Extreme values of queue lengths in M/G/1 and GI/M/1 systems
Mathematics of Operations Research
The Fourier-series method for inverting transforms of probability distributions
Queueing Systems: Theory and Applications - Numerical computations in queues
Strong approximations of open queueing networks
Mathematics of Operations Research
Heavy-traffic extreme value limits for Erlang delay models
Queueing Systems: Theory and Applications
On extreme values in open queueing networks
Mathematical and Computer Modelling: An International Journal
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We consider the maximum waiting time along the first n customers in the G1/G/1 queue. We use strong approximations to prove, under regularity conditions, convergence of the normalized maximum wait to the Gumbel extreme-value distribution when the traffic intensity @r approaches 1 from below and n approaches infinity at a suitable rate. The normalization depends on the interarrival-time and service-time distributions only through their first two moments, corresponding to the iterated limit in which first @r approaches 1 and then n approaches infinity. We need n to approach infinity sufficiently fast so that n(1 - @r)^2 - ~. We also need n to approach infinity sufficiently slowly: If the service time has a pth moment for @r 2, then it suffices for (1 - @r)n^1^p to remain bounded; if the service time has a finite moment generating function, then it suffices to have (1 - @r)log n - 0. This limit can hold even when the normalized maximum waiting time fails to converge to the Gumbel distribution as n - ~ for each fixed @r. Similar limits hold for the queue-length process.