Answer
$v_\infty=\begin{bmatrix} \dfrac{1}{3}&\dfrac{1}{2}&\dfrac{1}{6} \end{bmatrix}$
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: $[x~~y~~z] \begin{bmatrix}
0 & 1&0 \\ 1/3& 1/3&0\\1&0&0 \\ \end{bmatrix}
=[x~~y~~z]$
We can have the following equations:
$$1/3y+z=x\\ x+1/3 y=y\\1/3 y=z$$
or, $$-x+1/3y+z=0\\ x-2/3y=0\\1/3y-z=0$$
Also, we have: $x+y+z=1$
So, the new system of equations are:
$x+y+z=1\\ x-2/3y=0 \\ 1/3y- z=0 $
After solving the above equations, we get:
$x=1/3; y=1/2 ; z=1/6$
Thus, the required steady-state distribution vector $v_\infty$
$v_\infty=\begin{bmatrix} \dfrac{1}{3}&\dfrac{1}{2}&\dfrac{1}{6} \end{bmatrix}$