The maximum queue length for heavy tailed service times

TL;DR

This paper analyzes the maximum queue length in an M/G/1 queue with heavy-tailed service times, showing that the Foreground Background discipline leads to exponentially decreasing tail probabilities and deriving asymptotics for overflow times.

Contribution

It demonstrates that for heavy-tailed service times with logconvex density, the Foreground Background discipline optimally reduces maximum queue length tail probabilities and provides asymptotic overflow estimates.

Findings

Maximum queue length tail decreases exponentially under the Foreground Background discipline.

Asymptotic behavior of maximum queue length over time is characterized.

Results applied to buffer overflow time estimation in stable and unstable queues.

Abstract

In this paper we study the maximum queue length $M$ (in terms of the number of customers present) in a busy cycle in the M/G/1 queue. Assume that the service times have a logconvex density. For such (heavy-tailed) service-time distributions the Foreground Background service discipline is optimal. This discipline gives service to the customer(s) that have received the least amount of service so far. It is shown that under this discipline $M$ has an exponentially decreasing tail. From the behaviour of $M$ we obtain asymptotics of the maximum queue length $M(t)$ over the interval $(0,t)$ for $t\to\infty$. These are applied to calculate the time to overflow of a buffer, both in stable and unstable queues.