  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.
