On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case

TL;DR

This paper investigates the transition point between heavy-traffic and heavy-tailed behaviors in the steady-state waiting time distribution of the M/G/1 queue with regularly varying processing times, providing new unified approximations.

Contribution

It identifies a sharp threshold for the tail behavior transition and introduces new uniform approximations that bridge the heavy-traffic and heavy-tailed regimes.

Findings

Determines the threshold in terms of tail value and traffic intensity.

Provides unified approximations valid across regimes.

Enhances understanding of queue behavior with heavy-tailed processing times.

Abstract

Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $\rho$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-\rho)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of $(1-\rho)$, that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold.