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

Abstract

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. 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 between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

On a first hit distribution of the running maximum of Brownian motion

“…The processes of the type (L t − t) t≥0 has been introduced and analyzed as models for fluid queues. For this , see, in particular, [7], where (L t ) t≥0 is the local time at 0 of a reflecting Brownian motion with negative drift, and [3], where a more general setting is considered and also further references can be found. In these articles the main interest is in finding the distribution of the length of a busy period (and also of the idle period) under the stationary probability measure associated with the underlying process.…”