MAD dispersion measure makes extremal queue analysis simple

Abstract

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival and service time distributions. We address this extremal queue problem by measuring dispersion in terms of Mean Absolute Deviation (MAD) instead of variance, making available recently developed techniques from Distributionally Robust Optimization (DRO). Combined with classical random walk theory, we obtain explicit expressions for the extremal interarrival time and service time distributions, and hence the best possible upper bounds, for all moments of the waiting time. {We also apply the DRO techniques to obtain tight lower bounds that together with the upper bounds provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp, also when the mean and MAD are not known precisely, but estimated based on available data instead.