On the excursions of reflected local time processes and stochastic fluid queues

TL;DR

This paper analyzes the behavior of reflected local time processes with negative drift, deriving joint laws for excursions, and applies these results to model stochastic fluid queues, providing new insights into their probabilistic structure.

Contribution

It introduces a novel analysis of excursions of reflected local time processes with drift and applies it to stochastic fluid queue modeling.

Findings

Derived joint law of excursion duration, maximum, and inter-excursion time.

Provided a stationary process framework for analysis.

Included practical examples illustrating the theoretical results.

Abstract

This paper extends previous work by the authors. We consider the local time process of a strong Markov process, add negative drift, and reflect it \`a la Skorokhod. The resulting process is used to model a fluid queue. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time distance between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.