Customers arrive at a facility at random according to a Poisson process of rate λ. There is a waiting time cost of c per customer per unit time. The customers gather at the facility and are processed or dispatched in groups at fixed times T, 2T, 3T, .... There is a dispatch cost of K. The process is depicted in the following graph.
(a) What is the total dispatch cost during the first cycle from time 0 to time T?
(b) What is the mean total customer waiting cost during the first cycle?
(c) What is the mean total customer waiting + dispatch cost per unit time during the first cycle?
(d) What value of T minimizes this mean cost per unit time?
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