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

TL;DR

The paper analyzes the Gittins policy in the M/G/1 queue with unknown job sizes, focusing on the asymptotic tail of the response time $T$. By embedding Gittins in the SOAP framework, it derives a heavy-tailed tail-optimality condition and proves that Gittins is tail-optimal for nicely heavy-tailed job sizes; for light-tailed sizes, it classifies tail behavior by the worst-age point $a^*$ and demonstrates that small perturbations of Gittins can yield tail-optimal or intermediate performance while preserving near-optimal mean response time. The results reveal a dichotomy: in the heavy-tailed regime Gittins is always tail-optimal, while in the light-tailed regime its tail behavior can be optimal, pessimal, or intermediate depending on the distribution; a modest modification can eliminate pessimal tails. Collectively, the work provides a unified, policy-agnostic framework for diagnosing and improving tail performance of SOAP-based scheduling in M/G/1 with unknown job sizes, with implications for designing near-optimal schedulers that balance mean and tail performance in practice.

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.

Figures (9)

• Figure 1: Illustration of worst ever rank $w_x$ (purple dotted line) and maximal $w_x$-intervals (orange regions) for a SOAP policy given by rank function $r$ (cyan curve). A tagged job of size $x$ always has rank $w_x$ or better, so if another job has priority over the tagged job, that other job's age must be in a $w_x$-interval.
• Figure 2: Illustration of \ref{['cond:exponents']}(i), with examples showing the roles $ζ$ and $θ$ play. Roughly speaking, one should think of $ζ + θ$ as characterizing how far apart different "peaks" of the rank function are, and one should think of $ζ/(ζ + θ)$ as characterizing how "steep" the rank function is between peaks. Both (a) and (b) have the same peaks, as reflected by (a) and (b) having the same value of $ζ + θ$. But the slopes are much steeper in (a) than in (b), as reflected by the larger value of $ζ/(ζ + θ)$ in (a) than in (b).
• Figure 3: The rank functions of the Gittins policy (translucent cyan curve) and a $q$-approximate Gittins policy (dotted yellow-green curve) for a hyperexponential distribution with rates $μ_1$ and $μ_2$. By \ref{['thm:light_soap']}, these have different tail asymptotics. The Gittins rank function attains its supremum of $1/μ_2$ in the $a ∞$ limit, so it is tail-pessimal. But the $q$-approximate Gittins rank function, described in \ref{['eq:light_gittins_repair']}, attains its supremum at a finite age $ã$, so it is tail-intermediate.
• Figure 4: Illustration of $y_x$ and $z_x$ (\ref{['def:yz']})
• Figure 5: Illustration of \ref{['eq:heavy_gittins_goal']}: any $w_x$-interval (orange regions), namely any interval where the Gittins rank function (cyan curve) is better than the worst ever rank $w_x$ of a job of size $x$, has length $O(x)$.

Theorems & Definitions (73)

• definition 1
• definition 2
• definition 3: Heavy-Tailed Job Size Distribution
• definition 4: Tail Optimality in Heavy-Tailed Case
• definition 5
• definition 6
• theorem 1
• theorem 2
• definition 7
• definition 8: Light-Tailed Job Size Distribution