I have a transition matrix $$ P = \begin{Vmatrix} 1 & 0 & 0 &0\\ .3& 0 &.7& 0\\ 0& .1 & 0 & .9 \\ 0& 0 & 0 &1 \end{Vmatrix}$$ on states $\{0,1,2,3\}$. I'm trying to calculate the mean time until absorption starting at state 1. I set up a system of equations and got $v_0 = v_3 = 0$, and \begin{align*} v_1 &= .7v_2+1\\ v_2&= .1v_1+1. \end{align*} Solving for $v_1$ gives $\frac{170}{93}$.
Then I'm asked to use results regarding random walks to solve this, but I don't have any.
My questions are:
- Is my system of equations correct?
- What results can I use about random walks to solve this? Thanks!