AMC-SS: Stochastic Networks -- Analysis, Control and Applications

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。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。具体的例子包括高科技制造、客户服务系统、电信、计算机系统和基因调控网络。该项目涉及研究一些数学问题，这些问题源于分析和控制这种随机网络的挑战。一些问题涉及到广泛的随机网络的一般理论的发展，而其他的问题则集中在直接由特定的应用所激发的数学问题上。由于随机网络的复杂性通常排除了详细的“微观”模型的精确分析，这里的重点是近似模型。两个层次的近似被认为是：第一阶近似称为流体模型，和第二阶近似，这往往是扩散models.Mathematical问题正在解决包括严格的理由，这些近似，分析和控制的行为的近似模型和解释的结果，为原来的微观模型。一个重要的子主题是理解近似水平之间的相互作用。正在研究的四个主题：（i）随机处理网络的动态调度，（ii）处理器共享网络的分析，（iii）数据网络的连接水平模型，（iv）具有延迟动态和状态约束的随机系统。这些主题的一些随机过程方面包括奇异扩散控制问题的研究，用于跟踪剩余工作大小的测量值过程的分析，反映过程的基础问题，以及具有自然状态约束的泛函随机微分方程的渐近性质。目前正在研究的具体应用包括互联网拥塞控制和生化反应网络。随机网络是复杂系统的数学模型，涉及动态相互作用的不确定性。这种网络出现在科学和工程的各种应用中，特别是在运筹学、计算机科学、电子工程和生物科学/生物工程中。该补助金资助因分析和控制此类随机网络而产生的数学问题的研究。这类网络的两个基本问题是：（a）识别和理解稳定系统的机制;（B）量化系统在这种稳定机制下的性能。由于其复杂性和异质性，这些网络提出了具有挑战性的数学问题。该项目涉及新的数学理论和技术的发展，以及该理论在研究特定问题，如互联网拥塞控制和理解基因调控中的应用。与熟悉应用领域的研究人员合作，研究生研究人员的培训，以及通过在同行评审期刊上发表和在跨学科研究会议上发表研究成果的传播是该项目的组成部分。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Stochastic models of complex networks with dynamic interactions arise in a wide variety of applications in science and engineering. Specific instances include high-tech manufacturing, customer service systems, telecommunications, computer systems, and gene regulatory networks. This project involves the study of a number of mathematical problems stemming from the challenges of analysing and controlling such stochastic networks. Some of the problems involve 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 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.Mathematical questions being addressed include rigorous justification of these approximations, analysing 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 the levels of approximation.Four topics are being studied:(i) dynamic scheduling for stochastic processing networks,(ii) analysis of processor sharing networks,(iii) connection level models for data networks,(iv) stochastic systems with delayed dynamics and state constraints. Some stochastic process aspects of these topics include study of singular diffusion control problems, analysis of measure-valued processes used to keep track of residual job sizes, foundational questions for reflected processes, and the asymptotic properties of functional stochastic differential equations with natural state constraints. Specific applications being investigated include Internet congestion control and biochemical reaction networks.Stochastic networks are mathematical models for complex systems involving dynamic interactions subject to uncertainty. Such networks arise in a wide variety of applications in science and engineering, especially in operations research, computer science, electrical engineering and bioscience/bioengineering. This grant funds research on mathematical problems arising from the need to analyse and control such stochastic networks. Two fundamental problems for such networks are(a) to identify and understand mechanisms that stabilize the systems, and(b) to quantify the performance of the systems under such stabilizing mechanisms.The networks under study are substantially more general than those that have been rigorously studied to date. Through their complexity and heterogeneity, these networks present challenging mathematical problems. This project involves the development of new mathematical theory and techniques as well as the application of this theory in studying specific problems such as Internet congestion control and understanding gene regulation. Collaborations with researchers familiar with areas of application, the training of graduate student researchers, and the dissemination of research results through publication in peer reviewed journals and presentations at cross-disciplinary research conferences are integral parts of the project.