Mathematical Sciences: Interacting Particle Systems and Queueing Networks

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Bramson The investigator's research is in stochastic processes, primarily in the areas of interacting particle systems and queueing networks. Although typically appealing to different groups of researchers, both areas deal with large systems of objects, such as particles or customers, which are connected together by some interaction rule. Interacting particle systems research usually deals with lattice-valued random systems which update according to some local rule. Such systems model various complex random systems and exhibit a wide range of spatial behavior; they frequently occur in the context of mathematical physics and mathematical biology. One is interested in the long-time behavior of such systems, such as the behavior of the associated equilibria. The investigator intends to work on topics such as the structure of multitype particle systems, systems with annihilation and creation, the equilibria of certain semi-infinite systems, and interface dynamics. Queueing networks research studies the evolution of queues of individuals under different rules for routing and assigned priorities. As time evolves, individuals enter the system, move from one queue to the next, and, upon completion of service, exit from the system. Depending on specifics, such a system may or may not be stable; the goal of much recent research has been to analyze such behavior. A recent important tool for such problems is the use of fluid limits, with which problems on stability of networks can be reduced to the stability of the associated fluid models. The investigator intends to investigate the properties of some of the main classes of fluid models, and apply this analysis to the study of stability questions for networks and to the related topic of heavy traffic limits for networks. The investigator's research lies primarily in two areas of probability theory, interacting particle systems and queueing networks. The field of interacting particle systems typically deals with large random systems. The compon ents of these systems can represent, for example, particles, cells, or various organisms. Such systems model various complex random systems and exhibit a wide range of behavior; they frequently occur in the context of mathematical physics and mathematical biology. One is typically interested in the long-time behavior of such systems. The topics the investigator intends to work on include biological models for the diversity and stability of different species, and physical models for the evolution of chemical reactions. The field of queueing networks studies the behavior of lines of individuals in general settings. These individuals can, for example, be customers waiting to be served, or components involved in some manufacturing process. As time evolves, individuals enter the system, move from one line to the next, and, upon completion of service, exit from the system. A general unsolved problem is to find which systems are stable, that is, the lengths of their lines do not tend to grow as time increases. This information can then be used to indicate more efficient algorithms for the service of customers, with the goal of saving time and materials. A recent important tool for the analysis of queueing networks is the use of fluid models, which are nonrandom simplifications of the original networks. The investigator intends to continue his work in the area by studying and applying the properties of fluid models.