Answer
$v_\infty=\begin{bmatrix} \dfrac{5}{14} & \dfrac{1}{28} & \dfrac{17}{28} \end{bmatrix}$
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: $[x~~y~~z] \begin{bmatrix}
0.1 & 0.1 &0.8 \\0.5& 0&0.5\\0.5&0&0.5 \\ \end{bmatrix}
=[x~~y~~z]$
We can have the following equations:
$$0.1x+0.5y+0.5z=x\\ 0.1 x=y\\ 0.8x+0.5y+0.5z=z$$
or, $$-0.9x+0.5y+0.5z=0\\ 0.1 x-y=0\\ 0.8x+0.5y-0.5z=0$$
Also, we have: $x+y+z=1$
So, the new system of equations are:
$x+y+z=1\\ 0.1 x-y=0 \\ 0.8x+0.5y-0.5z=0$
After solving the above equations, we get:
$x=\dfrac{5}{14} ; y=\dfrac{1}{28} ; z=\dfrac{17}{28}$
Thus, the required steady-state distribution vector $v_\infty$ is:
$v_\infty=\begin{bmatrix} \dfrac{5}{14} & \dfrac{1}{28} & \dfrac{17}{28} \end{bmatrix}$
