Analysis of Toll Switching Networks

In a well-known 1955 article,1 C. Y. Lee introduced simplified methods for the analysis of switching network characteristics, such as blocking probability. Using a probability linear-graph (hereafter called Lee graph) to represent the network and an assumption of independent link occupancies, he described ways to quickly obtain approximate expressions in many cases of interest. As an example of possible inaccuracy, Lee pointed out a three-stage network that is known to be nonblocking but is assigned a nonzero blocking probability by his method. The present work introduces two different techniques for extending Lee's method to avoid certain inconsistent independence assumptions, which are the source of the inaccuracies he noted. The extended methods will, for instance, reproduce M. Karnaugh's more accurate expression2 for blocking probability of a three-stage network, but with less mathematical labor. When applied to a "generalized" threestage network that can model the No. 4 ESS,3 the techniques yield formulas that greatly simplify expressions currently in use. The appendix lists a computer routine to calculate blocking for the case of generalized three-stage networks. II. FIRST METHOD 2.1 Generalized three-stage switching network