RUI: Scaling Limits of Infinite Dimensional Queueing Models

这个项目需要调查一些数学问题，出现在分析某些模型的性能。随机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、运输和高科技制造中出现的网络。 网络结构通常是确定性的，并且调度策略通常是指定的。 随机性是由外部的到达时间、服务时间和内部路由决定的。 反馈和非头的线调度策略是常见的，在这样的网络。 这些局部动态相互作用，产生复杂的聚合行为，往往逃避封闭形式的分析。 因此，需要易于处理的近似。 在本项目中，PI将指定和验证各种模型近似，分析其性能和/或最优控制，并解释原始系统的结果。本研究计划为研究生及本科生提供研究训练的机会，主要研究在一般分布假设下的三个排队模型，这些模型具有独特的数学挑战性：（1）在存在反馈的情况下，建立处理器共享排队网络的扩散近似;（2）通过对流场和扩散控制问题的研究，得到了多类多服务台放弃排队的渐近最优调度策略;（3）证明极限定理，证明流不变态近似于随机化负载均衡算法的平稳分布。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，到达间隔、服务和/或放弃时间呈指数分布。然而，这样的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，在非马尔可夫分布假设的存在下，这种系统的性能可能会显着不同。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，以表示系统状态。例如，必须为系统中的每个作业跟踪剩余服务时间、服务年龄和/或系统年龄。这导致了一个无限维的系统，测量值的状态描述符提供了一个有效的表示。尽管有这个共同的描述符，由于不同的系统动态，每个模型的数学挑战是不同的。对于处理器共享网络，将开发用于分析流体模型解决方案的长时间行为的新方法。预计这一方法将适用于其他分时系统。对于多类队列的控制，非指数分布的放弃时间的流体控制问题中出现的非线性为证明渐近最优性提出了新的挑战。预计这种非线性会在二阶扩散控制问题的分析中带来更多需要克服的困难。对于随机负载平衡算法，将开发一种证明流体模型解在时间接近无穷大时收敛到不变状态的方法。这里的一个挑战是设计策略，配备处理可数系统的耦合测量值方程满足流体模型的解决方案。该奖项反映了NSF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.