Scaling Limits for Infinite-server Systems in a Random Environment

Abstract

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.

Exact Asymptotics for a Multitimescale Model with Applications in Modeling Overdispersed Customer Streams

“…Observe that the 'two-timescale random walk' (3) we introduced can be seen as the discrete counterpart of the two-timescale Lévy processes considered in the present paper. In [16] the above two-timescale random walk model is studied under certain scalings (comparable to the scalings with ϕ n and ψ n that are imposed in the present paper). The results in [16] include logarithmic asymptotics, which are for special cases refined to exact asymptotics in [15].…”

“…The reason for developing such models was the observation that in various types of service systems [5,21] the customer arrival process is intrinsically more variable than the traditionally used Poisson process. An approach to overcome the lack of overdispersion was proposed in [16]: extra variability is produced by periodically resampling the Poisson arrival rate. As a consequence, when resampling every unit of time, the number of arrivals in [0, m] denoted by C(m) (for m ∈ N) is Poisson distributed with (random) parameter m i=1 Λ i , where the Λ i are i.i.d.…”

“…In [16] a mechanism is proposed that produces overdispersion. The main idea is that the arrival rate is random rather than deterministic; this is achieved by resampling the arrival rate periodically.…”

A ruin model with a resampled environment

“…Evidently, in principle also the other model primitives (i.e., premium rate and claim-size distribution) can be periodically resampled. In [22], for a different class of models, a similar mechanism to introduce parameter fluctuations has been proposed.…”

Exact asymptotics in an infinite-server system with overdispersed input

“…[1,5]). An alternative is to follow a scaling approach, as advocated by [7]: after rescaling the random variables Λ k and the sampling interval ∆ in terms of a parameter n, explicit characterizations of the distribution of N (K∆) can be derived in the asymptotic regime where n → ∞. More specifically, after an appropriate centering and normalization a diffusion limit has been established, as well as rough tail asymptotics (in terms of an exponential decay rate).…”