  We study ergodic properties of a class of Markov-modulated general
birth-death processes under fast regime switching. The first set of results
concerns the ergodic properties of the properly scaled joint Markov process
with a parameter that is taken large. Under very weak hypotheses, we show that
if the averaged process is exponentially ergodic for large values of the
parameter, then the same applies to the original joint Markov process. The
second set of results concerns steady-state diffusion approximations, under the
assumption that the 'averaged' fluid limit exists. Here, we establish
convergence rates for the moments of the approximating diffusion process to
those of the Markov modulated birth-death process. This is accomplished by
comparing the generator of the approximating diffusion and that of the joint
Markov process. We also provide several examples which demonstrate how the
theory can be applied.
