A Functional Equation Arising in a Queue with a Gating Mechanism

Our primary aim in this paper is to study a functional equation that arises in a problem of queueing. Consider a queue with compound Poisson arrivals and general service times with a gating mechanism. The gating mechanism takes in at most m (= proportional to) customers at a time for service and serves these customers according to the processor sharing discipline. In this paper, we examine various performance characteristics for this queue. The characteristics include waiting time distribution, queue length distribution, time spent with the server and batch size distribution for service. The generating function of the batch size distribution (for m= proportional to) for service gives rise to a functional equation which is the main focus of this paper. For this functional equation, we find an exact solution from which the batch size distribution can be obtained by numerical inversion. For a special case, we find an explicit analytic solution. For other cases, we propose an easily computable approximation whose performance we compare against the exact solution. The approximation proposed in the paper is closely related to a one dimensional continuous time branching process. In addition, we find the queue length distribution for this queue. The queue studied in this paper has applications to computer and communications systems.