affiliation:
Clausthal University of Technology
Werner Sandmann*
Affiliation:
Clausthal University of Technology
*
∗ Postal address: Department of Applied Stochastics and Operations Research, Clausthal University of Technology, Erzstr. 1, D-38678, Clausthal-Zellerfeld, Germany.
∗ Postal address: Department of Applied Stochastics and Operations Research, Clausthal University of Technology, Erzstr. 1, D-38678, Clausthal-Zellerfeld, Germany.
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Abstract

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Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

Keywords

Level-dependent quasi-birth-and-death processmatrix-analytic computationlong-run average additive functionalstationary expectationmemory-efficient algorithm

MSC classification

Primary: 60J22: Computational methods in Markov chains
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 151 - 165
Copyright
Copyright © Applied Probability Trust 2013 

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