Programming and control problems arising from optimal routing in telephone networks

In many circumstances a telephone call can be completed through a connecting network in several ways. Hence, there naturally arise problems of optimal routing, that is, of making the choices of routes so as to achieve extrema of one or more measures of system performance, such as the loss (probability of blocking) or the carried load. As is customary in traffic theory, a Markov process is used to describe network operation with complete information. The controlled system is described by linear differential equations with the control functions (expressing the routing method being used) among the coefficients. Restricting attention to asymptotic behavior leads to a problem of maximizing a bilinear form subject to a linear equality constraint whose matrix is itself constrained to lie in a given convex set. An alternative approach first shows that minimizing the loss, and maximizing the fraction of events that are successful attempts to place a call, are equivalent. This fact permits a dynamic programming formulation, which, in turn, leads to a very large linear programming problem. Two small examples are treated numerically by this method. It is particularly important to try to verbalize, and then mechanize, the optimal routing strategies. In this endeavor, the linear programming formulation is of limited usefulness. Therefore, in the latter half of the work we