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Performance bounds for queueing networks and scheduling policies

机译：排队网络和调度策略的性能界限

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Except for the class of queueing networks and scheduling policies admitting a product form solution for the steady-state distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), then there are no existing results on performance, or even stability. In most applications such as manufacturing systems, however, one has to choose a control or scheduling policy, i.e., a priority discipline, that optimizes a performance objective. In this paper the authors introduce a new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies. Assuming stability, and examining the consequence of a steady state for general quadratic forms, the authors obtain a set of linear equality constraints on the mean values of certain random variables that determine the performance of the system. Further, the conservation of time and material gives an augmenting set of linear equality and inequality constraints. Together, these allow the authors to bound the performance, either above or below, by solving a linear program. The authors illustrate this technique on several typical problems of interest in manufacturing systems. For an open re-entrant line modeling a semiconductor plant, the authors plot a bound on the mean delay (called cycle-time) as a function of line loading. It is shown that the last buffer first serve policy is almost optimal in light traffic. For another such line, it is shown that it dominates the first buffer first serve policy. For a set of open queueing networks, the authors compare their lower bounds with those obtained by another method of Ou and Wein (1992). For a closed queueing network, the authors bracket the performance of all buffer priority policies, including the suggested priority policy of Harrison and Wein (1990). The authors also study the asymptotic heavy traffic limits of the lower and upper bounds. For a manufacturing system with machine failures, it is shown how the performance changes with failure and repair rates. For systems with finite buffers, the authors show how to bound the throughput. Finally, the authors illustrate the application of their method to GI/GI/1 queues. The authors obtain analytic bounds which improve upon Kingman's bound (1970) for E2/M/1 queues
机译：除了排队产品的类别和允许产品形成稳态分配解决方案的调度策略外，对此类系统的性能知之甚少。例如，如果部件的优先级取决于其类别（例如，部件所在的缓冲区），则在性能甚至稳定性上都不会存在现有结果。然而，在诸如制造系统之类的大多数应用中，必须选择一种控制或调度策略，即优先原则，以优化性能目标。在本文中，作者介绍了一种用于获得马尔可夫排队网络和调度策略的性能上下限的新技术。假定稳定性，并检查一般二次形式的稳态结果，作者对确定系统性能的某些随机变量的平均值获得了一组线性相等约束。此外，时间和材料的节省给出了线性等式和不等式约束的增加集合。总之，这些方法使作者可以通过求解线性程序来限制性能（高于或低于）。作者在制造系统中关注的几个典型问题上说明了该技术。对于半导体工厂的开放式折返线建模，作者绘制了平均延迟（称为循环时间）随线负载变化的界限。结果表明，在轻流量中，最后一个缓冲区优先服务策略几乎是最佳的。对于另一条这样的行，表明它支配了第一个缓冲区先服务策略。对于一组开放排队网络，作者将其下界与通过Ou和Wein（1992）的另一种方法获得的下界进行了比较。对于一个封闭的排队网络，作者将所有缓冲区优先级策略的性能括起来，包括建议的Harrison和Wein（1990）的优先级策略。作者还研究了上下限的渐近交通限制。对于具有机器故障的制造系统，显示了性能如何随故障和维修率而变化。对于具有有限缓冲区的系统，作者展示了如何限制吞吐量。最后，作者说明了他们的方法在GI / GI / 1队列中的应用。作者获得了分析界线，该界线改进了E2 / M / 1队列的Kingman界线（1970）

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