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Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Quan-Lin Li  and
Yiqiang Q. Zhao
Quan-Lin Li*
Affiliation:
Tsinghua University
Yiqiang Q. Zhao*
Affiliation:
Carleton University
*
∗ Postal address: Department of Industrial Engineering, Tsinghua University, Beijing, 100084, P. R. China.
∗∗ Postal address: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6, Email address: zhao@math.carleton.ca
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Abstract

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In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Keywords

Markov chain of GI/G/1 typeMarkov chain of GI/M/1 typeMarkov chain of M/G/1 typeasymptotic analysislight tailcensoring methodRG-factorizationbatch Markov arrival process

MSC classification

Primary: 60K25: Queueing theory 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60J22: Computational methods in Markov chains
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 1075 - 1093
Copyright
Copyright © Applied Probability Trust 2005 

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