Polling in a Closed Network

Abstract

We consider a closed queueing network with a fixed number of customers, where a single server moves cyclically between N stations, rending service in each station according to some given discipline (Gated, Exhaustive, or the Globally Gated regime). When service of a customer (message) ends in station j, it is routed to station k with probability Pjk. We derive explicit expressions for the probability generating function and the moments of the number of customers at the various queues at polling instants and calculate the mean cycle duration and throughput for each service discipline. We then obtain the first moments of the queues' length at an arbitrary point in time. A few examples are given to illustrate the analysis. Finally, we address the problem of optimal dynamic control of the order of stations to be served.