Analysis of backoff protocols for multiple access channels

In this paper, we analyze the stochastic behavior of backoff protocols for multiple access channels such as the Ethernet. In particular, we prove that binary exponential backoff is unstable if the arrival rate of new messages at each station is $\tfrac{\lambda }{N}$ for any $\lambda > \frac{1}{2}$ and the number of stations $N$ is sufficiently large. For small $N$, we prove that $\lambda \geqslant \lambda _0 + \frac{1}{{4N - 2}}$` implies instability, where $\lambda _0 \approx .567$. More importantly, we also prove that any superlinear polynomial backoff protocol (e.g., quadratic backoff) is stable for any set of arrival rates that sum to less than one and any number of stations. The results significantly extend the previous work in the area and provide the first examples of acknowledgment-based protocols known to be stable for a nonnegligible overall arrival rate distributed over an arbitrarily large number of stations. The results also disprove a popular assumption that exponential backoff is the best choice among acknowledgment-based protocols for systems with large overall arrival rates. Finally, we prove that any linear or sublinear backoff protocol is unstable if the arrival rate at each station is $\frac{\lambda }{N}$ for any fixed $\lambda $ and sufficiently large $N$.