IE_425_H6_Applications_of_Queueing_Theor (1).pdf

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1 APPLICATIONS OF QUEUEING THEORY Problem 1 Customers arrive to a fast-food restaurant with one server according to a Poisson process at a mean rate of 30 per hour. The server has just resigned, and the two candidates for the replacement are X (fast but expensive) and Y (slow but inexpensive). Both candidates would have an exponential distribution of service times, with X having a mean of 1.2 minutes and Y having a mean of 1.5 minutes. Restaurant revenue per month is given by $6,000/W, where W is the expected waiting time (in minutes) of a customer in the system. Determine the upper bound on the difference in their monthly compensations that would justify hiring X rather than Y. Solution: Candidate X (M/M/1): 30 , 50 X , 6 . 0 X , . min 3 . hr 20 1 30 50 1 1 W X X Candidate Y (M/M/1): 30 , 40 Y , 75 . 0 Y , . min 6 . hr 10 1 30 40 1 1 W Y Y Thus, Monthly compensation difference between candidates X and Y that justifies hiring X should be difference in monthly revenue for hiring X instead of Y . month / 000 , 1 $ 6 1 3 1 000 , 6
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2 Problem 2 A single crew is provided for unloading each truck that arrives at the loading dock of a warehouse. These trucks arrive according to a Poisson input process at the rate of 1 per hour. The time required by a crew to unload and/or load a truck has an exponential distribution (regardless of the crew size). The expected time required by a one-person crew is 1 hour. The cost of providing for each individual member of the crew is $10 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 $15 per hour. (a) Assume that the mean service rate of the crew is proportional to its size. What should the size be to minimize the total cost per hour? (b) Assume that the mean service rate of the crew is proportional to the square root of its size.
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