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Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/n queues

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Zhenting Hou  and
Yuanyuan Liu
Zhenting Hou*
Affiliation:
Central South University, Changsha
Yuanyuan Liu
Affiliation:
Central South University, Changsha
*
∗ Postal address: School of Mathematics, Central South University, Changsha, Hunan 410075, P. R. China. Email address: zthou@csu.edu.cn
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Abstract

This paper investigates the rate of convergence to the probability distribution of the embedded M/G/1 and GI/M/n queues. We introduce several types of ergodicity including l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity and strong ergodicity. The usual method to prove ergodicity of a Markov chain is to check the existence of a Foster–Lyapunov function or a drift condition, while here we analyse the generating function of the first return probability directly and obtain practical criteria. Moreover, the method can be extended to M/G/1- and GI/M/1-type Markov chains.

Keywords

Queueing theoryl-ergodicitygeometric ergodicityuniformly polynomial ergodicitystrong ergodicitygenerating functionanalytic

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 778 - 790
Copyright
Copyright © Applied Probability Trust 2004 

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