Answer
$v_\infty=\begin{bmatrix} 0&1&0\end{bmatrix}$
Work Step by Step
The steady-state distribution vector $v_\infty$ can be written as: $v_\infty P=v_\infty$
where, $v_\infty=[x~~y~~z]$
This gives: $[x~~y~~z] \begin{bmatrix}
0.9 & 0.1 &0 \\0& 1&0\\0&0.2&0.8 \\ \end{bmatrix}
=[x~~y~~z]$
We can have the following equations:
$$0.9x=x\\ 0.1 x+y+0.2 z=y\\ 0.8z=z$$
or, $$-0.1x=0\\ 0.1 x+0.2 z=0\\ -0.2z=0$$
Also, we have: $x+y+z=1$
So, the new system of equations are:
$x+y+z=1\\ (1/2) x-(1/2) y=y\\ (-\dfrac{1}{2}) z=0 $
After solving the above equations, we get:
$x=0; y=1 ; z=0$
Thus, the required steady-state distribution vector $v_\infty$ is:
$v_\infty=\begin{bmatrix} 0&1&0\end{bmatrix}$
