RAP-modulated fluid processes: First passages and the stationary distribution

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) in Asmussen and Bladt (1999) which we call the RAP-modulated fluid process. As opposed to the classic Markov-modulated 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 Markov-modulated fluid process, such as first passage probabilities and the stationary distribution of its queue, translate to its RAP-modulated counterpart.