Stochastic bounds for the queue GI/G/1 in heavy traffic

It is often of greater practical value to have results about queueing theory which involve probabilities rather than characteristic functions. To quote Kendall (4), section 5, ‘These results of Prabhu, further exploited by himself and Takács, are rapidly raising the Laplacian curtain which has hitherto obscured much of the details of the queue-theoretic scene’. In this paper we derive the exponential limit formula for the equilibrium waiting time distribution function G, for the queue GI/G/1 in heavy traffic, using stochastic bounds which are asymptotically sharp as the traffic intensity (defined below) increases to unity (which has not been done before to the author's knowledge). This formula was derived by Kingman (5), (6) and (9), using characteristic functions, who, in section 9 of the latter paper, stressed the need for improving the precision of the approximation ‘by giving inequalities, bounds for errors, and generally by setting the theory on a more elegant and rigorous basis’. Kingman (6) and (9) also sketched a proof of the same result using a Brownian approximation, which was done in detail by Viskov (18); but here again the same difficulties are present in practical interpretation, error bounds etc.