When Does the Gittins Policy Have Asymptotically Optimal Response Time Tail?

TL;DR

This paper investigates the tail behavior of the Gittins scheduling policy in M/G/1 queues with unknown job sizes, revealing conditions under which it is asymptotically optimal or suboptimal, and proposing modifications for improved tail performance.

Contribution

It provides the first comprehensive analysis of Gittins's asymptotic tail behavior and introduces a modified policy to ensure near-optimal tail performance.

Findings

Gittins has asymptotically optimal tail for heavy-tailed job sizes.

Gittins's tail can be pessimal for light-tailed job sizes.

A modified Gittins policy achieves near-optimal mean response time and avoids pessimal tail behavior.

Abstract

We consider scheduling in the M/G/1 queue with unknown job sizes. It is known that the Gittins policy minimizes mean response time in this setting. However, the behavior of the tail of response time under Gittins is poorly understood, even in the large-response-time limit. Characterizing Gittins's asymptotic tail behavior is important because if Gittins has optimal tail asymptotics, then it simultaneously provides optimal mean response time and good tail performance. In this work, we give the first comprehensive account of Gittins's asymptotic tail behavior. For heavy-tailed job sizes, we find that Gittins always has asymptotically optimal tail. The story for light-tailed job sizes is less clear-cut: Gittins's tail can be optimal, pessimal, or in between. To remedy this, we show that a modification of Gittins avoids pessimal tail behavior while achieving near-optimal mean response time.