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，例如在客户服务系统、计算和电信、以及运输和高科技制造业中出现的网络。网络结构通常是确定的，并且通常指定调度策略。随机性来自外部到达时间、服务时间和内部路由。反馈和非排队调度策略在这种网络中很常见。这些局部动态相互作用产生了复杂的聚合行为，并且经常逃避封闭形式的分析。因此，需要易于处理的近似值。该项目包括指定和验证各种模型近似，分析其性能和/或最优控制，并为原始系统解释这些结果。更具体地说，本项目涉及三种排队模型的研究，它们具有不同的特征，呈现出独特的数学挑战，如下：(1)在反馈存在的一般分布假设下，为处理器共享队列网络开发扩散近似；(2)采用非标准的、分布相关的缩放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放弃的多类队列的渐近最优调度策略。每个模型都以各种形式进行了分析，其中包括一些马尔可夫分布假设（即，指数分布的到达间隔、服务和/或放弃时间）。这样的假设对于现代应用程序的行为建模来说并不特别现实。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来跟踪系统状态。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。这自然导致了无限维系统，其中测量值状态描述符为跟踪系统状态提供了有效的工具。尽管采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，将开发一种分析流体模型解的长时间行为的新策略。我们预计，这种方法将转化为其他系统，其中分时存在。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，以考虑在繁忙流量中队列长度和工作负载进程之间的数量级差异。在其他调度策略中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分析一般分布式放弃时间下的流体控制问题。这些进展将进一步有助于扩散控制问题的分析。

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.