Heavy traffic limits for queues with many deterministic servers
Consider a sequence of stationary GI/D/N queues indexed by N up arrow infinity, with servers' utilization 1 - beta/root(N) over bar, beta > 0. For such queues we show that the scaled waiting times root(N) over barW(N) converge to the ( finite) supremum of a Gaussian random walk with drift -beta. 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 root(N) over bar 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.
gi/d/n gaussian random walk multi-server queue deterministic service time heavy-traffic quality and efficiency driven (qed) or halfin-whitt regime telephone call or contact centers economies of scale spitzer's identities