A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase-Type Service Times

TL;DR

This paper develops a matrix-analytic method to analyze randomized load balancing models with phase-type service times, extending previous exponential time models and providing insights into their fixed points and convergence behavior.

Contribution

It introduces a matrix-analytic framework for supermarket models with phase-type service times, addressing the complexity of non-exponential distributions.

Findings

Derived a doubly exponential solution for the fixed point.

Proved exponential convergence to the fixed point.

Numerical examples demonstrate the method's effectiveness.

Abstract

In this paper, we provide a matrix-analytic solution for randomized load balancing models (also known as \emph{supermarket models}) with phase-type (PH) service times. Generalizing the service times to the phase-type distribution makes the analysis of the supermarket models more difficult and challenging than that of the exponential service time case which has been extensively discussed in the literature. We first describe the supermarket model as a system of differential vector equations, and provide a doubly exponential solution to the fixed point of the system of differential vector equations. Then we analyze the exponential convergence of the current location of the supermarket model to its fixed point. Finally, we present numerical examples to illustrate our approach and show its effectiveness in analyzing the randomized load balancing schemes with non-exponential service requirements.