Approximations for time-dependent distributions in Markovian fluid models

TL;DR

This paper introduces new approximation methods for the distributions of levels in Markovian fluid queues and continuous-time random walks, using Erlang-distributed time approximations and alternative computational techniques.

Contribution

It proposes a novel approach to approximate time-dependent distributions in Markovian fluid models, offering probabilistic insights and numerical methods beyond traditional Laplace transform techniques.

Findings

Effective approximation of distributions at random times

Probabilistic interpretation of the equations

Numerical illustrations demonstrating accuracy

Abstract

In this paper we study the distribution of the level at time $\theta$ of Markovian fluid queues and Markovian continuous time random walks, the maximum (and minimum) level over $[0,\theta]$, and their joint distributions. We approximate $\theta$ by a random variable $T$ with Erlang distribution and we use an alternative way, with respect to the usual Laplace transform approach, to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration.