Well, it started with an homework exercise, but this is all about my curiosity, because in my opinion is a great thing trying to solve problems in different ways.

It's too complicated

No, here I'm asking another probability. I want to know the probability that the customer B is still in the system when $S_1$ finishes to serve $X$. In that question we wondered the $P(T_1 < T_2)$, here we are asking for $P(T_A+ T_B > T_1 )$

In this case, given two service times, I need to have T1 < T2. But from the theory I know that the probability $P(X_1 < X_2)$ converges to $\frac{\mu_1}{\mu_1 + \mu_2}$. Thus is there therefore the possibility that those integrals converge to this value? Thank you

Then I misunderstood the problem, since I supposed that the N in $X_N$ was only an index

Sorry, my fault.