RUI: Analysis and Control of Infinite Dimensional Queueing Models

这个项目需要研究在某些排队模型的行为分析中出现的一些数学问题。排队模型是一种概率模型，它捕捉了各种现代网络中固有的随机性，例如客户服务系统、计算和电信以及运输和高科技制造中出现的那些网络。网络结构通常是确定性的，并且调度策略通常是指定的。随机性是由外部到达时间、服务时间和内部路线造成的。反馈和非队头(HL)调度策略在这样的网络中很常见。这些局部动态相互作用，产生复杂的聚合行为，通常会逃避封闭形式的分析。因此，需要易于处理的近似值。该项目涉及指定和验证各种模型近似，分析它们的性能和/或最优控制，并解释原始系统的这些结果。更具体地说，该项目涉及三个具有独特数学挑战的排队模型的研究：(1)在存在反馈的一般分布假设下，发展处理器共享队列网络的扩散近似；(2)使用非标准的分布相关的尺度来获得最短剩余处理时间队列的扩散近似；(3)通过对流体控制问题的研究，得到了在一般分布假设下具有丢弃的多类别排队的渐近最优调度策略，分析了每个模型的各种形式，其中包含一些马尔可夫分布假设(即到达间隔、服务和/或放弃时间的指数分布)。这样的假设对于现代应用程序的行为建模并不是特别现实。此外，在存在非马尔可夫分布假设的情况下，这种系统的性能可能会有很大的不同。因此，需要更全面地了解系统性能。从数学的角度来看，一般的分布假设导致需要跟踪更多的信息，以便跟踪系统状态。例如，必须以某种形式跟踪系统中每个作业的剩余服务时间或剩余放弃时间。这自然会导致无限维系统，其中测量值状态描述符提供了跟踪系统状态的有效工具。尽管使用了这种通用的建模工具，但由于不同的系统动力学，每个模型的数学挑战的性质是不同的。对于处理器共享网络，将开发一种新的策略来分析流体模型解的长时间行为。我们预计，这种方法将适用于存在分时功能的其他系统。对于剩余处理时间最短的队列，需要非标准分布相关的缩放，以考虑高流量中队列长度和工作负载处理之间的数量级差异。对于其他调度策略，还没有观察到这种行为。对于多类队列的控制，需要开发新的框架来提供对一般分布的放弃时间的流体控制问题的分析。这些进展应进一步有助于分析扩散控制问题。

This project entails investigating some mathematical questions that emerge in the behavioral analysis of certain queueing models. Queueing models are probabilistic models that capture the inherent randomness in a variety of modern networks, such as those that arise in customer service systems, computing and telecommunications, as well as transportation and hi-tech manufacturing. The network structure is typically deterministic and the scheduling policy is usually specified. Randomness results from exogenous arrival times, service times, and internal routing. Feedback and non-head-of-the-line (HL) scheduling policies are common in such networks. These local dynamics interact to produce aggregate behavior that is complex and often evades closed form analysis. Hence, tractable approximations are needed. The project involves specifying and validating various model approximations, analyzing their performance and/or optimal control and interpreting those results for the original system.More specifically, this project concerns the study of three queueing models with distinct features presenting unique mathematical challenges as follows:(1) to develop a diffusion approximation for networks of processor sharing queues, under general distributional assumptions in the presence of feedback;(2) to employ nonstandard, distribution dependent scaling to obtain a diffusion approximation for shortest remaining processing time queues;(3) to obtain asymptotically optimal scheduling policies for multiclass queues with abandonment under general distributional assumptions through the study of fluid control problems.Each model has been analyzed in various forms that include some Markovian distributional assumptions (i.e., exponentially distributed interarrival, service, and/or abandonment times). Such assumptions aren't particularly realistic for modeling the behavior of modern applications. Furthermore, the performance can be dramatically different for such systems in the presence of non-Markovian distributional assumptions. Therefore, system performance needs to be understood more fully. From a mathematical point of view, general distributional assumptions result in the need to track significantly more information in order to track the system state. For example, residual service times or residual abandonment times for each job in the system must be tracked in some form. This naturally leads to an infinite dimensional system where measure-valued state descriptors provide an effective tool for tracking the system state. In spite of employing this common modeling tool, the nature of the mathematical challenges are distinct for each model due to the distinct system dynamics. For processor sharing networks, a new strategy for analyzing the long time behavior of fluid model solutions will be developed. We anticipate that this methodology will translate to other systems where time-sharing is present. For shortest remaining processing time queues, nonstandard distribution dependent scaling is required to account for the order of magnitude difference between the queue length and workload processes in heavy traffic. Such behavior has not been observed for other scheduling policies. For the control of multiclass queues, new frameworks need to be developed to provide an analysis of the fluid control problem for generally distributed abandonment times. Such advances should further help in the analysis of a diffusion control problem.