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.

中文翻译：

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上界算法在 GI/GI/1 队列中平均等待时间

长期以来，人们一直推测，在给定到达时间间隔和服务时间分布的前两个时刻的情况下，GI/GI/1 队列中平均稳态等待时间的紧上界是通过两点分布渐近获得的。到达间隔时间的两点分布在 0 处有一个质点，但服务时间分布有限制；有一个质点处于高值，但上质点必须增加到无穷大，而该点的概率必须适当降低到 0。在本文中，我们开发了有效的数值和模拟算法来计算这个推测的紧界的值。通过将具有极值间隔时间和极值服务时间分布的特殊队列减少到 D/GI/1 和 GI/D/1 模型来辅助算法。结合这些减少产生了 D/RS(D)/1 离散时间模型的总体表示，该模型涉及确定性随机变量的几何随机和 (RS(D))，其中模型中的两个确定性随机变量可能具有不同的值，因此极值稳态等待时间不需要具有晶格分布。开发了有效的计算方法。计算结果表明，推测的紧上限比既定的边界有显着的改进。开发了有效的计算方法。计算结果表明，推测的紧上限比既定的边界有显着的改进。开发了有效的计算方法。计算结果表明，推测的紧上限比既定的边界有显着的改进。