Limiting Behavior of Queueing Networks and Interacting Particle Systems

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Professor Bramson's main research on stochastic processes continues to be in the areas of queueing networks and interacting particle systems. Two major themes in the first topic are the stability of queueing networks and their heavy traffic limits. A queueing network is said to be stable when the associated Markov process is positive recurrent. The heavy traffic limit of a sequence of queueing networks is the diffusive limit obtained by applying the usual central limit scaling. Results over the last decade have led to a better understanding of queueing networks, although a general theory is lacking. The researcher plans to investigate the behavior of queueing networks under certain well-known disciplines, such as processor sharing and earliest-due-date, first-served. Other related topics in queueing theory include dynamic scheduling in heavy traffic and the stability of self-managed models for the internet. Interacting particle systems consist of systems of many particles that evolve according to some interaction law. Such systems are frequently motivated by models from the physical and biological sciences. The researcher plans to investigate models that have creation and annihilation of mass, as well the stationary measures of certain exclusion processes. Professor Bramson's current research is in two areas of probability theory, queueing networks and interacting particle systems. In the first area, one studies the behavior of large systems of individuals who proceed from one server to the next, until their needs are satisfied and they leave the system. "Individuals" here should be interpreted in a general sense, and may be customers waiting to be served (for instance, at a bank or at a service counter), jobs in some manufacturing process (such as the production of semiconductor wafers) or information packets on the internet. In practice, one is often faced with the choice of whom to serve first. This choice can substantially affect the efficiency of a system, and sometimes even its stability. That is, based on how good a service rule is in a given setting, the number of customers waiting in line may typically remain small over extended periods of time, or may tend to grow without bound. Work over the last decade has shown that such systems can be unstable for unexpected reasons. The researcher plans to continue his work in this area and investigate the stability of the network under certain well-known service rules. The area of interacting particle systems is motivated by models from the physical and biological sciences consisting of large numbers of "individuals" who interact according to some rule. For instance, individuals might be molecules moving about according to some rule or cancer cells spreading along a surface. The behavior of these systems can depend strongly on the nature of their interactions. The researcher plans to continue his research on certain of these models.