= Alternating split of a Poisson process
{title2=$X_B(t)=\lceil N(t)/2\rceil,\quad X_C(t)=\lfloor N(t)/2\rfloor$}
Alternately routing a rate-$r$ <Poisson process> produces non-Poisson streams with common interarrival law Erlang$(2,r)$. The odd-numbered stream has an initial exponential delay, while the even-numbered stream is an ordinary renewal process. Their equilibrium delay survivor is $e^{-rt}(1+rt/2)$. The probability of no odd-stream point in $(s,s+t]$ is $e^{-rt}[1+rt(1-e^{-2rs})/2]$; for the even stream replace the minus by a plus. Conditioning on the parity of $N(s)$ proves these formulas.
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