Dynamic Stochastic Networks: Analysis, Control and Applications

复杂网络的随机动力学模型在科学和工程中有着广泛的应用。具体实例包括高科技制造业、电信、计算机系统、服务系统和生化反应网络。该项目涉及研究一些数学问题，这些问题源于分析和控制这种动态随机网络的挑战。提出了平衡的议题。有些主题涉及广泛的网络类的一般理论的发展，而其他人则专注于直接由特定应用激发的数学问题。由于这些网络的复杂性通常排除了详细的微观模型的精确分析，这里的重点是近似模型。要解决的数学问题包括这些近似的严格理由，分析和控制近似模型的行为，并解释原始微观模型的结果。建议研究四个专题：（i）随机处理网络的动态控制，（ii）处理器共享网络的分析，（iii）数据网络中的拥塞控制和资源夹带，（iv）具有延迟动态和状态约束的随机系统。这些主题的一些随机过程方面包括奇异扩散控制问题的研究，测度值过程的分析，反射过程的基础问题，以及具有自然状态约束的泛函随机微分方程的渐近性质。动态随机网络在科学和工程中广泛用作数学模型，特别是在运营管理中，计算机科学、电子工程和生物工程。该补助金资助研究开发新的成果和技术，以解决因分析和控制此类动态随机网络而产生的数学问题。所研究的网络比迄今为止已经严格研究的网络要普遍得多。结果将是基本的数学兴趣，以及感兴趣的应用研究人员在相关领域的科学和工程。数学博士生的培训，通过在同行评审的期刊上发表和在跨学科会议上介绍研究成果来传播成果，都是该项目的组成部分。

Stochastic dynamic models of complex networks arise in a wide variety of applications in science and engineering. Specific instances include high-tech manufacturing, telecommunications, computer systems, service systems and biochemical reaction networks. This project involves the study of a number of mathematical problems stemming from the challenges of analyzing and controlling such dynamic stochastic networks. A balance of topics is proposed. Some topics involve the development of general theory for broad classes of networks, while others focus on mathematical problems directly motivated by specific applications. Since the complexity of these networks usually precludes exact analysis of detailed microscopic models, the focus here is on approximate models. Mathematical questions to be addressed include rigorous justification of these approximations, analyzing and controlling the behavior of the approximate models and interpreting the results for the original microscopic models. Four topics are proposed for study:(i) Dynamic control of stochastic processing networks,(ii) Analysis of processor sharing networks,(iii) Congestion control and resource entrainment in data networks,(iv) Stochastic systems with delayed dynamics and state constraints.Some stochastic process aspects of these topics include study of singular diffusion control problems, analysis of measure-valued processes, foundational questions for reflected processes, and the asymptotic properties of functional stochastic differential equations with natural state constraints.Dynamic stochastic networks are used as mathematical models in a wide variety of applications in science and engineering, particularly in operations management, computer science, electrical engineering and bioengineering. This grant funds research to develop new results and techniques for addressing mathematical problems stemming from the need to analyze and control such dynamic stochastic networks. The networks under study are substantially more general than those that have been rigorously studied to date. The results will be of fundamental mathematical interest as well as of interest for applied researchers in relevant areas of science and engineering. The training of mathematics PhD students, dissemination of results through publication in peer-reviewed journals and presentations of the research at cross-disciplinary conferences, are all integral parts of the project.