I have drawn a certain Markov chain with a weird transition matrix. Here's the drawing:
And here's the transition matrix:
My problem is that I don't quite know how to calculate the steady state probabilities of this chain, if it exists. I need a bit of help, because just getting the $\pi$'s, they don't sum to 1 for me, which is weird.
I've assembled this Markov chain using some data and conditions given to me and, right now, I would insist that it is the inevitable result of a very long process of solving. But now I hit a brick wall and can't continue. Please help?
Here's my solution to the problem, using the ordinary ways of calculating steady states.
As you can see, it's a little bit problematic. The probabilities don't add up to 1. I've turned it upside down several times. Is it really just something about my arithmetic, or is this simply a weird Markov chain?