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Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/$n$ queues. (English) Zbl 1065.60134
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:
 60K25 Queueing theory 90B22 Queues and service (optimization)