The project is in probability, and revolves around a primary goal: develop a single framework for proving scaling limit theorems for certain types of measure-valued processes. The immediate application is to the theory of queueing networks, but other applications will be demonstrated. Measure-valued processes are finding increasing use in the modern study of stochastic systems. In particular for the study of queueing networks in the last several years, they are changing the way problems are approached. The project will develop a novel way to unify the treatment of an important class of scaling limit results that must currently be proved one at a time using ad hoc arguments. In the process, interesting connections to functional analysis and differential equations will be made. Several ancillary problems will be investigated as well; each of these supports the main focus of the project by providing special cases, possible avenues for generalization, or alternative areas of application. These supporting problems are: (a) prove a diffusion limit theorem for bandwidth-sharing networks; (b) obtain fluid and diffusion limits for Least Attained Service queues; (c) study a model of social news aggregators.
This project aims to improve the theoretical tools available for analysis of complex random systems, such as computer and telecommunication networks, social networks, and biological processes. For example, the project will produce theorems that enable a practitioner to understand the variability of congestion in a proposed computer network design. Such work has tangible benefits to society. The designs of many components of the internet for example -- switches, routing protocols, load balancing systems -- have been informed by theoretical work of this kind. In our ever more networked world, we continue to create new structures and interactions, such as the models under study in this project. There is a clear benefit to deepening our understanding of these complex systems. The project also integrates research training activities for both graduate and undergraduate students, enhancing research infrastructure for the future.
