Queueing System with Selection of the Shortest of Two Queues: An Asymptotic Approach

Consider a system that consists of $N$ servers with a Poisson input flow of demands of intensity $N\lambda$. Each demand arriving to the system randomly selects two servers and is instantly sent to the one with the shorter queue. The service time is distributed exponentially with mean 1. It turns out that for $\lambda<1$ it is possible to investigate the asymptotic distribution of the queue lengths as $N\to\infty$. In the limit the queue length probability decreases superexponentially as the queue length increases.