Mathematical Sciences: Several Questions in Probability

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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.