The max-plus algebra approach in modelling of queueing networks

TL;DR

This paper applies max-plus algebra to model queueing networks with single-server fork-join nodes, providing explicit state equations and methods to compute the transition matrix from service times.

Contribution

It introduces a max-plus algebra framework for representing queueing network dynamics with explicit state equations and matrix computation methods.

Findings

Derived a common dynamic state equation for queueing networks

Presented a method to calculate the transition matrix from service times

Provided examples of matrices for specific network types

Abstract

A class of queueing networks which consist of single-server fork-join nodes with infinite buffers is examined to derive a representation of the network dynamics in terms of max-plus algebra. For the networks, we present a common dynamic state equation which relates the departure epochs of customers from the network nodes in an explicit vector form determined by a state transition matrix. We show how the matrix may be calculated from the service time of customers in the general case, and give examples of matrices inherent in particular networks.