Heavy-traffic asymptotics for the single-server queue with random order of service

We consider the waiting time distribution of the Gl/Gl /1 queue where customers are served in random order; inter-arrival and service times may have finite or infinite variance. Our main result shows that the waiting time in heavy traffic can be written as a product of two random variables. Our proof is based on the intuitively appealing fact that in heavy traffic, the queue length stays constant during the sojourn time of a customer. For the special finite variance case, our result settles a conjecture of Kingman (1982). 2000 Mathematics Subject Classification: 60K25.