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

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.