I have this problem, I figured out the part (a) but I'm having a little trouble with part (b) if anyone can help me with that.
Two repairmen serve three machines (that is, at most two machines can be under repair at the same time). The time until a particular machine needs repair is exponential with mean $1/3$ and independent of other repairs and failures. Each repair time is exponential with mean $1/2$ and independent of other repairs and failures. Let $X_t$ be the number of machines under repair or needing repair at time $t$.
(a) Determine the birth and death intensities of the birth and death process $(X_t)$, $t≥0$.
(b) Determine $\lim_{t \to \infty} \mathbb{P}_i(X_t=0)$ for $i = 0,1,2,3$.
From a I got this matrix $$Q = \begin{bmatrix} -9 & 9 & 0 & 0 \\ 2 & -8 & 6 & 0 \\ 0 & 4 & -7 & 3 \\ 0 & 0 & 2 & -2 \end{bmatrix} $$ But I am not sure how to do (b), I was trying to do $\lambda Q=0$ but I was not getting the right solution.
From $\pi Q = 0$ we have \begin{align} -9\pi_0 + 2\pi_1 = 0\\ 9\pi_0-8\pi_1+4\pi_2=0\\ 6\pi_1-7\pi_2+2\pi_3=0\\ 3\pi_2-2\pi_3=0 \end{align} Note that this system of equations is linearly independent, since the right-hand side are all zero. So arbitrarily choose an equation to remove, and add the equation $\sum_{i=0}^3\pi_i=1$. This yields the solution $$ \pi_0 = \frac8{179},\quad \pi_1 = \frac{36}{179},\quad \pi_2 = \frac{54}{179},\quad \pi_3 = \frac{81}{179}. $$ Alternatively we could look at a row of $\lim_{t\to\infty}e^{tG}$. But $e^{tG}$ is not trivial to compute, so using the relation $\pi Q=0$ is highly recommended.