Queueing systems with many servers: Null controllability in heavy traffic

Abstract

A queueing model has $J\ge2$ heterogeneous service stations, each consisting of many independent servers with identical capabilities. Customers of $I\ge2$ classes can be served at these stations at different rates, that depend on both the class and the station. A system administrator dynamically controls scheduling and routing. We study this model in the central limit theorem (or heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model on $\mathbb {R}^I$ with a singular control term that describes the scaling limit of the queueing model. The singular term may be used to constrain the diffusion to lie in certain subsets of $\mathbb {R}^I$ at all times $t>0$. We say that the diffusion is null-controllable if it can be constrained to $\mathbb {X}_-$, the minimal closed subset of $\mathbb {R}^I$ containing all states of the prelimit queueing model for which all queues are empty. We give sufficient conditions for null controllability of the diffusion. Under these conditions we also show that an analogous, asymptotic result holds for the queueing model, by constructing control policies under which, for any given $0<\epsilon <T<\infty$, all queues in the system are kept empty on the time interval $[\epsilon, T]$, with probability approaching one. This introduces a new, unusual heavy traffic ``behavior'': On one hand, the system is critically loaded, in the sense that an increase in any of the external arrival rates at the ``fluid level'' results with an overloaded system. On the other hand, as far as queue lengths are concerned, the system behaves as if it is underloaded.