Source: J. Appl. Probab.
Volume 41, Number 3
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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