  A common assumption when modeling queuing systems is that arrivals behave
like a Poisson process with constant parameter. In practice, however, call
arrivals are often observed to be significantly overdispersed. This motivates
that in this paper we consider a mixed Poisson arrival process with arrival
rates that are resampled every $N^{a}$ time units, where $a> 0$ and $N$ a
scaling parameter. In the first part of the paper we analyse the asymptotic
tail distribution of this doubly stochastic arrival process. That is, for large
$N$ and i.i.d. arrival rates $X_1, \dots, X_N$, we focus on the evaluation of
$P_N(A)$, the probability that the scaled number of arrivals exceeds $NA$.
Relying on elementary techniques, we derive the exact asymptotics of $P_N(A)$:
For $a< \frac{1}{3}$ and $a > 3$ we identify (in closed-form) a function
$\tilde{P}_N(A)$ such that $P_N(A) / P_N(A)$ tends to $1$ as $N \to \infty$.
For $a \in [\frac{1}{3},\frac{1}{2})$ and $a\in [2, 3)$ we find a partial
solution in terms of an asymptotic lower bound. For the special case that the
$X_i$s are gamma distributed, we establish the exact asymptotics across all $a>
0$. In addition, we set up an asymptotically efficient importance sampling
procedure that produces reliable estimates at low computational cost. The
second part of the paper considers an infinite-server queue assumed to be fed
by such a mixed Poisson arrival process. Applying a scaling similar to the one
in the definition of $P_N(A)$, we focus on the asymptotics of the probability
that the number of clients in the system exceeds $NA$. The resulting
approximations can be useful in the context of staffing. Our numerical
experiments show that, astoundingly, the required staffing level can actually
decrease when service times are more variable.
