RUI: Scaling Limits of Infinite Dimensional Queueing Models

这个项目需要研究在分析某些排队模型的性能时出现的一些数学问题。排队模型是概率模型，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、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。在这个项目中，PI将指定和验证各种模型近似，分析它们的性能和/或最优控制，并为原始系统解释这些结果。该项目为研究生和本科生提供研究训练机会。本研究项目涉及在一般分布假设下运行的三种排队模型的研究，这些模型具有不同的特征，提出了以下独特的数学挑战：(1)为存在反馈的处理器共享队列网络建立扩散近似；(2)通过对流体和扩散控制问题的研究，得到了具有放弃的多类多服务器队列的渐近最优调度策略；(3)证明极限定理来证明流体不变状态作为随机负载平衡算法的平稳分布的近似。这些模型已经以各种形式进行了分析，包括马尔可夫分布假设，即指数分布的到达间隔、服务和/或放弃时间。然而，这些假设对于现代计算机、通信和客户服务系统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表示系统状态。例如，必须跟踪系统中每个作业的剩余服务时间、服务年龄和/或系统年龄。这导致了一个无限维系统，其中测量值状态描述符提供了一个有效的表示。尽管有这个通用的描述符，但由于不同的系统动力学，每个模型的数学挑战是不同的。对于处理器共享网络，将开发一种新的方法来分析流体模型解的长期行为。预计这一方法将适用于采用分时制的其他系统。对于多类队列的控制，非指数分布放弃时间的流体控制问题所产生的非线性对证明渐近最优性提出了新的挑战。在二阶扩散控制问题的分析中，这样的非线性将会引入更多需要克服的困难。对于随机负载平衡算法，将开发一种方法来证明流体模型解在时间趋于无穷时不变状态的收敛性。这里的一个挑战是设计策略来处理由流体模型解满足的一对测量值方程组成的可数系统。这样的策略预计将与其他具有负载平衡的模型的分析相关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project entails investigating some mathematical questions that emerge in analyzing the performance 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, 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 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. In this project the PI will specify and validate various model approximations, analyzing their performance and/or optimal control, and interpreting those results for the original system. The project provides research training opportunities for graduate and undergraduate students.This research project concerns the study of three queueing models operating under general distributional assumptions with distinct features presenting unique mathematical challenges as follows: (1) Develop a diffusion approximation for networks of processor sharing queues in the presence of feedback; (2) Obtain asymptotically optimal scheduling policies for multi-class many server queues with abandonment through the study of fluid and diffusion control problems; and (3) Prove limit theorems to justify fluid invariant states as approximations of stationary distributions for randomize load balancing algorithms. These models have been analyzed in various forms that include Markovian distributional assumptions, i.e., exponentially distributed inter-arrival, service, and/or abandonment times. However, such assumptions are not particularly realistic for modeling the behavior of modern computers, communications, and customer service systems. 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 represent the system state. For example, residual service times, age-in-service, and/or age-in-system must be tracked for each job in the system. This leads to an infinite dimensional system where measure-valued state descriptors provide an effective representation. Despite this common descriptor, the mathematical challenges are different for each model due to distinct system dynamics. For processor sharing networks, a new methodology for analyzing the long-time behavior of fluid model solutions will be developed. It is anticipated that this methodology will translate to other systems where time sharing is present. For the control of multi-class queues, non-linearity that arises in the fluid control problem for non-exponentially distributed abandonment times presents new challenges for demonstrating asymptotic optimality. Such non-linearities are expected to introduce further difficulties to be overcome in the analysis of a second order, diffusion control problem. For randomized load balancing algorithms, a methodology for proving the convergence of fluid model solutions to invariant states as time approaches infinity will be developed. A challenge here is to devise strategies equipped to handle the countable system of couple measure-valued equations satisfied by fluid model solutions. Such strategies are expected to be relevant for the analysis of other models with load balancing.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.