Abstract
Consider a sequence of stationary GI/D/Nqueues indexed by N↑∞, with servers' utilization \(1 - \beta /\sqrt N ,\beta >0\). For such queues we show that the scaled waiting times \(\sqrt N W_N \) 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.
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Jelenković, P., Mandelbaum, A. & Momčilović, P. Heavy Traffic Limits for Queues with Many Deterministic Servers. Queueing Systems 47, 53–69 (2004). https://doi.org/10.1023/B:QUES.0000032800.52494.51
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DOI: https://doi.org/10.1023/B:QUES.0000032800.52494.51