# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1908.04461