On Reliable Topological Structures for Message-Switching Communication Networks

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The synthesis of optimal reliable (invulnerable) topological structures for message-switching communication networks is considered. The connectivity of the underlying graphs is used as a measure of the network invulnerability. The maximal average message delay value is utilized as the network delay measure. Simultaneously with choosing the topological structure, optimal line capacities are assigned. Therefore, the performance measure of a given network structure is chosen to be given by its delay-capacity product function, incorporating the product of the prescribed network maximal delay value and the associated minimal overall line capacity value. The latter involves a distance-independent link cost function incorporating the line capacity. A general routing discipline is used to account for dynamic updating of fixed routing procedures, needed to accomodate terminal traffic flow fluctuations. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -node <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -connected graphs yielding networks with minimal delaycapacity product functions are characterized and realized. Complete networks (utilizing direct dedicated lines) are shown to be optimal if the resulting lines have a high average line utilization value. Otherwise (under appropriate symmetry conditions on the network traffic matrix), the optimal message-switching reliable network structures are characterized by a family of graphs of diameter two. The latter thus allow between any pair of nodes a route which is either a direct line or contains a single intermediate node. Also noted is a family of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -connected networks, for which the delay-capacity product function is not increased by more than twice upon the failure of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k-1)</tex> or less nodes or lines.