Mathematical Sciences: Research on Stochastic Processes and Optimization

本研究分为两个部分。 第一部分考虑大偏差理论的问题。 该理论旨在估计和描述随机系统中的罕见事件。 工作的重点是扩展一些最近开发的方法。 在这种新的方法中，大偏差问题是重铸的一系列随机变分问题的极限行为。 一个特别感兴趣的问题是分析排队系统的大偏差性质。 提案的第二部分讨论了所谓“资源系统的交互用户”的随机过程模型的发展。“这些模型描述了有限资源集有大量用户的情况。 经济、金融和运输领域都有这样的例子。 工作的目标是发展的模型，极限定理，模型简化。 本研究分为两个部分。 第一部分考虑大偏差理论的问题。 该理论旨在估计和描述随机系统中的罕见事件。 这种罕见的事件可以发挥关键作用，在确定随机过程建模的许多系统的性能。 一个令人感兴趣的问题是分析排队系统的大偏差特性，特别是那些出现在通信系统。 同时，大偏差理论在高速网络（包括ATM网络）的分析和设计中也具有很大的潜力。 工作的重点是扩展一些最近开发的方法，可以处理这类困难的问题。 研究的第二部分涉及我们所谓的“资源系统的交互用户”的随机过程模型的发展。“这些模型描述了有限资源集有大量用户的情况。 经济、金融和运输领域都有这样的例子。 需要这种动态模型来预测用户群体将如何适应环境的变化。这部分提案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.