On Crossbar Switching Networks

For networks that exhibit neither concentration nor expansion, the well-known probabilistic model of C. Y. Lee is refined so as to take account of the dependence of events in different stages. For series-parallel networks, the refined model yields exact expressions for the point-to-point blocking probability. These expressions bear the same relationship to the refined model as the KittredgeMolina expressions do to Lee's model. Comparison between the two sets of expressions shows that Lee's model tends to overestimate the blocking probability. Asymptotic analysis of the new expressions leads to improved upper bounds on the cost: it is shown that a network that carries N erlangs with a blocking probability at most \epsilon > 0 can be built with 6N \log_{2} N + O(N \log \log 1/\epsilon) contacts.