Consider a harbor with a single dock for unloading ships. The ships arrive according to a Poisson process at a mean rate of λ ships per week, and the service-time distribution is exponential with a mean rate of μ unloadings per week. Assume that harbor facilities are owned by the shipping company, so that the objective is to balance the cost associated with idle ships with the cost of running the dock. The shipping company has no control over the arrival rate (that is, is fixed); however, by changing the size of the unloading crew, and so on, the shipping company can adjust the value of μ as desired.
Suppose that the expected cost per unit time of running the unloading dock is Dμ. The waiting cost for each idle ship is some constant (C) times the square of the total waiting time (including loading time). The shipping company wishes to adjust μ so that the expected total cost (including the waiting cost for idle ships) per unit time is minimized. Derive this optimal value of μ in terms of D and C.