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。这种复杂的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。其中一些工作涉及发展广泛类别的随机网络的一般理论，而另一些工作则专注于直接由特定应用驱动的数学问题。由于随机网络的复杂性通常排除了对详细的“微观”模型的准确分析，所以重点是制定和分析更容易处理的近似。将开发新的技术和成果，使其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议的期刊上发表、在加州大学的开放获取网站上张贴以及在数学、科学和工程会议上发表演讲来传播。PI将通过与早期职业研究人员的合作，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的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证明，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。考虑了两个层次的近似：称为流体模型的一阶近似和通常是扩散模型的二阶近似。一个重要的副主题是理解近似级别之间的相互作用。提出了五个研究课题：(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拒绝的网络，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反射扩散过程在密度依赖马氏链逼近中的收敛速度，资源共享随机网络模型中用于跟踪剩余作业大小或作业年龄的度量过程的分析，奇异扩散控制问题，反射过程的基本问题，以及非光滑区域中反射扩散过程的数值逼近。该奖项将通过对一名女研究生的直接支持，以及通过国际和平协会与一名早期职业研究人员和一名在非博士学位授予机构工作的女性研究人员的合作，支持早期职业研究人员和代表性不足的少数群体的培训。

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.