ALOHA throughput bound (${\max }_{p}Np(1-p{)}^{N-1}=(1-1/N{)}^{N-1}$)

For $N\ge 2$ backlogged stations, the largest one-slot success probability is $(1-1/N{)}^{N-1}$, attained at $p=1/N$. Its limit is $e^{-1}$; the exact finite-$N$ maximum is larger, and for $N=1$ it is one.