Dynamics of Stochastic Networks: Approximation, Analysis, and Control

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Stochastic models of complex networks with dynamic interactions arise in a wide variety of applications in science and engineering. Specific instances include biochemical reaction networks, high-tech manufacturing, computer systems, telecommunications, transportation, and business service systems. This project addresses mathematical questions stemming from the challenges of analyzing and controlling such stochastic networks. The research involves the development of general theory for some broad classes of stochastic networks and the study of questions directly motivated by specific applications. Since the complexity of stochastic networks usually precludes exact analysis of detailed “microscopic” models, the focus here is on approximate models. Two levels of approximation are considered: first-order approximations called fluid models, and second-order approximations, which frequently are diffusion models. New techniques and results will be developed with an eye toward application areas. The investigator will help train a diverse mathematics research workforce through collaboration with early career researchers and women researchers. The project also provides training opportunities for graduate students. This project will address mathematical questions associated with the analysis and control of stochastic network dynamics. Topics to be addressed include rigorous justification of approximations, analyzing and controlling the behavior of the approximate models, and interpreting the results for the original microscopic models. An important subtheme is understanding the interplay between levels of approximation. Five topics are to be studied:(i) Diffusion Approximations for (Bio)Chemical Reaction Networks and Nearly Density-Dependent Markov Chains;(ii) Analysis of Processor Sharing Networks;(iii) Congestion Control and Resource Entrainment in Data Networks;(iv) Networks with Random Order of Service and Reneging; and(v) Dynamic Control of Stochastic Processing Networks.Some stochastic process aspects of these topics include error quantification in the approximation of nearly density-dependent Markov chains by reflected diffusion processes, analysis of measure-valued processes used to track residual job sizes or patience times in stochastic networks with resource sharing and reneging, singular diffusion control problems, foundational questions for reflected processes, and numerical approximation of reflected diffusion processes in non-smooth domains.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.