Growth, Complexity, and Performance of Telephone Connecting Networks

The relationships between traffic carried and traffic lost, between load and loss, have always been at the center of interest in telephone traffic theory. Since the time of Erlang,1 over fifty years ago, the principal problems of traffic theory have been analytical: to predict mathematically, from the structure and mode of use of a switching or connecting network, and from the assumed stochastic behavior of the customers, how much traffic the network will carry on the average, 499 and how much it will lose as a result of blocking, overload, suboptimal routing, or incomplete searches for paths. As telephone networks have become larger, two more design parameters of interest have emerged and now command attention: the size of the network as measured by the number of crosspoints, and its complexity as measured, for example, by the number of stages of switching it has. The probabilistic principle that it is very unlikely for more than a moderate number of customers to want to talk simultaneously has been the theoretical basis of traffic theory since its start. We can view it as an unrefined analog of the principle in information theory that separates a relatively small class of events that exhaust most of the probability from a remaining large class of very unlikely events. This principle has led quite naturally to the use of concentrators, and of networks in which blocking, mismatch, and overflow all can and do occur as it were by design. It is a function of traffic theory to articulate this principle in mathematical models for operating telephone networks, and to use such models to examine its implications for the growth and complexity of networks, as well as their loads and losses.