Scaling limits for measure-valued processes with applications to stochastic networks

这个项目是概率论的，并且围绕着一个主要目标：为某些类型的测量值过程开发一个单一的框架来证明缩放极限定理。直接的应用是排队网络理论，但其他应用将被演示。测量值过程在随机系统的现代研究中得到越来越多的应用。特别是最近几年对排队网络的研究，它们正在改变解决问题的方式。该项目将开发一种新的方法来统一处理一类重要的缩放极限结果，这些结果目前必须使用临时参数一次证明一个。在这个过程中，我们会发现与泛函分析和微分方程之间有趣的联系。还将研究几个辅助问题；其中每一个都通过提供特殊案例、可能的推广途径或应用程序的替代领域来支持项目的主要焦点。这些支持问题是：(a)证明了带宽共享网络的扩散极限定理；(b)获得最小可达服务队列的流体和扩散极限；(c)研究社会新闻聚合器模型。该项目旨在改进用于分析复杂随机系统的理论工具，如计算机和电信网络、社会网络和生物过程。例如，该项目将产生定理，使从业者能够理解拟议的计算机网络设计中拥塞的可变性。这样的工作对社会有实实在在的好处。例如，互联网的许多组件的设计——交换机、路由协议、负载平衡系统——都受到了这种理论工作的启发。在我们日益网络化的世界中，我们继续创造新的结构和相互作用，例如本项目中正在研究的模型。加深我们对这些复杂系统的理解显然有好处。该项目还整合了研究生和本科生的研究培训活动，为未来加强研究基础设施。

The project is in probability, and revolves around a primary goal: develop a single framework for proving scaling limit theorems for certain types of measure-valued processes. The immediate application is to the theory of queueing networks, but other applications will be demonstrated. Measure-valued processes are finding increasing use in the modern study of stochastic systems. In particular for the study of queueing networks in the last several years, they are changing the way problems are approached. The project will develop a novel way to unify the treatment of an important class of scaling limit results that must currently be proved one at a time using ad hoc arguments. In the process, interesting connections to functional analysis and differential equations will be made. Several ancillary problems will be investigated as well; each of these supports the main focus of the project by providing special cases, possible avenues for generalization, or alternative areas of application. These supporting problems are: (a) prove a diffusion limit theorem for bandwidth-sharing networks; (b) obtain fluid and diffusion limits for Least Attained Service queues; (c) study a model of social news aggregators. This project aims to improve the theoretical tools available for analysis of complex random systems, such as computer and telecommunication networks, social networks, and biological processes. For example, the project will produce theorems that enable a practitioner to understand the variability of congestion in a proposed computer network design. Such work has tangible benefits to society. The designs of many components of the internet for example -- switches, routing protocols, load balancing systems -- have been informed by theoretical work of this kind. In our ever more networked world, we continue to create new structures and interactions, such as the models under study in this project. There is a clear benefit to deepening our understanding of these complex systems. The project also integrates research training activities for both graduate and undergraduate students, enhancing research infrastructure for the future.