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Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Allan J. Motyer  and
Peter G. Taylor
Allan J. Motyer*
Affiliation:
University of Melbourne
Peter G. Taylor*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
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Abstract

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We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Keywords

Decay rateQBD processcountable phasetridiagonal block generatorstationary distributiontwo-node Jackson network

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 522 - 544
Copyright
Copyright © Applied Probability Trust 2006 

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