Fairness and efficiency for polling models with the κκ-gated service discipline

We study a polling model in which we want to achieve a balance between the fairness of the waiting times and the efficiency of the system. For this purpose, we introduce a novel service discipline: the κκ-gated service discipline. It is a hybrid of the classical gated and exhausted disciplines, and consists of using κiκi consecutive gated service phases at queue ii before the server switches to the next queue. The advantage of this discipline is that the parameters κiκi can be used to balance fairness and efficiency. We derive the distributions and means of the waiting times, a pseudo conservation law for the weighted sum of the mean waiting times, and the fluid limits of the waiting times. Our goal is to optimize the κiκi so as to minimize the differences in the mean waiting times, i.e. to achieve maximal fairness, without giving up too much on the efficiency of the system. From the fluid limits we derive a heuristic rule for setting the κiκi. In a numerical study, the heuristic is shown to perform well in most cases.