Queueing and communication networks governed by generalised Lindley-Loynes equations.

Abstract

Several decades after A.K. Erlang originated the theory of queues and queueing networks, D.V. Lindley added impetus to the development of this field by determining a recursive relation for waiting times. Part I of this thesis provides a theoretical treatment of single-server and multiserver queues described by the basic Lindley relation and its extensions, which are referred to collectively as Lindley-Loynes equations. The concepts of stability, and minimal and maximal solutions are investigated. The interdependence of theory and practice becomes evident in Part II, where the results of recent and current research are highlighted. While the main aim of the first part of the thesis is to provide a firm theoretical framework for the sequel, the objective in Part II is to derive generalised forms of the Lindley-Loynes equations from different network protocols. Such protocols are regulated by different switching rules and synchronization constraints. Parts I and II of the thesis are preceded by Chapter 0 in which several fundamental ideas (including those on notation and probability) are described. It is in this chapter too that a more detailed overview of the concept of the thesis is provided.