This thesis is devoted to the construction and analysis of models which can be used to evaluate the performance and reliability of distributed systems. The general object of the research therefore is to extend the types of queueing models with breakdowns which have been solved, with a particular interest in networking structures. The systems that are studied involve various collections of servers and their associated queues. These range from isolated nodes, though parallel nodes coupled by the effect of breakdowns on arrivals, to pipelines of such parallel stages and more general networks. The issues that are explored include the influence of breakdowns and repairs on delays, job losses and optimal routeing. Obtaining performance measures for interacting queues is difficult, however a degree of abstraction has been used here which allows long run averages to be calculated (exactly in many cases) for quite complex systems. A variety of different techniques are used in order to obtain solutions to these models, including exact equations, exact numerical and approximate numerical techniques.