82) 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 total time an average customer spends
in the system will be
A) 37 minutes.
B) 2.1 minutes.
C) 9.6 minutes.
D) 33.3 minutes.
E) 24.6 minutes.
Table 13–2
M/M/2
Mean Arrival Rate:
5 occurrences per minute
Mean Service Rate:
3 occurrences per minute
Number of Servers:
2
Queue Statistics:
Mean Number of Units in the System:
5.455
Mean Number of Units in the Queue:
3.788
Mean Time in the System:
1.091 minutes
Mean Time in the Queue:
0.758 minutes
Service Facility Utilization Factor:
0.833
Probability in No Units in System:
0.091
83) According to the information provided in Table 13–2, which presents a queuing problem solution, on average,
how many units are in the line?
A) 5.455
B) 3.788
C) 1.091
D) 0.758
E) 0.833
84) According to the information provided in Table 13–2, which presents a queuing problem solution, what
proportion of time is at least one server busy?
A) 0.833
B) 0.758
C) 0.091
D) 0.909
E) None of the above
85) According to the information provided in Table 13–2, which presents a queuing problem solution, there are
two servers in this system. Counting each person being served and the people in line, on average, how many
people would be in this system?
A) 5.455
B) 3.788
C) 9.243
D) 10.900
E) None of the above
Table 13–3
M/M/3
Mean Arrival Rate:
4 occurrences per minute
Mean Service Rate:
2 occurrences per minute
Number of Servers:
3
Queue Statistics:
Mean Number of Units in the System:
2.889
Mean Number of Units in the Queue:
0.889
Mean Time in the System:
0.722 minutes
Mean Time in the Queue:
0.222 minutes
Service Facility Utilization Factor:
0.667
Probability in No Units in System:
0.111
86) According to the information provided in Table 13–3, which presents a queuing problem solution, what
proportion of time is the system totally empty?
A) 0.111
B) 0.333
C) 0.889
D) 0.667
E) None of the above
87) According to the information provided in Table 13–3, which presents a queuing problem solution, on average,
how long does each customer spend waiting in line?
A) 0.333 minute
B) 0.889 minute
C) 0.222 minute
D) 0.722 minute
E) 0.111 minute
88) According to the information provided in Table 13–3, which presents a queuing problem solution what is the
utilization rate of the service facility?
A) 0.111
B) 0.889
C) 0.222
D) 0.722
E) 0.667
89) Little’s Flow Equations are transferable to a production environment. Which of the following would be a
proper interpretation of Little’s Flow Equations?
A) Flow Rate = Inventory × Flow Time
B) Flow Time = Inventory × Flow Rate
C) Inventory = Flow Rate × Flow Time
D) Time to Take an Order = Flow Rate × Flow Time
E) Flow Rate = Time to Take an Order × Flow Time
90) If everything else remains constant, including the mean arrival rate and service rate, except that the service
time becomes constant instead of exponential,
A) the average queue length will be halved.
B) the average waiting time will be doubled.
C) the average queue length will be doubled.
D) There is not enough information to know what will happen to the queue length and waiting time.
E) None of the above
91) At an automatic car wash, cars arrive randomly at a rate of 7 cars every 30 minutes. The car wash takes
exactly 4 minutes (this is constant). On average, what would be the length of the line?
A) 8.171
B) 7.467
C) 6.53
D) 0.467
E) None of the above
92) At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4
minutes (this is constant). On average, how long would each car spend at the car wash?
A) 28 minutes
B) 32 minutes
C) 17 minutes
D) 24 minutes
E) None of the above
93) At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4
minutes (this is constant). On average, how long would each driver have to wait before receiving service?
A) 28 minutes
B) 32 minutes
C) 17 minutes
D) 24 minutes
E) None of the above
94) At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4
minutes (this is constant). On average, how many customers would be at the car wash (waiting in line or being
serviced)?
A) 8.17
B) 7.46
C) 6.53
D) 0.46
E) None of the above
95) A(n) ________ state is the normal operating condition of the queuing system.
A) primary
B) transient
C) NOC
D) balanced
E) steady
96) At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food joint takes
exactly 2 minutes (this is constant). The average total time in the system is
A) 5.4 minutes.
B) 6.0 minutes.
C) 8.0 minutes.
D) 2.5 minutes.
E) None of the above
Table 13–4
M/D/1
Mean Arrival Rate:
3 occurrences per minute
Constant Service Rate:
4 occurrences per minute
Queue Statistics:
Mean Number of Units in the System:
1.875
Mean Number of Units in the Queue:
1.125
Mean Time in the System:
0.625 minutes
Mean Time in the Queue:
0.375 minutes
Service Facility Utilization Factor:
0.750
Probability in No Units in System:
0.250
97) According to the information provided in Table 13–4, which presents a queuing problem solution for a
queuing problem with a constant service rate, on average, how much time is spent waiting in line?
A) 1.875 minutes
B) 1.125 minutes
C) 0.625 minute
D) 0.375 minute
E) None of the above
98) According to the information provided in Table 13–4, which presents a queuing problem solution for a
queuing problem with a constant service rate, on average, how many customers are in the system?
A) 1.875
B) 1.125
C) 0.625
D) 0.375
E) None of the above
99) According to the information provided in Table 13–4, which presents a queuing problem solution for a
queuing problem with a constant service rate, on average, how many customers arrive per time period?
A) 3
B) 4
C) 1.875
D) 1.125
E) None of the above
100) According to Table 13–4, which presents a queuing problem with a constant service rate, on average, how
many minutes does a customer spend in the service facility?
A) 0.375 minutes
B) 4 minutes
C) 0.625 minutes
D) 0.25 minutes
E) None of the above
Table 13–5
M/D/1
Mean Arrival Rate:
5 occurrences per minute
Constant Service Rate:
7 occurrences per minute
Queue Statistics:
Mean Number of Units in the System:
1.607
Mean Number of Units in the Queue:
0.893
Mean Time in the System:
0.321 minutes
Mean Time in the Queue:
0.179 minutes
Service Facility Utilization Factor:
0.714
101) According to the information provided in Table 13–5, which presents the solution for a queuing problem with
a constant service rate, on average, how much time is spent waiting in line?
A) 1.607 minutes
B) 0.714 minute
C) 0.179 minute
D) 0.893 minute
E) None of the above
102) According to the information provided in Table 13–5, which presents the solution for a queuing problem with
a constant service rate, on average, how many customers are in the system?
A) 0.893
B) 0.714
C) 1.607
D) 0.375
E) None of the above
103) According to the information provided in Table 13–5, which presents a queuing problem solution for a
queuing problem with a constant service rate, on average, how many customers arrive per time period?
A) 5
B) 7
C) 1.607
D) 0.893
E) None of the above
104) According to the information provided in Table 13–5, which presents the solution for a queuing problem with
a constant service rate, on average, how many minutes does a customer spend in the system?
A) 0.893 minute
B) 0.321 minute
C) 0.714 minute
D) 1.607 minutes
E) None of the above
105) According to the information provided in Table 13–5, which presents the solution for a queuing problem with
a constant service rate, what percentage of available service time is actually used?
A) 0.217
B) 0.643
C) 0.321
D) 0.179
E) None of the above
106) According to the information provided in Table 13–5, which presents the solution for a queuing problem with
a constant service rate, the probability that the server is idle is ________.
A) 0.217
B) 0.643
C) 0.286
D) 0.714
E) None of the above
107) The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5
hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15
minutes to replace the toner cartridge. What is the probability the system is empty?
A) 1.1500
B) 1.1658
C) .8578
D) .8696
E) None of the above
108) Which of the following is not an assumption for the M/M/1 model with finite population source?
A) There is only one server.
B) The population of units seeking service are finite.
C) Arrivals follow a Poisson distribution and service times are exponential.
D) Customers are served on a first come, first served basis.
E) The length of the queue is independent of the arrival rate.
109) The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5
hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15
minutes to replace the toner cartridge. What is the average number of fax machines in the queue?
A) 1.1658 fax machines
B) 2.9904 fax machines
C) .1563 fax machine
D) .0142 fax machine
E) None of the above
110) The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5
hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15
minutes to replace the toner cartridge. What is the average number of fax machines in the system?
A) .0142 fax machine
B) .1563 fax machine
C) .0249 fax machine
D) .2749 fax machine
E) None of the above
111) The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5
hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15
minutes to replace the toner cartridge. What is the average waiting time in the queue?
A) .0142 hour
B) .1563 hour
C) .0249 hour
D) .2749 hour
E) None of the above
112) The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5
hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15
minutes to replace the toner cartridge. What is the average time spent in the system?
A) .0142 hour
B) .1563 hour
C) .0249 hour
D) .2749 hour
E) None of the above
113) The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5
hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15
minutes to replace the toner cartridge. What is the probability that 2 fax machines need toner at the same time?
A) .8576
B) .1286
C) .0129
D) .1415
E) None of the above
114) A new shopping mall is considering setting up an information desk manned by one employee. Based upon
information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of
15 per hour. It takes an average of two minutes to answer a question. It is assumed that arrivals are Poisson and
answer times are exponentially distributed.
(a) Find the probability that the employee is idle.
(b) Find the proportion of time that the employee is busy.
(c) Find the average number of people receiving and waiting to receive information.
(d) Find the average number of people waiting in line to get information.
(e) Find the average time a person seeking information spends at the desk.
(f) Find the expected time a person spends waiting in line to have his question answered.
115) A new shopping mall is considering setting up an information desk manned by one employee. Based upon
information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of
15 per hour. It takes exactly two minutes to answer each question. It is assumed that arrivals are Poisson.
(a) Find the probability that the employee is idle.
(b) Find the proportion of time that the employee is busy.
(c) Find the average number of people receiving and waiting to receive information.
(d) Find the average number of people waiting in line to get information.
(e) Find the average time a person seeking information spends at the desk.
(f) Find the expected time a person spends waiting in line to have his question answered.
116) Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog
every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a
rate of 1 dog every 6 minutes, according to a Poisson distribution. Also assume that Sam’s vaccinating times are
exponentially distributed.
(a) Find the probability that Sam is idle.
(b) Find the proportion of time that Sam is busy.
(c) Find the average number of dogs receiving or waiting to be vaccinated.
(d) Find the average number of dogs waiting to be vaccinated.
(e) Find the average time a dog waits before getting vaccinated.
(f) Find the average amount (mean) of time a dog spends between waiting in line and getting vaccinated.
117) Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog
every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a
rate of 1 dog every 4 minutes, according to a Poisson distribution. Also assume that Sam’s vaccinating times are
exponentially distributed.
(a) Find the probability that Sam is idle.
(b) Find the proportion of time that Sam is busy.
(c) Find the average number of dogs receiving or waiting to be vaccinated.
(d) Find the average number of dogs waiting to be vaccinated.
(e) Find the average time a dog waits before getting vaccinated.
(f) Find the average amount (mean) of time a dog spends between waiting in line and getting vaccinated.
118) A dry cleaner has a single drive–thru window for customers. The arrival rate of cars follows a Poisson
distribution, while the service time follows an exponential distribution. The average arrival rate is 16 per hour
and the average service time is three minutes.
(a) What is the average number of cars in the line?
(b) What is the average time spent waiting to get to the service window?
(c) What percentage of time is the dry cleaner‘s drive–thru window idle?
(d) What is the probability there are more than 2 cars at the drive–thru window?
119) A dry cleaner has a single drive–thru window for customers. The arrival rate of cars follows a Poisson
distribution, while the service time follows an exponential distribution. The average arrival rate is 16 per hour
and the average service time is three minutes. If the dry cleaner wants to accommodate (have enough room for)
All of the waiting cars at least 96 percent of the time, how many car–lengths should they make the driveway
leading to the window?
120) Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog
every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a
rate of 1 dog every 6 minutes, according to a Poisson distribution. Also assume that Sam’s vaccinating times are
exponentially distributed. Sam would like to have each waiting dog placed in a holding pen. If Sam wants to be
certain he has enough cages to accommodate all dogs at least 90 percent of the time, how many cages should he
prepare?
121) 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.
122) Cars arrive at a parking lot entrance at the rate of 20 per hour. The average time to get a ticket and proceed
to a parking space is two minutes. There are two lot attendants at the current time. The Poisson and exponential
distribution appear to be relevant in this situation.
(a) What is the probability that an approaching auto must wait?
(b) What is the average waiting time?
(c) What is the average number of autos waiting to enter the garage?
123) Bank Boston has a branch at Bryant College. The branch is busiest at the beginning of the college year when
freshmen and transfer students open accounts. This year, freshmen arrived at the office at a rate of 40 per day (8–
hour day). On average, it takes the Bank Boston staff person about ten minutes to process each account
application. The bank is considering having one or two tellers. Each teller is paid $12 per hour and the cost of
waiting in line is assumed to be $8 per hour.
(a) What is the total daily waiting cost for the single teller model?
(b) What is the total daily waiting cost for the two–teller model?
(c) What is the total daily service cost for the single teller model?
(d) What is the total daily service cost for the two–teller model?
(e) Which model is preferred?
124) At the start of 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?
125) Consider a single–server queuing system with Poisson arrivals of 10 units per hour and a constant service
time of 2 minutes per unit. How long will the customer waiting time be in seconds, on average?
126) 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?
127) A professor decides to hold a three hour window open for student advising. The professor has 15 advisees.
He’s unsure if each advisee will show up, and its possible that a single advisee could show up multiple times
during the 3 hour window. On average, the professor will spend 12 minutes with each student. He usually
expects each student to arrive once during the 3 hour window.
(a) What proportion of the time can the professor expect to be busy during the window?
(b) What is the expected number of students in queue during the window?
(c) What is the expected wait time in queue during the window?
128) With regard to queuing theory, define what is meant by balking.
129) With regard to queuing theory, define what is meant by reneging.
130) What is meant by a single–channel queuing system?
131) What is meant by a multichannel queuing system?
132) What is meant by a single–phase system?
133) What is meant by a multiphase system?