Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling

TL;DR

This paper introduces a new nonstandard spatial scaling for SRPT queues in heavy traffic, revealing a nonzero diffusion limit for queue length that differs from the workload process, especially for unbounded processing times.

Contribution

It develops a novel spatial scaling approach that captures the nonzero limit of queue length in SRPT queues under heavy traffic, extending previous diffusion limit results.

Findings

Standard diffusion scaling yields zero limit for queue length with unbounded processing times.

The new scaling produces a nonzero reflected Brownian motion limit for queue length.

The spatial scale factor characterizes the difference between queue length and workload processes.

Abstract

We develop a heavy traffic diffusion limit theorem under nonstandard spatial scaling for the queue length process in a single server queue employing shortest remaining processing time (SRPT). For processing time distributions with unbounded support, it has been shown that standard diffusion scaling yields an identically zero limit. We specify an alternative spatial scaling that produces a nonzero limit. Our model allows for renewal arrivals and i.i.d. processing times satisfying a rapid variation condition. We add a corrective spatial scale factor to standard diffusion scaling, and specify conditions under which the sequence of unconventionally scaled queue length processes converges in distribution to the same nonzero reflected Brownian motion to which the sequence of conventionally scaled workload processes converges. Consequently, this corrective spatial scale factor characterizes the order of magnitude difference between the queue length and workload processes of SRPT queues in heavy traffic. It is determined by the processing time distribution such that the rate at which it tends to infinity depends on the rate at which the tail of the processing time distribution tends to zero. For Weibull processing time distributions, we restate this result in a manner that makes the resulting state space collapse more apparent.