Week 8
Impaler_2019
Stevenson_CH18_Accessible.pptxChapter 18
Waiting Lines
© McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
1
Learning Objectives
You should be able to:
18.1 What imbalance does the existence of a waiting line reveal?
18.2 What causes waiting lines to form, and why is it impossible to eliminate them completely?
18.3 What metrics are used to help managers analyze waiting lines?
18.4 What very important lesson does the constant service time model provide for managers?
18.5 What are some psychological approaches to managing lines, and why might a manager want to use them?
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.1
Waiting Lines
Waiting lines occur in all sorts of service systems
Wait time is non-value added
Wait time ranges from the acceptable to the emergent
Short waits in a drive-thru
Sitting in an airport waiting for a delayed flight
Waiting for emergency service personnel
Waiting time costs
Lower productivity
Reduced competitiveness
Wasted resources
Diminished quality of life
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.1
Queuing Theory
Queuing theory
Mathematical approach to the analysis of waiting lines
Applicable to many environments
Call centers
Banks
Post offices
Restaurants
Theme parks
Telecommunications systems
Traffic management
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.2
Why Is There Waiting?
Waiting lines tend to form even when a system is not fully loaded
Variability
Arrival and service rates are variable
Services cannot be completed ahead of time and stored for later use
18-‹#›
© McGraw-Hill Education.
Waiting Lines: Managerial Implications
Why waiting lines cause concern:
The cost to provide waiting space
A possible loss of business when customers leave the line before being served or refuse to wait at all
A possible loss of goodwill
A possible reduction in customer satisfaction
Resulting congestion may disrupt other business operations and/or customers
18-‹#›
© McGraw-Hill Education.
Waiting Line Management
The goal of waiting line management is to minimize total costs:
Costs associated with customers waiting for service
Capacity cost
18-‹#›
© McGraw-Hill Education.
Waiting Line Characteristics
The basic characteristics of waiting lines
Population source
Number of servers (channels)
Arrival and service patterns
Queue discipline
18-‹#›
© McGraw-Hill Education.
Simple Queuing System
18-‹#›
© McGraw-Hill Education.
Population Source (1 of 2)
Infinite source
Customer arrivals are unrestricted
The number of potential customers greatly exceeds system capacity
18-‹#›
© McGraw-Hill Education.
Population Source (2 of 2)
Finite source
The number of potential customers is limited
18-‹#›
© McGraw-Hill Education.
Channels and Phases
Channel
A server in a service system
It is assumed that each channel can handle one customer at a time
Phases
The number of steps in a queuing system
18-‹#›
© McGraw-Hill Education.
Common Queuing Systems
18-‹#›
© McGraw-Hill Education.
Arrival and Service Patterns
Arrival pattern
Most commonly used models assume the arrival rate can be described by the Poisson distribution
Arrivals per unit of time
Equivalently, interarrival times are assumed to follow the negative exponential distribution
The time between arrivals
Service pattern
Service times are frequently assumed to follow a negative exponential distribution
18-‹#›
© McGraw-Hill Education.
Poisson and Negative Exponential
18-‹#›
© McGraw-Hill Education.
Queue Discipline
Queue discipline
The order in which customers are processed
Most commonly encountered rule is that service is provided on a first-come, first-served (FCFS) basis
Non FCFS applications do not treat all customer waiting costs as the same
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.3
Waiting Line Metrics
Managers typically consider five measures when evaluating waiting line performance:
The average number of customers waiting (in line or in the system)
The average time customers wait (in line or in the system)
System utilization
The implied cost of a given level of capacity and its related waiting line
The probability that an arrival will have to wait for service
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.3
Waiting Line Performance
The average number waiting in line and the average time customers wait in line increase exponentially as the system utilization increases
18-‹#›
© McGraw-Hill Education.
Queuing Models: Infinite Source
Four basic infinite source models
All assume a Poisson arrival rate
Single server, exponential service time
Single server, constant service time
Multiple servers, exponential service time
Multiple priority service, exponential service time
18-‹#›
© McGraw-Hill Education.
Infinite-Source Symbols
18-‹#›
© McGraw-Hill Education.
Basic Relationships (1 of 3)
System Utilization
Average number of customers being served
18-‹#›
© McGraw-Hill Education.
Basic Relationships (2 of 3)
Little’s Law
For a stable system the average number of customers in line or in the system is equal to the average customer arrival rate multiplied by the average time in the line or system
18-‹#›
© McGraw-Hill Education.
Basic Relationships (3 of 3)
The average number of customers
Waiting in line for service:
In the system:
The average time customers are
Waiting in line for service:
In the system
18-‹#›
© McGraw-Hill Education.
Single Server, Exponential Service Time
M/M/1
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.4
Single Server, Constant Service Time
M/D/1
If a system can reduce variability, it can shorten waiting lines noticeably
For, example, by making service time constant, the average number of customers waiting in line can be cut in half
Average time customers spend waiting in line is also cut by half.
Similar improvements can be made by smoothing arrival rates (such as by use of appointments)
18-‹#›
© McGraw-Hill Education.
Multiple Servers (M/M/S)
Assumptions:
A Poisson arrival rate and exponential service time
Servers all work at the same average rate
Customers form a single waiting line (in order to maintain FCFS processing)
18-‹#›
© McGraw-Hill Education.
M/M/S
18-‹#›
© McGraw-Hill Education.
Cost Analysis
Service system design reflects the desire of management to balance the cost of capacity with the expected cost of customers waiting in the system
Optimal capacity is one that minimizes the sum of customer waiting costs and capacity or server costs
18-‹#›
© McGraw-Hill Education.
Total Cost Curve
18-‹#›
© McGraw-Hill Education.
Maximum Line Length
An issue that often arises in service system design is how much space should be allocated for waiting lines
The approximate line length, Lmax, that will not be exceeded a specified percentage of the time can be determined using the following:
18-‹#›
© McGraw-Hill Education.
Multiple Priorities
Multiple priority model
Customers are processed according to some measure of importance
Customers are assigned to one of several priority classes according to some predetermined assignment method
Customers are then processed by class, highest class first
Within a class, customers are processed by FCFS
Exceptions occur only if a higher-priority customer arrives
That customer will be processed after the customer currently being processed
18-‹#›
© McGraw-Hill Education.
Multiple–Server Priority Model (1 of 3)
Performance Measure: System Utilization
Formula:
Formula Number: (18-15)
Performance Measure: Intermediate values(Lq from Table 18.4)
18-‹#›
© McGraw-Hill Education.
Multiple–Server Priority Model (2 of 3)
Performance Measure: Average waiting time in line for units in kth priority class
Formula:
Formula Number: (18-18)
Performance Measure: Average time in the system for units in the Kth priority class
Formula:
Formula Number: (18-19)
18-‹#›
© McGraw-Hill Education.
Multiple–Server Priority Model (3 of 3)
Performance Measure: Average number waiting in line for units in the Kth priority class
Formula:
Formula Number: (18-20)
18-‹#›
© McGraw-Hill Education.
Finite-Source Model (1 of 5)
Appropriate for cases in which the calling population is limited to a relatively small number of potential calls
Arrival rates are required to be Poisson
Unlike the infinite-source models, the arrival rate is affected by the length of the waiting line
The arrival rate of customers decreases as the length of the line increases because there is a decreasing proportion of the population left to generate calls for service
Service times are required to be exponential
18-‹#›
© McGraw-Hill Education.
Finite-Source Model (2 of 5)
Procedure:
Identify the values for
N, population size
M, the number of servers/channels
T, average service time
U, average time between calls for service
Compute the service factor, X=T/(T + U)
Locate the section of the finite-queuing tables for N
Using the value of X as the point of entry, find the values of D and F that correspond to M
Use the values of N, M, X, D, and F as needed to determine the values of the desired measures of system performance
18-‹#›
© McGraw-Hill Education.
Finite-Source Model (3 of 5)
Table 18.6 Finite-source queuing formulas and notation
| Performance Measure | Formulas | Notation† | |
| Service factor | (18-21) | D = Probability that a customer will have to wait in line | |
| Average number waiting | (18-22) | F = Efficiency factor 1 – Percentage waiting in line | |
| Average waiting time | (18-23) | H = Average number of customers being served | |
| Average number running | (18-24) | J = Average number of customers not in line or in service |
18-‹#›
© McGraw-Hill Education.
Finite-Source Model (4 of 5)
| Performance Measure | Formulas | Notation† | |
| Average number being served | H=FNX | (18-25) | L = Average number of customers waiting for service |
| Number in population | N=J+L+H | (18-26) | M = Number of service channels N = Number of potential customers T = Average service time U = Average time between customer service requirements per customer W = Average time customers wait in line X = Service factor |
18-‹#›
© McGraw-Hill Education.
Finite-Source Model (5 of 5)
18-‹#›
© McGraw-Hill Education.
Constraint Management
Managers may be able to reduce waiting lines by actively managing one or more system constraints:
Fixed short-term constraints
Facility size
Number of servers
Short-term capacity options
Use temporary workers
Shift demand
Standardize the service
Look for a bottleneck
18-‹#›
© McGraw-Hill Education.
Learning Objective 18.5
Psychology of Waiting
If those waiting in line have nothing else to occupy their thoughts, they often tend to focus on the fact they are waiting in line
They will usually perceive the waiting time to be longer than the actual waiting time
Steps can be taken to make waiting more acceptable to customers
Occupy them while they wait
In-flight snack
Have them fill out forms while they wait
Make the waiting environment more comfortable
Provide customers information concerning their wait
18-‹#›
© McGraw-Hill Education.
Operations Strategy
Managers must carefully weigh the costs and benefits of service system capacity alternatives
Options for reducing wait times:
Work to increase processing rates, instead of increasing the number of servers
Use new processing equipment and/or methods
Reduce processing time variability through standardization
Shift demand
18-‹#›
© McGraw-Hill Education.
End of Presentation
© McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
18-‹#›
line
in
waiting
number
expected
maximum
The
(channels)
servers
of
number
The
system
the
in
units
of
y
probabilit
The
system
the
in
units
zero
of
y
probabilit
The
time
Service
system
the
in
spend
customers
time
average
The
line
in
wait
customers
time
average
The
n
utilizatio
system
The
served
being
customers
of
number
average
The
system
the
in
customer
of
number
average
The
service
for
waiting
customers
of
number
average
The
server
per
rate
Service
rate
arrival
Customer
=
=
=
=
=
=
=
=
=
=
=
=
=
max
n
0
s
q
s
q
L
M
n
P
P
μ
1
W
W
ρ
r
L
L
μ
λ
M
μ
λ
ρ
=
μ
λ
r
=
q
λW
q
L
s
λW
s
L
=
=
]
dependent.
[Model
q
L
r
L
L
q
s
+
=
λ
L
W
q
q
=
λ
L
μ
W
W
s
q
s
=
+
=
1
(
)
n
n
n
0
n
0
2
q
μ
λ
P
μ
λ
P
P
μ
λ
P
λ
μ
μ
λ
L
÷
÷
ø
ö
ç
ç
è
æ
-
=
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
-
=
-
=
á
1
1
(
)
λ
μ
μ
λ
L
2
q
-
=
2
(
)
(
)
s
q
w
s
M
n
M
n
0
0
M
q
W
W
P
λ
M
μ
W
M
ν
λ
M
μ
λ
n
μ
λ
P
P
λ
M
μ
M
μ
λ
λμ
L
=
-
=
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ë
é
÷
ø
ö
ç
è
æ
-
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
-
-
÷
÷
ø
ö
ç
ç
è
æ
=
-
-
=
å
1
1
!
!
!
1
1
1
0
2
(
)
ρ
L
K
ρ
K
ρ
K
L
q
max
-
-
=
=
1
percentage
specified
1
where
ln
ln
or
log
log
μ
λ
ρ
M
=
(
)
1
17)
-
(18
μ
λ
1
16)
-
(18
ρ
1
λ
1
c
=
-
=
-
=
å
=
0
k
c
k
q
B
M
B
L
A
Number
Formula
Formula
k
k
k
B
B
A
W
×
×
=
-
1
1
μ
1
+
=
k
W
W
k
k
k
W
λ
L
´
=
U
T
T
X
+
=
(
)
F
N
L
-
=
1
(
)
(
)
XF
F
T
L
N
U
T
L
W
-
=
-
+
=
1
(
)
X
NF
J
-
=
1
H
L
J
H
J
F
T
H
W
L
U
J
+
+
·
+
=
time
Average
|
|
:|
number
Average
Cycle
served
Being
Waiting
served
being
for
waiting
Not
4
4
4
4
4
8
4
4
4
4
4
7
6
