Asymptotic analysis of a fluid model modulated by an $M/M/1$ queue

TL;DR

This paper performs an asymptotic analysis of a differential-difference equation from a Markov-modulated fluid model using singular perturbation, ray methods, and asymptotic matching to understand its behavior.

Contribution

It introduces a novel asymptotic analysis approach for a differential-difference equation in a Markov-modulated fluid context, employing advanced perturbation techniques.

Findings

Derived asymptotic solutions for the differential-difference equation

Applied singular perturbation and ray methods effectively

Provided insights into the fluid model's long-term behavior

Abstract

We analyze asymptotically a differential-difference equation, that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.