The S-Aloha capacity: Beyond the e−1 myth

The stability and throughput of the Slotted Aloha protocol have been studied at length, yielding results that depend on the environment and channel assumptions, in many cases indicating e-1 as the S-Aloha capacity. When users can detect only their own collisions, and the number of users N goes to infinity, no definite capacity result exists. Approximated models have been introduced to study the exponential back-off mechanism, which seem to indicate an asymptotic capacity of ln(2)/2 when binary back-off is used, and again e-1 when the exponential base is optimized. Here we introduce a more accurate and flexible model that shows that past results miss their mark. In fact, we prove that with binary back-off the capacity is practically 0.370, slightly greater than e-1; furthermore, and more important, we prove that using 1.35 as exponential back-off base, the capacity reaches 0.4303 with an infinite number of users, and up to 0.496 with N = 2 users.