MAD dispersion measure makes extremal queue analysis simple

TL;DR

This paper introduces a novel approach using MAD dispersion measures and DRO techniques to simplify extremal queue analysis, providing explicit bounds on waiting times in GI/G/1 queues.

Contribution

It develops a new method combining MAD and DRO to derive explicit, computationally tractable bounds for queue performance under uncertainty.

Findings

Explicit formulas for extremal distributions derived

Upper and lower bounds on waiting times obtained

Bounds remain sharp with estimated parameters

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.