Some of the most important applications of probabilistic modelling techniques are in the area of distributed systems. The term ‘distributed’ means, in this context, that various tasks that are somehow related can be carried out by different servers which may or may not be in different geographical locations. Such a broad definition covers a great variety of applications, in the areas of manufacturing, transport, computing and communications. To study the behaviour of a distributed system, one normally needs a model involving a number of service centres, with jobs arriving and circulating among them according to some random or deterministic routeing pattern. This leads in a natural way to the concept of a network of queues.

A queueing network can be thought of as a connected directed graph whose nodes represent service centres. The arcs between those nodes indicate one-step moves that jobs may make from service centre to service centre (the existence of an arc from nodei to nodej does not necessarily imply one from j to i). Each node has its own queue, served according to some scheduling strategy. Jobs may be of different types and may follow different routes through the network. An arc without origin leading into a node (or one without destination leading out of a node) indicates that jobs arrive into that node from outside (or depart from it and leave the network). Figure 4.1 shows a five-node network, with external arrivals into nodes 1 and 2, and external departures from nodes 1 and 5. At this level of abstraction, only the connectivity of the nodes is specified; nothing is said about their internal structure, nor about the demands that jobs place on them.