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By Lowell W. Beineke, Robin J. Wilson

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1). We shall normally suppose that stage 1 corresponds to inlets, and stage / corresponds to outlets in the practical network, and that a legitimate path from inlet to outlet traverses one and only one edge from each subset. The adjacency matrix is readily adapted to the multi-stage situation. Let the element of a matrix A represent the adjacency of inlet i and outlet j of an individual stage. Then the product of such matrices represents the tandem connection of stages, each element giving the number of paths between an inlet and an outlet.

Consider a single-input switch which is required to select one of N outlets. To define an outlet, at least log2 binary choices are required, 44 K. W. CATTERMOLE and consequently an address must contain at least log2N bits of information and require at least log2A bits of storage. Next consider a rearrangeable switch, as defined in Section 4, which is required to connect a set of N inlets to a set of N outlets. If information about each connection is stored separately, then at least N log2A bits are required.

The Enumeration of States A connecting network of whatever complexity has a finite number of distinct states in each of which a distinct set of links and crosspoints is busy. In a given state, certain potential connections may be blocked and others not. A complete enumeration of the states, with a state­ ment of their blocking implications, would be the foundation of an exact theory of congestion. The number of states is in fact so large that complete enumeration, in the sense of listing, is impracticable.

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