Mathematical Sciences: Research on Stochastic Processes and Optimization

本研究分为两部分。第一部分从大偏差理论出发考虑问题。这个理论试图估计和描述随机系统中的罕见事件。这项工作的重点是扩展一些最近开发的方法。在这种新方法中，大偏差问题被重新定义为随机变分问题序列的极限行为。一个特别有趣的问题是对排队系统的大偏差特性的分析。提案的第二部分讨论了所谓的“资源系统的交互用户”的随机过程模型的发展。这些模型描述了有大量用户使用有限资源的情况。这样的例子出现在经济、金融和交通领域。工作的目标是开发模型、极限定理和模型简化。本研究分为两部分。第一部分从大偏差理论出发考虑问题。这个理论试图估计和描述随机系统中的罕见事件。这种罕见的事件在决定许多由随机过程建模的系统的性能方面起着关键作用。一个引人注目的问题是对排队系统的大偏差特性的分析，特别是那些出现在通信系统中的问题。同时，大偏差理论被认为在高速网络（包括ATM网络）的分析和设计中具有很大的潜力。这项工作的重点是扩展一些最近开发的可以处理这类难题的方法。研究的第二部分涉及随机过程模型的开发，我们称之为“资源系统的交互用户”。这些模型描述了有大量用户使用有限资源的情况。这样的例子出现在经济、金融和交通领域。需要这样的动态模型来预测用户群体如何适应环境的变化。这部分建议的主要目标是开发模型和简化模型。

This research is divided into two parts. The first part considers problems from the theory of large deviations. This theory seeks to estimate and characterize rare events in stochastic systems. The focus of the work is on extending some recently developed methods. In this new approach the large deviation problem is recast in terms of the limiting behavior of a sequence of stochastic variational problems. A problem of particular interest is the analysis of large deviation properties of queuing systems. The second part of the proposal discusses the development of stochastic process models for what are called "interacting users of a resource system." These models describe situations in which there are a large number of users of a finite set of resources. Examples occur in economics, finance and transportation. The goals of the work are development of the models, limit theorems, and model simplification. This research is divided into two parts. The first part considers problems from the theory of large deviations. This theory seeks to estimate and characterize rare events in stochastic systems. Such rare events can play a critical role in determining the performance of many systems modeled by stochastic processes. A problem of compelling interest is the analysis of large deviation properties of queuing systems, and in particular those that arise in communication systems. At the same time, large deviation theory is thought to have great potential in the analysis and design of high speed networks (including ATM networks). The focus of the work is on extending some recently developed methods that can handle this difficult class of problems. The second part of the research involves the development of stochastic process models for what we call "interacting users of a resource system." These models describe situations in which there are a large number of users of a finite set of resources. Examples occur in economics, finance and transportation. Such dynamical models are needed t o predict how the population of users would adapt to changes in the environment. The main goal of this part of the proposal is the development of the models and model simplification.