sumer's resulting net benefit equals because the consumer's expected payment is pl. Let 2 = Xi(p, W) be the solution to the consumer's problem. The consumer's choice of usage equates the consumer's 12 It can be demonstrated that U(A) is an increasing and concave function. Taking a derivative with respect to a yields So U'(2) > 0 because u(k) is increasing. Taking the second derivative with respect to A, by similar arguments we obtain Thus, U"(2) < 0 because u(k) is concave. 13 The expected number of calls is EQ= = Pricing Network Services Marginal benefit U2(A,W1) U2(A,W2) Figure 3.11. The consumer's marginal benefit is reduced by an increase in waiting time, which represents a reduction in network service quality, W1. marginal benefit of usage, for a given waiting time W, to the price of usage: Suppose that there are I consumers. Given the demand for capacity of each con- sumer, the total demand for capacity can be obtained by summing consumer demands: This is the total demand for capacity given a common waiting time W. Recall that with an independent memoryless arrival processes, the arrival rate of the combined traffic is also a memoryless arrival process. The arrival rate of the combined traffic is the sum of the arrival rates of the individual components. Thus, the aggregate demand in our model is the arrival rate of traffic coming to the network. In the short run, the service capacity of the network is fixed and equal to . Therefore, more traffic on the network creates congestion that reduces service quality. Recall the production function, as already specified in equation (8) in Chapter 2: Because the capacity of the network is fixed, an increase in the amount of service 2 can be achieved only by reducing service quality W. Networks in Telecommunications W*(p,U) D(p, W) = 2* (p, lu) Figure 3.12. The equilibrium level of network congestion and capacity utilization with a fixed price p and a fixed network capacity . For any price p, consumers will choose their amount of demand 2i based on their expectations 