= Permanent collisions in constant-probability slotted ALOHA
{title2=$\mathbb P(\exists J:\ Z_t=*\ \forall t\geq J)=1$}
With independent Poisson arrivals of any positive rate and a fixed retry probability $0<f\leq1$, the infinite-population <slotted ALOHA> model eventually has collisions in every slot almost surely. For $f<1$, an <exponential-supermartingale escape bound> proves transience of the backlog chain. Bounded martingale increments then give backlog growth at the arrival rate, and the <Conditional Borel-Cantelli lemma> makes the exponentially unlikely noncollision slots finite in number. At $f=1$, two old packets suffice for permanent collisions. At $f=0$ the assertion is false.
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