Heavy-traffic extreme-value limits for queues

We consider the maximum waiting time among the first n customers in the GI/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 p 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 p approaches 1 and then n approaches infinity. We need n to approach infinity sufficiently fast so that n(l p)2 ~ oo. We also need n to approach infinity sufficiently slowly: If the service time has a pth moment for p > 2, then it suffices for (1 p)n TM to remain bounded; if the service time has a finite moment generating function, then it suffices to have (1 p)logn ~ 0. This limit can hold even when the normalized maximum waiting time fails to converge to the Gumbel distribution as n ~ ~ for each fixed p. Similar limits hold for the queue-length process.