Heavy-traffic single-server queues and the transform method

TL;DR

This paper extends the transform method to analyze heavy-traffic queues, providing convergence of moments and applying it to nearly deterministic queues, with numerical validation of approximations.

Contribution

It advances the transform method for heavy-traffic analysis by including moment convergence and error bounds, and applies it to nearly deterministic queues.

Findings

Transform method yields moment convergence with error bounds.

Applicable to nearly deterministic queues in different heavy-traffic regimes.

Numerical results confirm accuracy of heavy-traffic approximations.

Abstract

Heavy-traffic limit theory deals with queues that operate close to criticality and face severe queueing times. Let $W$ denote the steady-state waiting time in the ${\rm GI}/{\rm G}/1$ queue. Kingman (1961) showed that $W$, when appropriately scaled, converges in distribution to an exponential random variable as the system's load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of $W$ (Pollaczek's contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.