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Countable-state-space Markov chains with two time scales and applications to queueing systems

Part of: Asymptotic theory Markov processes

Published online by Cambridge University Press:  01 July 2016

G. Yin  and
Hanqin Zhang
G. Yin*
Affiliation:
Wayne State University
Hanqin Zhang*
Affiliation:
Academia Sinica, Beijing
*
∗ Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: gyin@math.wayne.edu
∗∗ Postal address: Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, China.
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Abstract

Motivated by various applications in queueing systems, this work is devoted to continuous-time Markov chains with countable state spaces that involve both fast-time scale and slow-time scale with the aim of approximating the time-varying queueing systems by their quasistationary counterparts. Under smoothness conditions on the generators, asymptotic expansions of probability vectors and transition probability matrices are constructed. Uniform error bounds are obtained, and then sequences of occupation measures and their functionals are examined. Mean square error estimates of a sequence of occupation measures are obtained; a scaled sequence of functionals of occupation measures is shown to converge to a Gaussian process with zero mean. The representation of the variance of the limit process is also explicitly given. The results obtained are then applied to treat Mt/Mt/1 queues and Markov-modulated fluid buffer models.

Keywords

Markov chainsingular perturbationcountable state spaceasymptotic expansionoccupation measurefunctional central limit theoremMt/Mt/1 queuefluid model

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 34E05: Asymptotic expansions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 662 - 688
Copyright
Copyright © Applied Probability Trust 2002 

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