D
Waiting-Line Models
PowerPoint presentation to accompany
Heizer and Render
Operations Management, 10e
Principles of Operations Management, 8e
PowerPoint slides by Jeff Heyl
D-1
Outline
 Queuing Theory
 Characteristics of a Waiting-Line System
 Arrival Characteristics
 Waiting-Line Characteristics
 Service Characteristics
 Measuring a Queue’s Performance
 Queuing Costs
 Queuing Models
 Model A(M/M/1): Single-Channel Queuing Model
with Poisson Arrivals and Exponential Service
Times
 Little’s Law
2
Learning Objectives
1. Describe the characteristics of arrivals,
waiting lines, and service systems
2. Apply the single-channel queuing model
equations
3. Conduct a cost analysis for a waiting
line
3
Queuing Theory
 The study of waiting lines
 Waiting lines are common
situations
 Useful in both
manufacturing
and service
areas
4
Common Queuing
Situations
Situation
Supermarket
Arrivals in Queue
Grocery shoppers
Service Process
Checkout clerks at cash
register
Collection of tolls at booth
Highway toll booth
Automobiles
Doctor’s office
Patients
Computer system
Programs to be run
Telephone company
Callers
Bank
Customer
Switching equipment to
forward calls
Transactions handled by teller
Machine
maintenance
Harbor
Broken machines
Repair people fix machines
Ships and barges
Dock workers load and unload
Treatment by doctors and
nurses
Computer processes jobs
Table D.1
5
Characteristics of WaitingLine Systems
1. Arrivals or inputs to the system
 Population size, behavior, statistical
distribution
2. Queue discipline, or the waiting line itself
 Limited or unlimited in length, discipline of
people or items in it
3. The service facility
 Design, statistical distribution of service
times
6
Parts of a Waiting Line
Population of
dirty cars
Arrivals
from the
general
population …
Queue
(waiting line)
Service
facility
Dave’s
Car Wash
Enter
Arrivals to the system
Arrival Characteristics
 Size of the population
 Behavior of arrivals
 Statistical distribution
of arrivals
Exit the system
In the system
Waiting Line
Characteristics
 Limited vs.
unlimited
 Queue discipline
Exit
Exit the system
Service Characteristics
 Service design
 Statistical distribution
of service
Figure D.1
7
Arrival Characteristics
1. Size of the population
 Unlimited (infinite) or limited (finite)
2. Pattern of arrivals
 Scheduled or random, often a Poisson
distribution
3. Behavior of arrivals
 Wait in the queue and do not switch lines
 No balking or reneging
8
Waiting-Line Characteristics
 Limited or unlimited queue length
 Queue discipline - first-in, first-out
(FIFO) is most common
 Other priority rules may be used in
special circumstances
9
Service Characteristics
 Queuing system designs
 Single-channel system, multiplechannel system
 Single-phase system, multiphase
system
 Service time distribution
 Constant service time
 Random service times, usually a
negative exponential distribution
10
Queuing System Designs
A family dentist’s office
Queue
Service
facility
Arrivals
Departures
after service
Single-channel, single-phase system
A McDonald’s dual window drive-through
Queue
Arrivals
Phase 1
service
facility
Phase 2
service
facility
Departures
after service
Single-channel, multiphase system
Figure D.3
11
Queuing System Designs
Most bank and post office service windows
Service
facility
Channel 1
Queue
Arrivals
Service
facility
Channel 2
Departures
after service
Service
facility
Channel 3
Multi-channel, single-phase system
Figure D.3
12
Queuing System Designs
Some college registrations
Queue
Phase 1
service
facility
Channel 1
Phase 2
service
facility
Channel 1
Phase 1
service
facility
Channel 2
Phase 2
service
facility
Channel 2
Arrivals
Departures
after service
Multi-channel, multiphase system
Figure D.3
13
Measuring Queue
Performance
1. Average time that each customer or object
spends in the queue
2. Average queue length
3. Average time each customer spends in the
system
4. Average number of customers in the system
5. Probability that the service facility will be idle
6. Utilization factor for the system
7. Probability of a specific number of customers
in the system
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Queuing Costs
Cost
Minimum
Total
cost
Total expected cost
Cost of providing service
Cost of waiting time
Low level
of service
Optimal
service level
High level
of service
Figure D.5
15
Queuing Models
Model
Name
Example
A
Single-channel
system
(M/M/1)
Information counter
at department store
Number Number
of
of
Channels Phases
Arrival
Rate
Pattern
Service
Time
Pattern
Single
Poisson
Exponential
Single
Population Queue
Size
Discipline
Unlimited
FIFO
Table D.2
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Model A – Single-Channel
1. Arrivals are served on a FIFO basis and
every arrival waits to be served
2. Arrivals are independent of preceding
arrivals
3. Arrivals are random and come from an
infinite population
4. Service times are variable
5. The service rate is faster than the
arrival rate
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Model A – Single-Channel
 = Mean number of arrivals per time period
µ = Mean number of units served per time period
Ls = Average number of units (customers) in the
system (waiting and being served)

=
µ–
Ws = Average time a unit spends in the system
(waiting time plus service time)
1
=
µ–
Table D.3
18
Model A – Single-Channel
Lq = Average number of units waiting in the
queue
2

=
µ(µ – )
Wq = Average time a unit spends waiting in the
queue

=
µ(µ – )
 = Utilization factor for the system

=
µ
Table D.3
19
Model A – Single-Channel
P0 = Probability of 0 units in the system (that is,
the service unit is idle)

µ
= Probability of more than k units in the
system, where n is the number of units in
the system
= 1–
Pn > k
=

µ
k+1
Table D.3
20
Single-Channel Example
 = 2 cars arriving/hour
µ = 3 cars serviced/hour
2

Ls =
=
= 2 cars in the system on average
3
2
µ–
1
1
Ws =
=
= 1 hour average waiting time in
µ–
3-2
the system
22
2
Lq =
=
= 1.33 cars waiting in line
3(3
2)
µ(µ – )
21
Single-Channel Example
 = 2 cars arriving/hour
Wq
µ = 3 cars serviced/hour
2

=
=
= 2/3 hour = 40 minute
3(3
2)
µ(µ – )
average waiting time
 = /µ = 2/3 = 66.6% of time mechanic is busy

P0 = 1 = .33 probability there are 0 cars in the
µ
system
22
Single-Channel Example
Probability of more than k Cars in the System
k
0
1
2
3
4
5
6
7
Pn > k = (2/3)k + 1
.667  Note that this is equal to 1 - P0 = 1 - .33
.444
.296
.198  Implies that there is a 19.8% chance that
more than 3 cars are in the system
.132
.088
.058
.039
23
Single-Channel Economics
Customer dissatisfaction
and lost goodwill
Wq
Total arrivals
Mechanic’s salary
Total hours
customers spend
waiting per day
=
= $10 per hour
= 2/3 hour
= 16 per day
= $56 per day
2
2
(16) = 10
hours
3
3
Customer waiting-time cost = $10 10
2
3
= $106.67
Total expected costs = $106.67 + $56 = $162.67
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In-Class Problems from the
Lecture Guide Practice Problems
Problem 1:
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 a rate of 20 per hour. It takes
an average of 2 minutes to answer a question. It is assumed that the arrivals
follow a Poisson distribution and answer times are exponentially distributed.
(a) Find the probability that the employee is idle.
(b) Find the proportion of the time that the employee is busy.
(c) Find the average number of people receiving and waiting to receive some information.
(d) Find the average number of people waiting in line to get some information.
(e) Find the average time a person seeking information spends in the system.
(f) Find the expected time a person spends just waiting in line to have a question
answered (time in the queue).
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In-Class Problems from the
Lecture Guide Practice Problems
Problem 2:
Assume that the information desk employee in Problem 1 earns $10 per
hour. The cost of waiting time, in terms of customer unhappiness with
the mall, is $12 per hour of time spent waiting in line. Find the total
expected costs over an 8-hour day.
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