Blocking States in Connecting Networks Made of Square Switches Arranged in Stages

Connecting networks made of rectangular switches have long been an important feature of telephone central offices, and their combinatorial properties still command substantial practical as well as theoretical interest. The probability of blocking, calculated or measured in a suitable setting, is a principal measure of the performance of a network, and it is of particular interest to identify and study those combinatorial features which give rise to blocked calls. To be sure, blocking is known to be an unfortunate mismatch of available resources. But precisely, what structural features lead to it? What presages blocking? What are some tell-tale signs and necessary concomitants of blocking? When and how can blocking be avoided? To try to answer these questions, we focus on the blocking states of the network, those in which some call is blocked. Although in a century of telephony much work has gone into calculations and simulations for blocking probability, there are (except for a few examples) virtually no results on the basic causes and nature of blocking. For networks made of square switches arranged in stages, this paper will partly remedy this lack by introducing some simple concepts suggested by examples; theorems based on these bear on the position and structure of the blocking states within the set (semilattice) 511