Operations and Supply Chain Management The Core

234 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
an average exponentially distributed time of two minutes. Assume the population
is infinite, arrivals are Poisson, and queue length is infinite with the FCFS
discipline.
a. Calculate the percentage utilization of the operator.
b. Calculate the average number of documents in the system.
c. Calculate the average time in the system.
d. Calculate the probability of four or more documents being in the system.
e. If another clerk were added, the document origination rate would increase to
30 per hour. What would the expected average number of documents in the
system become? Show why.
25. A study-aid desk staffed by a graduate student has been established to answer
students’ questions and help in working problems in your OSCM course. The desk
is staffed eight hours per day. The dean wants to know how the facility is working.
Statistics show that students arrive at a rate of four per hour and the distribution
is approximately Poisson. Assistance time averages 10 minutes, distributed
exponentially. Assume population and line length can be infinite and queue
discipline is FCFS.
a. Calculate the percentage utilization of the graduate student.
b. Calculate the average number of students in the system, excluding the graduate
student service.
c. Calculate the average time in the system.
d. Calculate the probability of four or more students being in line or being served.
e. Before a test, the arrival of students increases to six per hour, on average. What
will the new average line length be?
26. At a border inspection station, vehicles arrive at a rate of 10 per hour in a Poisson
distribution. For simplicity in this problem, assume there is only one lane and one
inspector, who can inspect vehicles at the rate of 12 per hour in an exponentially
distributed fashion.
a. What is the average length of the waiting line?
b. What is the average time that a vehicle must wait to get through the system?
c. What is the utilization of the inspector?
d. What is the probability that when you arrive there will be three or more
vehicles ahead of you?
27. During the campus Spring Fling, the bumper car amusement attraction has a
problem with cars becoming disabled and needing repair. Repair personnel can
be hired at the rate of $20 per hour, but they work only as one team. So, if one
person is hired, he or she works alone; two or three people work together on the
same repair.
One repairer can fix cars in an average time of 30 minutes. Two repairers take
20 minutes, and three take 15 minutes. While these cars are down, lost income is
$40 per hour. Cars tend to break down at the rate of two per hour.
How many repairers should be hired?
28. A toll tunnel has decided to experiment with the use of a debit card for the
collection of tolls. Initially, only one lane will be used. Cars are estimated to
arrive at this experimental lane at the rate of 750 per hour. It will take exactly four
seconds to verify the debit card.
a. How much time would you expect a customer to wait in line, pay with a debit
card, and leave?
b. How many cars would you expect to see in the system?
29. A local fast-food restaurant wants to analyze its drive-through window. At this
time, the only information known is the average number of customers in the system
(4.00) and the average time a customer spends at the restaurant (1.176 minutes).
What are the arrival rate and the service rate?