Asymptotic Behavior of the Stationary Distribution for a Closed Queueing System

We consider a closed queueing system consisting of $M$ identical servers with fixed unit service time. The number of customers is fixed and equal to $N$. Each served customer is instantaneously routed with equal probability to one of $M$ servers in the system (or is enqueued if the server is busy). An asymptotic result is proved for the stationary distribution of the queueing process as $N,M\to\infty$, $N/M\to\nu=\mathrm{const}$, and also a result on deterministic approximation of the process on a finite time interval.