A Decomposition Theorem for Polling Models: The Switchover Times are Effectively Additive

Abstract

We consider the classical polling model: queues served in cyclic order with either exhaustive or gated service, each with its own distinct Poisson arrival stream, service-time distribution, and switchover-time (the server's travel time from that queue to the next) distribution. Traditionally, models with zero switchover times (the server travels at infinite speed) and nonzero switchover times have been considered separately because of technical difficulties reflecting the fact that in the latter case the mean cycle time approaches zero as the travel speed approaches infinity. We argue that the zero-switchover-times model is the more fundamental model: the mean waiting times in the nonzero-switchover-times model decompose (reminiscent of vacation models) into a sum of two terms, one being a simple function of the sum of the mean switchover times, and the other the mean waiting time in a “corresponding” model obtained from the original by setting the switchover times to zero and modifying the service-time variances. This generalizes a recent result of S. W. Fuhrmann for the case of constant switchover times, where no variance modification is necessary. The effect of these studies is to reduce computation and to improve theoretical understanding of polling models.

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