ge time spent waiting for a dial tone. 11 11 For a general approach to pricing multiple services with quality-of-service requirements, see Keon and Anandalingam (2003). For a related discussion of congestion on the Internet, see Mackie-Mason and Varian (1995a). Networks in Telecommunications Suppose that each consumer obtains a benefit from the number of completed phone calls Q. The number of phone calls is integer-valued. The consumer's benefit is a standard increasing and concave utility function u(Q). Suppose that the num- ber of phone calls that the consumer makes is random due to exogenous factors that affect the necessity of communicating with others. Suppose further that the consumer chooses the intensity of usage A, which determines the average rate at which calls are made within a period of time, but not the actual number of calls. Within a given period of time, the consumer's calls have an arrival rate that does not depend on prior arrivals. This is referred to as a Poisson arrival rate. The consumer's expected utility is then a function of the rate of making phone calls, The expected utility has the standard properties of a utility function.12 Suppose that the network's capacity to complete calls constrains demand, which leads to the possibility of blocked calls. Let W be the average waiting time for service for consumers seeking to initiate a call, as described in the previous chapter. Suppose that the consumer's benefit from the number of calls is affected by the average waiting time, u = u(Q, W). An increase in average waiting time reduces the consumer's total and marginal benefit from service. The consumer's expected benefit also will depend on the waiting time, U = U(A, W). An increase in waiting time reduces the marginal expected benefit from service; see Figure 3.11. The expected number of calls per period is equal to the arrival rate, EQ=2.13 The consumer pays a price p per completed call. The consumer chooses the rate of usage to maximize the expected net benefit from network services. The con