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The RG-Factorizations
in Stochastic Models
Dr. Quan-Lin Li
Department of Industrial Engineering
Tsinghua University
Beijing 100084, P.R. China
Outline of this talk
Why to need the RG-factorizations
How to construct the RG-factorizations
How to apply the RG-factorizations
Promising issues in the future
Why to need
From 1996 to 2000, my research focuses on quasistationary distributions of stochastic models
Our main problem is described as follows:
P
Discrete
P is a transition probability matrix
Q
Continuous
Q is an infinitesimal generator
e 1
Our Question: How to compute?
Why to need
When the size of the matrix P is finite, this
computation is similar to that for the stationary
probability vectors of the finite-state Markov
chains by using systems of linear equations
-classification of state;
solving ; is the convengence radious
0 for all the other cases
Why to need
When the size of the matrix P is infinite, this
computation will become different and difficult
Need to consider the existence
Need to consider the uniqueness:
There are over one quasi-stationary distributions
No available expression
No effective approach
Why to need
Our work from 1996 to 1999 was to
develop the LU-block-decomposition
for Markov chains of M/G/1 type
and GI/M/1 type
Our method is different from
that used by Bean, Latouche,
Taylor etc.
Why to need
B1
B2
P B3
B4
B1
B0
P
B0
A1
A0
A2
A3
A1
A2
A0
A1 A0
B2
B3
B4
A1
A0
A2
A1
A0
A3
A2
A1
GI/M/1 type
M/G/1 type
Why to need
The LU-block-decomposition is
I P L U
Q L U
Discrete
Continuous
Our computation is given by
L U 0
Let x L . Then xU 0
Based on this, we can give a solution
Our Question: Such a solution is OK?
Why to need
For a special Markov chain, we
obtained two different LU-blockdecompositions, which lead to two
different expressions
One of them is correct and is the
same as that in the literature; while
another is wrong
Why?
Why to need
We
analyzed many real examples
and then found the main reasons
These computations motivate us to
LU-block-decomposition
to the RG-factorization
extend the
Why to need
For an arbitary irreducible Markov chain, the RG-factorization
is given by
I P I RU I D I GL Discrete
Q I RU D I GL
Continuous
Two different LU-decompositions are
L I RU I D ,
L I RU ,
U I GL ;
U I D I GL
Our computation is given by
I RU I D I GL 0
Why to need
For this computation
I RU I D I GL 0
How to take the vector
?
I RU
x
I RU I D
A key observation:
?
When P is -positive recurrent with = , we should use
x I RU
All the other cases, we should use
x I RU I D
Our Comparisons
Utility of the RG-factorization is
related to the classification of
state by means of the diagonal
matrix, and keep effective
computations
Better than LU-decomposition
How to construct
Consider a discrete-time Markov chain
P0,1
P0,0
P1,0
P1,1
P
PN ,0 PN ,1
where N or N
Let E 0,
P0, N
P1, N
PN , N
, n and E c n 1,
P E
Ec
E
Ec
T
V
U
W
, N . Then
How to construct
We can have two types of censored Markov chains:
UL-type: To the level set E
n
k
P T U W V
k 0
which leads to the UL-type RG-factorization
LU-type: To the level set E c
P
n
W V I T U
1
which yields the LU-type RG-factorization
The UL-type RG-factorization
We write
n
0,0
n
1,0
n
P
n
n ,0
We define
U-measure:
n
0,1
n
1,1
n ,1n
0, nn
1,n
n ,nn
n
n n , n , n 0,
n
j
I
1
R-measure:
Ri , j i , j
G-measure:
Gi , j I i i, j , 0 j i.
j
1
i
, 0 i j,
The UL-type RG-factorization
For an abitrary irreducible Markov chain P, the UL-type
RG-factorization is given by
I P I RU I D I GL ,
where
0 R0,1 R0,2 R0,3
0
R1,2 R1,3
RU
0
R2,3
D diag 0 , 1 , 2 , 3 ,
0
G1,0
GL G2,0
G3,0
0
G2,1
G3,1
0
G3,2
0
The UL-type RG-factorization
Important Properties for Censoring Structure:
0 is irreducible if P is irreducible;
0 is positive recurrent if P is recurrent;
0 is transient if P is transient;
k is transient for all k 1.
Some special cases
The QBD processes
0 R0,1
0
R1,2
RU
0
The M/G/1 type
R2,3
0
G
GL 1,0
The GI/M/1 type
0
RU
0
, G G1,0
L
0
G2,1
0
R0,1
0
R1,2
0
R2,3
0
G2,1
0
Some special cases
The GI/G/1 type
0 R0,1 R0,2 R0,3
0
R1
R2
RU
0
R1
D diag 0 , , , ,
0
G1,0
GL G2,0
G3,0
0
G1
G2
0
G1 0
How to apply
I RU I D I GD 0
Let x I RU . Then x I D I GL 0.
Observating a non-zero nonnegative solution
x x0 , 0, 0, 0,
,
where x0 is the stationary probability vector of 0
Therefore, = x0 , 0, 0, 0,
I RU
0 x0 ,
k 1
k i Rk i , k 1.
i 0
1
yields
Remarks
Computing the stationary probability vector of the
Markov chain P with a huge state space or an infinite
state space is decomposited into two steps:
Step one: Computing the stationary probability
vector of the censored chain 0 with a smaller
state space
Step two: Computing the R-measure Ri , j for 0 i j
by using the above iterative relations.
A crucial advance
Infinite states
Finite states
Huge
The UL-type RG-factorization
Finite states
Smaller
The LU-type RG-factorization
We write
n n,n
n n,n1 n n,n 2
n
n
n
n
n 1, n 1
n 1, n 2
P n 1,n
n
n
n
n 2,n
n 2, n 1
n 2, n 2
We define
U-measure:
n n n,n , n 0,
R-measure:
R i , j i , j I j , 0 j i ,
G-measure:
G i , j I i i , j , 0 i j.
1
j
1
i
The LU-type RG-factorization
For an abitrary irreducible Markov chain P, the LU-type
RG-factorization is given by
I P I R L I D I GU ,
where
0
0
R1,0
R L R 2,0 R 2,1
0
R 3,0 R 3,1 R 3,2 0
D diag 0 , 1 , 2 , 3 ,
GU
0
G 0,1
G 0,2
G 0,3
0
G1,2
0
G1,3
G 2,3
The LU-type RG-factorization
k is transient for all k 0.
The matrix I P or Q of size must be invertible,
I P
I D I RL
1
1
Q 1 I GU D 1 I R L
1
I GU
1
1
An example,
1
2
1
P2
1
2
1
2
2
1
2
3
1
2
2
1
2
3
1
2
2
1
2
3
1
Some special cases
The QBD processes
0
R
0
RU 1,0
R 2,1
The M/G/1 type
0
0 G 0,1
0 G1,2
,G
L
0
0 G 0,1
0 G1,2
GL
0
The GI/M/1 type
0
R
RU 1,0
0
R 2,1
0
How to apply
The LU-type RG-factorization is different from the UL-type
case. It may be used to deal with the first passage times and
the sojourn times. In addition, we provide a better example:
Consider a perturbed Markov chain P P . Let and
be the stationary probability vectors of P and P, respectively.
Then
P
d
| 0 I P
d
which leads to
d
1
1
1
| 0 I GU I D I R L
d
Comparison for UL- and LU-type
Systems of linear equations
xA 0
or
Ax 0
Systems of linear equations
xA b (b 0)
or
Ax b (b 0)
UL-type RG-factorization
LU-type RG-factorization
Our work on the RG-factorizations
Theory
Applications
Promising problems (1)
For the RG-factorizations:
1. It is interesting to consider the d-period for
the R-, U- and G-measures. For example
(1) A = A0 + A1 + A2 is irreducible and is d-period,
the two matrices R and G are d-period
?
(2) For a Markov chain of GI/G/1 type, what
happen to
?
Such a work is useful for tailed analysis
Discrete time
R R Ak ,
k
k 0
G Ak G
k
k 0
Continuous time
R
k 0
k
Ak 0,
AG
k 0
k
k
0
Promising problems (2)
For the RG-factorizations:
1. It is interesting to consider spectral analysis
for the R-, U- and G-measures.
When A = A0 + A1 + A2 is irreducible and is
infinite size, how to analyze the spectral of
the two matrices R and G
?
2. For a Markov chain of GI/G/1 type, what
happen to
?
Promising problems (3)
To construct the RG-factorization, we have formed
many useful relations such as Winner-Holp equations
Ri , j I j Pi , j
R I G
k j 1
I i Gi , j Pi , j
k j 1
n Pn ,n
i ,k
k
k, j
Ri ,k I k Gk , j
R I G
k n 1
n ,k
k
k ,n
Effective algorithms are necessary to compute the
R-, U- and G-measures, and then compute performance
measures of a stochastic models.
Promising problems (4)
Transient Performance:
Continuous-time Markov chain: Q or Q t
d
t t Q
dt
t 0 exp Qt
d
t t Q t
dt
t 0 exp
Q u du
t
0
Continuous-time Markov reward process: Q, f X t or R
t f X u du, H i t , x P t x, X t i , t , x
t
0
t, x
t, x R t, x Q
t
x
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Thanks for you
and
questions ?