Stationary distributions for fluid flow models with or without Brownian noise

The paper considers a process with reflection at the origin and paths which are piecewise linear or Brownian, with the drift and variance constants being determined by the state of an underlying finite Markov process; the purely linear case corresponds to fluid flow models of current interest in telecommunications engineering. It is shown that the stationary distribution is phase-type, and various algorithms for computing the phase representation are given, some iterative with each step involving a matrix inversion and some based upon spectral expansion of the phase generator. Mathematically, the point of view is Markov additive processes, and some key tools are time-reversal and auxiliary Markov processes obtained by observing the underlying Markov process when the additive component is at a maximum.