Answer
$v_\infty=\begin{bmatrix} 0.4& 0.2&0.4 \end{bmatrix}$
Work Step by Step
The steady-state distribution vector $v_\infty$ can be written as: $v_\infty P=v_∞$
where, $v_\infty=[x~~y~~z]$
This gives: $[x~~y~~z] \begin{bmatrix}
0.5 & 0&0.5 \\ 1 & 0&0\\0&0.5&0.5 \\ \end{bmatrix}
=[x~~y~~z]$
We can have the following equations:
$$0.5x+0.5z=x\\ y=y \\ 0.5y+0.5 z=z$$
or, $$-0.5x+0.5z=0\\ 0=0 \\ 0.5y-0.5 z=0$$
Also, we have: $x+y+z=1$
So, the new system of equations are:
$x+y+z=1\\ -0.5x+0.5z=0 \\ 0.5y-0.5 z=0 $
After solving the above equations, we get:
$x=0.4; y=0.2 ; z=0.4$
Thus, the required steady-state distribution vector $v_\infty$
$v_\infty=\begin{bmatrix} 0.4& 0.2&0.4 \end{bmatrix}$
