A Study of Several Stochastic Models

Cornell University, Ithaca NY

Investigators

Abstract

The principal investigator plans to work on several topics in probability theory. The aim of the first project is to study transience/recurrence behavior of edge reinforced random walks on graphs. The purpose of the second project is to characterize stability and optimality of batch service policies for queueing systems. The third project is a rigorous study of the NK model of protein evolution. The main interest here is in determining the exact or asymptotic behavior of various quantities in the corresponding fitness landscape. The fourth project involves a sequence of interacting Moran model particle systems and their scaling limit, the interacting Fisher-Wright diffusions. The goal here is to develop tools and techniques that will help one answer questions about fine properties of the interacting diffusions. Random processes with reinforcements appear in models of biological systems. A particle performing an edge reinforced random walk on a graph has memory and prefers to traverse edges that were traversed before. It is known that some reinforced random walks exhibit behavior very different from that of the classical random walks without memory. The aim of the first project is to show that edge reinforced walks for which each edge is reinforced only once, by a fixed amount, are not so different from the classical random walks. The batch service queueing system is a model of a retail store delivery van, or a terminal in a computer network which downloads software packages. The purpose of the second project is to describe a large class of batch service policies that keep the queueing system stable, and to find easy to implement policies in this class, that are close to optimal with respect to a specific cost function. The NK model is an idealization of protein evolution in the presence of selection. The interacting Moran models represent a subdivided population with a resampling mechanism within each subpopulation, and migration of individuals to neighboring subpopulations. The properties of both models are of interest to geneticists.