Transitory Stochastic Models: Analysis and Optimization

随机模型被广泛用于分析和优化医疗保健、通信、大规模计算和运输系统。现有的理论倾向于关注更容易分析的同构模型，这些模型假设所有用户的服务需求都是相同的。最近的技术趋势使得收集大量数据成为可能，这些数据表明用户的需求通常是异构的。因此，需要一种非齐次随机模型理论。该项目的目标是发展一种“临时随机模型”理论，该理论明确地结合了用户的非同质性，并同时考虑了这些模型的优化和控制。如果成功，该项目将有助于对非同质随机系统的理论理解，并且理论结果的应用可能会在许多服务系统的运营管理中带来显着的性能提升。此外，本项目的思想将被纳入一门新的本科生随机建模课程，该课程将理论、数据建模和优化相结合。PI还将利用暑期研究项目，为少数民族和女性本科生提供研究经验。本研究的理论目标是发展用于临时随机模型性能分析和优化的随机过程和解析近似。所考虑的模型包括具有许多服务器的系统、排队网络以及具有大量服务时间和非平稳相关流量的系统。从某种意义上说，系统是暂时的，因为我们对有限的、但在操作上重要的时间范围内的预测感兴趣，而且模型需要纯粹的瞬态分析，即使在更简单的情况下，这也不是微不足道的。该研究有两个相关的焦点：首先，这项工作将确定新的缩放机制，其中可以以功能中心极限定理，强近似和条件极限定理的形式建立流量和工作负载过程的“通用”随机过程近似。其次，研究还将考虑使用这些随机过程近似来识别底层系统的最优控制。这里的重点将是开发预期成本函数的解析近似，并确定伴随的控制的渐近最优性概念。

Stochastic models are widely used to analyze and optimize healthcare, communication, large-scale computing, and transportation systems. Existing theory has tended to focus on easier-to-analyze homogeneous models that assume that service requirements are identical for all users. Recent technological trends have made it possible to collect large amounts of data that show that users' requirements are often heterogeneous. Thus, a theory of nonhomogeneous stochastic models is needed. The goal of this project is develop a theory of 'transitory stochastic models' that explicitly incorporates users' inhomogeneity and, concomitantly, considers the optimization and control of these models. If successful, this project will contribute to the theoretical understanding of nonhomogeneous stochastic systems, and the application of the theoretical results can potentially result in significant performance gains in the operational management of many service systems. Furthermore, ideas from this project will be incorporated into a new stochastic modeling course for undergraduates that integrates theory, data modeling and optimization. The PI will also leverage summer research programs to facilitate research experiences for minority and female undergraduate students.The theoretical objective of this research is to develop stochastic process and analytical approximations for the performance analysis and optimization of transitory stochastic models. The models considered include systems with many servers, queueing networks, and systems with heavy-tailed service times and non-stationary, correlated traffic. The systems are transitory in the sense that we are interested in predictions over a finite, but operationally significant time horizon, and the models require a purely transient analysis, which is non-trivial even in the simpler cases. The research has twin related foci: first, this effort will identify novel scaling regimes in which "universal" stochastic process approximations to the traffic and workload processes can be established, in the form of functional central limit theorems, strong approximations and conditioned limit theorems. Second, the research will also consider the identification of optimal controls for the underlying systems using these stochastic process approximations. The focus here will be on developing analytical approximations to the expected cost functions and identifying an accompanying notion of asymptotic optimality of controls.