Stochastic Convexity in Queueing Networks and Its Applications

这个项目是与哥伦比亚大学的David D. Yao合作的。主要研究人员发展了强随机凸性和定向凸性的概念，这是基于样本路径的结构性质，对于一般随机离散事件系统的分析和优化设计至关重要，特别是排队网络。这个项目关注的是排队网络中与结构和策略变化相关的属性。为了模拟和理解这些变化，单调性和凸性的概念并不直接适用。沿着随机重排和随机多数化路线的新概念和新机器将得到发展。这些概念为在调度、控制和资源分配中广泛使用的“成对交换”参数提供了数学支持。初步研究表明它们在处理结构和政策变化方面有用和有效。此外，这些性质是随机凸性的自然推广。在排队网络中，一种新的、统一的（基于样本路径的）二阶属性层次结构似乎正在出现。总之，这些属性将支持排队网络的（静态）参数化设计，以及动态调度和控制。

This project is collaborative with David D. Yao of Columbia University. The principal investigators developed the notions of strong stochastic convexity and directional convexity which are sample-path based structural properties essential to the analysis and optimal design of stochastic discrete event systems in general and queueing networks in particular. This project focuses on properties relating to structural and policy changes in queueing networks. To model and understand these changes, notions of monotonicity and convexity do not apply directly. New concepts and new machineries along the lines of stochastic rearrangement and stochastic majorization will be developed. These notions provide mathematical support for "pairwise interchange" arguments that are widely used in scheduling, control, and resource allocation. Preliminary studies have demonstrated their usefulness and effectiveness in dealing with structural and policy changes. Furthermore, these properties are natural extension of stochastic convexity. A new, unified hierarchy of (sample-path based) second-order properties in queueing networks appears to be now emerging. Together, these properties will support the (static) parametric design of queueing networks, and dynamic scheduling and control.