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

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.