Problem 3 (Queuing Theory, 20 Points): In a drilling shop the materials that must be processed arrive randomly with an average rate of 70 items per hour. Based on queuing theory you are supposed to decide how many drilling machines are required. Suppose that a M/M/1 model can be applied. The following data is given: A shop consisting of 4 machines can process the materials with an average rate of 80 items per hour. A shop consisting of 5 machines can process the materials with an average rate of 100 items per hour. A shop consisting of 6 machines can process the materials with an average rate of 120 items per hour. The operation of one drilling machine costs 35 per hour. a. Calculate for each number of drilling machines specified above the average number of items waiting for processing. b. Calculate for each number of drilling machines specified above the average waiting time of an item before it is processed. C. What is the probability (for each number of drilling machines specified above) that an item cannot be processed after its arrival? d. Suppose that the accounting calculates with penalty costs of 0,90 for every minute waiting time. Which number of drilling machines (4, 5 or 6) causes the lowest average costs