Some Examples of Comparisons of Connecting Networks

In the theory of telephone traffic it is of interest to compare the performance of connecting networks, as measured by the probability of blocking, when they are subjected to the same traffic sources. Naturally, there are cases in which the result of this comparison is independent of the calling rate X.1 In this connection, H. O. Pollak has raised the question whether there are examples of pairs of networks, with the same number of crosspoints, the first of which is better than the second at one value of X, while the second is better than the first at another value of X. The existence of such examples would establish the principle that some network configurations (in particular, some switch sizes) are inherently more efficient at some traffic levels than at others, so that the "excellence" of a network is not necessarily a purely combinatorial notion independent of offered traffic. We shall exhibit examples of the above phenomenon which are nontrivial in that they do not involve only very small networks. 1829