How large delays build in GI/G/1 queue

Let Wk denote the waiting time, of customer k, k ?? 0, in an initially empty GI/G/1 queue. Fix a > 0. We prove weak limit theorems describing the behavior of Wk/n, 0 ?? k ?? n, given Wn > na. Let X have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold for X, and queues for which X has regularly varying positive tail.