3. (3 pts.) Consider a harbor with single dock for unloading ships. The ships ar

Question

3. (3 pts.) Consider a harbor with single dock for unloading ships. The ships arrive accord- ing to a Poisson process at a mean rate of ships per week, and the service time distribution is exponential with a mean rate of 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 company can adjust the value of as desired. Suppose the expected cost per unit time of running the unloading dock is D. The waiting cost for each idle ship is some constant C times the square of the total waiting time (includ- ing loading time). The shipping company wishes to adjust so that the expected total cost (including the waiting cost for idle ships) per unit time is minimized. (i) Derive this optimal value of in terms of D and C

Explanation / Answer

(i) Average waiting time (including loading time), W = 1/(-)

Total cost per unit time TC = D + C*W2

TC = D + C*(1/(-))2*

= D + C/(-)2

To minimize the total cost,

d(TC)/d = d(D + C/(-)2)/d = 0

=> D - 2C/(-)3 = 0

=> = (2C/D)1/3 +

Optimal value of = (2C/D)1/3 +

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