An ISP tries to even out the capacity demand over day and night by rewarding its users for delaying their data transmission. Suppose there are just two types of users, type A and type B, which have different levels of willingness to delay their sessions. Originally the demand during day-time is in total vA,day + vB,day = 14 GB, which consists of vA,day = 8 GB from type A users and vB,day = 6 GB from type B users. The demand during night-time is in total vA, night+vB, night = 5 GB, which consists of vA, night = 2 GB from type A users and vB,night = 3 GB from type B users. Suppose the ISP has capacity C = 10 GB and the marginal cost of exceeding capacity is $1 per GB. It provides a reward of $p per GB for day-time users to delay their data transfer until night-time. Let wA(p) and wB(p) be the proportions of data from type A and type B users, respectively, to be delayed from day-time to night-time:
wA(p) = 1 − exp(−p/pA),
wB(p) = 1 − exp(−p/pB),
where parameters pA = 4 and pB = 2.
The ISP wishes to find the reward price p that minimizes its total cost, i.e., the sum of the cost due to the demand exceeding the capacity and the rewards given out.
(a) What is the formulation of the minimization problem?
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