Chapter 11 – Waiting Line Models
True / False
1. For an M/M/1 queuing system, if the service rate, µ, is doubled, the average wait in the system, W, is cut in half.
a.
True
b.
False
False
1
Operating characteristics
2. A waiting line situation where every customer waits in the same line before being served by the same server is called a
single server waiting line.
a.
True
b.
False
3. Use of the Poisson probability distribution assumes that arrivals are not random.
a.
True
b.
False
False
1
Distribution of arrivals
4. Queue discipline refers to the assumption that a customer has the patience to remain in a slow moving queue.
a.
True
b.
False
False
1
Queue discipline
5. For all waiting lines, P0 + Pw = 1.
a.
True
b.
False
False
1
Operating characteristics
6. Before waiting lines can be analyzed economically, the arrivals’ cost of waiting must be estimated.
a.
True
b.
False
True
1
Economic analysis
Chapter 11 – Waiting Line Models
7. In a multiple channel system it is more efficient to have a separate waiting line for each channel.
a.
True
b.
False
False
1
8. Little’s flow equations indicate that the relationship of L to Lq is the same as that of W to Wq.
a.
True
b.
False
False
1
9. If some maximum number of customers is allowed in a queuing system at one time, the system has a finite calling
population.
a.
True
b.
False
False
1
10. When blocked customers are cleared, an important decision is how many channels to provide.
a.
True
b.
False
True
1
11. If service time follows an exponential probability distribution, approximately 63% of the service times are less than
the mean service time.
a.
True
b.
False
True
1
12. Queue discipline refers to the manner in which waiting units are arranged for service.
a.
True
b.
False
True
1
13. Waiting line models describe the transient-period operating characteristics of a waiting line.
Chapter 11 – Waiting Line Models
a.
True
b.
False
False
1
14. For a single-channel waiting line, the utilization factor is the probability that an arriving unit must wait for service.
a.
True
b.
False
True
1
15. When a waiting system is in steady-state operation, the number of units in the system is not changing.
a.
True
b.
False
False
1
16. Adding more channels always improves the operating characteristics of the waiting line and reduces the waiting cost.
a.
True
b.
False
True
1
17. In developing the total cost for a waiting line, waiting cost takes into consideration both the time spent waiting in line
and the time spent being served.
a.
True
b.
False
True
1
18. In waiting line systems where the length of the waiting line is limited, the mean number of units entering the system
might be less than the arrival rate.
a.
True
b.
False
True
1
19. A multiple-channel system has more than one waiting line.
a.
True
Chapter 11 – Waiting Line Models
b.
False
False
1
Multiple-channel waiting line models
20. For an M/M/k system, the average number of customers in the system equals the customer arrival rate times the
average time a customer spends waiting in the system.
a.
True
b.
False
True
1
Multiple-channel waiting line model operating characteristics
21. For a single-server queuing system, the average number of customers in the waiting line is one less than the average
number in the system.
a.
True
b.
False
False
1
Single-channel waiting line model operating characteristics
22. In waiting line applications, the exponential probability distribution indicates that approximately 63 percent of the
service times are less than the mean service time.
a.
True
b.
False
True
1
Distribution of service times
23. With no waiting allowed, operating characteristics Lq and Wq are automatically zero regardless of the number of
servers.
a.
True
b.
False
True
1
Multiple-channel model with no waiting line
24. Little’s flow equations apply to any waiting line model.
a.
True
b.
False
True
1
Some general realtionships for waiting line models
Multiple Choice
Chapter 11 – Waiting Line Models
25. Decision makers in queuing situations attempt to balance
a.
operating characteristics against the arrival rate.
b.
service levels against service cost.
c.
the number of units in the system against the time in the system.
d.
the service rate against the arrival rate.
b
1
Introduction
26. Performance measures dealing with the number of units in line and the time spent waiting are called
a.
queuing facts.
b.
performance queues.
c.
system measures.
d.
operating characteristics.
d
1
Introduction
27. If arrivals occur according to the Poisson distribution every 20 minutes, then which is NOT true?
a.
λ = 20 arrivals per hour
b.
λ = 3 arrivals per hour
c.
λ = 1/20 arrivals per minute
d.
λ = 72 arrivals per day
1
Distribution of arrivals
28. The manner in which units receive their service, such as FCFS, is the
a.
queue discipline.
b.
channel.
c.
steady state.
d.
operating characteristic.
a
1
Queue discipline
29. In a waiting line situation, arrivals occur, on average, every 10 minutes, and 10 units can be received every hour. What
are λ and μ?
a.
λ = 10, μ = 10
b.
λ = 6, μ = 6
c.
λ = 6, μ = 10
d.
λ = 10, μ = 6
c
1
Structure of a waiting line system
Chapter 11 – Waiting Line Models
30. Operating characteristics formulas for the single-channel queue do NOT require
a.
λ ≥ μ.
b.
Poisson distribution of arrivals.
c.
an exponential distribution of service times.
d.
an FCFS queue discipline.
1
Operating characteristics
31. In a multiple channel system
a.
each server has its own queue.
b.
each server has the same service rate.
c.
μ > λ
d.
All of the alternatives are correct.
b
1
Multiple channel waiting line
32. Little’s flow equations
a.
require Poisson and exponential assumptions.
b.
are applicable to any waiting line model.
c.
require independent calculation of W, L, Wq, and Lq.
d.
All of the alternatives are correct.
b
1
General relationships for waiting line models
33. The total cost for a waiting line does NOT specifically depend on
a.
the cost of waiting.
b.
the cost of service.
c.
the number of units in the system.
d.
the cost of a lost customer.
d
1
Economic analysis
34. Models with a finite calling population
a.
have an arrival rate independent of the number of units in the system.
b.
have a service rate dependent on the number of units in the system.
c.
use the size of the population as a parameter in the operating characteristics formulas.
d.
All of the alternatives are correct.
1
Chapter 11 – Waiting Line Models
35. Which of the following can NOT be found by the queuing formulas presented in the textbook?
a.
the probability that no units are in the system.
b.
the average number of units in the system.
c.
the maximum time a unit spends in the system.
d.
the average time a unit spends in the system.
c
1
Operating characteristics
36. The arrival rate in queuing formulas is expressed as
a.
the mean time between arrivals.
b.
the minimum number of arrivals per time period.
c.
the mean number of arrivals per channel.
d.
the mean number of arrivals per time period.
d
1
Distribution of arrivals
37. What queue discipline is assumed by the waiting line models presented in the textbook?
a.
first-come first-served.
b.
last-in first-out.
c.
shortest processing time first.
d.
No discipline is assumed.
a
1
Queue discipline
38. For many waiting line situations, the arrivals occur randomly and independently of other arrivals and it has been found
that a good description of the arrival pattern is provided by
a.
a normal probability distribution.
b.
an exponential probability distribution.
c.
a uniform probability distribution.
d.
a Poisson probability distribution.
d
1
Distribution of arrivals
39. The assumption of exponentially distributed service times indicates that
a.
37% of the service times are less than the mean service time.
b.
50% of the service times are less than the mean service time.
c.
63% of the service times are less than the mean service time.
d.
service time increase at an exponential rate as the waiting line grows.
c
Finite calling population
Chapter 11 – Waiting Line Models
40. Single-booth ticket sales at a theater would be an example of which queuing model?
a.
single-channel, Poisson service rate distribution, unlimited queue length.
b.
single-channel, Poisson service rate distribution, limited queue length.
c.
single-channel, constant service rate distribution, unlimited queue length.
d.
single-channel, normal service rate distribution, unlimited queue length.
41. The machine repair problem is an application of the M/M/1 model with
a.
no waiting line.
b.
arbitrary service times.
c.
a finite calling population.
d.
blocked customers cleared.
1
Finite calling populations
42. The equations provided in the textbook for computing operating characteristics apply to a waiting line operating
a.
at start-up.
b.
at steady-state.
c.
at peak-demand times.
d.
in transition
b
1
Operating characteristics
Subjective Short Answer
43. During summer weekdays, boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of 3 per
hour. In a 2-hour period,
a.
what is the probability that no boats arrive?
b.
what is the probability that 2 boats arrive?
c.
what is the probability that 8 boats arrive?
a.
P(0) = .0025
b.
P(2) = .0446
c.
P(8) = .1033
1
Distribution of arrivals
44. The time to process a registration at the Sea View Resort follows the exponential distribution and has a mean of 6
minutes.
a.
What is the probability of a registration time shorter than 3 minutes?
1
Distribution of service times
Chapter 11 – Waiting Line Models
b.
What is the probability of a registration time shorter than 6 minutes?
c.
What is the probability of a registration time between 3 and 6 minutes?
a.
P(time ≤ 3) = 1 − .6065 = .3935
b.
P(time ≤ 6) = 1 − .3679 = .6321
c.
P(3 ≤ time ≤ 6) = .6321 − .3935 = .2386
1
Distribution of service times
45. The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a movie can be sold
its ticket at the rate of 6 per minute. For the theater’s normal offerings of older movies, customers arrive at the rate of 3 per
minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.
a.
What is the average number of customers waiting in line?
b.
What is the average time a customer spends in the waiting line?
c.
What is the average number of customers in the system?
d.
What is a customer’s average time in the system?
e.
What is the probability that someone will be buying tickets when an arrival occurs?
The Grand has booked the Stars Wars Trilogy and expects more customers. From conversations with other theater owners,
it estimates that the arrival rate will increase to 10 per minute. Output is supplied for a two-cashier and a three-cashier
system.
Number of Channels
2
3
Arrival Rate
10
10
Service Rate
6
6
Probability of No Units in System
.0909
.1727
Average Waiting Time
.3788
.0375
Average Time in System
.5455
.2041
Average Number Waiting
3.7879
.3747
Average Number in System
5.4545
2.0414
Probability of Waiting
.7576
.2998
Probability of 11 in System
.0245
less than .0088
f.
The Grand has space for ten customers to wait indoors to buy tickets. Which system will be
better?
g.
Do you think it is more sensible for them to continue the one-cashier system?
c.
L = 1
d.
W = .3333
The three-cashier system is probably too good and not cost effective.
The one cashier system won’t work because now the arrival rate is faster than the service
rate.
1
Operating characteristics
46. The Arctic Flyers minor league hockey team has one box office clerk. On average, each customer that comes to see a
game can be sold a ticket at the rate of 8 per minute. For normal games, customers arrive at the rate of 5 per minute.
Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.
Chapter 11 – Waiting Line Models
a.
What is the average number of customers waiting in line?
b.
What is the average time a customer spends in the waiting line?
c.
What is the average number of customers in the system?
d.
What is a customer’s average time in the system?
e.
What is the probability that someone will be buying tickets when an arrival occurs?
The Flyers are playing in the league playoffs and anticipate more fans, estimating that the arrival rate will increase to 12
per minute. Output is supplied for a two-cashier and a three-cashier system.
Number of Channels
2
3
Arrival Rate
12
12
Service Rate
8
8
Probability of No Units in System
.1429
.2105
Average Waiting Time
.1607
.0197
Average Time in System
.2857
.1447
Average Number Waiting
1.9286
.2368
Average Number in System
3.4286
1.7368
Probability of Waiting
.6429
.2368
Probability of 7 in System
.0381
.0074
f.
The rink has space for six customers to wait indoors to buy tickets. Which system will be
better?
g.
Do you think it is more sensible for them to continue the one cashier system?
c.
L = 1.665
f.
The three-cashier system is probably too good and not cost effective.
The one cashier system won’t work because now the arrival rate is faster than the service
rate.
1
Operating characteristics
47. In a waiting line situation, arrivals occur at a rate of 2 per minute, and the service times average 18 seconds. Assume
the Poisson and exponential distributions.
a.
What is λ?
b.
What is μ?
c.
Find probability of no units in the system.
d.
Find average number of units in the system.
e.
Find average time in the waiting line.
f.
Find average time in the system.
g.
Find probability that there is one person waiting.
h.
Find probability an arrival will have to wait.
a.
b.
d.
L = 3
Chapter 11 – Waiting Line Models
48. In a waiting line situation, arrivals occur around the clock at a rate of six per day, and the service occurs at one every
three hours. Assume the Poisson and exponential distributions.
a.
What is λ?
b.
What is μ?
c.
Find probability of no units in the system.
d.
Find average number of units in the system.
e.
Find average time in the waiting line.
f.
Find average time in the system.
g.
Find probability that there is one person waiting.
h.
Find probability an arrival will have to wait.
a.
λ = 6
b.
d.
L = 3
f.
W = .5
g.
P(2) = .1406
1
Operating characteristics: M/M/1
49. The Sea View Resort uses a multiple-channel queue registration system. If the average service time is 8 minutes, there
are three registration clerks, and guests arrive at the rate of one every 5 minutes, find
a.
λ and μ.
b.
the probability all three clerks are idle.
c.
the probability a guest will have to wait.
d.
the average time a customer is in line.
e.
the average number of customers in line.
a.
λ = 12 per hour, μ = 7.5 per hour
1
Operating characteristics: M/M/3
50. The post office uses a multiple channel queue, where customers wait in a single line for the first available window. If
the average service time is 1 minute and the arrival rate is 7 customers every five minutes, find, when two service
windows are open,
a.
the probability both windows are idle.
b.
the probability a customer will have to wait.
f.
W = 1.00 min.
g.
P(2) = .140625
1
Operating characteristics: M/M/1
Chapter 11 – Waiting Line Models
c.
the average time a customer is in line.
d.
the average time a customer is in the post office.
d.
W = .3922 (of a five-minute period)
1
Operating characteristics: M/M/2
51. Two new checkout scanning systems are under consideration by a retail store. Arrivals to the checkout stand follow
the Poisson distribution with λ = 2 per minute. The cost for waiting is $18 per hour. The first system has an exponential
service rate of 5 per minute and costs $10 per hour to operate. The second system has an exponential service rate of 8 per
minute and costs $20 per hour to operate. Which system should be chosen?
First system costs .37 per minute, second system costs .43 per minute. Choose first system.
1
Economic analysis
52. Circle Electric Supply is considering opening a second service counter to better serve the electrical contractor
customers. The arrival rate is 10 per hour. The service rate is 14 per hour. If the cost of waiting is $30 and the cost of each
service counter is $22 per hour, then should the second counter be opened?
1
Economic analysis
53. For an M/G/1 system with λ = 6 and μ = 9, with σ = .03, find
a.
the probability the system is idle.
b.
the average length of the queue.
c.
the average number in the system.
c.
L = 1.3819
1
Operating characteristics: M/G/1
54. For an M/G/1 system with λ = 20 and μ = 35, with σ = .005, find
a.
the probability the system is idle.
b.
the average length of the queue.
c.
the average number in the system.
c.
L = .964
1
Operating characteristics: M/G/1
55. Arrivals at a box office in the hour before the show follow the Poisson distribution with λ = 7 per minute. Service
Chapter 11 – Waiting Line Models
times are constant at 7.5 seconds. Find the average length of the waiting line.
1
Operating characteristics: M/D/1
56. The 8 students in a seminar class must come to the professor’s office to turn in a paper and give a 5-minute oral
summary. Assume there is a service rate of 10 per hour and adequate time is available for all. The arrival rate for each unit
is 5 per hour. What is the probability there is no one in the office or waiting when you come?
1
Finite calling population
57. Andy Archer, Ph.D., is a training consultant for six mid-sized manufacturing firms. On the average, each of his six
clients calls him for consulting assistance once every 25 days. Andy typically spends an average of five days at the client’s
firm during each consultation.
Assuming that the time between client calls follows an exponential distribution, determine the following:
a.
the average number of clients Andy has on backlog
b.
the average time a client must wait before Andy arrives to it
c.
the proportion of the time Andy is busy
1
Finite calling population
58. The Quick Snap photo machine at the Lemon County bus station takes four snapshots in exactly 75 seconds.
Customers arrive at the machine according to a Poisson distribution at the mean rate of 20 per hour. On the basis of this
information, determine the following:
a.
the average number of customers waiting to use the photo machine
b.
the average time a customer spends in the system
c.
the probability an arriving customer must wait for service.
b.
W = .028 hour = 1.7 minutes
1
Constant service times
59. Quick Clean Rooter cleans out clogged drains. Due to the competitive nature of the drain cleaning business, if a
customer calls Quick Clean and finds the line busy, they immediately try another company and Quick Clean loses the
business.
Quick Clean management estimates that on the average, a customer tries to call Quick Clean every three minutes and the
average time to take a service order is 200 seconds. The company wishes to hire enough operators so that at most 4% of
its potential customers get the busy signal.
a.
How many operators should be hired to meet this objective?
Chapter 11 – Waiting Line Models
b.
Given your answer to a), what is the probability that all the operators are idle?
1
M/G/k model with blocked customers cleared
60. A company has tool cribs where workmen draw parts. Two men have applied for the position of distributing parts to
the workmen. George Fuller is fresh out of trade school and expects a $6 per hour salary. His average service time is 4
minutes. John Cox is a veteran who expects $12 per hour. His average service time is 2 minutes. A workman’s time is
figured at $10 per hour. Workmen arrive to draw parts at an average rate of 12 per hour.
a.
What is the average waiting time a workman would spend in the system under each
applicant?
b.
Which applicant should be hired?
George: W = 20 minutes; John: W = 3 1/3 minutes
b.
Total Cost: George = $46.00, John = $18.67. Hire John
1
Economic analysis of waiting lines
61. The insurance department at Shear’s has two agents, each working at a mean speed of 8 customers per hour.
Customers arrive at the insurance desk at a mean rate of one every six minutes and form a single queue. Management
feels that some customers are going to find the wait at the desk too long and take their business to Word’s, Shear’s
competitor.
In order to reduce the time required by an agent to serve a customer Shear’s is contemplating installing one of two
minicomputer systems: System A which leases for $18 per day and will increase an agent’s efficiency by 25%; or, System
B which leases for $23 per day and will increase an agent’s efficiency by 50%. Agents work 8-hour days.
If Shear’s estimates its cost of having a customer in the system at $3 per hour, determine if Shear’s should install a new
minicomputer system, and if so, which one.
1
Economic analysis of waiting lines
62. The postmaster at the Oak Hill Post Office expects the mean arrival rate of people to her customer counter will soon
increase by fifty percent due to a large apartment complex being built. Currently, the mean arrival rate is 15 people per
hour. The postmaster can serve an average of 25 people per hour. By what percentage must the postmaster’s mean service
rate increase when the apartment complex is completed in order that the average time spent at the post office remains at its
current value?
30 percent
1
M/M/1 waiting line model
Essay
63. Discuss the importance of the utilization factor in a queuing system and the assumptions made about its value.
Answer not provided.
1
64. How can a system be changed to improve the service rate?
65. Diagram the servers and arrivals in the single and multiple channel models. Designate the line and the system.
66. Explain what is meant by the following statement, “operating characteristics are non-optimizing.”
67. Give examples of systems you have seen in which a) blocked arrivals are cleared, and b) there is a finite calling
population.
68. List six steady-state operating characteristics for a single-channel waiting line with Poisson arrivals and exponential
service times.