Problem: There are $n$ identical components in a system that operate independently. When a component fails, it undergoes repair, and after repair is placed back into the system. Assume that for a component the operating times between successive failures are i.i.d. exponential with mean $\frac{1}{\lambda}$, and that these are independent of the successive repair times, which are i.i.d. exponential with mean $\frac{1}{\mu}$. The state of the system is the number of components in operation. Determine the infinitesimal generator of this Markov process.

I try to find the transition probabilities $\mathbb{P}(X_t = j|X_s = i)$. First, consider the cases when $i>j$, which means more components failed in the time period than repaired. Let $k$ be the number of components that fails during $[s,t]$, then the rest of $i$ components didn't fail. Also $k - (i-j)$ components were repaired. I got this equation: $$ \begin{align} \mathbb{P}(X_t = j|X_s = i) &= p_{ij}(t-s)\\ &=\sum_{k=i-j}^{i}{i \choose k}(\int_0^{t-s}\lambda e^{-\lambda u}du)^k(\int_{t-s}^{\infty}\lambda e^{-\lambda u}du)^{i-k}{n-i+k \choose n-j}(\int_0^{t-s}\mu e^{-\mu u}du)^{k-(i-j)}(\int_{t-s}^{\infty}\mu e^{-\mu u}du)^{n-j} \end{align} $$ But this doesn't seem right, because then I get $q_{ij} = p'_{ij}(0) = 0$. Help is much appreciated.



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