Dynamics of Stochastic Networks: Approximation, Analysis, and Control

随机网络动力学：近似、分析和控制

• 批准号: 2153866
• 负责人: Ruth Williams
• 金额: $ 23.19万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153866&HistoricalAwards=false
• 关键词: Dynamics; Stochastic; Networks; Approximation; Analysis

Stochastic models of complex networks with dynamic interactions arise in a wide variety of applications in science and engineering. Specific instances include biochemical reaction networks, high-tech manufacturing, computer systems, telecommunications, transportation, and business service systems. This project addresses mathematical questions stemming from the challenges of analyzing and controlling such stochastic networks. The research involves the development of general theory for some broad classes of stochastic networks and the study of questions directly motivated by specific applications. Since the complexity of stochastic networks usually precludes exact analysis of detailed “microscopic” models, the focus here is on approximate models. Two levels of approximation are considered: first-order approximations called fluid models, and second-order approximations, which frequently are diffusion models. New techniques and results will be developed with an eye toward application areas. The investigator will help train a diverse mathematics research workforce through collaboration with early career researchers and women researchers. The project also provides training opportunities for graduate students. This project will address mathematical questions associated with the analysis and control of stochastic network dynamics. Topics to be addressed include rigorous justification of approximations, analyzing and controlling the behavior of the approximate models, and interpreting the results for the original microscopic models. An important subtheme is understanding the interplay between levels of approximation. Five topics are to be studied:(i) Diffusion Approximations for (Bio)Chemical Reaction Networks and Nearly Density-Dependent Markov Chains;(ii) Analysis of Processor Sharing Networks;(iii) Congestion Control and Resource Entrainment in Data Networks;(iv) Networks with Random Order of Service and Reneging; and(v) Dynamic Control of Stochastic Processing Networks.Some stochastic process aspects of these topics include error quantification in the approximation of nearly density-dependent Markov chains by reflected diffusion processes, analysis of measure-valued processes used to track residual job sizes or patience times in stochastic networks with resource sharing and reneging, singular diffusion control problems, foundational questions for reflected processes, and numerical approximation of reflected diffusion processes in non-smooth domains.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.

具有动态相互作用的复杂网络的随机模型在科学和工程中的各种应用中都出现。特定实例包括生化反应网络，高科技制造，计算机系统，电信，运输和商业服务系统。该项目解决了分析和控制此类随机网络的挑战所引起的数学问题。该研究涉及一些广泛的随机网络的一般理论的发展以及对特定应用程序直接动机的问题的研究。由于随机网络的复杂性通常会规定详细的“微观”模型的精确分析，因此这里的重点是近似模型。考虑了两个级别的近似值：一阶近似值称为流体模型，以及二阶近似值，通常是扩散模型。将开发新技术和结果，以注重应用领域。研究人员将通过与早期职业研究人员和女性研究人员的合作来帮助培训潜水员数学研究人员。该项目还为研究生提供了培训机会。该项目将解决与随机网络动态分析和控制相关的数学问题。要解决的主题包括近似值的严格理由，分析和控制近似模型的行为以及解释原始微观模型的结果。一个重要的子主题是了解近似水平之间的相互作用。将研究五个主题：（i）（BIO）化学反应网络和几乎密度依赖的Markov链的扩散近似；（ii）分析处理器共享网络的分析；（iii）数据网络中的拥挤控制和资源夹带；（IV）具有随机的服务顺序和服务和重新组织；（IV）； （v）对随机处理网络的动态控制。这些主题的几个随机过程各个方面包括通过反射差异过程在几乎密度依赖的马尔可夫链的近似中进行错误量化，分析用于跟踪剩余的作业量或耐心时间的测量值的分析，用于跟踪资源编织和验证的问题，以控制和纠纷的差异问题，并在数量上差异，这些问题是对差异的差异。反映了非平滑域中的差异过程。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响标准来评估，以诚实的方式表示支持。