Analysis of stochastic fluid queues driven by local time processes

TL;DR

This paper models a stochastic fluid queue driven by the local time of a Markov process, analyzing its stationary behavior and performance metrics using Lévy process theory and Palm calculus.

Contribution

It introduces a rigorous construction of the stationary version of the local time-driven fluid queue and derives its steady-state performance distributions.

Findings

Explicit stationary distribution of buffer content

Distribution of idle and busy periods

Application of Lévy process theory and Palm calculus

Abstract

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a L\'evy process (a subordinator) hence making the theory of L\'evy processes applicable. Another important ingredient in our approach is the Palm calculus coming from the point process point of view.