Exact asymptotic for infinite-server queues

Abstract

Infinite-server queues are often used as a coupling tool for analyzing many-server systems that arise in telecommunications and call centers. Here we derive an exact asymptotic, up to a constant factor, for the tail probability of the number of customers in a G/G/∞ queue at a fixed time under heavy traffic. We are interested in the scenario when this exceedence is a rare event. In particular, the exceedance level for the number of customers is scaled with the arrival rate such that both of them go to infinity at a fixed ratio, and that the exceedance level is proportionately larger than the mean number of customers. Our main analytical technique is by obtaining fine estimates for the rate of convergence of the corresponding Gartner-Ellis limit via non-homogeneous renewal-theoretic bounds. Using such approach, the refined asymptotic can be seen to resemble the standard form for sum of i.i.d. random variables, as a result of the Gaussian diffusion approximation as suggested by Pang and Whitt (2010) [Two-parameter heavy-traffic limits for infinite-server queues, QUESTA, 65, 325--364]. This result extends the logarithmic asymptotic of Glynn (1995) [Large deviations for the infinite server queue in heavy traffic, IMA Vol. 71, 387--395].