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. The analysis and control of such complex stochastic networks require solving challenging mathematical problems. This award supports research on solving such mathematical problems. Some of the work involves the development of general theory for broad classes of stochastic networks, while others focus on mathematical problems directly motivated by specific applications. Since the complexity of stochastic networks usually precludes exact analysis of detailed "microscopic" models, the focus is on formulating and analyzing more tractable approximations. New techniques and results will be developed in such a way that they can be used by applied researchers in areas of application. The results of this research will be disseminated through publication in peer reviewed journals, by posting on the University of California's open access website and by presentations at mathematics, science and engineering conferences. The PI will help train new mathematics researchers through collaboration with early career researchers.
The research addresses mathematical problems 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. Two levels of approximation are considered: first order approximations called fluid models, and second order approximations which frequently are diffusion models. An important subtheme is understanding the interplay between levels of approximation. Five topics are proposed for study: diffusion approximations for (bio)chemical reaction networks, analysis of processor sharing networks, congestion control and resource entrainment in data networks, networks with random order of service and reneging, and dynamic control of stochastic processing networks. Some stochastic process aspects of these topics include rates of convergence in the approximation of density dependent Markov chains by reflected diffusion processes, analysis of measure-valued processes used to track residual job sizes or ages of jobs in stochastic network models with resource sharing, singular diffusion control problems, foundational questions for reflected processes, and numerical approximation of reflected diffusion processes in non-smooth domains. This award will support the training of early career researchers and underrepresented minorities through direct support of a female graduate student, and through collaboration of the PI with one early career researcher and one female researcher who is at a non-PhD granting institution.
