Analysis of the asymmetric shortest queue problem with threshold jockeying

In this paper we study a system consisting of two parallel servers with possibly 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, he joins the first queue with probability 1 - a and the second one with probability a, where a is an arbitrary number between 0 and 1. If the difference between the lengths of both queues reaches some threshold value T, then one job switches from the longest to the shortest queue. It is shown that the equilibrium probabilities of the queue lengths satisfy a product form for states where the longer queue exceeds the threshold value T. Furthermore, it is shown that for a sensible partitioning of the state space the matrix-geometric approach is essentially equivalent with our approach