 traffic on route I can be assessed from the following rule of thumb: 5.3.4 Network bounds an additional call offered to route r will be accepted with probability 1-L; if accepted it will earn w, directly, but at a cost Ck for each In this section the aim is to obtain bounds on the performance of any link ker. The costs C measure the knock-on effects of accepting a dynamic routeing scheme under a fixed stationary pattern of offered call upon the other routes in the network. From Equation 5.43 it traffic. Consider two approaches. The first makes minimal assump- follows that the costs calso have an interpretation as shadow prices, tions concerning the stochastic structure of the system, and within with C1 measuring the sensitivity of the rate of return to the capacity this rather weak framework the bounds are the best possible. The Ck of link k. Note that Equation 5.39 can be rewritten in the form second approach considers various ways to improve these bounds shown in Equation 5.44 where S. is given by Equation 5.45. when the offered traffic can be assumed Poisson. (5.44) 5.3.4.1 Max-flow bound For (2ij:i<j), let F(A) be the maximum attained in the linear (5.45) programme of Equation 5.48, subject to Equations 5.49 to 5.51. maximise F (a) (5.48) Here s the surplus value of a call on route T. (5.49) 5.3.3 Applications of implied costs 5.3.3.1 Network dimensioning (5.50) The Equations 5.42 and 5.43 can be used directly as the basis for an efficient hill climbing algorithm to maximise the expression (5.37). Ai,j,k (5.51) Through analogies with deterministic network flow the shadow price interpretation of Equation 5.43 could be used in algorithms to aid capacity expansion decisions (see Key, 1988). The overall The interpretation of this problem is as follows. Regard as the approach also has important economic implications for pricing direct flow between nodes i and j along the link (i,j), and XIK as policies and for the apportionment of revenue between different flow between nodes i and j through the tandem node k. Reg