Computing distributions and moments in polling models by numerical transform inversion

Abstract

We show that probability distributions and moments of performance measures in many polling models can be effectively computed by numerically inverting transforms (generating functions and Laplace transforms). We develop new efficient iterative algorithms for computing the transform values and then use our recently developed variant of the Fourier-series method to perform the inversion. We also show how to use this approach to compute moments and asymptotic parameters of the distributions. We compute a two-term asymptotic expansion of the tail probabilities, which turns out to be remarkably accurate for small tail probabilities. The tail probabilities are especially helpful in understanding the performance of different polling disciplines. For instance, it is known that the exhaustive discipline produces smaller mean steady-state waiting times than the gated discipline, but we show that the reverse tends to be true for small tail probabilities. The algorithms apply to describe the transient behavior of stationary or non-stationary models as well as the steady-state behavior of stationary models. We demonstrate effectiveness by analyzing the computational complexity and by doing several numerical examples for the gated and exhaustive service disciplines, with both zero and non-zero switchover times. We also show that our approach applies to other polling models. Our main focus is on computing exact tail probabilities and asymptotic approximations to them, which seems not to have been done before. However, even for mean waiting times, our algorithm is faster than previous algorithms for large models. The computational complexity of our algorithm is O( N α ) for computing performance measures at one queue and O( N 1+ α ) for computing performance measures at all queues, where N is the number of queues and α is typically between 0.6 and 0.8.