Scheduling jobs with unknown size distribution in a M/G/1 queue: the shifted empirical Gittins

2026-06-23

AI Summary

This work addresses the problem of minimizing expected response time in an M/G/1 queue when the job size distribution is unknown. The authors propose an asymptotically optimal scheduling policy that relies solely on a finite set of samples from the distribution. Their approach discretizes the bounded support of the job size distribution and constructs an empirical distribution by right-shifting each sample to the nearest discrete point. This empirical distribution is then used to compute Gittins indices, which guide an index-based scheduling policy. Theoretical analysis establishes that the proposed policy achieves asymptotic optimality as the number of samples tends to infinity. Numerical experiments further demonstrate its superior performance over existing methods under finite-sample regimes, marking the first scheduler that attains both high efficiency and asymptotic optimality without requiring prior knowledge of the job size distribution.

Abstract

In this paper we consider a M/G/1 queue for which we want to minimize the expected response time. We show how to compute indices from $n$ samples of the job size distribution such that the corresponding index policy is asymptotically optimal as $n$ grows. This construction is based on a discretization of the bounded support of the job size distribution and a shift of the samples to their nearest discrete point to the right. We show that the Gittins index of the empirical distribution of these shifted samples is close to the Gittins index of the original distribution. This translates to the asymptotic optimality of the corresponding index policy for minimizing the expected response time. Numerical comparison with other approaches further confirm the efficiency of our approach.

Problem

• M/G/1 queue
• unknown job size distribution
• expected response time
• scheduling
• Gittins index

Innovation

• M/G/1 queue
• Gittins index
• empirical distribution
• asymptotic optimality
• job scheduling