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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 (aspects of probability theory)
90B22 Queues and service in operations research
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