Asymptotic representations in the ergodic theorem for generalized Markov chains and their applications. (English. Russian original) Zbl 0632.60076

Kartashov, N. V.

Theory Probab. Math. Stat. 32, 131-139 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 113-121 (1985).

Let \(X=(X_ t\), \(t\geq 0)\) be a Markov chain with values in a measurable space E. The author, who introduced the concept of a uniform ergodic chain X with respect to a norm [ibid. 30, 65-81 (1984; Zbl 0562.60070); English translation in Theory Probab. Math. Stat. 30, 71-89 (1985)], obtains here asymptotic representations for the transition probabilities of general Markov chains in terms of exponential order of decrease. He shows that, in the general case, the exponent cannot be made smaller and gives applications of these results to imbedded Markov chains in M/G/1 and G/M/1 queueing systems.
Very clearly exposed and full of important questions, this work is of real interest for specialists.

Reviewer: G.G.Vrânceanu

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60K15	Markov renewal processes, semi-Markov processes
60K25	Queueing theory (aspects of probability theory)
90B22	Queues and service in operations research
28D99	Measure-theoretic ergodic theory

Keywords:

uniform ergodic chain; asymptotic representations for the transition probabilities

Citations:

Zbl 0562.60070