Stochastic Network Dynamics: Approximation, Analysis and Control

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有着广泛的应用。具体的例子包括生化反应网络、高科技制造、计算机系统、电信、运输和商业服务系统。分析和控制这种复杂的随机网络需要解决具有挑战性的数学问题。该奖项支持解决这类数学问题的研究。其中一些工作涉及发展广泛类别的随机网络的一般理论，而另一些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排除了对详细的“微观”模型的精确分析，因此重点是制定和分析更易于处理的近似。新技术和新成果将被应用研究人员用于应用领域。这项研究的结果将通过在同行评议的期刊上发表、在加州大学的开放获取网站上发布以及在数学、科学和工程会议上发表报告等方式传播。PI将通过与早期职业研究人员的合作，帮助培养新的数学研究人员。该研究解决了与随机网络动力学分析和控制相关的数学问题。要解决的主题包括对近似的严格论证，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。考虑两种近似：称为流体模型的一阶近似和通常是扩散模型的二阶近似。一个重要的次要主题是理解近似水平之间的相互作用。提出了5个研究课题：生物化学反应网络的扩散逼近、处理器共享网络的分析、数据网络中的拥塞控制和资源占用、随机服务顺序和违约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在密度依赖的马尔可夫链近似中的收敛速度，用于跟踪资源共享随机网络模型中剩余作业大小或作业年龄的测量值过程的分析，奇异扩散控制问题，反射过程的基础问题，以及非光滑域中反射扩散过程的数值近似。该奖项将通过直接支持一名女研究生，以及通过PI与一名早期职业研究人员和一名在非博士学位授予机构的女研究人员的合作，支持对早期职业研究人员和未被充分代表的少数民族的培训。

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.