Customer sojourn time in GI/G/1 feedback queue in the presence of heavy tails

TL;DR

This paper analyzes the tail behavior of customer sojourn times in a GI/GI/1 queue with feedback, focusing on heavy-tailed service times, and derives asymptotic formulas for different arrival scenarios.

Contribution

It provides new tail asymptotics for sojourn times in GI/GI/1 queues with regularly varying service times, including precise formulas in the Poisson case.

Findings

Tail asymptotics for sojourn times in different regimes

Asymptotics for busy period distribution with heavy tails

Principle-of-a-single-big-jump characterizations

Abstract

We consider a single-server GI/GI/1 queueing system with feedback. We assume the service times distribution to be (intermediate) regularly varying. We find the tail asymptotics for a customer's sojourn time in two regimes: the customer arrives in an empty system, and the customer arrives in the system in the stationary regime. In particular, in the case of Poisson input we use the branching processes structure and provide more precise formulae. As auxiliary results, we find the tail asymptotics for the busy period distribution in a single-server queue with an intermediate varying service times distribution and establish the principle-of-a-single-big-jump equivalences that characterise the asymptotics.