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、大规模计算和交通系统。现有的理论倾向于关注更容易分析的同质模型，这些模型假设所有用户的服务需求都是相同的。最近的技术趋势使收集大量数据成为可能，这些数据表明用户的需求往往是不同的。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“暂态随机模型”理论，该模型明确地包含了用户的异质性，并同时考虑了这些模型的优化和控制。如果成功，这个项目将有助于从理论上理解非齐次随机系统，理论结果的应用可能会在许多服务系统的运行管理中产生显著的性能收益。此外，来自该项目的想法将被融入到一门新的本科生随机建模课程中，该课程集理论、数据建模和优化于一体。PI还将利用暑期研究计划，促进少数族裔和女性本科生的研究体验。本研究的理论目标是开发随机过程和分析近似，用于暂态随机模型的性能分析和优化。所考虑的模型包括具有多个服务器的系统、排队网络，以及具有重尾服务时间和非平稳相关流量的系统。从某种意义上说，这些系统是暂时的，因为我们对有限但具有重要操作意义的时间范围内的预测感兴趣，并且模型需要纯粹的瞬时分析，即使在更简单的情况下，这也不是微不足道的。这项研究有两个相关的焦点：首先，这项工作将确定新的标度机制，在这种机制中，可以以函数中心极限定理、强逼近和条件极限定理的形式建立对交通和工作负荷过程的“普适”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基础系统的最优控制。这里的重点将是开发预期成本函数的解析近似，并确定伴随而来的控制的渐近最优概念。

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.