Strong stability of a Markov chain embedded in the system M/G/1. (Russian) Zbl 0577.60088

Ajssani, D.; Kartashov, N. V.

Teor. Veroyatn. Mat. Stat. 29, 3-7 (1983).

The authors consider the Markov chain \(X_ n=(q_ n,\gamma_ n)\) in the system G/G/1, where \(q_ n\) is the number of customers in the moment, when the nth one is served, and, starting from this moment, \(\gamma_ n\) means the time to arrival of the next customer. It is shown, that the characteristics of \(X_ n\) are close to the characteristics of the referring chain in M/G/1, under the condition of a Poisson input and the same serving time distributions in both systems.
Obtained is the following theorem: If a condition of geometrical ergodicity is fulfilled in M/G/1, than the distribution of \(X_ n\) in G/G/1 is stationary for every G in some neighbourhood of the exponential distribution E. An estimate for the deviation of the distribution of \(q_ n\) for \(G\neq E\) is given. These results follow from the strong stability of \(X_ n\) in M/G/1.

Reviewer: C.Baldauf

MSC:

60K25	Queueing theory (aspects of probability theory)
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

geometrical ergodicity; strong stability