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Queueing Systems

, 63:13 | Cite as

Heavy-traffic extreme value limits for Erlang delay models

  • Guodong Pang
  • Ward WhittEmail author
Article

Abstract

We consider the maximum queue length and the maximum number of idle servers in the classical Erlang delay model and the generalization allowing customer abandonment—the M/M/n+M queue. We use strong approximations to show, under regularity conditions, that properly scaled versions of the maximum queue length and maximum number of idle servers over subintervals [0,t] in the delay models converge jointly to independent random variables with the Gumbel extreme value distribution in the quality-and-efficiency-driven (QED) and ED many-server heavy-traffic limiting regimes as n and t increase to infinity together appropriately; we require that t n →∞ and t n =o(n 1/2−ε ) as n→∞ for some ε>0.

Keywords

Erlang models Many-server queues Extreme values Heavy traffic Diffusion approximations Strong approximations Limit theorems 

Mathematics Subject Classification (2000)

60K25 60F05 60G70 90B22 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.IEOR DepartmentColumbia UniversityNew YorkUSA

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