racteristics of network cost functions are important for regulatory policy and legal aspects of networks. To illustrate the network cost function and its relation to the production func- tion, consider the basic example of a single-server network. As before, traffic arrives at the network at rate A and is served at rate . Waiting time in a queue, W, is the quality of the network's output. The production function as specified in equation (8) is Suppose that the cost of network capacity is (u). Then the amount of capacity needed to provide the quantity of service a at quality W is obtained by inverting the production function: 13 Gasmi et al. (2002) construct network cost functions by examining the costs of different out- put combinations and using econometric models to estimate a cost function based on these simulations. Networks in Telecommunications Waiting Service Figure 2.12. Combining traffic flows into a network. Substitute for u in the capacity and function to obtain the network cost function for output 2 and parameterized by the quality of service W: C(2)=c(2/2+(22/4+)12) = Consider next a generalization to multiple outputs. A useful property of Poisson arrival processes is that the sum of independent Poisson processes is itself a Poisson process. Suppose that the network receives many types of traffic with independent arrival rates Aj, j = 1,..., J. Then total traffic received by the network is a Poisson process with an arrival rate that is the sum of these arrival rates: This immediately generalizes the cost function in equation (14) to a multiproduct cost function, C(21, 22, = This type of multiproduct cost function shows the effects of congestion. To avoid increasing costs, while maintaining a given quality of service in terms of queuing time, increases in the rate of traffic flow of any type would need to be balanced by reductions in the flow rates of other types of traffic. In general, traffic flows may be correlated, either positively or negatively, rather than being independent. Also, traffic flows