Research in Large Deviations and Applications to Queueing Networks

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9700852 Ellis This research will focus on two broad areas: (1) a weak convergence approach to the theory of large deviations and (2) large deviations for jump Markov processes and applications to queueing networks. In area (1) the research is based in part on a new book written jointly with Paul Dupuis. It consists of the following: (a) deriving representation formulas for new classes of processes, including jump Markov processes; (b) proving the Laplace principle for diffusions and other processes having discontinuous statistics with various geometries; (c) proving the Laplace principle for a random walk model with state- dependent noise; (d) proving the Laplace principle for empirical measures of Markov chains and related processes, including measure-valued processes with state dependencies; (e) generalizing a new formula on the Legendre-Fenchel transform of compositions of convex functions. Research in area (2) consists of the following: (a) using a new representation formula for large deviation probabilities together with weak convergence methods to obtain an explicit formula for the rate function for a number of queueing models of interest; (b) extending a recent paper on the large deviation analysis of general queueing systems to other queueing models, including state-dependent queues; (c) evaluating the exact large deviation rates and determining the overflow paths corresponding to certain overflow events in a number of queueing networks. This research program in large deviations is motivated by applications to queueing networks, which are of fundamental importance in communication and high performance computing. The work of the Principal Investigator over the past few years, which culminated in a number of important papers and a research-level book, develops a set of new and powerful techniques for studying the behavior of such networks. For example, a critical problem in the design of high speed networking technologies is to size the buffe rs at communications switches within these networks so that certain overflow probabilities are suitably small. Another critical problem in the design of very large multiprocessor systems is to estimate the probability that the system will fail in some time interval. This research will be adapted to studying these and related problems.