Example 18 T= [ ²1₁ IfS=TS,, then S, equals A D 0.8 0.3 0.2 07 [19 1000 1000 is a transition matrix and Sy= 5-1 1150 850 [₁ :] B 1090 940 E [ {] 1175 825 1150 850 is a state matrix. с 1100 900
Example 18 T= [ ²1₁ IfS=TS,, then S, equals A D 0.8 0.3 0.2 07 [19 1000 1000 is a transition matrix and Sy= 5-1 1150 850 [₁ :] B 1090 940 E [ {] 1175 825 1150 850 is a state matrix. с 1100 900
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Transition matrices and Recurrence Matrices
Using recursion to generate state matrices
In financial modelling, we were able to model growth in investments year by year. We can use the same concept to
solve transition problems.
Construct a matrix recurrence relation
A recurrence relation must have a starting point-initial state matrix, So
Example 16
In a country town, people only have the choice of doing their food shopping at a store called Marks
(M) or at a newly opened store called Foodies (F). In the first week that Foodies opened, only 300 of
the town's 800 shoppers did their food shopping at Marks. The remainder did their food shopping at
Foodies.
A state matrix S, that can be used to represent this situation is
]
]
300 M
500 F
A S
DS-
300
800
M
F
BS=
E S
500 M
300 F
800 M
500 F
C S
Generating the next state Sn, S1, S2, etc
To generate the next state matrix, S₁ in the sequence you use: S₁ = T So
To generate S₂, follow the same pattern: S₂ = TS₁
To generate S3, follow the same pattern: S3 = T 2₁ and so on.
Recurrence relation
So initial value, Sn+1 = TSn
Where T is the transition matrix and Sn is the state matrix
800 M
300
[VCAA 2009 1MQ7]
15 | Page
Example 17
A number of train carriages start each day at one
of two depots, A or B. By the end of the day, the
train carriages end up at either of the two depots
according to this transition matrix.
At the start of today, there are 81 train carriages
at depot A and 49 train carriages at depot B.
How many train carriages are at each depot after 3 days?
Example 18
T=
0.8 0.3
0.2 0.7
D
If S=TS, then S, equals
^ [1 3]
A 1000
1000
[
:]
is a transition matrix and S, 2
1150
850
B
E
1090
940
1175
825
1150
850
T=
Start of this day
A B
с
0.8 0.15
0.2 0.85
is a state matrix.
1100
900
]
A Start of next day
B
16 | Page](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2ad6f6b-c489-48ba-b53b-a9acb582a8af%2F30ed2c8e-f1a1-4337-a57f-27409b2e92ef%2F16c226o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Transition matrices and Recurrence Matrices
Using recursion to generate state matrices
In financial modelling, we were able to model growth in investments year by year. We can use the same concept to
solve transition problems.
Construct a matrix recurrence relation
A recurrence relation must have a starting point-initial state matrix, So
Example 16
In a country town, people only have the choice of doing their food shopping at a store called Marks
(M) or at a newly opened store called Foodies (F). In the first week that Foodies opened, only 300 of
the town's 800 shoppers did their food shopping at Marks. The remainder did their food shopping at
Foodies.
A state matrix S, that can be used to represent this situation is
]
]
300 M
500 F
A S
DS-
300
800
M
F
BS=
E S
500 M
300 F
800 M
500 F
C S
Generating the next state Sn, S1, S2, etc
To generate the next state matrix, S₁ in the sequence you use: S₁ = T So
To generate S₂, follow the same pattern: S₂ = TS₁
To generate S3, follow the same pattern: S3 = T 2₁ and so on.
Recurrence relation
So initial value, Sn+1 = TSn
Where T is the transition matrix and Sn is the state matrix
800 M
300
[VCAA 2009 1MQ7]
15 | Page
Example 17
A number of train carriages start each day at one
of two depots, A or B. By the end of the day, the
train carriages end up at either of the two depots
according to this transition matrix.
At the start of today, there are 81 train carriages
at depot A and 49 train carriages at depot B.
How many train carriages are at each depot after 3 days?
Example 18
T=
0.8 0.3
0.2 0.7
D
If S=TS, then S, equals
^ [1 3]
A 1000
1000
[
:]
is a transition matrix and S, 2
1150
850
B
E
1090
940
1175
825
1150
850
T=
Start of this day
A B
с
0.8 0.15
0.2 0.85
is a state matrix.
1100
900
]
A Start of next day
B
16 | Page
![Example 19
A fleet of trucks starts each day at one of two depots, A or B. By the end of the day, the trucks end
up at either of the two depots according to the following transition matrix.
T-[a
T=
Start of this day
A B
T-[
0.95 0.15
0.05 0.85
T=
At the start of a particular day, there are 40 trucks at depot A and 100 trucks at depot B.
a How many trucks are at each depot after 6 days?
b How many trucks are at each depot after 10 days?
Long term trends
It is often possible to make statement about long term trends directly form transition matrices.
This day
A B
0.6 0.7
3]
0.4 0.3
:
]
911 912
Start of next day
For a 2 x 2 transition matrix T where
Current state
A B
921 922
Next day
B
The percentage of those moving form B to A (70%) is greater than the percentage
of those moving from A to B (40 %)
This means in the long term there will be more moving to A than to B.
Next state
.
• if an>a, in the long term more will
transition to B than to A.
. if a> a, in the long term more will
transition to A than to B.
19 | Page
The steady-state or equilibrium-state solution
Although transition matrices involve regular change and movement, often in the long term the state matrix gets to a
point where it isn't changing from one transition to the next.
By taking higher and higher powers of the transition matrix, T there will be a point where there is no longer
noticeable change from one state matrix to the next. This is called the equilibrium state matrix or steady-state
matrix.
The steady state matrix, S, represents the equilibrium state of a system. In other words, the number in each category
of the state matrix will converge on a particular value and will stabilise.
Estimating the steady state solution
If So is the intital state matrix, then the steady-state matrix, S, is given by
S=T" So
as n tends to infinity ().
To find the steady state solution:
Make n a large number, i.e. n = 50 (But test the next n value just to make sure the matrix is unchanged).
To find how long it takes to reach steady state, correct to 2 dp,:
then keep increasing the value of n until the matrix remains unchanged for consecutive matrices.
Note: If a transition matrix contains a zero element, it will not be able to reach a steady state
Example 20
A coffee shop sells three types of coffee, Brazilian (B), Italian (I) and Kenyan (K). The regular
customers buy one cup of coffee each per day and choose the type of coffee they buy according
to the following transition matrix, T.
Choose today
BIK
0.8 0.1 0.1 B
]
0.2 0.1 0.8 K
T= 0 0.8 0.1
I
D Brazilian 89
Italian
Kenyan
89
83
Choose tomorrow
88
On a particular day, 84 customers bought Brazilian coffee, 96 bought Italian coffee and 81 bought
Kenyan coffee. If these same customers continue to buy one cup of coffee each per day, the number
of these customers who are expected to buy each of the three types of coffee in the long term is
A Brazilian 85
B Brazilian.
87
Italian
58
Kenyan 116
E Brazilian 116
Italian
Kenyan
85
Italian
Kenyan
C Brazilian
Italian
Kenyan
86
91
87
89
58
IN
20 | Pag](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2ad6f6b-c489-48ba-b53b-a9acb582a8af%2F30ed2c8e-f1a1-4337-a57f-27409b2e92ef%2Fc2tl2qv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example 19
A fleet of trucks starts each day at one of two depots, A or B. By the end of the day, the trucks end
up at either of the two depots according to the following transition matrix.
T-[a
T=
Start of this day
A B
T-[
0.95 0.15
0.05 0.85
T=
At the start of a particular day, there are 40 trucks at depot A and 100 trucks at depot B.
a How many trucks are at each depot after 6 days?
b How many trucks are at each depot after 10 days?
Long term trends
It is often possible to make statement about long term trends directly form transition matrices.
This day
A B
0.6 0.7
3]
0.4 0.3
:
]
911 912
Start of next day
For a 2 x 2 transition matrix T where
Current state
A B
921 922
Next day
B
The percentage of those moving form B to A (70%) is greater than the percentage
of those moving from A to B (40 %)
This means in the long term there will be more moving to A than to B.
Next state
.
• if an>a, in the long term more will
transition to B than to A.
. if a> a, in the long term more will
transition to A than to B.
19 | Page
The steady-state or equilibrium-state solution
Although transition matrices involve regular change and movement, often in the long term the state matrix gets to a
point where it isn't changing from one transition to the next.
By taking higher and higher powers of the transition matrix, T there will be a point where there is no longer
noticeable change from one state matrix to the next. This is called the equilibrium state matrix or steady-state
matrix.
The steady state matrix, S, represents the equilibrium state of a system. In other words, the number in each category
of the state matrix will converge on a particular value and will stabilise.
Estimating the steady state solution
If So is the intital state matrix, then the steady-state matrix, S, is given by
S=T" So
as n tends to infinity ().
To find the steady state solution:
Make n a large number, i.e. n = 50 (But test the next n value just to make sure the matrix is unchanged).
To find how long it takes to reach steady state, correct to 2 dp,:
then keep increasing the value of n until the matrix remains unchanged for consecutive matrices.
Note: If a transition matrix contains a zero element, it will not be able to reach a steady state
Example 20
A coffee shop sells three types of coffee, Brazilian (B), Italian (I) and Kenyan (K). The regular
customers buy one cup of coffee each per day and choose the type of coffee they buy according
to the following transition matrix, T.
Choose today
BIK
0.8 0.1 0.1 B
]
0.2 0.1 0.8 K
T= 0 0.8 0.1
I
D Brazilian 89
Italian
Kenyan
89
83
Choose tomorrow
88
On a particular day, 84 customers bought Brazilian coffee, 96 bought Italian coffee and 81 bought
Kenyan coffee. If these same customers continue to buy one cup of coffee each per day, the number
of these customers who are expected to buy each of the three types of coffee in the long term is
A Brazilian 85
B Brazilian.
87
Italian
58
Kenyan 116
E Brazilian 116
Italian
Kenyan
85
Italian
Kenyan
C Brazilian
Italian
Kenyan
86
91
87
89
58
IN
20 | Pag
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