How large delays build up in a GI/G/1 queue

Let denote the waiting time of customer≥ 0, in an initially empty GI/G/1 queue. Fix> 0. We prove weak limit theorems describing the behaviour of/, 0≤≤, given W > Let have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold for, and queues for which has regularly varying positive tail.The results can also be interpreted as conditional limit theorems, conditional on large maxima in the partial sums of random walks with negative drift.