Use this information to answer the following questions.
A small suburban post office is frequented by customers at the rate of 15 per hour. The post office has a
single employee who typically processes each customer’s request at an average rate of 3 minutes per
transaction. Assume that arrival rate follows a Poisson distribution and service time follows an
exponential distribution.
3) To determine the efficiency of operations, the postmaster wishes to examine several queue operating
characteristics.
a. What is the utilization rate of this service system?
b. What is the average number of customers in line?
c. What is the average time that each customer spends in the queue?
d. What is the average time that each customer spends in the queue and being serviced?
e. What is the probability that the post office employee will be idle?
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4) Refer to the information above. The postmaster has been receiving complaints from customers about
long waiting times. The postmaster is considering hiring an additional employee who will also be able
to process each customer’s request at the same rate of 3 minutes per transaction. Assume that the post
office pays its employees $12 per hour and that the cost of customer waiting time, in terms of customer
dissatisfaction and goodwill, is $10 per hour. Assume that arrival rate follows a Poisson distribution and
service time follows an exponential distribution. Should the postmaster hire an additional employee?
Compare the total costs of this two-employee system with the old single-server queuing system.
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Use this information to answer the following questions.
A fast food restaurant currently has 2 cashiers. Upon arrival, customers form a single line and place
their food order at the next available register. Assume that customers arrive at the rate of 35 per hour. It
takes an average of 3 minutes to place and process each customer’s order. Assume that arrival rate
follows a Poisson distribution and service time follows an exponential distribution.
5) Refer to the information above. To determine the efficiency of operations, the cafeteria manager
wishes to examine several queue operating characteristics. Based upon the analysis, the manager
believes that if the average waiting time per customer in the system is greater than 5 minutes, then a
request for an additional employee should be made. Should the manager make a request to hire an
additional employee?
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6) Refer to the information above. Assume that each employee earns an hourly wage of $6.50/hour.
Also, assume that the cost of customer waiting time is $10/hour. What would be the impact of hiring an
additional employee on customers’ waiting time and total costs?
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Use this information to answer the following questions.
Cars arrive at Sparkling Clean automatic car wash at the rate of 15 per hour and follow the Poisson
distribution. It takes exactly 3 minutes for each car to go through the automated process of wetting,
soaping, scrubbing, and drying.
7) Refer to the information above.
a. What is the average time that each car spends in the queue?
b. What is the average time that each car spends in the queue and gets washed?
c. What is the probability that the next car that arrives will not have to wait?
8) Refer the information above. Assume that the owners of Sparkling Clean decide that in order to
remain competitive in the automatic car wash business, each car should not spend more than 6 minutes
in the system. The company that had installed the car wash system told the owners that they would be
happy to tweak the timing of the automated washing process to achieve the desired goal of the owners.
What should the service rate be? Hint: Use Excel’s Goal Seek function.
Use this information to answer the following questions.
Cars arrive at Speedy Lube at an average rate of 4 per hour. Arrivals can be assumed to follow the
Poisson distribution. The sole mechanic who works at Speedy Lube spends an average of 12 minutes
changing each car’s oil and filter. The standard deviation of service time is 2 minutes and service time
distribution is arbitrary.
9) Refer to the information above. Calculate the operating characteristics of the queuing system at
Speedy Lube. What is the probability that an arriving customer will have to wait for service?
10) Refer to the information above. Assume that Speedy Lube has been acquired by a new owner who is
planning an advertisement campaign that claims that each car would spend an average of 20 minutes in
the shop inclusive of waiting time and service time. What should the mechanic’s service rate be so as
not to accuse the new owner of false advertising? Hint: Use Excel’s Goal Seek function.
11) A technician who works in a computer lab at a local college has the responsibility of maintaining 30
desktop computers. Past records indicate that each desktop needs repair after about 40 hours of use.
Breakdowns have been found to be Poisson distributed. The one technician on duty can repair a desktop
in an average of 1 hour, following an exponential distribution.
a. How many desktops are waiting for service (i.e., in the repair queue), on average?
b. What is the average waiting time in the queue?
c. What is the average wait in the system?
d. What is the probability that all desktops are in good working condition and not needing any repairs?
12) A professor is holding office hours for a class of 15 students. Each student arrives on average once
per hour. The arrival process follows a Poisson distribution. The time the professor spends with each
student follows an exponential distribution with an average of 5 minutes. (Assume that each student may
seek help and may decide to make multiple trips to the professor’s office.) Answer the questions below.
a. How many students are waiting to see the professor on average?
b. What is the average waiting time in the queue?
c. What is the average wait in the system?