Stochastic Networks: Analysis and Control

该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。两个层次的近似被认为是：第一阶（功能大数定律）近似称为流体模型，和二阶（功能中心极限定理）近似，这是经常扩散模型。这两个层次之间的相互作用是贯穿始终的一个重要副主题。三个主要议题是解决：（i）随机处理网络的动态调度，（ii）分析处理器共享网络，（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是相当普遍的比传统的多类排队网络下运行的头的线（HL）的服务纪律。虽然有一个相当发达的理论稳定性和重型交通扩散近似后者，有许多具有挑战性的开放问题与控制和分析更一般的随机网络模型正在研究这里。随机过程的一些方面是，主题（i）涉及的近似约束扩散控制问题的解决方案的分析和解释，主题（ii）使用测量值的过程，以跟踪网络中的所有作业的剩余服务时间，和主题（iii）涉及反映利维过程时，文档大小有沉重的尾巴。 该补助金资助运筹学、计算机科学和电气工程学科中应用数学问题的研究。具体而言，与随机网络的控制和分析相关的问题正在研究中。这种网络被用作复杂制造、电信和计算机系统的模型。这样的网络的两个基本问题是（a）了解常见的政策对拥塞和延迟的影响，和（B）设计这些系统的“好”的控制器。所研究的网络比迄今为止已经严格研究的网络要普遍得多。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的新理论的发展，并使用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.