Collision-path coupling for slotted ALOHA (${\mathbb{P}}_b(\text{a future noncollision})\le \epsilon (b),\epsilon (b)\to 0$)

In slotted ALOHA, let old packets retry independently with fixed probability $0<p<1$, and let independent Poisson random variables of mean $\nu >0$ transmit once on arrival. Write $q=1-p$ and $r=e^{-\nu p}$. A hypothetical process which never removes a packet has backlog $b+A_t$, where $A_t$ is the cumulative arrival count over $t$ slots. Couple its attempts to those of the actual process until the first noncollision. Its one-slot noncollision probability at backlog $n$ is $h(n)=e^{-\nu }[(1+\nu )q^n+np{q}^{n-1}]$. The union bound and the Poisson distribution giveIndeed $\mathbb{E}[q^{A_t}]=r^t$ and $\mathbb{E}[A_t{q}^{A_t}]=\nu tq{r}^t$. Thus $\epsilon (b)\to 0$. From every sufficiently large backlog, permanent collisions have probability at least $1/2$. Independent arrival batches reach that threshold almost surely; after each failed attempt the Strong Markov property permits another attempt. The probability of $k$ consecutive failures is at most $2^{-k}$, proving almost-sure eventual permanent collisions without assuming in advance that the backlog escapes.