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Zbl 1065.60134
Hou, Zhenting; Liu, Yuanyuan
Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/$n$ queues. (English)
[J] J. Appl. Probab. 41, No. 3, 778-790 (2004). ISSN 0021-9002

Summary: 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.

MSC 2000:: *60K25 Queueing theory
90B22 Queues and service

Keywords: $l$-ergodicity; geometric ergodicity; uniformly polynomial ergodicity; strong ergodicity; generating function

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