Can anyone help me with this problem?
A data centre is equipped with $M = 5$ servers, completely interchangeable. Each of them can fail independently of the others. The time to failure of a running server is a negative exponential random variable with mean $\frac1v = 7$ days. There is only one technician repairing failed servers. Repair times have negative exponential pdf with mean $\frac{1}{\mu} = 1 $ day.
Calculate the probability that no server is operating (all are down) and the mean number of operational servers.
Find the minimum value of $M$ that guarantees that the unavailability of the system is $10^{-3}$.
Note: unavailability = probability that all servers are down).
Let $N$ be the number of working machines. Note that $N$ is a continuous-time Markov chain (a birth--and--death process even). The transition rates are
\begin{align} q_{i,i + 1} &= (M - i)v, \quad 0 \le i \le M - 1, \\ q_{i,i - 1} &= \mu, \quad 1 \le i \le M. \end{align}
I assume you are interested in the probabilities in the stationary case. Let us call the equilibrium probabilities $\pi_i$. You are interested in $\pi_M$ and $\mathbb{E}[N]$.
Can you continue from here?