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

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.