Heavy Traffic Scaling Limits for shortest remaining processing time queues with heavy tailed processing time distributions

TL;DR

This paper investigates the asymptotic behavior of SRPT queues with heavy-tailed processing times, revealing a new limit process without state space collapse, contrasting previous diffusive and light-tailed results.

Contribution

It introduces a novel non-standard scaling for heavy-tailed processing time distributions with finite second moments, deriving explicit limit processes without state space collapse.

Findings

Measure-valued process converges to a limit described by a Brownian motion-based random field.

No state space collapse occurs in the heavy-tailed setting under the new scaling.

Convergence of workload and queue length processes is established, with differences diminishing as tails become lighter.

Abstract

We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, the server preemptively serves the job with the shortest remaining processing time first. In this work we are interested in studying the asymptotic behavior of suitably scaled measure-valued state descriptors that describe the evolution of a sequence of SRPT queuing systems. Gromoll, Kruk, and Puha (2011) have studied this problem under diffusive scaling. In the setting where the processing time distributions have unbounded support, under suitable conditions, they show that the diffusion scaled measures converge in distribution to the process that is identically zero. In Puha (2015) for the setting where the processing time distributions have unbounded support and light tails, a non-standard scaling of the queue length process is shown to give rise to a form of state space collapse that results in a nonzero limit. In the current work we consider the case where processing time distributions have finite second moments and regularly varying tails. We show that the measure valued process, under a non-standard scaling, converges in distribution in the space of paths of measures. In sharp contrast with previous results, there is no state space collapse. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain $\mathbb{R}_+$ valued random field which is determined from a single Brownian motion. Along the way we establish convergence of suitably scaled workload and queue length processes. We also show that as the tail of the distribution of job processing times becomes lighter in an appropriate fashion, the difference between the limiting queue length process and the limiting workload process converges to zero, thereby approaching the behavior of state space collapse.