= Mean delay under independent exponential-backoff failures
{title2=$\mathbb ET=[p_0(1-2q)]^{-1}\quad(q<1/2)$}
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 $\mathbb ET=p_0^{-1}\sum_{k\ge0}(2q)^k$, 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
Back to article page
