  We extend to Markov-modulated Brownian motion (MMBM) the renewal approach
which has been successfully applied to the analysis of Markov-modulated fluid
models. It has recently been shown that MMBM may be expressed as the limit of a
parameterized family of Markov-modulated fluid models. We prove that the weak
convergence also holds for systems with two reflecting boundaries, one at zero
and one at $b >0$, and that the stationary distributions of the approximating
fluid models converge to the stationary distribution of the two-sided reflected
MMBM. Thus, we obtain a new representation for the stationary distribution,
effectively separating the limiting behaviour of the process at the boundaries
from its behaviour in the interior of $(0,b)$.
