# Markov Chains: Long Run Behavior

In a large metropolitian area, communters either deive along (A), carpool (C) or take public transportation (T). A study showed that transportation changes according to the following transition matrix: $$\begin{matrix} & A & C & T\\ A & 0.8 & 0.15 & 0.05\\ C & 0.05 & 0.9 & 0.05\\ T & 0.05 & 0.1 & 0.85 \end{matrix}$$ In the long run what fraction of the communters will use the three types of transportations?

I know that this problem can be solved by finding the stationary distribution (as it exists and this matrix will tend to it), but that is a lengthy calculation and I have a theorem that says the stationary distribution $\pi(y)=\frac{1}{(E_{y}T_{y})}$. I am wondering if this method would be any easier/faster, and if so, how would I go about beginning this?

• No, it's not any faster to relate the stationary distribution back to the return time. It's also really not that long of a calculation to get the stationary distribution. You have to solve a system of three equations. Since the equations are in fact redundant, you can neglect one of them. Then you just normalize. – Ian Oct 7 '15 at 22:35
• If I'm solving by hand wouldn't I need to go through the whole $\pi*q=\pi$ calculation, which in a 3x3 matrix is kinda ugly? – Studying Hard Oct 7 '15 at 22:38
• That's right. But because you know the eigenvalue is $1$, you therefore know that there is a nontrivial kernel, so it's really a system of just two equations in three unknowns. It's not that bad. – Ian Oct 7 '15 at 22:44