 
Summary: Queueing Systems, 5 (1988) 345368 345
HOW LARGE DELAYS BUILD UP IN A GI/G/1 QUEUE *
V. ANANTHARAM
School of Electrical Engineering, Phillips Hall, Cornell University, Ithaca, NY 14853, U.S.A.
Received 1 April 1988; revised 12 June 1989
Abstract
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 behaviour of Wk/n, 0 <~k <~n,
given W~> 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.
The results can also be interpreted as conditional limit theorems, conditional on large
maxima in the partial sums of random walks with negative drift.
Keywords:Weak limit theorems, Cramer type conditions, random walks with negative drift,
waiting time.
1. Introduction
Consider an initially empty GI/G/1 queue. Customer 0 arrives at time 0 and
customer k at time Aa +... +A~, where At, i = 1, 2,...are i.i.d.. Let Bi, i =
1, 2,... be i.i.d, service times, Bk+ 1 being the service time of customer k. If we let
Xk=B~Ak, then X,., i= 1, 2.... are i.i.d.. Let EX 1=~, /~> 0. The waiting
