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业、客户服务系统、电信、计算机系统和基因调控网络。这个项目涉及一些数学问题的研究，这些问题源于分析和控制这些随机网络的挑战。其中一些问题涉及到广泛的随机网络的一般理论的发展，而另一些问题则集中在由特定应用直接引起的数学问题上。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，这里的重点是近似模型。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。正在解决的数学问题包括这些近似的严格证明，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。一个重要的次要主题是理解近似水平之间的相互作用。目前正在研究四个主题：(i)随机处理网络的动态调度，（ii）处理器共享网络的分析，（iii）数据网络的连接级模型，（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，用于跟踪剩余作业大小的测量值过程的分析，反射过程的基本问题，以及具有自然状态约束的泛函随机微分方程的渐近性质。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。随机网络是涉及不确定性的动态相互作用的复杂系统的数学模型。这种网络在科学和工程领域有着广泛的应用，特别是在运筹学、计算机科学、电气工程和生物科学/生物工程领域。这项拨款用于研究由于需要分析和控制这种随机网络而产生的数学问题。这种网络的两个基本问题是(a)识别和理解稳定系统的机制，以及(b)量化在这种稳定机制下系统的性能。所研究的网络比迄今为止严格研究过的网络要普遍得多。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，以及将这些理论应用于研究互联网拥塞控制和理解基因调控等具体问题。与熟悉应用领域的研究人员合作，培训研究生研究人员，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来传播研究成果，这些都是项目的组成部分。

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.