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

Zhenting Hou and Yuanyuan Liu
Source: J. Appl. Probab. Volume 41, Number 3 (2004), 778-790.

#### 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.

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Primary Subjects: 60K25
Secondary Subjects: 90B22
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Permanent link to this document: http://projecteuclid.org/euclid.jap/1091543425
Digital Object Identifier: doi:10.1239/jap/1091543425
Mathematical Reviews number (MathSciNet): MR2074823
Zentralblatt MATH identifier: 1065.60134

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