Deck 9: Queuing Models

Question 1

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.What is the average number of customers in the queue plus the number being served?

A)0.70
B)2.33
C)1.63
D)0.23
E)0.33

Answer 1

2.33

Question 2

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.Assuming a Poisson distribution arrival rate and an exponential distribution service rate,what is the Kendall notation of this system?

A)M/M/1
B)M/D/1
C)M/G/1
D)D/M/1
E)G/M/1

Answer 2

M/M/1

Question 3

Reneging refers to customers who:

A)do not join a queue
B)switch queues
C)join a queue but abandon their shopping carts before checking out
D)join a queue but are dissatisfied
E)join a queue and complain because of long lines

Answer 3

C

Question 4

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.What is the average number of customers waiting to be served?

A)0.23
B)2.33
C)0.33
D)1.63
E)0.70

Question 5

A service system has a constant service time,Poisson arrival rates and 1 server.What is the Kendall notation for this system?

A)M/M/1
B)M/D/1
C)M/G/1
D)D/M/1
E)G/M/1

Question 6

A queuing system has an arrival rate of 5 customers per hour and a service rate of 8 customers per hour.What is the utilization factor (ρ)of the system?

A)40
B)1)6
C)4
D)5
E)0)625

Question 7

Use this information to answer the following questions.
$$\begin{array} { l | c } \hline \text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 8.00 \\\text { Service Rate } & 10.00 \\\text { Standard deviation of servicetime } & 0.09 \\\hline\end{array}$$

-Refer to the table.What is the average number of customers in the queue plus the number being served?

A)0.12
B)1.76
C)2.12
D)2.90
E)3.30

Question 8

Service time in queuing systems often follows the Poisson distribution.

Question 9

Customers arrive at a grocery store following a Poisson distribution at an average rate of 70 per hour.On average,how many customers arrive per minute?

A)1)2 customers per minute
B)7 customers per minute
C)0)86 customers per minute
D)0)02 customers per minute
E)0)7 customers per minute

Question 10

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.What is the probability that the service facility will be idle?

A)0.23
B)0.70
C)0.33
D)0.30
E)0.233

Question 11

This queuing system configuration is referred to as a:

A)single-server,multiphase system
B)multiple-server,multiphase system
C)multiple-server,single-phase system
D)single-server,single-phase system
E)single-server,parallel phase system

Question 12

A beauty salon employs three hairdressers.All customers get their hair washed in one area of the salon before being taken to another area to get their hair cut.Customer arrival rate and service time follows the Poisson and exponential distributions,respectively.What is the Kendall notation for this system?

A)M/M/1
B)D/M/3
C)M/D/3
D)M/G/3
E)M/M/3

Question 13

This queuing system configuration is referred to as a:

A)single-server,multiphase system
B)multiple-server,multiphase system
C)multiple-server,single-phase system
D)single-server,single-phase system
E)single-server,parallel phase system

Question 14

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.If a second server was added with the same service rate,what would be the revised utilization factor of the system?

A)70%
B)50%
C)49%
D)35%
E)25%

Question 15

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.What is the average time a customer spends waiting in line and being served?

A)0.23
B)2.33
C)0.33
D)1.63
E)0.70

Question 16

The average time each customer spends in the queue is referred to as:

A)W
B)Wq
C)L
D)Lq
E)ρ

Question 17

What is the Kendall notation for the queuing system that develops when students from a single class wait to see a professor for office hours,assuming Poisson arrivals and exponential service time?

A)M/M/1/∞/N
B)M/M/1
C)M/G/1
D)M/D/1
E)G/M/1

Question 18

In a drive-in fast food restaurant,customers form a single lane,place their order and pay their bill at one window,and then pick up their food at a second window.This queuing configuration is referred to as:

A)single-server,multiphase system
B)multiple-server,multiphase system
C)multiple-server,single-phase system
D)single-server,single-phase system
E)single-server,parallel phase system

Question 19

Use this information to answer the following questions.
$$\begin{array} { l | c } \hline \text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 8.00 \\\text { Service Rate } & 10.00 \\\text { Standard deviation of servicetime } & 0.09 \\\hline\end{array}$$

-Refer to the table.What is the average server utilization?

A)125%
B)100%
C)80%
D)10%
E)9%

Question 20

Use this information to answer the following questions.
$\begin{array}{l|c}\text { Number of Servers } & 1.0 \\\text { Arrival Rate } & 7.00 \\\text { Service Rate } & 10.00 \\\mathrm{P}(0) \text {, probability that there are no customers in the system } & 30 \% \\\mathrm{Lq} \text {, average length of the queue } & 1.63 \\\mathrm{~W} \text {, average time in the system } & 0.33 \\\mathrm{~L} \text {, average number of customers in the system } & 2.33 \\\mathrm{Wq}, \text { average time in the queue } & 0.23 \\\text { Utilization factor of the system } & 70 \% \\\hline\end{array}$


-Refer to the table.What percent of the time is the server busy?

A)10%
B)33%
C)23%
D)70%
E)30%

Question 21

Cars arriving at a highway tollbooth would be an example of a limited (finite)population.

Question 22

The cost of providing service is generally easier to quantify than the cost of waiting time.

Question 23

A college cafeteria has a single checkout lane manned by a single cashier.Students arrive at the cash register at the rate of 2 per minute during peak lunch hour.It takes about 20 seconds to check out each student's tray.Assume that arrival rate follows a Poisson distribution and service time follows an exponential distribution.To determine the efficiency of operations,the cafeteria manager wishes to examine several queue operating characteristics.
a.What is the utilization rate of this service system?
b.What is the average number of students in line?
c.What is the average time that each student spends in the queue?
d.What is the average time that each student spends in the queue and being checked out?
e.What is the probability that the cashier will be idle?

Question 24

The utilization factor (ρ)represents the proportion of time that the service facility is idle.

Question 25

One of the M/M/1 queuing model assumptions is that the average number of arrivals (the arrival rate)does not change over time.

Question 26

The two primary components of queuing system costs are the cost of providing service and the cost of wages.

Question 27

An M/M/1 system denotes a single server queuing system with exponential arrival and Poisson service time distributions.

Question 28

The Exponential distribution is used in many queuing models to represent service time patterns.

Question 29

If the average arrival rate (λ)is greater than the average service rate (µ),the queue length will grow indefinitely.

Question 30

The Poisson distribution is quite often used to represent arrival rates in queuing systems.

Question 31

An M/M/s system implies there is a single number of servers.

Question 32

A multiphase queuing system is one in which service is received from more than one station,one after the other.

Question 33

A local bank has a single drive-in teller window.Customers arrive at the drive-in window following a Poisson distribution at an average rate of 20 per hour.Assume that service is provided at the drive-in window at a rate of 30 customers per hour and that service time follows an exponential distribution.Moreover,assume that the population size is infinite and the queue length is unrestricted.To determine the efficiency of operations,the bank 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 system?
e.What is the probability that the drive-in window will be idle?

Question 34

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.
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.

Question 35

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.
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?

Question 36

Service time distribution can be either constant or random.

Question 37

As service levels increase,the cost of providing service also increases,but the cost of customer dissatisfaction decreases.

Question 38

The service time pattern of a single lane automated car wash service would most likely be exponential.

Question 39

Kendall's three-symbol notation is sometimes extended to include 5 symbols.

Question 40

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.
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?

Question 41

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.
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?

Question 42

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.
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.

Question 43

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.
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.

Question 44

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.
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?

Question 45

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?

Question 46

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?

Question 47

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.
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?