  In this paper we study the Markov-modulated M/M/$\infty$ queue, with a focus
on the correlation structure of the number of jobs in the system. The main
results describe the system's asymptotic behavior under a particular scaling of
the model parameters in terms of a functional central limit theorem. More
specifically, relying on the martingale central limit theorem, this result is
established, covering the situation in which the arrival rates are sped up by a
factor $N$ and the transition rates of the background process by $N^\alpha$,
for some $\alpha>0$. The results reveal an interesting dichotomy, with
crucially different behavior for $\alpha>1$ and $\alpha<1$, respectively. The
limiting Gaussian process, which is of the Ornstein-Uhlenbeck type, is
explicitly identified, and it is shown to be in accordance with explicit
results on the mean, variances and covariances of the number of jobs in the
system.
