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 Nᵃ 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 X₁,, Xₙ, we focus on the evaluation of Pₙ(A), the probability that the scaled number of arrivals exceeds NA. Relying on elementary techniques, we derive the exact asymptotics of Pₙ(A): For a< 1/3 and a > 3 we identify (in closed-form) a function Pₙ(A) such that Pₙ(A) / Pₙ(A) tends to 1 as N → ∞. For a ∈ [1/3,1/2) and a∈ [2, 3) we find a partial solution in terms of an asymptotic lower bound. For the special case that the Xᵢs 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 Pₙ(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.