Queueing network models are widely used in the study of telecommunication networks, manufacturing systems, and service systems. Performance analysis is usually the first step towards the understanding of the queueing network models and the systems they model. The stability and diffusion approximation of queueing networks are important issues in the performance analysis and have attracted a lot of attention in recent years. In this thesis, we focus on the stability and diffusion approximation of some open multiclass queueing networks....[ Read more ]

Queueing network models are widely used in the study of telecommunication networks, manufacturing systems, and service systems. Performance analysis is usually the first step towards the understanding of the queueing network models and the systems they model. The stability and diffusion approximation of queueing networks are important issues in the performance analysis and have attracted a lot of attention in recent years. In this thesis, we focus on the stability and diffusion approximation of some open multiclass queueing networks.

This thesis contains three parts. First, we prove that a necessary and sufficient condition for the stability of a generic fluid model is the existence of a Lyapunov function for its fluid level processes. This general result is then applied to various specific fluid networks (such as fluid networks under all work-conserving disciplines, a priority service discipline, or a first-in-first-out service discipline), fluid limit models and a linear Skorohod problem. In the second part of the thesis, we propose a class of piecewise linear Lyapunov functions to construct some sufficient conditions for the stability of a fluid network under a priority service discipline. These sufficient conditions improve all known sufficient conditions, and verifying these new sufficient conditions amounts to solving a set of linear program problems. In the final part of the thesis, we establish a new existence condition for the (conventional) diffusion approximation of multiclass queueing networks under priority service disciplines. Applying this sufficient condition, we establish the existence of the diffusion approximation for a reentrant line queueing network under a last-buffer- first-served discipline. We also study a three-station network example, and observe that the diffusion approximation may not exist, even if the "proposed" limiting semi-martingale reflected Brownian motion exists.