Algorithms for the upper bound mean waiting time in the GI/GI/1 queue

It has long been conjectured that the tight upper bound for the mean steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions is attained asymptotically by two-point distributions. The two-point distribution for the interarrival time has one mass point at 0, but the service-time distribution involves a limit; there is one mass point at a high value, but that upper mass point must increase to infinity while the probability on that point must decrease to 0 appropriately. In this paper, we develop effective numerical and simulation algorithms to compute the value of this conjectured tight bound. The algorithms are aided by reductions of the special queues with extremal interarrival-time and extremal service-time distributions to D/GI/1 and GI/D/1 models. Combining these reductions yields an overall representation in terms of a D/RS(D)/1 discrete-time model involving a geometric random sum of deterministic random variables (the RS(D)), where the two deterministic random variables in the model may have different values, so that the extremal steady-state waiting time need not have a lattice distribution. Efficient computational methods are developed. The computational results show that the conjectured tight upper bound offers a significant improvement over established bounds.