Consider a telegraph system consisting of two independently operating wires, each capable of processing one message at a time. The time that each cable remains in operation until it breaks down is a random variable with exponential distribution with parameter $\lambda$. The repair time of each cable has an exponential distribution with parameter $\mu$, however, for each $t \geq 0$, we denote by $X(t)$ the number of cables operating at the same time. Define the stochastic process we are considering and its states, later:
a) Determine the infinitesimal generator of the chain
I've tried this:
The infinitesimal generator of the chain is: $$ Q= \begin{bmatrix} -\mu & \mu & 0 \\ \lambda & -(\lambda+\mu) & \mu \\ 0 & \lambda & -\lambda \\ \end{bmatrix} $$
Assuming that each wire is repaired independently, the matrix should be as follows:
$$ Q= \begin{bmatrix} -2\mu & 2\mu & 0 \\ \lambda & -(\lambda+\mu) & \mu \\ 0 & 2\lambda & -2\lambda \\ \end{bmatrix} $$
Assuming that only one wire can be repaired at a time, then only the third row would differ. Otherwise you are correct.