Beyond Heavy-Traffic Regimes: Universal Bounds and Controls for the Single-Server Queue

Abstract

Brownian approximations are widely studied because of their tractability relative to the original queueing models. Their stationary distributions are used as proxies for those of the original queues that they model and the convergence of suitably scaled and centered processes provides mathematical support for the use of these Brownian models. To establish convergence, one must impose assumptions directly on the primitives or, indirectly, on the parameters of a related optimization problem. These assumptions reflect an interpretation of the underlying parameters -- a classification into so-called heavy-traffic regimes that specify a scaling relationship between the utilization and the arrival rate. From a heuristic point of view, though, there is an almost immediate Brownian analogue of the queueing model that is derived directly from the primitives and requires no (limit) interpretation of the parameters. In this paper we prove that for the fundamental M/GI/1 GI queue, the direct intuitive (limitless approach) in fact works. The Brownian model is universally (i.e., across regimes and patience distributions) accurate. It maintains the tractability and appeal of the limit approximations while avoiding many of the assumptions that facilitate them. In the process of building mathematical support for the accuracy of this model, we introduce a framework built around queue families that allows us to treat various patience distributions simultaneously, and uncovers the role of a concentration property of the queue.