I'm looking for techniques or tricks to solve a system of linear equations you get where you want to find the limiting probabilities.
The system is this:
$\pi_0 = 0.7\pi_0 + 0.2\pi_1 + 0.1\pi_2$ ( EQUATION 2)
$\pi_1 = 0.2\pi_0 + 0.6\pi_1 + 0.4\pi_2$ (EQUATION 2)
$\pi_2 = 0.1\pi_0 + 0.2\pi_1 + 0.5\pi_2 $ (EQUATION 3)
$\pi_0 + \pi_1 + \pi_2 = 1$
What's an easy way to solve this?
I Reeeeeally don't want to put it into a matrix and do reduce row echelon form on it. Please tell me there is another way. I remember other folks solving it differently
I have figured it out. Maybe if one of yall are searching for answers on how to solve these Markov chains it will help.
First step is this:
$\pi_2 = 1 - \pi_0 - \pi_1$
Now I substitute this $\pi_2$ into equation 1 and 2 above.
For equation 1 I get:
$\pi_0 = 0.6\pi_0 + 0.1\pi_1 + 0.1$
For equation 2 I get:
$\pi_1 = -0.2\pi_0 + 0.2\pi_1 + 0.4$
For equation 1 and 2 we move the the $\pi_0$ and $\pi_1$ to the other sides of the equation setting both to zero:
$0 = -0.4\pi_0 + 0.1\pi_1 + 0.1$ EQUATION 1
$0 = -0.2\pi_0 - 0.8\pi_1 + 0.4$ EQUATION 2
Adding them together by multiplying constants can solve for both A and B. And once you know those 2 you can solve for C. I hope this was useful to anyone who has stumbled upon it