Dynamic Stochastic Networks: Analysis, Control and Applications

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Stochastic dynamic models of complex networks arise in a wide variety of applications in science and engineering. Specific instances include high-tech manufacturing, telecommunications, computer systems, service systems and biochemical reaction networks. This project involves the study of a number of mathematical problems stemming from the challenges of analyzing and controlling such dynamic stochastic networks. A balance of topics is proposed. Some topics involve the development of general theory for broad classes of networks, while others focus on mathematical problems directly motivated by specific applications. Since the complexity of these networks usually precludes exact analysis of detailed microscopic models, the focus here is on approximate models. Mathematical questions to be addressed include rigorous justification of these approximations, analyzing and controlling the behavior of the approximate models and interpreting the results for the original microscopic models. Four topics are proposed for study:(i) Dynamic control of stochastic processing networks,(ii) Analysis of processor sharing networks,(iii) Congestion control and resource entrainment in data networks,(iv) Stochastic systems with delayed dynamics and state constraints.Some stochastic process aspects of these topics include study of singular diffusion control problems, analysis of measure-valued processes, foundational questions for reflected processes, and the asymptotic properties of functional stochastic differential equations with natural state constraints.Dynamic stochastic networks are used as mathematical models in a wide variety of applications in science and engineering, particularly in operations management, computer science, electrical engineering and bioengineering. This grant funds research to develop new results and techniques for addressing mathematical problems stemming from the need to analyze and control such dynamic stochastic networks. The networks under study are substantially more general than those that have been rigorously studied to date. The results will be of fundamental mathematical interest as well as of interest for applied researchers in relevant areas of science and engineering. The training of mathematics PhD students, dissemination of results through publication in peer-reviewed journals and presentations of the research at cross-disciplinary conferences, are all integral parts of the project.