Customers arrive at a server with rate $\lambda$ and are served at rate $\mu$. The server breaks down with rate $\gamma$, which causes all customers to leave. New customers can only arrive once the server has been repaired, which happens at rate $\delta$. Model as a MC give balance equations, and rewrite those so that they can be recursively solved (i.e. express for instance $\pi_k$ in $\pi_{k-n}$ ($n=1,\dots k$), $k=1,2,\dots$. You don't have to solve the equations.
My approach. Let $X(t)$ be the number of customers in the system and let the state space be $A=\{d, 0, 1, \dots\}$ where $d$ denotes a state in which the server is down.
The transition rates are then: $q_{d,0}=\delta$, $q_{i,d}=\gamma$, $q_{i,i+1}=\lambda$, $q_{i,i-1}=\mu$.
Then \begin{align*}\delta \pi_d &= \gamma \quad[\text{is this correct or should it be } \delta \pi_d = \gamma \pi_i, \ i =0,1,\dots?]\\ \lambda \pi_0 &= \mu \pi_1 + \delta \pi_d\\ (\lambda + \mu)\pi_i &= \lambda \pi_{i-1} + \mu \pi_{i+1}, \quad i \geq 2\\ \sum_i \pi_i&=1\end{align*}
Is my definition of the Markov chain correct and are the balance equations correct (specifically the first equation)? Also, how can I then rewrite the system so that it can be recursively solved like the question asks?