On Large Delays in Multi-Server Queues with Heavy Tails

TL;DR

This paper derives bounds for the tail distribution of waiting times in multi-server queues with heavy-tailed service times, revealing the dominant 'big jumps' causing delays and providing conditions for moments' existence.

Contribution

It establishes precise bounds and a 'big jumps' principle for heavy-tailed queues, advancing understanding of delay behavior and moments in multi-server systems.

Findings

Bounds depend on traffic load and tail distribution

'Big jumps' dominate large delays in heavy-tailed queues

Conditions for moments' existence are characterized

Abstract

We present upper and lower bounds for the tail distribution of the stationary waiting time $D$ in the stable $GI/GI/s$ FCFS queue. These bounds depend on the value of the traffic load $\rho$ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a `principle of $s-k$ big jumps' in this case (here $k$ is the integer part of $\rho$), which gives the most probable way for the stationary waiting time to be large. Another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time.