  This paper studies the effect of an overdispersed arrival process on the
performance of an infinite-server system. In our setup, a random environment is
modeled by drawing an arrival rate $\Lambda$ from a given distribution every
$\Delta$ time units, yielding an i.i.d. sequence of arrival rates
$\Lambda_1,\Lambda_2, \ldots$. Applying a martingale central limit theorem, we
obtain a functional central limit theorem for the scaled queue length process.
We proceed to large deviations and derive the logarithmic asymptotics of the
queue length's tail probabilities. As it turns out, in a rapidly changing
environment (i.e., $\Delta$ is small relative to $\Lambda$) the overdispersion
of the arrival process hardly affects system behavior, whereas in a slowly
changing random environment it is fundamentally different; this general finding
applies to both the central limit and the large deviations regime. We extend
our results to the setting where each arrival creates a job in multiple
infinite-server queues.
