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 erlangs with a blocking probability at most can be built with contacts.