pends on total capacity. Now, plug the shadow price back into individual demands to obtain the output allocation rule, (25) X K), Finally, calculate the output that a consumer of type i expects to receive, df(b1)...df(bi-1)df(bi+1)..df(b This is the reliability of service for a consumer of type i. To find the reliability schedule, find the inverse of the reliability function, b* = = Q*-1 (Q).The optimal price schedule for reliability is then It is easy to see that R'(Q) = b*(Q). The consumer chooses a reliability level that equates the marginal payment to the willingness to pay for a unit of reliability. It follows that b' = R'(Q) = b*(Q 6 The term (1 - F(b))/f(b) is the reciprocal of the hazard rate. Networks in Telecommunications R'(Q) R'(Q) Figure 3.10. Consumers pay a premium for greater reliability of network service. The consumer's choice of reliability will be the efficient level; that is, Qi=Q*(bi). = Thus, the consumer's choice of reliability Qi reveals the consumer's type, because i=b*(Q*(b))). Consider the properties of the reliability payment schedule. Because higher- willingness-to-pay consumers choose more reliability, the marginal payment sched- ule is increasing, R"(Q) = '(Q) > 0. This means that consumers pay a quantity premium to obtain greater reliability. This is shown in Figure 3.10. Moreover, the average payment per unit of reliability is increasing in the level of reliability. From equation (27), Compare the outcome for the reliability pricing mechanism with the outcome of the Vickrey auction. In both cases, consumers with the K highest willingness-to- pay levels each receive a unit of service. This suggests that the likelihood of winning the Vickrey auction for a particular consumer exactly equals that consumer's choice of reliability of service with reliability pricing. This can be demonstrated using basic order statistics. Let S1-kbethe - I - Korder statistic if the sample of consumer types is in increasing b1. Thus, SI-K is the type of the losing consumer with the highest willingness to