The Seabuck and Roper Company has a large warehouse in southern California to store its inventory of goods until they are needed by the company’s many furniture stores in that area. A single crew with four members is used to unload and/or load each truck that arrives at the loading dock of the warehouse. Management currently is downsizing to cut costs, so a decision needs to be made about the future size of this crew.
Trucks arrive at the loading dock according to a Poisson process at a mean rate of 1 per hour. The time required by a crew to unload and/or load a truck has an exponential distribution (regardless of crew size). The mean of this distribution with the fourmember crew is 15 minutes. If the size of the crew were to be changed, it is estimated that the mean service rate of the crew (now μ = 4 customers per hour) would be proportional to its size.
The cost of providing each member of the crew is $20 per hour. The cost that is attributable to having a truck not in use (i.e., a truck standing at the loading dock) is estimated to be $30 per hour.
(a) Identify the customers and servers for this queueing system. How many servers does it currently have?
(b) Use the appropriate Excel template to find the various measures of performance for this queueing system with four members on the crew. (Set t = 1 hour in the Excel template for the waiting-time probabilities.)
(c) Repeat (b) with three members.
(d) Repeat part (b) with two members.
(e) Should a one-member crew also be considered? Explain.
(f) Given the previous results, which crew size do you think management should choose?
(g) Use the cost figures to determine which crew size would minimize the expected total cost per hour.
(h) Assume now that the mean service rate of the crew is proportional to the square root of its size. What should the size be to minimize expected total cost per hour?