Markov Chain Expected Time

Question

I am studying Markov Chain Expected time. I came across this formula. However, it wasn't state explicitly when would the formula apply.

In the following photo, do you guys think the formula above also applied on markov chain below?

I want to calculate the expected time to return back to state 0 if started from state 0.

Here is the result of apply the formula to my problem above. Wolfram told me no solution exists.

Answer 1

Here is the correct set of equations for your problem. We define $T=\inf(t\geq 1: X_t=0)$ and define $S_j=\mathbb{E}_j(T).$ Then first step analysis gives \begin{eqnarray*} S_0&=&1+S_1\\ S_1&=&1+(1/3) 0+ (2/3)S_2\\ S_2&=&1+(2/3)S_1+(1/3)S_3\\ S_3&=&1+S_2 \end{eqnarray*}

Notice that you should replace $S_0$ in your second equation with $0$ (zero).