Matrix-geometric analysis of the shortest queue problem with threshold jockeying

In this paper we study a system consisting of c parallel servers with possibly different service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. An arriving job joins the shortest queue, where in case of multiple shortest queues, one of these queues is selected according to some arbitrary probability distribution. If the maximum difference between the lengths of the c queues exceeds some threshold value T, then one job switches from the longest to the shortest queue, where in case of multiple longest queues, the queue loosing a job is selected according to some arbitrary probability distribution. It is shown that the matrix-geometric approach is very well suited to find the equilibrium probabilities of the queue lengths. The interesting point is that a proper choice for the state space partitioning depends on the aspect one is interested in. Using one partitioning of the state space an explicit ergodicity condition can be derived from Neuts' mean drift condition and using another partitioning the associated R-matrix can be determined explicitly. Moreover, both partitionings used are different from the one suggested by the conventional way of applying the matrix-geometric approach. Therefore, the paper can be seen as a plea for giving more attention to the question of the selection of a partitioning in the matrix-geometric approach.