Matrix-analytic solution of system of integral equations in three tandem servers

TL;DR

This paper introduces a matrix-analytic approach to solve integral equations in a three-server tandem system, transforming them into algebraic equations for closed-form solutions, with discussion of solution properties.

Contribution

It presents a novel matrix-analytic method for solving integral equations in multi-server systems, enabling closed-form solutions and analysis of their properties.

Findings

Successfully transforms integral equations into algebraic equations

Derives closed-form solutions for the system

Discusses properties of the solutions

Abstract

A matrix-analytic method is proposed for solving a system of linear integral equations arises in three tandem servers. The approach is by modelling the cumulative distribution function (cdf) of the service time as a matrix exponential function. The method transforms the system of linear integral equations into a system of linear algebraic equations, hence can produce closed-form solutions. Properties of the solution are discussed.