Fairness and efficiency for polling models with the k-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 @k-gated service discipline. It is a hybrid of the classical gated and exhausted disciplines, and consists of using @k"i consecutive gated service phases at queue i before the server switches to the next queue. The advantage of this discipline is that the parameters @k"i 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 @k"i 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 @k"i. In a numerical study, the heuristic is shown to perform well in most cases.