Help on Markov chains to calculate the mean time to reach a particular state

Mar 2013
10
0
Japan

Hello guys,

I have a 2D markov chain as shown above with 4 states. I need to calculate the mean time to reach the state '11' from state '00'. Could please someone help me with this.

Thanks and Regards,
Koustubh
 

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Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
Hello, Koustubh!

Without numbers, the problem is virtually impossible.
Even with numbers, the solution would fill volumes.


I have a 2D markov chain with 4 states.
I need to calculate the mean time to reach the state '11' from state '00'.
Could please someone help me with this?

According to your diagram, we seem to have this transition chart.

. . \(\displaystyle \begin{array}{c|cccc|} & 00 & 01 & 10 & 11 \\ \hline 00 & i & a & g & 0 \\ 01 & b & j & 0 & c \\ 10 & h & 0 & l & e \\ 11 & 0 & d & f & k \\ \hline\end{array}\)


Let \(\displaystyle A \:=\:\begin{pmatrix}i&a&g&0 \\ b&j&0&c \\ h&0&l&e \\ 0&d&f&k \end{pmatrix}\)


We want the probability of going from \(\displaystyle 00\) to \(\displaystyle 11\) in \(\displaystyle n\) steps.

\(\displaystyle \begin{array}{ccccc}n=2\!: & \text{ Calculate }A^2. & \text{ Examine the term }a_{14} \\ n=3\!: & \text{ Calculate }A^3. & \text{ Examine the term }a_{14} \\ n=4\!: & \text{ Calculate }A^4. & \text{ Examine the term }a_{14} \\ n=5\!: & \text{ Calculate }A^5. & \text{ Examine the term }a_{14} \\ \vdots & \vdots & \vdots \end{array}\)


Do you see the difficulty?