Analysis of the asymmetric shortest queue problem : Part 1: theoretical analysis

In this paper we study a system consisting of two parallel servers with different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths. be joins the first queue with probability q and the second one with probability 1 - q. where q is an arbitrary number between 0 and 1. In a previous paper we showed that for the symmetric problem, that is for equal service rates and q = ½, the equilibrium distribution of the lengths of the two queues can be represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of this paper is to prove this product form result for the asymmetric problem by using a generalization of the compensation procedure. In a companion paper we show that the product form representation leads to a numerically highly attractive algorithm