Abstract. 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 Λ from a given distribution every ∆ time units, yielding an i.i.d. sequence of arrival rates Λ 1 , Λ 2 , . . .. 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., ∆ is small relative to Λ) 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.

In this paper we study the probability ξn(u) := P (Cn un), with Cn := A(ψnB(ϕn)) for Lévy processes A(·) and B(·), and ϕn and ψn non-negative sequences such that ϕnψn = n and ϕn → ∞ as n → ∞. Two timescale regimes are distinguished: a 'fast' regime in which ϕn is superlinear and a 'slow' regime in which ϕn is sublinear. We provide the exact asymptotics of ξn(u) (as n → ∞) for both regimes, relying on change-of-measure arguments in combination with Edgeworth-type estimates. The asymptotics have an unconventional form: the exponent contains the commonly observed linear term, but may also contain sublinear terms (the number of which depends on the precise form of ϕn and ψn). To showcase the power of our results we include two examples, covering both the case where Cn is lattice and non-lattice. Finally we present numerical experiments that demonstrate the importance of taking into account the doubly stochastic nature of Cn in a practical application related to customer streams in service systems; they show that the asymptotic results obtained yield highly accurate approximations, also in scenarios in which there is no pronounced timescale separation.We proceed by explaining that there are two timescale regimes. (i) If ϕ n is superlinear, it is anticipated that B(ϕ n )/ϕ n is close to b, such that ξ n (u) resembles P(A(bn) un). We refer to this AMS 2000 subject classifications: 60F10, 60G51, 60K37

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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 −α time units, where α > 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 X1, . . . , XN , we focus on the evaluation of the probability that the scaled number of arrivals exceeds N a,

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