This question asks about a variant of an alternating renewal process.
I am sitting at a cafe watching men and women walk by. The interarrival time $X$ between successive men is iid with distribution $F$, while the interarrival time $Y$ between successive women is iid with distribution $G$. Both $F$ and $G$ are nonlattice. Unlike the usual alternate renewal process, here we have two independent renewal processes, one for each sex.
Is there a nice way to compute the asymptotic probability that the most recent person seen is a man? (The usual theorem doesn't quite apply, since the interarrival times are within-gender only, not from person to person independent of gender.)
I'm interested both in general, and for the specific case that F and G are gamma distributions...