= Backlog-aware ALOHA with arrival-adjusted attempt probabilities
{title2=$p_b=\min\{1,(1-\eta)/b\}$}
In infinite-population <slotted ALOHA>, suppose new arrivals form an independent Poisson variable of mean $\eta$ per slot and attempt immediately. At backlog $b\ge1$, let old packets attempt independently with probability $p_b$. Their total attempted load tends to $1-\eta$, so the total offered attempts tend to one and the success probability tends to $e^{-1}$. For $0<\eta<e^{-1}$ the backlog has strictly negative drift outside a finite set, and <Foster theorem> proves a <positive recurrent Markov chain>. This mathematical benchmark requires exact backlog information; a practical feedback estimate needs its own analysis.
// Target: probability-and-statistics.bigb
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