An asymmetric shortest queue problem

In this paper we study a system consisting of two identical servers, each with exponentially distributed service times. Jobs arrive according to a Poisson stream. On arrival a job joins the shortest queue and in case both queues have equal length. he joins queue 1 say with probability q and queue 2 with probability 1-q, where q is not necessarily equal to ½ (which corresponds to the symmetric shortest queue problem) but arbitrary between 0 and 1. We show that the stationary queue length distribution can be represented by an infinite sum of product form solutions, which satisfy nice recurrence relations. Due to the recurrence relations. the successive terms of the infinite sum can be easily calculated. Moreover, the convergence of the infinite sum is exponentially fast. Based on these properties, a numerically highly attractive algorithm is obtained.