Analysis of fluid flow models

TL;DR

This paper reviews the analysis of Markov-modulated fluid flow models, emphasizing matrix-analytic methods, computational improvements, and extensions to more general processes in various applications.

Contribution

It provides a high-level overview of matrix-analytic techniques for fluid queues, discusses computational advancements, and explores extensions to broader classes of processes.

Findings

Improved algorithms for solving Riccati equations in fluid queue analysis.

Enhanced numerical accuracy and convergence speed in computational methods.

Extension of analysis techniques to more general Markov additive processes.

Abstract

Markov-modulated fluids have a long history. They form a simple class of Markov additive processes, and were initially developed in the 1950s as models for dams and reservoirs, before gaining much popularity in the 1980s as models for buffers in telecommunication systems, when they became known as fluid queues. More recent applications are in risk theory and in environmental studies. In telecommunication systems modelling, the attention focuses on determining the stationary distribution of the buffer content. Early ODE resolution techniques have progressively given way to approaches grounded in the analysis of the physical evolution of the system, and one only needs now to solve a Riccati equation in order to obtain several quantities of interest. To the early algorithms proposed in the Applied Probability literature, numerical analysts have added new algorithms, improved in terms of convergence speed, numerical accuracy, and domain of applicability. We give here a high-level presentation of the matrix-analytic approach to the analysis of fluid queues, briefly address computational issues, and conclude by indicating how this has been extended to more general processes.