Diffusion limits for shortest remaining processing time queues

TL;DR

This paper analyzes the behavior of a single-server queue under heavy traffic conditions using SRPT policy, deriving diffusion limits for the queue length and measure-valued state descriptors, and discussing the policy's optimality and trade-offs.

Contribution

It provides the first diffusion limit theorem for SRPT queues with general service time distributions, enhancing understanding of queue length behavior under heavy traffic.

Findings

Diffusion limit theorem for measure-valued state descriptor

Diffusion limit for queue length process

Insights into queue length optimality and trade-offs

Abstract

We present a heavy traffic analysis for a single server queue with renewal arrivals and generally distributed i.i.d. service times, in which the server employs the Shortest Remaining Processing Time (SRPT) policy. Under typical heavy traffic assumptions, we prove a diffusion limit theorem for a measure-valued state descriptor, from which we conclude a similar theorem for the queue length process. These results allow us to make some observations on the queue length optimality of SRPT. In particular, they provide the sharpest illustration of the well-known tension between queue length optimality and quality of service for this policy.