A small lumberyard is supplied by a fleet of 10 trucks. One overhead crane is available to unload the long logs from the trucks. It takes an average of 1 hour to unload a truck. After unloading, a truck takes an average of 3 hours to get the next load of logs and return to the lumberyard.

(a) Certain distributional assumptions are needed to analyze this problem with the models of this chapter. State them and discuss their reasonableness. 

(b) With one crane, what is the average number of trucks waiting to be unloaded? On average, how many trucks arrive at the yard each hour? What percentage of trucks upon arrival find the crane busy? Is this the same as the long-run proportion of time the crane is busy? 

(c) Suppose that a second crane is installed at the lumberyard. Answer the same questions as in part (b). Make a chart comparing one crane to two cranes. 

(d) If the value of the logs brought to the yard is approximately $200 per truck load and if long-run crane costs are $50 per hour per crane (whether busy or idle), compute an optimal number of cranes on the basis of cost per hour. 

(e) In addition to the costs in part (d), if management decides to consider the cost of idle trucks and drivers, what is the optimal number of cranes? A truck and its driver together are estimated to cost approximately $40 per hour and are considered to be idle when they are waiting for a crane.