Post on 17-May-2018
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Matrix-Geometric Analysis and ItsApplications
John C.S. Lui
Department of Computer Science & EngineeeringThe Chinese University of Hong Kong
Email: cslui@cse.cuhk.edu.hk
Summer Course at Tsinghua, 2005.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
IntroductionWhy we need the Matrix-Geometric Technique?
Matrix-Geometric in ActionKey Idea
General Matrix-Geometric SolutionGeneral Concept
Application of Matrix-GeometricPerformance Analysis of Multiprocessing System
Properties of SolutionsProperties
Computational Properties of RAlgorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Why we need the Matrix-Geometric Technique?
OutlineIntroduction
Why we need the Matrix-Geometric Technique?Matrix-Geometric in Action
Key IdeaGeneral Matrix-Geometric Solution
General ConceptApplication of Matrix-Geometric
Performance Analysis of Multiprocessing SystemProperties of Solutions
PropertiesComputational Properties of R
Algorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Why we need the Matrix-Geometric Technique?
Motivation
� Closed-form solution is hard to obtain.� Need to seek efficient, numerical stable solutions.� Can be viewed as a generalization of conventionalqueueing analysis.� A Special way to solve a Markov Chain.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Why we need the Matrix-Geometric Technique?
Motivation
� Closed-form solution is hard to obtain.� Need to seek efficient, numerical stable solutions.� Can be viewed as a generalization of conventionalqueueing analysis.� A Special way to solve a Markov Chain.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Why we need the Matrix-Geometric Technique?
Motivation
� Closed-form solution is hard to obtain.� Need to seek efficient, numerical stable solutions.� Can be viewed as a generalization of conventionalqueueing analysis.� A Special way to solve a Markov Chain.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Why we need the Matrix-Geometric Technique?
Motivation
� Closed-form solution is hard to obtain.� Need to seek efficient, numerical stable solutions.� Can be viewed as a generalization of conventionalqueueing analysis.� A Special way to solve a Markov Chain.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
OutlineIntroduction
Why we need the Matrix-Geometric Technique?Matrix-Geometric in Action
Key IdeaGeneral Matrix-Geometric Solution
General ConceptApplication of Matrix-Geometric
Performance Analysis of Multiprocessing SystemProperties of Solutions
PropertiesComputational Properties of R
Algorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Key Ideas� It is a technique to solve stationary state probability forvector state Markov processes. Two parts:
1. Boundary set2. Repetitive set� Example: a modified M
�M�1 ����� if the system is empty,
else � . Customers require two exponential stages ofservice, � 1 � and � 2
S ��� i � s �� i � 0 and it is the no. of customer in the queue, �s is the current stage of service, s ��� 1 � 2 ��� s � 0 if no customer in the system� Well, let us proceed to specify the state transition diagram,
then the Q matrix.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Key Ideas� It is a technique to solve stationary state probability forvector state Markov processes. Two parts:
1. Boundary set2. Repetitive set� Example: a modified M
�M�1 ����� if the system is empty,
else � . Customers require two exponential stages ofservice, � 1 � and � 2
S ��� i � s �� i � 0 and it is the no. of customer in the queue, �s is the current stage of service, s ��� 1 � 2 ��� s � 0 if no customer in the system� Well, let us proceed to specify the state transition diagram,
then the Q matrix.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Key Ideas� It is a technique to solve stationary state probability forvector state Markov processes. Two parts:
1. Boundary set2. Repetitive set� Example: a modified M
�M�1 ����� if the system is empty,
else � . Customers require two exponential stages ofservice, � 1 � and � 2
S ��� i � s �� i � 0 and it is the no. of customer in the queue, �s is the current stage of service, s ��� 1 � 2 ��� s � 0 if no customer in the system� Well, let us proceed to specify the state transition diagram,
then the Q matrix.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Key Ideas� It is a technique to solve stationary state probability forvector state Markov processes. Two parts:
1. Boundary set2. Repetitive set� Example: a modified M
�M�1 ����� if the system is empty,
else � . Customers require two exponential stages ofservice, � 1 � and � 2
S ��� i � s �� i � 0 and it is the no. of customer in the queue, �s is the current stage of service, s ��� 1 � 2 ��� s � 0 if no customer in the system� Well, let us proceed to specify the state transition diagram,
then the Q matrix.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Key Ideas� It is a technique to solve stationary state probability forvector state Markov processes. Two parts:
1. Boundary set2. Repetitive set� Example: a modified M
�M�1 ����� if the system is empty,
else � . Customers require two exponential stages ofservice, � 1 � and � 2
S ��� i � s �� i � 0 and it is the no. of customer in the queue, �s is the current stage of service, s ��� 1 � 2 ��� s � 0 if no customer in the system� Well, let us proceed to specify the state transition diagram,
then the Q matrix.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Key Ideas� It is a technique to solve stationary state probability forvector state Markov processes. Two parts:
1. Boundary set2. Repetitive set� Example: a modified M
�M�1 ����� if the system is empty,
else � . Customers require two exponential stages ofservice, � 1 � and � 2
S ��� i � s �� i � 0 and it is the no. of customer in the queue, �s is the current stage of service, s ��� 1 � 2 ��� s � 0 if no customer in the system� Well, let us proceed to specify the state transition diagram,
then the Q matrix.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
0,0
0,1
0,2
1,1
1,2
2,1
2,2
3,1
3,2
λ λ λ λ
λ λ λ λ
λ*
µ2
µ2
µ1 µ1 µ1 µ1µ2 µ2
..........
..........
Let ai ������� i , i � 1 � 2. Arrange states lexicographically,� 0 � 0 ���� 0 � 1 ���� 0 � 2 ���� 1 � 1 ���� 1 � 2 �������� .John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
The transition rate matrix Q is:
Q ��������������� ��� ��� 0 0 0 0 0 0 0 � � 0 � a1 � 1 � 0 0 0 0 0 � � � 2 0 � a2 0 � 0 0 0 0 � � 0 0 0 � a1 � 1 � 0 0 0 � � 0 � 2 0 0 � a2 0 � 0 0 � � 0 0 0 0 0 � a1 � 1 � 0 � � 0 0 0 � 2 0 0 � a2 0 � � � ...
......
......
......
...... � �
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Let us re-write the Q in matrix form:
A0 �"! � 00 �$#&% A1 �"! � a1 � 1
0 � a2 #&% A2 �"! 0 0� 2 0 #B00 ')(*,+.-0/ -0/ 0
0 + a1 1 11 2 0 + a2
2354B01 ')(* 0 0- 0
0 -2364
B10 '87 0 0 00 1 2 0 9
Q � �������B00 B01 0 0 0 � � B10 A1 A0 0 0 � � 0 A2 A1 A0 0 � � 0 0 A2 A1 A0 � � ...
......
...... � �
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Let us solve it. For the repetitive portion:j ; 1A0 � : jA1 � : j < 1A2 � 0 j � 2 � 3 ������� (1)
This is similar to the solution of M�M�1. Therefore, : j is a
function only of the transition rates between states with j � 1queued customers and states with j queued customers.:
j � :j ; 1R j � 2 � 3 �������
or :j � :
1Rj ; 1 j � 2 � 3 ������� (2)
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
Putting (2) into (1), we have::1Rj ; 2A0 � : 1R j ; 1A1 � : 1RjA2 � 0 j � 2 � 3 �������
Since it is true for j � 2 � 3 ������� , substitute j � 2, we have:
A0 � RA1 � R2A2 � 0
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
For the initial portion: :0B00 � : 1B10 � 0:
0B01 � : 1A1 � : 2A2 � 0
or = :0 � : 1 > ! B00 B01
B10 A1 � RA2 # � 0
We also need:
1 � :0e � : 1 ?@
j A 1
Rj ; 1e � : 0e � : 1 � I � R ; 1 e= :0 � : 1 > ! e B �00 B01� I � R ; 1e B �10 A1 � RA2 # � = 1 � 0 >
where M � is M with first column being eliminated.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Key Idea
BNq � E[queued customers]� ?@
j A 1
j : je � ?@j A 1
� j : 1Rj ; 1e � : 1 � I � R ; 2 e
Note: S � ?@j A 1
R j ; 1 � I � R � R2 �C � � SR � R � R2 � R3 �C � �
S � I � R D� I
S � I � I � R ; 1 �E� I � R ; 1
This is true only when the spectral radius of R is less than unity.John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
General Concept
OutlineIntroduction
Why we need the Matrix-Geometric Technique?Matrix-Geometric in Action
Key IdeaGeneral Matrix-Geometric Solution
General ConceptApplication of Matrix-Geometric
Performance Analysis of Multiprocessing SystemProperties of Solutions
PropertiesComputational Properties of R
Algorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
General Concept
Q ����������
B00 B01 0 0 0 � � B10 B11 A0 0 0 � � B20 B21 A1 A0 0 � � B30 B31 A2 A1 A0 � � B40 B41 A3 A2 A1 � �
......
......
......
We index the state by � i � j where i is the level, i � 0 and j is thestate within the level, 0 F j F m � 1 for i � 1.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
General Concept
For the repetitive portion,?@k A 0
:j ; 1 < k Ak � 0 j � 2 � 3 ������� (3):
j � :1R j ; 1 j � 2 � 3 ������� (4)
putting (4) to (3), we have:?@k A 0
Rk Ak � 0
For the boundary states, we have:= :0 � : 1 > ! B00 B01G ?k A 1 Rk ; 1Bk0
G ?k A 1 Rk ; 1Bk1 #John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
General Concept
Procedure (continue:)
Using the same normalization, we have= :0 � : 1 >IH e B �00 B01� I � R ; 1 e J G ?k A 1 Rk ; 1Bk0 K � G ?k A 1 Rk ; 1Bk1 L � = 1 � 0 >
Therefore, it boils down to
1. Solving R.
2. Solving the initial portion of the Markov process.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
General Concept
Procedure (continue:)
Using the same normalization, we have= :0 � : 1 >IH e B �00 B01� I � R ; 1 e J G ?k A 1 Rk ; 1Bk0 K � G ?k A 1 Rk ; 1Bk1 L � = 1 � 0 >
Therefore, it boils down to
1. Solving R.
2. Solving the initial portion of the Markov process.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
OutlineIntroduction
Why we need the Matrix-Geometric Technique?Matrix-Geometric in Action
Key IdeaGeneral Matrix-Geometric Solution
General ConceptApplication of Matrix-Geometric
Performance Analysis of Multiprocessing SystemProperties of Solutions
PropertiesComputational Properties of R
Algorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Multiprocessing System
We have a multiprocessing system in which� K homogeneous processors.� Each processor is subjected to failure with rate M .� A single repair facility with repair rate N .� Jobs arrive at a Poisson rate � .� Whenever there is no processor available, all jobs are lost.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Multiprocessing System
We have a multiprocessing system in which� K homogeneous processors.� Each processor is subjected to failure with rate M .� A single repair facility with repair rate N .� Jobs arrive at a Poisson rate � .� Whenever there is no processor available, all jobs are lost.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Multiprocessing System
We have a multiprocessing system in which� K homogeneous processors.� Each processor is subjected to failure with rate M .� A single repair facility with repair rate N .� Jobs arrive at a Poisson rate � .� Whenever there is no processor available, all jobs are lost.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Multiprocessing System
We have a multiprocessing system in which� K homogeneous processors.� Each processor is subjected to failure with rate M .� A single repair facility with repair rate N .� Jobs arrive at a Poisson rate � .� Whenever there is no processor available, all jobs are lost.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Multiprocessing System
We have a multiprocessing system in which� K homogeneous processors.� Each processor is subjected to failure with rate M .� A single repair facility with repair rate N .� Jobs arrive at a Poisson rate � .� Whenever there is no processor available, all jobs are lost.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Markov Model
0,K 1,K 2,K
0,3 1,3 2,3
0,2 1,2 2,2
0,1 1,1 2,1
0,0
λ
Κµ
λ
µK
λ λ
λ λ
λ λ
3µ µ3
2µ µ2
µ µ
α
α
α
α
α
α
α
ααα γK γK γK
γ3 γ3 γ3
γ2 γ2 γ2
γγ γ
λ
λ
λ
λ
µK
µ3
µ2
µ
.................
.................
.................
.................
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
Define bi ���O� i MP�QN for i � 1 � 2 �������R� K . We have:
B00 � = � N > % B01 � = N.� 0 �� � � S� 0 > % Bj0 � ����� M0...0
TVUUUW % B0 X j � 0 j � 2 � 3 �������R�B1 X 1 �
���������� b1 N 0 0 � � 0 0 02 M � b2 N 0 � � 0 0 00 3 M � b3 N � � 0 0 0...
......
......
......
...0 0 0 0 � � Y� K � 1 ZM � bK < 1 N0 0 0 0 � � 0 K M � bK
TVUUUUUUUWJohn C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
� The matrices of the repeating portion of the process are:
A0 �[� I % A1 � B1 X 1 % A2 ��������� 0 � � 0 00 2 � � � 0 0...
... � � ......
0 0 � � Y� K � 1 \� 00 0 � � 0 K �
TVUUUUUW� This can be solved numerically rather than using thetransform method.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Performance Analysis of Multiprocessing System
� The matrices of the repeating portion of the process are:
A0 �[� I % A1 � B1 X 1 % A2 ��������� 0 � � 0 00 2 � � � 0 0...
... � � ......
0 0 � � Y� K � 1 \� 00 0 � � 0 K �
TVUUUUUW� This can be solved numerically rather than using thetransform method.
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties
OutlineIntroduction
Why we need the Matrix-Geometric Technique?Matrix-Geometric in Action
Key IdeaGeneral Matrix-Geometric Solution
General ConceptApplication of Matrix-Geometric
Performance Analysis of Multiprocessing SystemProperties of Solutions
PropertiesComputational Properties of R
Algorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties� Informally, the stability of a process depends on the drift ofthe process for states in the repetitive portion.� For example, M
�M�1, the expected drift toward higher
states is �^]1. The expected drift toward lower states is�_� � 1 _� � � . The drift of the process is � � � . Process isstable if the total expected drift is NEGATIVE, or �a`Q� inour case.� Now suppose the process can go up by 1 and go down byat most K steps. Let the rate for l steps be r � l ,l � � K � � K � 1 �������R� 0 � 1.
r � 1 b� K@l A 1
� � l r � � l .c r � 1 d` K@l A 1
lr � l John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties� Informally, the stability of a process depends on the drift ofthe process for states in the repetitive portion.� For example, M
�M�1, the expected drift toward higher
states is �^]1. The expected drift toward lower states is�_� � 1 _� � � . The drift of the process is � � � . Process isstable if the total expected drift is NEGATIVE, or �a`Q� inour case.� Now suppose the process can go up by 1 and go down byat most K steps. Let the rate for l steps be r � l ,l � � K � � K � 1 �������R� 0 � 1.
r � 1 b� K@l A 1
� � l r � � l .c r � 1 d` K@l A 1
lr � l John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties� Informally, the stability of a process depends on the drift ofthe process for states in the repetitive portion.� For example, M
�M�1, the expected drift toward higher
states is �^]1. The expected drift toward lower states is�_� � 1 _� � � . The drift of the process is � � � . Process isstable if the total expected drift is NEGATIVE, or �a`Q� inour case.� Now suppose the process can go up by 1 and go down byat most K steps. Let the rate for l steps be r � l ,l � � K � � K � 1 �������R� 0 � 1.
r � 1 b� K@l A 1
� � l r � � l .c r � 1 d` K@l A 1
lr � l John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties� Analogous to the scalar case, we can think of the drift ofthe process in terms of levels. Assume that for therepetitive portion, we have m states, a transition from leveli , i efe 0, to level i � k , 1 F k F K� k
m@l A 1
Ak < 1 � j � l where Ak < 1 � j � l is the transition from state j in level i tostate l in level i � k .� Let fj , 0 F j F m � 1 be the probability that the process isin inter-level j of the repeating portion of the process oflevel i efe 0. The average drift from level i to level i � k is� k
m ; 1@j A 0
fjm ; 1@l A 0
Ak < 1 � j � l John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties� Analogous to the scalar case, we can think of the drift ofthe process in terms of levels. Assume that for therepetitive portion, we have m states, a transition from leveli , i efe 0, to level i � k , 1 F k F K� k
m@l A 1
Ak < 1 � j � l where Ak < 1 � j � l is the transition from state j in level i tostate l in level i � k .� Let fj , 0 F j F m � 1 be the probability that the process isin inter-level j of the repeating portion of the process oflevel i efe 0. The average drift from level i to level i � k is� k
m ; 1@j A 0
fjm ; 1@l A 0
Ak < 1 � j � l John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties
� To get the total drift, we sum the previous equation for all k ,0 F k F K � 1.� But what is fj? Let us define A � G K < 1
l A 0 Al , we havef �E� f0 � f1 �������g� fm ; 1 . Therefore:
f A � 0 h f e � 1� The stability condition is:
f A0e ` K < 1@k A 2
� k � 1 f Ak e
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties
� To get the total drift, we sum the previous equation for all k ,0 F k F K � 1.� But what is fj? Let us define A � G K < 1
l A 0 Al , we havef �E� f0 � f1 �������g� fm ; 1 . Therefore:
f A � 0 h f e � 1� The stability condition is:
f A0e ` K < 1@k A 2
� k � 1 f Ak e
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Properties
� To get the total drift, we sum the previous equation for all k ,0 F k F K � 1.� But what is fj? Let us define A � G K < 1
l A 0 Al , we havef �E� f0 � f1 �������g� fm ; 1 . Therefore:
f A � 0 h f e � 1� The stability condition is:
f A0e ` K < 1@k A 2
� k � 1 f Ak e
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
OutlineIntroduction
Why we need the Matrix-Geometric Technique?Matrix-Geometric in Action
Key IdeaGeneral Matrix-Geometric Solution
General ConceptApplication of Matrix-Geometric
Performance Analysis of Multiprocessing SystemProperties of Solutions
PropertiesComputational Properties of R
Algorithm for solving R
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R� It is an iterative method. Let
R � 0 i� 0
R � n � 1 i� � ?@l A 0 X l jA 1
R l � n AlA; 11 n � 0 � 1 � 2 �������� The iterative process halts whenever entries in R � n � 1
and R � n differ in absolute value by less than a givenconstant.� The sequence R � n �� are entry-wise non-decreasing andconverge monotonically to a non-negative matrix R.� the number of iteration needed for convergence increasesas the spectral radius of R increases. This is similar to thescalar case where klc 1. As the system utilizationincreases, it becomes computationally more difficult to getR. John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R� It is an iterative method. Let
R � 0 i� 0
R � n � 1 i� � ?@l A 0 X l jA 1
R l � n AlA; 11 n � 0 � 1 � 2 �������� The iterative process halts whenever entries in R � n � 1
and R � n differ in absolute value by less than a givenconstant.� The sequence R � n �� are entry-wise non-decreasing andconverge monotonically to a non-negative matrix R.� the number of iteration needed for convergence increasesas the spectral radius of R increases. This is similar to thescalar case where klc 1. As the system utilizationincreases, it becomes computationally more difficult to getR. John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R� It is an iterative method. Let
R � 0 i� 0
R � n � 1 i� � ?@l A 0 X l jA 1
R l � n AlA; 11 n � 0 � 1 � 2 �������� The iterative process halts whenever entries in R � n � 1
and R � n differ in absolute value by less than a givenconstant.� The sequence R � n �� are entry-wise non-decreasing andconverge monotonically to a non-negative matrix R.� the number of iteration needed for convergence increasesas the spectral radius of R increases. This is similar to thescalar case where klc 1. As the system utilizationincreases, it becomes computationally more difficult to getR. John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R� It is an iterative method. Let
R � 0 i� 0
R � n � 1 i� � ?@l A 0 X l jA 1
R l � n AlA; 11 n � 0 � 1 � 2 �������� The iterative process halts whenever entries in R � n � 1
and R � n differ in absolute value by less than a givenconstant.� The sequence R � n �� are entry-wise non-decreasing andconverge monotonically to a non-negative matrix R.� the number of iteration needed for convergence increasesas the spectral radius of R increases. This is similar to thescalar case where klc 1. As the system utilizationincreases, it becomes computationally more difficult to getR. John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
Replicated Database
� Poisson arrival with rate � .� Probability it is a read request: r .� A read request can be served by any server.� A write request has to be served by BOTH servers.� What is the proper state space?
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
Replicated Database
� Poisson arrival with rate � .� Probability it is a read request: r .� A read request can be served by any server.� A write request has to be served by BOTH servers.� What is the proper state space?
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
Replicated Database
� Poisson arrival with rate � .� Probability it is a read request: r .� A read request can be served by any server.� A write request has to be served by BOTH servers.� What is the proper state space?
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
Replicated Database
� Poisson arrival with rate � .� Probability it is a read request: r .� A read request can be served by any server.� A write request has to be served by BOTH servers.� What is the proper state space?
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
Replicated Database
� Poisson arrival with rate � .� Probability it is a read request: r .� A read request can be served by any server.� A write request has to be served by BOTH servers.� What is the proper state space?
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
The state space S �E� i � j where i is the number of queuedcustomers and j is the number of replications that are involvedin service. So i � 0 and j � 0 � 1 � 2.
λ
0,0
0,1
0,2
1,1
1,2
2,1
2,2
λr
µ
λ(1-r) λ
λr
λ λ λ
µ2 r µ2 r µ2 r
λ(1-r)
µ2
µ
µ2 (1-r) µ2 (1-r)
µ µ
...........
...........
John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
Q ' (mmmmmmmmm*+.- - r -on 1 + r p 0 0 0 q�q�q1 +rns-ut 1 p - r -vn 1 + r p 0 0 q�q�q0 2 1 +rns-wt 2 1 p 0 - 0 q�q�q0 0 1 +rnx-yt 1 p 0 - q�q�q0 0 2 1 r 2 1 n 1 + r p +rnx-ut 2 1 p 0 q�q�q0 0 0 0 1 +rns-wt 1 pzq�q�q...
......
......
... q�q�qB00 � �� � � � r �b� 1 � r � � �\���{�| � r
0 2 � � �\�O� 2 �| TW % B01 � �� 0 0�b� 1 � r 00 � TW
B10 �}! 0 0 �0 0 2 � r #&% B11 � A1 �"! � �\�O�{�| 0
2 �_� 1 � r � �\�O� 2 �| #&%John C.S. Lui Matrix-Geometric Analysis and Its Applications
logo
OutlineIntroduction
Matrix-Geometric in ActionGeneral Matrix-Geometric Solution
Application of Matrix-GeometricProperties of Solutions
Computational Properties of R
Algorithm for solving R
A0 �"! � 00 �$#&% A2 �"! 0 �
0 2 � r #To determine the stability:
A �}! � � �2 �_� 1 � r � 2 �_� 1 � r l# � A0 � A1 � A2
f1 � 2 � 1 � r 3 � 2r % f2 � 1
3 � 2r
f A0e ` K < 1@k A 2
� k � 1 f Ak e c~��` 2 �3 � 2r
John C.S. Lui Matrix-Geometric Analysis and Its Applications