Mathematical Sciences: Several Questions in Probability

这项研究涉及概率的数学理论及其应用，主要由两位首席研究员戴教授和希尔教授进行的独立调查组成。戴教授的研究将主要集中在排队网络，这是由计算机网络和制造系统产生的。它包括证明多类排队网络的布朗近似，理解网络在重载时的性能，以及计算反射的布朗运动的平稳分布。最终目标是开发一套强大的计算机软件工具，能够准确和有效地预测广泛类别的排队网络的性能。希尔教授的研究包括：最优停止问题，特别是顺序选择和极大极小规则；概率和统计学中的划分或公平划分问题；单边极限律和概率界，特别是平均值和分位数；概率的融合及其与膨胀、凸控制和鞅的关系。主要研究人员将在概率的理论和应用两个方面进行研究。更适用的主题包括调查来自计算机网络和制造系统的排队网络(由随机到达的具有随机服务需求的客户组成的互连服务设施)的性能，以及研究最优停止理论(对城市污染水平或股票价格等与时间相关的随机过程的顺序分析)。这项研究的一个极端是开发实用的计算机软件工具，可以准确有效地预测一大类随机系统的性能，另一个极端是在抽象空间和环境中发现新的基本数学定理。

This research involves the mathematical theory of probability and its application, and consists of largely independent investigations by two principal investigator, Professors Dai and Hill. Professor Dai's research will mainly focus on queueing networks, which arise from computer networks and manufacturing systems. It includes justification of the Brownian approximation for multiclass queueing networks, understanding network performance when it is heavily loaded, and computation of stationary distribution of a reflected Brownian motion. The ultimate goal is to develop a robust set of computer software tools that can accurately and efficiently predict the performance of a wide class of queueing networks. Professor Hill's research includes: optimal stopping problems, especially order selection and minimax rules; partitioning or fair-division problems in probability and statistics; one-sided limit laws and probability bounds, especially for means and quantiles; and fusions of probabilities and their relationship to dilation, convex domination and martingales. The principal investigators will carry out research in both theoretical and applied aspects of probability. The more applied topics include investigating the performance of queueing networks (inter-connected service facilities populated by randomly arriving customers with random amounts of service requirements) which arise from computer networks and manufacturing systems, and investigating optimal stopping theory (sequential analysis of time dependent random processes such as pollution levels in a city or stock prices). At one extreme, this research is aimed at developing practical computer software tools that can accurately and efficiently predict the performance of a wide class of random systems, and at the other extreme it is aimed at discovering new basic mathematical theorems in abstract spaces and settings.