Mean delay under independent exponential-backoff failures ($\mathbb{E}T=[p_0(1-2q){]}^{-1}(q<1/2)$)

For geometric-attempt exponential backoff, assume each attempted transmission fails independently with a fixed probability $q<1$. The probability of reaching stage $k$ is $q^k$, and its mean waiting time is $2^k/p_0$. The Tonelli theorem gives $\mathbb{E}T=p_0^{-1}{\sum }_{k\ge 0}(2q{)}^k$, finite exactly when $q<1/2$. The packet succeeds almost surely for every $q<1$, so almost-sure success does not guarantee finite mean delay. A fixed independent failure probability is a simplifying assumption, not a theorem about collisions in an interacting network.

// Target: probability-and-statistics.bigb