Geometric-attempt exponential backoff ($p_k=p_0{2}^{-k},0<p_0<1$)

A packet with $k$ previous collisions attempts independently in each slot with probability $p_k=p_0{2}^{-k}$. On failure it advances to stage $k+1$, and on success it leaves. The Markov chain state records the number of packets at every stage, rather than only the total backlog. This memoryless-window model is related to binary exponential backoff, but its geometric waiting times differ from uniform counters in finite retry windows.

// Target: probability-and-statistics.bigb

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