Response time in a pair of processor sharing queues with Join-the-Shortest-Queue scheduling

TL;DR

This paper derives a numerical solution for the response time distribution in two parallel processor sharing queues with Join-the-Shortest-Queue scheduling, providing insights into system performance.

Contribution

It introduces a novel numerical approach to solve PDEs for response times in JSQ queues with PS discipline, which was previously unsolved.

Findings

Numerical solutions match well with simulation results.

Derived response time density and moments with high accuracy.

Provides a new method for analyzing JSQ with PS in network systems.

Abstract

Join-the-Shortest-Queue (JSQ) is the scheduling policy of choice for many network providers, cloud servers and traffic management systems, where individual queues are served under processor sharing (PS) queueing discipline. A numerical solution for the response time distribution in two parallel PS queues with JSQ scheduling is derived for the first time. Using the generating function method, two partial differential equations (PDEs) are obtained corresponding to conditional response times, where the conditioning is on a particular traced task joining the first or the second queue. These PDEs are functional equations that contain partial generating functions and their partial derivatives, and therefore cannot be solved by commonly used techniques. We are able to solve these PDEs numerically with good accuracy and perform the deconditioning with respect to the queue-length probabilities by evaluating a certain complex integral. Numerical results for the density and the first four moments compare well against regenerative simulation with 500,000 regeneration cycles.