DeColores Paint Company owns ten trucks that it uses to deliver paint and decorating supplies to builders. On average, each truck returns to the company’s single loading dock at a rate of three times per eight-hour day (or at a rate of 3/8 = 0.375 times per hour). The times between arrivals at the dock follow an exponential distribution. The loading dock can service an average of four trucks per hour with actual service times following an exponential distribution.
a. What is the probability that a truck must wait for service to begin?
b. On average, how many trucks wait for service to begin at any point in time?
c. On average, how long must a truck wait before service begins?
d. If the company builds and staffs another loading dock, how would your answers to parts a, b, and c change?
e. The capitalized cost of adding a loading dock is $5.40 per hour. The hourly cost of having a truck idle is $50. What is the optimal number of loading docks that will minimize the sum of dock cost and idle truck cost?