  Ramaswami showed recently that standard Brownian motion arises as the limit
of a family of Markov-modulated linear fluid processes. We pursue this analysis
with a fluid approximation for Markov-modulated Brownian motion. Furthermore,
we prove that the stationary distribution of a Markov-modulated Brownian motion
reflected at zero is the limit from the well-analyzed stationary distribution
of approximating linear fluid processes. Key matrices in the limiting
stationary distribution are shown to be solutions of a new quadratic equation,
and we describe how this equation can be efficiently solved. Our results open
the way to the analysis of more complex Markov-modulated processes.
