This project involves the study of mathematical problems associated with the control and analysis of stochastic networks. Examples of application domains for these stochastic models include manufacturing, the service industry, telecommunications, computer systems and biochemical reactions. Some of the problems being studied involve the development of general theory for broad classes of stochastic networks, while others focus on mathematical questions directly motivated by specific applications. The following five topics are considered: (i) dynamic scheduling for stochastic processing networks, (ii) analysis of processor sharing networks, (iii) congestion control for data networks, (iv) input-queued switches, and (v) biochemical reaction networks. The stochastic network models associated with these topics are considerably more general than conventional multiclass queueing networks operating under a head-of-the-line (HL) service discipline. Although there is a fairly well developed theory of stability and heavy traffic diffusion approximation for the latter, there are many challenging open problems associated with the control and analysis of more general stochastic networks. Some stochastic process aspects of this project include the study of singular diffusion control problems, the use of measure-valued processes to keep track of residual job sizes, foundational questions for reflected Levy processes, and the use of an infinite server queue to model biochemical reactions with non-exponential reaction times.
This grant funds research on mathematical problems motivated by applications in the disciplines of operations research, computer science, electrical engineering and biochemistry. The specific problems being studied concern the control and analysis of stochastic networks which are models for complex systems involving dynamic interactions subject to uncertainty. Two fundamental problems for such networks are (a) to understand the behavior of these systems under natural control policies, and (b) to design 'good' controllers for these systems. The networks under study are substantially more general than those that have been rigorously studied to date. Through their complexity and heterogeneity, these networks present challenging mathematical problems. The project involves the development of new theory for stochastic processes and uses techniques from a variety of mathematical disciplines. Collaborations with researchers familiar with areas of application, the training of graduate student researchers, and the dissemination of research results through publication in peer reviewed journals and presentations at cross-disciplinary research conferences are integral parts of the project.
