4 research outputs found

    Rare-event analysis of mixed Poisson random variables, and applications in staffing

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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 NaN^{a} time units, where a>0a> 0 and NN 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 NN and i.i.d. arrival rates X1,…,XNX_1, \dots, X_N, we focus on the evaluation of PN(A)P_N(A), the probability that the scaled number of arrivals exceeds NANA. Relying on elementary techniques, we derive the exact asymptotics of PN(A)P_N(A): For a3a 3 we identify (in closed-form) a function P~N(A)\tilde{P}_N(A) such that PN(A)/PN(A)P_N(A) / P_N(A) tends to 11 as Nβ†’βˆžN \to \infty. For a∈[13,12)a \in [\frac{1}{3},\frac{1}{2}) and a∈[2,3)a\in [2, 3) we find a partial solution in terms of an asymptotic lower bound. For the special case that the XiX_is are gamma distributed, we establish the exact asymptotics across all a>0a> 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 PN(A)P_N(A), we focus on the asymptotics of the probability that the number of clients in the system exceeds NANA. 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

    Exact asymptotics in an infinite-server system with overdispersed input

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    This short communication considers an infinite-server system with overdispersed input. The objective is to identify the exact tail asymptotics of the number of customers present at a given point in time under a specific scaling of the model (which involves both the arrival rate and time). The proofs rely on a change-of-measure approach. The results obtained are illustrated by a series of examples.Comment: Short communicatio
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