Stochastic Network Dynamics: Approximation, Analysis and Control

随机网络动力学：近似、分析和控制

基本信息

批准号：
1712974
负责人：
Ruth Williams
金额：
$ 25万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712974&HistoricalAwards=false
关键词：
Stochastic Network Dynamics Approximation Analysis

项目摘要

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. The analysis and control of such complex stochastic networks require solving challenging mathematical problems. This award supports research on solving such mathematical problems. Some of the work involves the development of general theory for broad classes of stochastic networks, while others focus on mathematical problems directly motivated by specific applications. Since the complexity of stochastic networks usually precludes exact analysis of detailed "microscopic" models, the focus is on formulating and analyzing more tractable approximations. New techniques and results will be developed in such a way that they can be used by applied researchers in areas of application. The results of this research will be disseminated through publication in peer reviewed journals, by posting on the University of California's open access website and by presentations at mathematics, science and engineering conferences. The PI will help train new mathematics researchers through collaboration with early career researchers.The research addresses mathematical problems 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. Two levels of approximation are considered: first order approximations called fluid models, and second order approximations which frequently are diffusion models. An important subtheme is understanding the interplay between levels of approximation. Five topics are proposed for study: diffusion approximations for (bio)chemical reaction networks, analysis of processor sharing networks, congestion control and resource entrainment in data networks, networks with random order of service and reneging, and dynamic control of stochastic processing networks. Some stochastic process aspects of these topics include rates of convergence in the approximation of density dependent Markov chains by reflected diffusion processes, analysis of measure-valued processes used to track residual job sizes or ages of jobs in stochastic network models with resource sharing, singular diffusion control problems, foundational questions for reflected processes, and numerical approximation of reflected diffusion processes in non-smooth domains. This award will support the training of early career researchers and underrepresented minorities through direct support of a female graduate student, and through collaboration of the PI with one early career researcher and one female researcher who is at a non-PhD granting institution.

具有动态交互作用的复杂网络的随机模型出现在科学和工程的各种应用中。具体实例包括生化反应网络、高科技制造、计算机系统、电信、交通和商业服务系统。此类复杂随机网络的分析和控制需要解决具有挑战性的数学问题。该奖项支持解决此类数学问题的研究。其中一些工作涉及广泛的随机网络的一般理论的发展，而另一些工作则专注于由特定应用直接激发的数学问题。由于随机网络的复杂性通常妨碍对详细“微观”模型的精确分析，因此重点是制定和分析更容易处理的近似值。新技术和成果的开发方式应可供应用研究人员在应用领域使用。这项研究的结果将通过在同行评审期刊上发表、在加州大学的开放获取网站上发布以及在数学、科学和工程会议上进行演示来传播。 PI 将通过与早期职业研究人员合作帮助培训新的数学研究人员。该研究解决与随机网络动力学分析和控制相关的数学问题。要解决的主题包括近似的严格论证、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。考虑两个级别的近似：称为流体模型的一阶近似，以及通常是扩散模型的二阶近似。一个重要的副主题是理解近似级别之间的相互作用。提出了五个研究主题：（生物）化学反应网络的扩散近似、处理器共享网络的分析、数据网络中的拥塞控制和资源夹带、具有随机服务顺序和更新的网络以及随机处理网络的动态控制。这些主题的一些随机过程方面包括通过反射扩散过程近似密度相关马尔可夫链的收敛率、用于跟踪具有资源共享的随机网络模型中的剩余作业规模或作业年龄的测值过程分析、奇异扩散控制问题、反射过程的基本问题以及非光滑域中反射扩散过程的数值近似。该奖项将通过一名女性研究生的直接支持，以及通过 PI 与一名早期职业研究员和一名非博士学位授予机构的女性研究员的合作，支持对早期职业研究人员和代表性不足的少数群体的培训。