Stochastic bounds for a polling system

Abstract

In this note we consider two queueing systems: a symmetric polling system with gated service at all N queues and with switchover times, and a single-server singlequeue model with one arrival stream of ordinary customers and N additional permanently present customers. It is assumed that the combined arrival process at the queues of the polling system coincides with the arrival process of the ordinary customers in the single-queue model, and that the service time and switchover time distributions of the polling model coincide with the service time distributions of the ordinary and permanent customers, respectively, in the single-queue model. A complete equivalence between both models is accomplished by the following queue insertion of arriving customers. In the single-queue model, an arriving ordinary customer occupies with probability Pi a position at the end of the queue section behind the ith permanent customer, i = I,..., N. In the cyclic polling model, an arriving customer with probability Pi joins the end of the ith queue to be visited by the server, measured from its present position. For the single-queue model we prove that, if two queue insertion distributions {Pi, i = 1 ..... N} and {qi, i = 1 ..... N} are stochastically ordered, then also the workload and queue length distributions in the corresponding two single-queue versions are stochastically ordered. This immediately leads to equivalent stochastic orderings in polling models. Finally, the single-queue model with Poisson arrivals and Pl = 1 is studied in detail.