Dynamics of Stochastic Networks: 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。这个项目解决了由分析和控制这种随机网络的挑战而产生的数学问题。这项研究包括发展几类广泛的随机网络的一般理论，以及研究直接由特定应用驱动的问题。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，这里的重点是近似模型。我们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。将着眼于应用领域开发新技术和新成果。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，帮助培训一支多样化的数学研究队伍。该项目还为研究生提供了培训机会。这个项目将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。一个重要的副主题是理解近似级别之间的相互作用。将研究五个主题：(I)(生物)化学反应网络和几乎依赖密度的马氏链的扩散近似；(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获取；(Iv)具有随机服务顺序和放弃的网络；和(V)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近似密度依赖的马尔可夫链时的误差量化、用于跟踪资源共享和拒绝的随机网络中剩余作业大小或耐心时间的度量值过程的分析、奇异扩散控制问题、反射过程的基本问题以及非光滑领域中反射扩散过程的数值逼近。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.