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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...

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    `the probability that the last person seen is a man` is not clear for me. It's the probability that the most recent person is a man given... what? The past arrival times? Labeled?2011-07-03
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    You say "we have two independent renewal processes". That is quite a restriction, and is not implied by your earlier words.2011-07-03
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    Sorry if the statement wasn't clear. Let S_i be the arrival time of the i-th man, where S_i = S_{i-1} + X_i, and X_i is an independent variable with distribution F. (Take S_0 = 0.) Here F is a given distribution, such as a Gamma distribution with parameters k=2 and Theta=3. The arrival times for the women satisfy a similar distribution, with interarrival times Y_i satisfying another distribution G, such as a Gamma distribution with parameters k=4 and Theta=5. The men and women are arriving independently of each other; there is no interaction.2011-07-03
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    Additional clarification: Let $P(t)$ be the event that at time $t$ the last person to walk by was a man. The quantity of interest is then $\lim_{t\rightarrow\infty} P(t)$, assuming this is defined.2011-07-03

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