Estimating Tail Probabilities in Queues via Extremal Statistics

P. W. Glynn and A. Zeevi

Analysis of Communication Networks: Call Centres, Traffic, and Performance[D. McDonald, S. Turner, eds.]  Fields Institute Communications, American Mathematical Society, Rhode Island, 135-158 (2000)

We study the estimation of tail probabilities in a queue via a semi-parametric estimator based on the maximum value of the workload, observed over the sampled time interval. Logarithmic consistency and efficiency issues for such estimators are considered, and their performance is contrasted with the (non-parametric) empirical tail estimator. Our results indicate that in order to “successfully” estimate and extrapolate buffer overflow probabilities in regenerative queues, it is in some sense necessary to first introduce a rough model for the behavior of the tails. In the course of developing these results, we establish new almost sure limit theory, in the context of regenerative processes, for the maximal extreme value and related first passage times.