On the construction and stationary distributions of some spatial queueing and particle systems

Abstract

We consider a `non-greedy' queueing system on a circle. In Chapter 2 of this thesis we present a new and very simple proof of the stability of this system (under the appropriate condition) based on the average travel times between customers. Then we consider a polling network with finitely many service stations, in which the server has a `greedy' service strategy. Using the same idea, under the appropriate condition, we also give a new simple proof of the stability of this system. In Chapter 3 we consider variants of the non-greedy system, which we call k-systems. In these systems at most k customers are allowed on the circle. We prove that the stationary distributions of the k-systems converge to the stationary distribution of the non-greedy system, if k tends to infinity. In Chapter 4 and Chapter 5 we consider two specific interacting particle systems. Chapter 4 handles an interacting particle system on {0,1}^Z with non-local, unbounded flip rates. A zero flips to a one at a rate that depends on the number of ones to the right until we see a zero (the flip rate equals a constant l times one plus this number). A one flips to a zero at rate m. The system is constructed using monotonicity. We show that for l<m, the system has a unique non-trivial stationary distribution, which is strongly mixing and has a density of ones of l/m. For l=m and l>m the limit is degenerate at {1}^Z. Our main tool is an explicit formula for the density of ones at any given moment. We also show that the stationary distribution has positive correlations, and is not a product measure. Chapter 5 deals with the construction of a bricklayer system on Z^Z, where the rates obey certain conditions. This system has local, but unbounded rates and is not monotone.