= Finite-population exclusion of permanent ALOHA collisions
If a finite-population <slotted ALOHA> system has at most $M$ packets eligible to retry, uses independent retry probability $0<p<1$, and has conditional probability at least $\delta>0$ of no fresh attempts in any slot, then its conditional idle probability is at least $\delta(1-p)^M$. The probability that the next $k$ slots all collide is therefore at most $[1-\delta(1-p)^M]^k$. Letting $k\to\infty$ and taking a countable union over starting slots excludes eventual permanent collisions. Positive lower bounds on idle probability are essential: deterministic retries at probability one can create an absorbing collision state.
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