Software:

Excel QM Version 5.2 Or POMQM

Question 1
A new shopping mall is considering setting up an information desk operated by two employees. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour.  It takes an average of 4 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed.
(a)    Find the proportion of the time that the employees are busy.
(b)    Find the average number of people waiting in line to get some information.
(c)    Find the expected time a person spends just waiting in line to have his question answered.

Question 2
At the start of the football season, the ticket office gets busy the day before the first game.  Customers arrive at the rate of four every ten minutes.  A ticket seller can service a customer in four minutes. Traditionally, there are two ticket sellers working. The university is considering an automated ticket machine similar to the airlines’ e-ticket system. The automated ticket machine can service a customer in 2 minutes.

(a)    What is the average length of the queue for the in-person model?
(b)    What is the average length of the queue for the automated system model?
(c)    What is the average time in the system for the in-person model?
(d)    What is the average time in the system for the automated system model?
(e)    Assume the ticket sellers earn $8 per hour and the machine costs $20 per hour (amortized over 5 years). The wait time is only $4 per hour because students are patient. What is the total cost of each model?

Question 3
There is a tutoring lab at a university that holds 20 students. Due to the popularity of the 3 tutors, the lab is always at capacity. Each student approaches a tutor an average of 2 times per hour, and a tutor spends on average 5 minutes per question from a student. The service time follows an exponential distribution.
(a)    What proportion of the time are the tutors busy?
(b)    What is the average waiting time for students in queue?

Question 4
A professor holds office hours during final exam week and discovers much to his dismay that his sudden popularity affords him little time to engage in his favorite activity, watching cat videos. Fifteen students per hour arrive at his office and he patiently answers questions, averaging three minutes per student. If the average cat video is 60 seconds, how many videos can the professor watch in an hour?
A) 15
B) 18
C) 20
D) 24

Question 5
A professor holds office hours during final exam week and discovers much to his dismay that his sudden popularity affords him little time to engage in his favorite activity, watching cat videos. Fifteen students per hour arrive at his office and he patiently answers questions, averaging three minutes per student. On average, how many students are waiting to ask questions?
A) 2.00
B) 2.25
C) 2.50
D) 2.75

Question 6
Customers enter the waiting line at a cafeteria on a first-come, first-served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, what is the average number of customers in the system?
A) 0.43
B) 1.67
C) 0.57
D) 1.33

Question 7
Customers enter the waiting line to pay for food as they leave a cafeteria on a first-come, first-served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, on average, how much time will elapse from the time a customer enters the line until he/she leaves the cafeteria?
A) 0.67 minute
B) 0.50 minute
C) 0.75 minute
D) 0.33 minute

End of Assignment

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