RAP-modulated Fluid Processes: First Passages and the Stationary Distribution

TL;DR

This paper introduces a new class of stochastic fluid processes modulated by a piecewise deterministic Markov process inspired by rational arrival processes, extending classic fluid process results to a broader, continuous state space setting.

Contribution

It develops the theory of RAP-modulated fluid processes, including formulas for first passage probabilities and stationary distributions, generalizing known results to a continuous state space framework.

Findings

Derived first passage probability formulas for RAP-modulated processes.

Established stationary distribution expressions for the new process.

Demonstrated the applicability of classic fluid process techniques to RAP-modulated models.

Abstract

We construct a stochastic fluid process with an underlying piecewise deterministic Markov process (PDMP) akin to the one used in the construction of the rational arrival process (RAP), which we call the RAP-modulated fluid process. As opposed to the classic stochastic fluid process driven by a Markov jump process, the underlying PDMP of a RAP-modulated fluid process has a continuous state space and is driven by matrix parameters which may not be related to an intensity matrix. Through novel techniques we show how well-known formulae associated to the classic stochastic fluid process, such as first passage probabilities and the stationary distribution of its queue, translate to its RAP-modulated counterpart.