How to solve the markov transition matrix

[allamar2012]

Homework Statement

I design a markov model to study the availability of a node which will not work unless the network work: I make the following matrix. In which (i,j) means i is node up /down and j is network up/down. R1,R2 means the repair rate for node and network. F1,F2 means the failure rate for node and network.
(0,0) (1,0) (0,1) (1,1)
(0,0) 0 R1 R2 0
(1,0) F1 0 0 R2
(0,1) F2 0 0 R1
(1,1) 0 F2 F1 0

Homework Equations

I do the following equations to detemine the P(1,1)
(R1+R2)P(0,0)= F1 P(1,0) + F2 P(0,1)
(F1+R2)P(1,0) = R1 P(0,0) +F1 P(1,1)
(F2+R1)P(0,1) = R2 P(0,0) +F1 P(1,1)
(F2+F1)P(1,1) = R2 P(1,0) +R1 P(0,1)

P(0,0) + P(1,0) + P(0,1) + P(1,1) = 1

I do this symbolically in which it gives me long recursive and I couldn't complete it.

The Attempt at a Solution

is my equation right or wrong?
I need advise in how to solve this equations to get P(1,1).
I try MATLAB to calculate this equations as linear system
it gives me
[ 0
0
0
0 ]

Thanks

 

[Ray Vickson]

Homework Statement

I design a markov model to study the availability of a node which will not work unless the network work: I make the following matrix. In which (i,j) means i is node up /down and j is network up/down. R1,R2 means the repair rate for node and network. F1,F2 means the failure rate for node and network.
(0,0) (1,0) (0,1) (1,1)
(0,0) 0 R1 R2 0
(1,0) F1 0 0 R2
(0,1) F2 0 0 R1
(1,1) 0 F2 F1 0

Homework Equations

I do the following equations to detemine the P(1,1)
(R1+R2)P(0,0)= F1 P(1,0) + F2 P(0,1)
(F1+R2)P(1,0) = R1 P(0,0) +F1 P(1,1)
(F2+R1)P(0,1) = R2 P(0,0) +F1 P(1,1)
(F2+F1)P(1,1) = R2 P(1,0) +R1 P(0,1)

P(0,0) + P(1,0) + P(0,1) + P(1,1) = 1

I do this symbolically in which it gives me long recursive and I couldn't complete it.

The Attempt at a Solution

is my equation right or wrong?
I need advise in how to solve this equations to get P(1,1).
I try MATLAB to calculate this equations as linear system
it gives me
[ 0
0
0
0 ]

Thanks

The first 4 equations have a redundancy: given that any three of them hold, the 4th one also holds. So, keep any three of the first four equations, and append the normalization equation P(0,0) + ... + P(1,1) = 1. That 4x4 linear system does have a unique solution.

It is not too bad, just lengthy. You can get P(0,0) = [F1 P(1,0) + F2 P(0,1)]/(R1+R2) from the first equation. You can get P(1,1) = [R2 P(1,0) +R1 P(0,1)]/(F2+F1) from the fourth equation. Now you can substitute these expressions for P(0,0) and P(1,1) into the second (or third) equation and into the sum(P) = 1 equation to get two equations in the two unknowns P(0,1) and P(1,0). It is messy and time-consuming, but is nevertheless straightforward.

RGV

 

[allamar2012]

Hi Ray :)
Thank you very much, it's helpful.