I want to calculate the expected number of steps (transitions) needed for absorbtion. So the transition probability matrix $P$ has exactly one (lets say it is the first one) column with a $1$ and the rest of that column $0$ as entries.
$P = \begin{bmatrix} 1 & * & \cdots & * \\ 0 & \vdots & \ddots & \vdots \\ \vdots & & & \\ 0 & & & \end{bmatrix} \qquad s_0 = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 1 \\0 \\ \vdots \\ \\ \end{bmatrix}$
How can I now find the expected (mean) number of steps needed for the absorbtion for a given initial state $s_0$?
EDIT: An explicit example here:
$P = \begin{bmatrix} 1 & 0.1 & 0.8 \\ 0 & 0.7 & 0.2 \\ 0 & 0.2 & 0 \end{bmatrix}
\qquad s_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \implies s_1 = \begin{bmatrix} 0.8 \\ 0.2 \\ 0 \end{bmatrix} \implies s_2 = \begin{bmatrix} 0.82 \\ 0.14 \\ 0.04 \end{bmatrix} \ldots $
Best Answer
We conventionally read row by row, but it appears that in your matrix, the columns add to 1 instead. I'll write your matrix like this and proceed:
\begin{align*} P = \begin{bmatrix} 0 & 0.2 & 0.8 \\ 0.2 & 0.7 & 0.1 \\ 0 & 0 & 1 \end{bmatrix} \end{align*}
which is of the form
$$P=\left[ \begin{array}{c|c} Q & R \\ \hline 0 & 1 \end{array}\right] $$
and
\begin{align*} s_0 = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \end{align*}
For the expected time to absorption, we need to calculate
$\left(I-Q\right)^{-1} \cdot \begin{pmatrix} 1 \\ 1 \end{pmatrix} \approx \left(\begin{array}{r} 1.92307692307692 \\ 4.61538461538461 \end{array}\right) $