Growth, complexity, and performance of telephone connecting networks

In an effort to free telephone traffic theory from some of its dependence on independence assumptions, and to reap some benefit from its traditional state equations, a systematic search is made to find relationships between load, loss, size, structure, and other network parameters that are simple, universal, and informative. Three principal topics are covered: (i) A load-loss-size formula, linking some half-dozen network parameters by a rational function, and used repeatedly to give (ii) Lower bounds on the number X of crosspoints in networks (iii) Asymptotic results about blocking, growth, and complexity of selected network structures in passing from finite to “infinite” sources at constant load. The major results in (ii) imply that for all practical networks on N terminals, the crosspoint count X must grow like N log N, i.e., incurring loss by restricting access or concentrating cannot avoid the N log N growth rate known to be exacted by nonblocking networks. The chief result under (iii) is that as a constant load is spread over N terminals, then the number X of crosspoints needed to keep loss less than ∊ > 0 need grow only linearly with N, at a rate dependent on ∊, while the usage (erlangs carried per terminal) goes to zero.