Waiting Time Variance in Polling Systems

In this paper, we develop a general expression for the second moment of waiting time for polling systems with either an exhaustive or gated service policy. We obtain a closed form expression for the second moment of waiting time for the special case of systems with two identical stations. This result is used to show that the behavior of the first two moments can be very different. Specifically, we show that while the expected waiting time is always smaller under the exhaustive policy, the same does not hold for waiting time variance. In fact, we characterize conditions for which the waiting time variance under the exhaustive policy is infinitely larger than the variance under the gated policy. For systems with more than two stations, we present computational results to extend these conclusions. We use these results to show that the difference in waiting time variance between the two policies is not monotonic in the number of stations. For asymmetric systems, we also show that the gated policy can have, for certain stations, both a smaller mean and variance of waiting time.