  Many networking-related settings can be modeled by Markov-modulated
infinite-server systems. In such models, the customers' arrival rates and
service rates are modulated by a Markovian background process, additionally,
there are infinitely many servers (and consequently the resulting model is
often used as a proxy for the corresponding many-server model). The
Markov-modulated infinite-server model hardly allows any explicit analysis,
apart from results in terms of systems of (ordinary or partial) differential
equations for the underlying probability generating functions, and recursions
to obtain all moments. As a consequence, recent research efforts have pursued
an asymptotic analysis in various limiting regimes, notably the central-limit
regime (describing fluctuations around the average behavior) and the
large-deviations regime (focusing on rare events). Many of these results use
the property that the number of customers in the system obeys a Poisson
distribution with a random parameter.
  The objective of this paper is to develop techniques to accurately
approximate tail probabilities in the large-deviations regime. We consider the
scaling in which the arrival rates are inflated by a factor $N$, and we are
interested in the probability that the number of customers exceeds a given
level $Na$. Where earlier contributions focused on so-called $logarithmic$
$asymptotics$ of this exceedance probability (which are inherently imprecise),
the present paper improves upon those results in that $exact$ $asymptotics$ are
established. These are found in two steps: first the distribution of the random
parameter of the Poisson distribution is characterized, and then this knowledge
is used to identify the exact asymptotics. The paper is concluded by a set of
numerical experiments, in which the accuracy of the asymptotic results is
assessed.
