  We analyze the stationary distribution of regulated Markov modulated Brownian
motions (MMBM) modified so that their evolution is slowed down when the process
reaches level zero --- level zero is said to be {\em sticky}. To determine the
stationary distribution, we extend to MMBMs a construction of Brownian motion
with sticky boundary, and we follow a Markov-regenerative approach similar to
the one developed in past years in the context of quasi-birth-and-death
processes and fluid queues. We also rely on recent work showing that
Markov-modulated Brownian motions may be analyzed as limits of a parametrized
family of fluid queues.
  We use our results to revisit the stationary distribution of the well-known
regulated MMBM.
