Quantitative Analysis for Management, 11e (Render)
Chapter 13 Waiting Lines and Queuing Theory Models
1) A goal of many waiting line problems is to help a firm find the ideal level of services that minimize the cost of
waiting and the cost of providing the service.
2) One difficulty in waiting line analysis is that it is sometimes difficult to place a value on customer waiting time.
3) The goal of most waiting line problems is to identify the service level that minimizes service cost.
4) Two characteristics of arrivals are the line length and queue discipline.
5) Limited calling populations are assumed for most queuing models.
6) An “infinite calling population” occurs when the likelihood of a new arrival depends upon the number of past
arrivals.
7) On a practical note — if we were to study the waiting lines in a hair salon that had only five chairs for patrons
waiting, we should use an infinite queue waiting line model.
8) If we are studying the arrival of automobiles at a highway toll station, we can assume an infinite calling
population.
9) When looking at the arrivals at the ticket counter of a movie theater, we can assume an unlimited queue.
10) Arrivals are random when they are dependent on one another and can be predicted.
11) On a practical note — if we are using waiting line analysis to study customers calling a telephone number for
service, balking is probably not an issue.
12) On a practical note — if we are using waiting line analysis to study cars passing through a single tollbooth,
reneging is probably not an issue.
13) On a practical note — we should probably view the checkout counters in a grocery store as a set of single
channel systems.
14) A bank with a single queue to move customers to several tellers is an example of a single–channel system.
15) Service times often follow a Poisson distribution.
16) An M/M/2 model has Poisson arrivals exponential service times and two channels.
17) In a single–channel, single–phase system, reducing the service time only reduces the total amount of time spent
in the system, not the time spent in the queue.
18) The wait time for a single–channel system is more than twice that for a two–channel system using two servers
working at the same rate as the single server.
19) The study of waiting lines is called queuing theory.
20) The three basic components of a queuing process are arrivals, service facilities, and the actual waiting line.
21) In the multichannel model (M/M/m), we must assume that the average service time for all channels is the
same.
22) Queuing theory had its beginning in the research work of Albert Einstein.
23) The arrivals or inputs to the system are sometimes referred to as the calling population.
24) Frequently in queuing problems, the number of arrivals per unit of time can be estimated by a probability
distribution known as the Poisson distribution.
25) An automatic car wash is an example of a constant service time model.
26) Balking customers are those who enter the queue but then become impatient and leave without completing
the transaction.
Topic: CHARACTERISTICS OF A QUEUING SYSTEM
27) In a constant service time model, both the average queue length and average waiting time are halved.
28) A hospital ward with only 30 beds could be modeled using a finite population model.
29) A finite population model differs from an infinite population model because there is a random relationship
between the length of the queue and the arrival rate.
30) A transient state is the normal operating condition of the queuing system.
31) A queue system is in a transient state before the steady state is reached.
32) Little’s Flow Equations are applicable for single–channel systems only.
33) Little’s Flow Equations are advantageous because if one characteristic of the operating system is known, the
other characteristics can be easily found.
34) Using a simulation model allows one to ignore the common assumptions required to use analytical models.
35) If we are using a simulation queuing model, we still have to abide by the assumption of a Poisson arrival rate,
and negative exponential service rate.
36) Queuing theory had its beginning in the research work of ________.
A) Albert Einstein
B) A.K. Erlang
C) J.K. Rowling
D) P.K. Poisson
E) A.K. Cox
37) Assume that we are using a waiting line model to analyze the number of service technicians required to
maintain machines in a factory. Our goal should be to
A) maximize productivity of the technicians.
B) minimize the number of machines needing repair.
C) minimize the downtime for individual machines.
D) minimize the percent of idle time of the technicians.
E) minimize the total cost (cost of maintenance plus cost of downtime).
38) In queuing analysis, total expected cost is the sum of expected ________ plus expected ________.
A) service costs, arrival costs
B) facility costs, calling costs
C) calling cost, inventory costs
D) calling costs, waiting costs
E) service costs, waiting costs
39) In queuing theory, the calling population is another name for ________.
A) the queue size
B) the servers
C) the arrivals
D) the service rate
E) the market researchers
40) Which of the following is not a valid queuing model based on the Kendall notation?
A) M/M/3
B) D/D/2
C) D/M/1
D) M/M/0
E) G/D/1
6
Topic: CHARACTERISTICS OF A QUEUING SYSTEM
41) Which of the following is not true about arrivals?
A) Random arrivals are independent of each other.
B) Random arrivals cannot be predicted exactly.
C) The Poisson distribution is often used to represent the arrival pattern.
D) Service times often follow the negative exponential distribution.
E) The exponential distribution is often used to represent the arrival pattern.
42) An arrival in a queue that reneges is one who
A) after joining the queue, becomes impatient and leaves.
B) refuses to join the queue because it is too long.
C) goes through the queue, but never returns.
D) jumps from one queue to another, trying to get through as quickly as possible.
E) None of the above
43) The customer who arrives at a bank, sees a long line, and leaves to return another time is
A) balking.
B) cropping.
C) reneging.
D) blithering.
E) None of the above
44) The three major characteristics of the input source that generates arrivals or customers for the service system
are ________.
A) size, demographics, and behavior
B) size, pattern, and behavior
C) demographics, pattern, and behavior
D) size, demographics, and pattern
E) None of the above
45) The term queue “discipline” describes the
A) degree to which members of the queue renege.
B) sequence in which members of the queue arrived.
C) degree to which members of the queue are orderly and quiet.
D) sequence in which members of the queue are serviced.
E) All of the above
7
46) A vendor selling newspapers on a street corner is an example of a
A) single–channel, multiphase system.
B) single–channel, single–phase system.
C) multichannel, multiphase system.
D) multichannel, single–phase system.
E) None of the above
47) Lines at banks where customers wait for a teller window are usually representative of a
A) single–channel, multiphase system.
B) single–channel, single–phase system.
C) multichannel, multiphase system.
D) multichannel, single–phase system.
E) None of the above
48) Which of the following distributions is most often used to estimate the arrival pattern?
A) negative exponential
B) normal
C) Poisson
D) Erlang
E) beta
49) A single automatic car wash with one bay and a cycle time of 2.5 minutes is what type of model?
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/1
50) A single automatic car wash with one bay and a service time that is normally distributed is what type of
model?
A) M/M/1
B) M/M/2
C) M/D/2
8
D) M/D/1
E) M/G/1
51) Which of the following is not an assumption in common queuing mathematical models?
A) Arrivals come from an infinite, or very large, population.
B) Arrivals are Poisson distributed.
C) Arrivals are treated on a first–in, first–out basis and do not balk or renege.
D) Service rates follow the normal distribution.
E) The average service rate is faster than the average arrival rate.
52) The “utilization factor” is defined as the
A) percent of time the system is idle.
B) average percent of time the customers wait in line.
C) average time the service system is open.
D) percent of time that a single customer is in the system.
E) None of the above
53) A suburban specialty restaurant has developed a single drive–thru window. Customers order, pay, and pick
up their food at the same window. Arrivals follow a Poisson distribution, while service times follow an
exponential distribution. What type of queuing model is exhibited in this problem?
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/1
54) A suburban specialty restaurant has developed a single drive–thru window. Customers order, pay, and pick
up their food at the same window. Arrivals follow a Poisson distribution, while service times follow an
exponential distribution. If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,
what is the average number of customers in the system?
A) 0.50
B) 4.00
C) 2.25
D) 3.00
E) None of the above
55) A suburban specialty restaurant has developed a single drive–thru window. Customers order, pay, and pick
up their food at the same window. Arrivals follow a Poisson distribution while service times follow an
exponential distribution. If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,
what is the average number of customers waiting in line behind the person being served?
A) 0.50
B) 0.75
C) 2.25
D) 3.00
E) None of the above
56) A suburban specialty restaurant has developed a single drive–thru window. Customers order, pay, and pick
up their food at the same window. Arrivals follow a Poisson distribution while service times follow an
exponential distribution. If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,
what proportion of the time is the server busy?
A) 0.25
B) 0.50
C) 0.75
D) 2.25
E) 3.00
57) A suburban specialty restaurant has developed a single drive–thru window. Customers order, pay, and pick
up their food at the same window. Arrivals follow a Poisson distribution while service times follow an
exponential distribution. If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,
how much time will elapse (in hours) from the time a customer enters the line until he/she leaves the restaurant?
A) 0.50
B) 0.25
C) 0.75
D) 2.25
E) 3.00
58) 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
E) None of the above
59) 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 waiting in line behind the person being served?
A) 0.76
B) 0.19
C) 1.33
D) 1.67
E) None of the above
60) 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,
what proportion of the time is the server busy?
A) 0.43
B) 0.57
C) 0.75
D) 0.25
E) 0.33
61) 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
E) 1.33 minutes
62) At a local fast–food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food joint takes an
average of 2 minutes to serve each arrival. The utilization factor for this system is
A) 0.467.
B) 0.547.
C) 0.800.
D) 0.133.
E) None of the above
63) At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. Service times are random
(exponential) and average 2 minutes per arrival. The average time in the queue for each arrival is
A) 2 minutes.
B) 4 minutes.
C) 6 minutes.
D) 8 minutes.
E) 10 minutes.
64) Which of the following distributions is most often used to estimate service times?
A) negative exponential
B) normal
C) Poisson
D) Erlang
E) beta
65) Most systems use a queue discipline known as ________.
A) shortest processing time
B) longest processing time
C) FIFO
D) earliest due date
E) critical ratio
66) When waiting time is based on time in the queue, which of the following is the correct equation for total cost?
A) mCs + λWCw
B) mCw + λWCs
C) mCs + λWqCs
D) mCs + λWqCw
E) mCw + λWqCs
67) A queuing system with a normally distributed arrival pattern, exponential service times, and three servers
would be described as ________.
A) G/G/3
B) M/M/3
C) G/M/3
D) M/G/3
E) N/E/3
68) The initial state of a queuing system (for example, when a restaurant first opens), is referred to as the ________
state.
A) steady
B) primary
C) introductory
D) transient
E) start–up
69) A post office has a single line for customers waiting for the next available postal clerk. There are two postal
clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service
time follows an exponential distribution. What type of queuing model applies here?
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/1
70) A post office has a single line for customers waiting for the next available postal clerk. There are two postal
clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service
time follows an normal distribution. What type of queuing model applies here?
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/2
71) A post office has a single line for customers waiting for the next available postal clerk. There are two postal
clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service
time follows an exponential distribution. The average arrival rate is three per minute and the average service rate
is two per minute for each of the two clerks. What is the average length of the line?
A) 3.429
B) 1.929
C) 1.143
D) 0.643
E) None of the above
72) A post office has a single line for customers waiting for the next available postal clerk. There are two postal
clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service
time follows an exponential distribution. The average arrival rate is three per minute and the average service rate
is two per minute for each of the two clerks. How long does the average person spend waiting for a clerk?
A) 3.429 minutes
B) 1.929 minutes
C) 1.143 minutes
D) 0.643 minute
E) None of the above
73) A post office has a single line for customers waiting for the next available postal clerk. There are two postal
clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service
time follows an exponential distribution. The average arrival rate is three per minute and the average service rate
is two per minute for each of the two clerks. What proportion of time are both clerks idle?
A) 0.643
B) 0.250
C) 0.750
D) 0.143
E) None of the above
Table 13–1
M/M/2
Mean Arrival Rate:
9 occurrences per minute
Mean Service Rate:
8 occurrences per minute
Number of Servers:
2
Queue Statistics:
Mean Number of Units in the System:
1.646
Mean Number of Units in the Queue:
0.521
Mean Time in the System:
0.183 minutes
Mean Time in the Queue:
0.058 minutes
Service Facility Utilization Factor:
0.563
Probability in No Units in System:
0.280
74) According to the information provided in Table 13–1, on average, how many units are in the line?
A) 1.646
B) 0.563
C) 0.280
D) 1.125
E) 0.521
75) According to the information provided in Table 13–1, what proportion of time is at least one server busy?
A) 0.563
B) 0.437
C) 0.720
D) 0.280
E) None of the above
76) Using the information provided in Table 13–1 and counting each person being served and the people in line,
on average, how many people would be in this system?
A) 0.521
B) 1.646
C) 1.125
D) 0.183
E) None of the above
77) According to the information provided in Table 13–1, what is the average time spent by a person in this
system?
A) 0.058 minute
B) 1.646 minutes
C) 0.521 minute
D) 0.183 minute
E) None of the above
78) According to the information provided in Table 13–1, what percentage of the total available service time is
being used?
A) 72.0%
B) 28.0%
C) 56.3%
D) It could be any of the above, depending on other factors.
E) None of the above
79) Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exponentially
distributed with an average of 15 minutes. Jack Burns, the JLUBE owner, has decided to open a second work bay,
i.e., make the shop into a two–channel system. Under this new scheme, the average customer will wait in an
________ line
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/2
80) Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exactly 15 minutes.
Jack Burns, the JLUBE owner, has decided to open a second work bay, i.e., make the shop into a two–channel
system. Under this new scheme, the average customer will wait in an ________ line
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/2
81) Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exponentially
distributed with an average of 15 minutes. Jack Burns, the JLUBE owner, has decided to open a second work bay,
i.e., make the shop into a two–channel system. Under this new scheme, the average customer will wait in line
A) approximately 9.6 minutes.
B) approximately 2.5 minutes.
C) approximately 24.6 minutes.
D) approximately 2.1 minutes.
E) None of the above