A 3-Queue Polling System with Join the Shortest -- Serve the Longest Policy

TL;DR

This paper analyzes a three-queue system where customers join the shortest queue and the server prioritizes the longest queue, extending previous two-queue models with new analytical methods and demonstrating effective load balancing.

Contribution

It generalizes Cohen's two-queue models to a three-queue system with combined JSQ-SLQ policies, providing new analytical solutions for the system's probability distribution.

Findings

The system effectively balances queue lengths, minimizing disparities.

The joint JSQ-SLQ policy results in a Gini Index close to zero.

Numerical results validate the analytical approach.

Abstract

In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ) queueing system, where a single server attends two non-symmetric $M/G/1$-type queues, exercising a non-preemptive priority switching policy. Cohen further analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving customers follow the Join the Shortest Queue (JSQ) discipline. The current paper generalizes and extends Cohen's works by studying a combined JSQ-SLQ model, and by broadening the scope of analysis to a non-symmetric 3-queue system, where arriving customers follow the JSQ strategy and a single server exercises the preemptive priority SLQ discipline. The system states' multi-dimensional probability distribution function is derived while applying a non-conventional representation of the underlying process's state-space. The analysis combines both Probability Generating Functions and Matrix Geometric methodologies. It is shown that the joint JSQ-SLQ operating policy achieves extremely well the goal of balancing between queue sizes. This is emphasized when calculating the Gini Index associated with the differences between mean queue sizes: the value of the coefficient is close to zero. Extensive numerical results are presented.