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

Abstract

A common assumption when modeling queueing 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 analyze 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 Na, PN(a)≔PPoisNX¯Nα⩾Na,withX¯N≔1N∑i=1NXi.The logarithmic asymptotics of PN(a) are easily obtained from previous results; we find constants rP and γ such that N−γlogPN(a)→−rP as N→∞. Relying on elementary techniques, we then derive the exact asymptotics of PN(a): For α<13 and α>3 we identify (in closed-form) a function P˜N(a) such that PN(a)∕P˜N(a) tends to 1 as N→∞. For α∈[13,12) and α∈[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 α>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.