Heavy Traffic Limits for Queues with Many Deterministic Servers

Abstract

Consider a sequence of stationary GI/D/N queues indexed by N↑∞, with servers' utilization 1−β/$\sqrt{N}$, β>0. For such queues we show that the scaled waiting times $\sqrt{N}$WN converge to the (finite) supremum of a Gaussian random walk with drift −β. This further implies a corresponding limit for the number of customers in the system, an easily computable non-degenerate limiting delay probability in terms of Spitzer's random-walk identities, and $\sqrt{N}$ rate of convergence for the latter limit. Our asymptotic regime is important for rational dimensioning of large-scale service systems, for example telephone- or internet-based, since it achieves, simultaneously, arbitrarily high service-quality and utilization-efficiency.