14) Which of the following is NOT an assumption of the M/M/1 model?
A) The first customers to arrive are the first customers served.
B) Each arrival comes independently of the arrival immediately before and after that arrival.
C) The population from which the arrivals come is very large or infinite in size.
D) Customers do not renege.
E) Service times occur according to a normal probability distribution.
15) A single-phase waiting-line system meets the assumptions of constant service time or M/D/1. Units
arrive at this system every 12 minutes on average. Service takes a constant 8 minutes. What is the average
length of the queue Lq in units?
A) 0.67
B) 2.5
C) 4.5
D) 5.0
E) 7.5
16) A single-phase waiting-line system meets the assumptions of constant service time or M/D/1. Units
arrive at this system every 12 minutes on average. Service takes a constant 8 minutes. What is the average
number in the system Ls in units?
A) 2.25
B) 2.5
C) 3.0
D) 1.33
E) 5.0
17) A queuing model that follows the M/M/1 assumptions has λ = 2 and μ = 3. What is the average
number of units in the system?
A) 2/3
B) 1
C) 1.5
D) 2
E) 6
18) A queuing model that follows the M/M/1 assumptions has λ = 3 and μ = 2. What is the average
number of units in the system?
A) -3
B) 3
C) 0.667
D) 150 percent
E) Growing without limit, since λ is larger than μ.
19) Students arrive randomly at the help desk of the computer lab. There is only one service agent, and
the time required for inquiry varies from student to student. Arrival rates have been found to follow the
Poisson distribution, and the service times follow the negative exponential distribution. The average
arrival rate is 12 students per hour, and the average service rate is 20 students per hour. On average, how
long does it take to service each student?
A) 1 minute
B) 2 minutes
C) 3 minutes
D) 5 minutes
E) 20 minutes
20) A queuing model that follows the M/M/1 assumptions has λ = 10 and μ = 12. What is the average
number of units in the system?
A) 0.83
B) 2
C) 2.5
D) 5
E) 6
21) A queuing model that follows the M/M/1 assumptions has λ = 2 and μ = 8. The average number in the
system Ls is ________ and the utilization of the system is ________.
A) 3; 100 percent
B) 0.33; 25 percent
C) 4; 33 percent
D) 6; 25 percent
E) 4; 25 percent
22) Four of the most widely used waiting line models—M/M/1 or A, M/M/S or B, M/D/1 or C, and
Limited population or D–all share which of the following three characteristics?
A) normal arrivals, FIFO discipline, and normal service times
B) Poisson arrivals, FIFO discipline, and a single-service phase
C) Poisson arrivals, FIFO discipline, and exponential service times
D) Poisson arrivals, no queue discipline, and exponential service times
E) Poisson arrivals, single-server, and FIFO discipline
23) A queuing model that follows the M/M/1 assumptions has λ = 2 and μ = 3. What is the average time in
the system?
A) 2/3
B) 1
C) 1.5
D) 2
E) 6
24) Students arrive randomly at the help desk of the computer lab. There is only one service agent, and
the time required for inquiry varies from student to student. Arrival rates have been found to follow the
Poisson distribution, and the service times follow the negative exponential distribution. The average
arrival rate is 12 students per hour, and the average service rate is 20 students per hour. What is the
utilization factor?
A) 20%
B) 30%
C) 40%
D) 50%
E) 60%
25) A finite population waiting line model has an average service time T of 100 minutes and an average
time between service requirements U of 400 minutes. What is the service factor X?
A) 0.20
B) 0.25
C) 4
D) 5
E) 300 minutes
26) A finite population waiting line model has an average service time T of 200 minutes and an average
time between service requirements U of 300 minutes. What is the service factor X?
A) 0.20
B) 0.40
C) 0.60
D) 0.67
E) 2.5
27) Students arrive randomly at the help desk of the computer lab. There is only one service agent, and
the time required for inquiry varies from student to student. Arrival rates have been found to follow the
Poisson distribution, and the service times follow the negative exponential distribution. The average
arrival rate is 12 students per hour, and the average service rate is 20 students per hour. A student has
just entered the system. How long is she expected to stay in the system?
A) 0.125 minutes
B) 0.9 minutes
C) 1.5 minutes
D) 7.5 minutes
E) 0.075 hours
28) Students arrive randomly at the help desk of the computer lab. There is only one service agent, and
the time required for inquiry varies from student to student. Arrival rates have been found to follow the
Poisson distribution, and the service times follow the negative exponential distribution. The average
arrival rate is 12 students per hour, and the average service rate is 20 students per hour. How many
students, on average, will be waiting in line at any one time?
A) 0.9 students
B) 1.5 students
C) 3 students
D) 4 students
E) 36 students
29) A waiting-line system that meets the assumptions of M/M/S has λ = 5, μ = 4, and M = 2. For these
values, Po is approximately 0.23077, and Ls is approximately 2.05128. What is the average time a unit
spends in this system?
A) approximately 0.1603
B) approximately 0.2083
C) approximately 0.4103
D) approximately 0.8013
E) Cannot be calculated because λ is larger than μ.
30) A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ = 4. For this system, Po is
________ and utilization is ________.
A) 0.75; 0.25
B) 0.80; .20
C) -3; -4
D) 3; 4
E) none of the above
31) A waiting-line system that meets the assumptions of M/M/S has λ = 5, μ = 4, and M = 2. For these
values, Po is approximately 0.23077, and Ls is approximately 2.05128. What is the average number of
units waiting in the queue?
A) approximately 0.1603
B) approximately 0.4103
C) approximately 0.8013
D) approximately 1.0417
E) Cannot be calculated because λ is larger than μ.
32) A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ = 4. For this system, what is
the probability of more than two units in the system?
A) zero
B) 0.015625
C) 0.0625
D) 0.25
E) 0.9375
33) A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ = 4. For this system, what is
the probability of fewer than two units in the system?
A) 0.0625
B) 0.25
C) 0.75
D) 0.9375
E) 1.00
34) Little’s Law is not applicable in which of the following situations?
A) the opening of a toy store on Black Friday morning
B) a 24-hour supermarket
C) a gas station with 24-hour self-service pumps
D) B and C
E) It is applicable to A, B, and C.
35) Which of the following is a requirement for application of Little’s Law to be valid?
A) arrival rates that follow the Poisson distribution
B) FIFO queue discipline
C) steady state conditions
D) single-server system
E) multiphase system
36) The ________ of a waiting line and the probability that the queue is empty add to one.
37) In a finite or limited population waiting line, the ________ is calculated from the average service time
and average time between service requirements before the problem can be completed.
38) What are Ls and Lq, as used in waiting line terminology? Which is larger, Ls or Lq? Explain.
39) What are the six assumptions underlying the M/M/1 waiting line model? Which of these also hold for
the M/D/1 model?
40) You have seen that, in an M/D/1 problem, the average queue length is exactly one-half the average
queue length of an otherwise identical M/M/1 problem. Are all other performance statistics one–half as
large also? Explain.
41) Why must the service rate be greater than the arrival rate in a single-server system?
42) A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of
12 per hour. What is the probability that the waiting line is empty?
43) A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of
12 per hour. What is the average time a unit spends in the system and the average time a unit spends
waiting (in hours)?
44) A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of
12 per hour. What is the average time a unit spends in the system and the average time a unit spends
waiting (in hours)?
45) A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of
12 per hour. What is the probability that the waiting line is empty?
46) A crew of mechanics at the Highway Department Garage repair vehicles that break down at an
average of λ = 7.5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an
average of μ = 10 vehicles per day with a repair time distribution that approximates an exponential
distribution.
a. What is the utilization rate for this service system?
b. What is the average time before the facility can return a breakdown to service?
c. How much of that time is spent waiting for service?
d. How many vehicles are likely to be in the system at any one time?
47) A crew of mechanics at the Highway Department Garage repair vehicles that break down at an
average of λ = 7 vehicles per day (approximately Poisson in nature). The mechanic crew can service an
average of μ= 11 vehicles per day with a repair time distribution that approximates an exponential
distribution.
a. What is the utilization rate for this service system?
b. What is the average time before the facility can return a breakdown to service?
c. How much of that time is spent waiting for service?
d. How many vehicles are likely to be waiting for service at any one time?
48) A crew of mechanics at the Highway Department Garage repair vehicles that break down at an
average of λ = 5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an
average of μ= 10 vehicles per day with a repair time distribution that approximates an exponential
distribution.
a. What is the probability that the system is empty?
b. What is the probability that there is precisely one vehicle in the system?
c. What is the probability that there is more than one vehicle in the system?
d. What is the probability of 5 or more vehicles in the system?
49) A crew of mechanics at the Highway Department Garage repair vehicles that break down at an
average of λ = 8 vehicles per day (approximately Poisson in nature). The mechanic crew can service an
average of μ= 11 vehicles per day with a repair time distribution that approximates an exponential
distribution. The crew cost is approximately $300 per day. The cost associated with lost productivity from
the breakdown is estimated at $150 per vehicle per day (or any fraction thereof). What is the expected cost
of this system?
50) A crew of mechanics at the Highway Department garage repair vehicles that break down at an
average of λ = 8 vehicles per day (approximately Poisson in nature). The mechanic crew can service an
average of μ= 10 vehicles per day with a repair time distribution that approximates an exponential
distribution.
a. What is the probability that the system is empty?
b. What is the probability that there is precisely one vehicle in the system?
c. What is the probability that there is more than one vehicle in the system?
d. What is the probability of 5 or more vehicles in the system?
51) A crew of mechanics at the Highway Department Garage repair vehicles that break down at an
average of λ = 8 vehicles per day (approximately Poisson in nature). The mechanic crew can service an
average of μ= 11 vehicles per day with a repair time distribution that approximates an exponential
distribution. The crew cost is approximately $300 per day. The cost associated with lost productivity from
the breakdown is estimated at $150 per vehicle per day (or any fraction thereof). Which is cheaper, the
existing system with one service crew, or a revised system with two service crews?
52) A dental clinic at which only one dentist works is open only two days a week. During those two days,
patients arrive at the rate of three per hour. The doctor serves patients at the rate of one every 15 minutes.
a. What is the probability that the clinic is empty (except for the dentist and staff)?
b. What percentage of the time is the dentist busy?
c. What is the average number of patients in the waiting room?
d. What is the average time a patient spends in the office (wait plus service)?
e. What is the average time a patient waits for service?
53) A dental clinic at which only one dentist works is open only two days a week. During those two days,
the traffic arrivals follow a Poisson distribution with patients arriving at the rate of three per hour. The
doctor serves patients at the rate of one every 15 minutes.
a. What is the probability that the clinic is empty (except for the dentist and staff)?
b. What is the probability that there are one or more patients in the system?
c. What is the probability that there are four patients in the system?
d. What is the probability that there are four or more patients in the system?
54) At the order fulfillment center of a major mail-order firm, customer orders, already packaged for
shipment, arrive at the sorting machine to be sorted for loading onto the appropriate truck for the parcel’s
address. The arrival rate at the sorting machine is at the rate of 100 per hour following a Poisson
distribution. The machine sorts at the constant rate of 150 per hour.
a. What is the utilization rate of the system?
b. What is the average number of packages waiting to be sorted?
c. What is the average number of packages in the sorting system?
d. How long must the average package wait until it gets sorted?
e. What would Lq and Wq be if the service rate were exponential, not constant?
55) At the order fulfillment center of a major mail-order firm, customer orders, already packaged for
shipment, arrive at the sorting machine to be sorted for loading onto the appropriate truck for the parcel’s
address. The arrival rate at the sorting machine is at the rate of 140 per hour following a Poisson
distribution. The machine sorts at the constant rate of 150 per hour.
a. What is the utilization rate of the system?
b. What is the average number of packages waiting to be sorted?
c. What is the average number of packages in the sorting system?
d. How long must the average package wait until it gets sorted?
56) A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ= 4. Calculate Po. Build a table
showing the probability of more than 0, 1, 2, 3, 4, 5, 6, and 7 units in the system. Round to six decimal
places in your work
57) A finite population waiting line model with a single server has an average service time T of 200
minutes and an average time between service requirements U of 300 minutes. Calculate the service factor
X. If the population consists of 5 elements, what are the average number waiting, the average number
being serviced, and the average number running? Refer to Table D.8.
58) A finite population waiting line model with a single server has an average service time T of 50
minutes and an average time between service requirements U of 350 minutes. Calculate the service factor
X. If the population consists of 5 elements, what are the average number waiting, the average number
being serviced, and the average number running? Refer to Table D.8.
59) Suppose that a service facility has an average line of 2 customers that must wait, on average, 5
minutes for service. How many customers are arriving per hour?
60) Suppose that a fast food restaurant wants the average line to be 4 customers and that 80 customers
arrive each hour. How many minutes will the average customer be forced to wait in line?
61) A manufacturing plant is trying to determine how long the average line for a repair process will be. If
10 machines arrive each hour and must wait 6 minutes in the line, how long will the line be, on average?
Section 5 Other Queuing Approaches
1) Service times in an automobile repair shop tend to follow which probability distribution?
A) exponential
B) normal
C) triangular
D) binomial
E) Erlang