University of Washington - Department of Statistics
Many application-specific approaches have been proposed for modeling resource sharing. Dijkstra's Dining Philosophers Problem is one of the first systematic attempts in the area, later generalized by Chandy and Misra as the Drinking Philosophers Problem. We consider Markovian versions of these situations. The aim is to analyze the performance of the underlying systems through their behavior at equilibrium. New Markovian models for resource sharing are introduced. They can be viewed as interacting particle systems. Mathematical techniques of treatment such as reversibility and stochastic comparison are proposed for these models. For finite systems, explicit calculations of measures at equilibrium are given. The behavior of systems increasing in size and complexity is closely related to that of infinite systems. For such systems, a phase transition phenomenon may occur. We give necessary and sufficient conditions for phase transitions for systems modeled by graphs built from infinite regular trees.