Programming and Control Problems Arising from Optimal Routing in Telephone Networks*

Abstract

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 have attempted to use the special combinatorial structure imposed by the telephonic origins of the problem to shed light on the character of the optimal strategies. In particular, we show that for connecting networks with suitable combinatorial properties, the optimal route choices can be very simply described. Some of the remits obtained were suggested by, and verify, conjectures from the practical lore of telephone routing. The problem of routing calls falls into two parts: Which attempted calls should be accepted in which states? What route should an accepted call use? The first problem is very hard, and only sample numerical answers for small networks are obtained. We solve the second problem analytically for a large class of cases by appeal to combinatorial structure in the network. These cases can be described roughly as those in which the relative merit of states (as far as blocking is concerned) is consistent or continuous; i.e., if a state x is “better” than another y, then the neighbors of x are in the same sense “better” than the corresponding neighbors of y. An abundance of examples indicates that these cases are numerous and so warrant attention. In a network with this kind of combinatorial property, a policy which rejects no unblocked calls and minimizes the number of additional calls that are blocked by completing an attempted call differs from an optimal policy only in that the latter may reject some calls.

Routing in circuit-switched networks: optimization, shadow prices and decentralization

“…For recent work on dynamic routing see [8] and [30]. For other examples of work on routing in telephone networks see [2], [9] and [19].…”

“…Regard r(l) as the route first attempted by a call labelled r; if this route is blocked the route r(2) is attempted; if both routes are blocked then the call is lost. As before assume a call with label r which is carried generates a mean revenue of W r • For simplicity assume that the routes r(l) and r (2) do not have any links in common. Let v = (v., r E R) where v, is the arrival rate of calls labelled r. In the formalism of this section the parameter A= A(v) is Equation (7.10) shows that the cost of increasing the offered traffic with label r = (r(l), r(2)) can be assessed by the following rule of thumb: an additional call with this label will be accepted by route r(l) with probability 1 -L r (1 ) and by route r(2) with probability L r(l)(l -L r(2»); if accepted it will earn W r directly, but at a cost Ck for each link on the accepted route.…”

“…Equation (7.11) can be interpreted as earlier, but now the surplus values Sr(l) for first-choice routes are calculated by (7.9), an equation with an additional term. The additional term can be interpreted as follows: a call blocked on route r(1) is not necessarily lost; with probability 1 -L r (2) it will be carried on its second-choice route r(2), generating there a surplus value of Sr (2). The above interpretations suggest various generalizations of the adaptive routing scheme described in Section 5.…”

“…The above interpretations suggest various generalizations of the adaptive routing scheme described in Section 5. To estimate iteratively Equations (7.8) and (7.9) we need only replace Equation (5.6) by Sr(2)(n + 1) = w, -LA jr(2)cj(n) j Sr(l)(n + 1) = W r -LA jr(1)cj(n) -(1 -L r(2)(n ))Sr(2)(n + 1) j and allow r to range over all possible first or second choice routes r(1), r (2) in Equations (5.3) and (5.5). As earlier the costs Ck and surplus values Sr(l)' Sr (2) estimated by the scheme can be used for various purposes.…”

“…Additionally they can now be used to decide on whether to reroute blocked calls. For example, if Sr (2) becomes negative while Sr(l) remains positive the single route r(1) is preferred to the pair r = (r(1), r(2)): a call blocked on the first-choice route r(1) should be discarded rather than offered to the second-choice route r(2). Example 2.…”