Queueing networks

Abstract

Researchers have been intrigued by queues and waiting lines which occur in real life situations as diverse as manufacturing, telecommunication and computer systems for years. A theoretical basis guarantees a solution that is not too far from the real situation. In queueing theory, a system is represented as a queueing network made up of a number of stations (or nodes) that provide service to customers. Queueing networks are described by the queueing process characteristics at each station and the interaction of the stations. The analysis involves the determination of the stationary distribution of the network states and the system performance measures. When the stationary distribution has a product form solution the analysis reduces to analysing each node in isolation. The thesis presents the theory of Markov chains, counting processes, local balance, and reversibility which lead to the conditions for the existence of a stationary distribution, for decomposability that verifies product forms, and for quasi-reversibility which allows nodes to be analysed in isolation. Also the study of queueing networks with customer classes and service positions yields, and verifies, the conditions for the class of queueing networks with exact product form solutions, i.e., the Jackson and the BCMP networks. Real systems rarely satisfy the restrictive conditions for product form solutions. Approximations must then be resorted to. They all have limitations but they allow a solution to be approached. The theory was applied to a practical queueing system in a commercial manufacturing enterprise. The system was first studied under ideal conditions, which yielded bounds for the system performance measures. It was then modelled subject to machine failures, which destroy the product form solution. A validation for the assumptions made to restore the product form is given and the results from the model were compared to the actual system performance measures.