California State Polytechnic University, POMONA
Industrial and Manufacturing Engineering Department
Course: IE417 FINAL EXAM, Date: 3-19-2002
Last name: ...... First name: ...... Student ID: ......
Instructions: Read all questions carefully. Organize your work clearly, neatly, and write legibly. Show all details of your work. Identify your answer. Do not write on the back of any page. Closed-book and closed-note exam. You may have one sheet of information. Total score is 35 points.
Good Luck
Dr. Parisay
1- The interarrival time of buses to an station is 20 minutes or 30 minutes with equal probability. Jack will randomly arrive at this station. (4 points)
a) What is the average length of time Jack must wait for a bus?
b) If interarrival time of buses is exponentially distributed with mean 25 minutes, then what is the average length of time Jack must wait for a bus?
2- Some of you lectured some sections of queuing theory. Mention name of one of the students and just one shortcoming in his/her lecture. How would you suggest improving this shortcoming? (2 points)
3- Gotham City has 13,500 streetlights. City investigators have determined that at any given time, an average of 900 lights are burned out. A streetlight burns out after an average of 90 days of use. The city has hired MNO Company, to replace burned-out lamps. MNO’s contract states that they are supposed to replace a burned-out street lamp in an average of 8 days. (6 points)
a) Write the Kendall-Lee notation for this problem.
b) Do you think that MNO is living up to the contract?
4- A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 30 customers arrive each hour, and an average of 6 customers can be served by each cash register. Interarrival times and service times are exponential, and the cash registers are identical. It costs $24 per hour to operate a cash register, and a cost of 50 cents is assessed for each minute a customer spends in the cash register area. WinQSB is utilized to find performance measures under different number of cash registers and the result is in the following table. (10 points)
Performance Measure / Result1 / System: / M/M/6 / M/M/7 / M/M/8
2 / Customer arrival rate (lambda) per hour = / 30 / 30 / 30
3 / Service rate per server (mu) per hour = / 6 / 6 / 6
4 / Overall system effective arrival rate per hour = / 30 / 30 / 30
5 / Overall system effective service rate per hour = / 30 / 30 / 30
6 / Overall system utilization = / 0.83 / 0.71 / 0.63
7 / Average number of customers in the system (L) = / 7.9 / 5.8 / 5.3
8 / Average number of customers in the queue (Lq) = / 2.9 / 0.8 / 0.3
9 / Average number of customers in the queue for a busy system (Lb) = / 5.0 / 2.5 / 1.7
10 / Average time customer spends in the system (W) = / 0.26 hours / 0.19 hours / 0.18 hours
11 / Average time customer spends in the queue (Wq) = / 0.1 hours / 0.03 hours / 0.01 hours
12 / Average time customer spends in the queue for a busy system (Wb) = / 0.17 hours / 0.08 hours / 0.06 hours
13 / The probability that all servers are idle (Po) = / 0.5% / 0.6% / 0.6%
14 / The probability an arriving customer waits (Pw or Pb) = / 58.8% / 32.4% / 16.7%
a) Based on this table and the costs involved, how many registers should the store open?
b) The manager wants to ensure that no more than 5% of all customers will have to wait in line for more than 6 minutes. How many registers must be open?
c) Find the fraction of time that a particular register is idle in the case of M/M/6 queuing system.
d) Write a report for a manager discussing the important performance measures of this system and your suggestions.
5- Each airline passenger and his or her luggage must be checked to determine whether he or she is carrying weapons onto the airplane. Suppose that at Gotham City Airport, an average of 12 passengers per hour arrive. There is only one checkpoint to check passengers for weapons and it takes on average 3 minutes. A passenger will then go through another checkpoint for luggage and it will take 4.75 minutes to check each luggage. Assume each passenger has only one luggage. Assume that all the times are exponentially distributed. (10 points)
a) What is the probability that a passenger will have to wait before the first checkpoint for weapons?
b) On the average, how many passengers are waiting in the system to go through both checkpoints?
c) What is the probability that 20 passengers arrive during one hour?
d) What is the probability that an interarrival time is between 2 and 8 minutes?
e) Comment if this system is desirable and how it can be improved.
6- In your project you were supposed to find EVPI. How did you find it? (1 point)
7- Draw a rate diagram for a M/M/3/GD/6/6 queuing system. Assume service rate is 11 and arrival rate is 9. (2 points)