Combinatorial Solution to the Problem of Optimal Routing in Progressive Gradings

Abstract

The grading or graded multiple proposed by E. A. Gray is a certain kind of one-stage, two-sided, partial access telephone connecting network for switching customers' lines to trunks all having the same destination. Its essential feature is that traffic from lines not having identical access patterns can be offered to a common trunk, and so pooled. In a progressive grading the trunk groups are partially ordered in a hierarchy, i.e., some provide primary routes, others function as secondary routes which handle traffic overflowing from primary routes, as well as originating traffic, etc., up to final routes. A call which is using an overflow or “later” trunk when it could be using a primary or “earlier” group is said to make a “hole in the multiple”. It was recognized early in the development of gradings that such holes were undesirable. The problem of optimal routing in telephone networks, considered in general in the author's earlier work, is here specialized to progressive gradings. It had been shown that for networks with certain combinatorial properties the optimal choices of routes for accepted calls (so as to minimize the loss under perfect information) could be described in a simple and intuitive way in terms of these properties. The present paper gives a proof that all progressive gradings have such a combinatorial property, associated with the hierarchical nature of the grading. The optimal policy for routing accepted calls is related to the phenomenon of “holes in the multiple”, and can be paraphrased in the traditional telephone terminology thus: filling a hole in the multiple is preferable to using a final route, and filling an earlier hole is preferable to filling a later one.