Short Communications

The problem of the distribution of the maximal queue size and its application

O. P. Vinogradov

Moscow

Abstract: In this paper we deal with the system $GI/M/1$. Let $\tau_n$ be the first time when the size of the queue equals $n$. An expression for $\mathbf Me^{-s\tau}n$ is obtained and the asymptotic behaviour of $\mathbf M\tau_n$ as $n\to\infty$ is studied. We prove also limit theorems for $\tau_n/\mathbf M\tau_n$ ($n\to\infty$). These results enable to analyse the system which has at most $n$ customers simultaneously.

Received: 30.01.1967

 English version: Theory of Probability and its Applications, 1968, 13:2, 346–353

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