Customers arrive in a FIFO queue with rate $\lambda$ and are serviced with mean time $\dfrac1\mu$. The system occasionally fails with time distribution between failures with mean $\dfrac1\gamma$. When the system fails, all customers are dropped.
The Markov chain that I came up with is below. Is it correct?
Assuming the Markov chain above is correct, how can we write the balance equations? Below is what I have: $$ \begin{align*} \lambda P_0 &= \mu P_1 + \gamma(P_1 + P_2 + \cdots + P_n + \cdots) = \mu P_1 + \gamma(1 - P_0)\\ \implies P_1 &= \frac{\lambda + \gamma}\mu P_0 - \frac\gamma\mu \end{align*} $$ After this, the balance equations start to become very convoluted $$ \begin{align*} \lambda P_1 &= \mu P_2 + \gamma(P_2 + P_3 + \cdots + P_n + \cdots) = \mu P_2 + \gamma(1 - P_0 - P_1) \\ \implies P_2 &= \left(\frac{\lambda(\lambda + \gamma)}{\mu^2} + \frac\gamma\mu + \frac{\gamma(\lambda + \gamma)}{\mu^2}\right)P_0 - \frac{\lambda\gamma}{\mu} - \frac\gamma\mu-\frac{\gamma^2}{\mu^2} \end{align*} $$ You can imagine that finding $P_3 \cdots$ would be a nightmare. I cannot figure out a closed form expression for $P_n$. The most I can come up with is $$ P_n = \frac{\lambda P_{n-1} - \gamma\left(1 - \sum_{i = 0}^{n - 1}P_i\right)}{\mu} $$ How would I find $P_n$? I am given a hint to guess an expression of the form $P_k = (1 - \beta)\beta^k$ where $P_k$ represents the steady-state probability of finding $k$ customers in the system. Any tips?
Assuming the exponential assumptions from my above comments, your work looks good. Good things indeed seem to happen when substituting $P_i = (1-\beta)\beta^i$ into your balance equations. If everything works properly, while you might find multiple $\beta$ that work for your equations, you should be able to find one and only one value of $\beta$ that satisfies $0<\beta < 1$ (else, there would be two distinct probability vector solutions to the balance equations, which is impossible for irreducible CTMCs).