Stochastic Networks: Analysis and Control

这个项目的目的是研究与控制和分析随机网络性能相关的数学问题。由于这些网络的复杂性和异质性通常排除了精确的分析，这里的重点是近似模型。考虑两个级别的近似：称为流体模型的一阶（大数泛函定律）近似，以及通常是扩散模型的二阶（泛函中心极限定理）近似。这两个层面之间的相互作用是贯穿始终的重要副主题。主要讨论三个主题：(i)随机处理网络的动态调度，（ii）处理器共享网络的分析，（iii）现代通信网络的拥塞控制。在这些主题下考虑的随机网络模型比传统的多类排队网络在排队首（HL）服务规则下运行要普遍得多。尽管对于后者有一个相当发达的稳定性和大流量扩散近似理论，但与这里研究的更一般的随机网络模型的控制和分析相关的许多具有挑战性的开放问题。随机过程的一些方面是，主题(i)涉及对近似约束扩散控制问题的解决方案的分析和解释，主题（ii）使用测量值过程来跟踪网络中所有作业的剩余服务时间，主题（iii）涉及文档大小有重尾时的反射Levy过程。本基金资助由运筹学、计算机科学及电子工程等学科的应用所引发的数学问题的研究。具体地说，与随机网络的控制和分析有关的问题正在研究中。这种网络被用作复杂制造业、电信和计算机系统的模型。这类网络的两个基本问题是：(a)理解共同策略对拥塞和延迟的影响，以及(b)为这些系统设计“好的”控制器。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。该项目涉及随机过程新理论的发展，并使用了各种数学学科的技术。与熟悉应用领域的研究人员合作以及研究生研究人员的培训是该项目的组成部分。

The aim of this project is to study mathematical problems associated with controlling and analyzing the performance of stochastic networks. Since the complexity and heterogeneity of these networks usually preclude exact analysis, the focus here is on approximate models. Two levels of approximation are considered: first order (functional law of large numbers) approximations called fluid models, and second order (functional central limit theorem) approximations, which are frequently diffusion models. The interplay between these two levels is an important subtheme throughout. Three main topics are addressed: (i) Dynamic scheduling for stochastic processing networks, (ii) Analysis of processor sharing networks, (iii) Congestion control for modern communication networks. The stochastic network models considered under these topics are considerably more general than conventional multiclass queueing networks operating under a head-of-the-line (HL) service discipline. Although there is a fairly well developed theory of stability and heavy traffic diffusion approximation for the latter, there are many challenging open problems associated with the control and analysis of the more general stochastic network models being studied here. Some of the stochastic process aspects are that topic (i) involves the analysis and interpretation of the solution of an approximating constrained diffusion control problem, topic (ii) uses measure-valued processes to keep track of residual service times of all jobs in the network, and topic (iii) involves reflected Levy processes when document sizes have heavy tails. This grant funds research on mathematical problems motivated by applications in the disciplines of operations research, computer science and electrical engineering. Specifically, problems associated with the control and analysis of stochastic networks are being studied. Such networks are used as models for complex manufacturing, telecommunications and computer systems. Two fundamental problems for such networks are (a) to understand the effects of common policies on congestion and delay, and (b) to design 'good' controllers for these systems. 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. The project involves the development of new theory for stochastic processes and uses techniques from a variety of mathematical disciplines. Collaborations with researchers familiar with areas of application and the training of graduate student researchers are integral parts of the project.