Alice is quite popular. She gets called on her cell-phone in a Poisson$(\lambda)$ manner. She answers her calls when possible, and ignores them when in the middle of conversation. Since you know her personally, you know that the length of her conversation is distributed by r.v. $X$, independently of the discussion/person.
Now, the first phone-call she receives will surely get through to her w.p. 1. However after some time both distributions (exponential and $X$) will "mix", and also the probability that some call gets through to her should start converging towards some value $p$. I'm most interested in $p$ and in the so-called "mixing time".
The answer should be in terms of moments (mean, variance, etc.) of $X$, and of course $\lambda$. But if the question is too general, I'd also be happy with some specific answer, say when also $X$ is $\lambda'$-exponentially distributed.