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On the asymptotic behaviour of the stationary distribution of Markov chains of \(M/G/1\)-type. (English) Zbl 0791.60087
Summary: The asymptotic behaviour of the stationary distribution of Markov chains of the \(M/G/1\)-type is investigated. Necessary and sufficient conditions for an asymptotic geometric behaviour are given. In the case of Markov chains arising in single server models with a general Markovian arrival process and general service times these conditions turn out to be simple and the coefficients of an asymptotically geometric expansion can be calculated easily. Some numerical results concerning the stationary distribution of embedded Markov chains in \(Ph/G/1\) models are presented.

MSC:
60K25 Queueing theory (aspects of probability theory)
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
90B22 Queues and service in operations research
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