## 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 ρ 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 ρ approaches 1 and then n approaches infinity.
We need n to approach infinity sufficiently fast so that n(1-ρ) |