Tail Asymptotics for Delay in a Half-loaded GI/GI/2 Queue with Heavy-tailed Job Sizes

TL;DR

This paper derives asymptotic bounds for the tail distribution of steady-state waiting times in a half-loaded GI/GI/2 queue with heavy-tailed job sizes, revealing different effects based on variance finiteness.

Contribution

It provides new asymptotic bounds for tail probabilities in a half-loaded queue with heavy-tailed job sizes, highlighting phase transitions based on variance.

Findings

Two dominant effects for finite variance jobs: one big job or two big jobs.

Phase transition occurs when job sizes have infinite variance, with only one effect dominating.

Asymptotic bounds depend on the tail behavior of job size distributions.

Abstract

We obtain asymptotic bounds for the tail distribution of steady-state waiting time in a two server queue where each server processes incoming jobs at a rate equal to the rate of their arrivals (that is, the half-loaded regime). The job sizes are taken to be regularly varying. When the incoming jobs have finite variance, there are basically two types of effects that dominate the tail asymptotics. While the quantitative distinction between these two manifests itself only in the slowly varying components, the two effects arise from qualitatively very different phenomena (arrival of one extremely big job (or) two big jobs). Then there is a phase transition that occurs when the incoming jobs have infinite variance. In that case, only one of these effects dominate the tail asymptotics, the one involving arrival of one extremely big job.