 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.