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

LetW _k denote the waiting time of customerk, k 0, in an initially empty GI/G/1 queue. Fixa> 0. We prove weak limit theorems describing the behaviour ofW _k /n, 0kn, given W_n >na. LetX have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold forX, and queues for whichX 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.Research supported by the NSF under Grant NCR 8710840 and under the PYI Award NCR 8857731.