The stability region of the finite-user slotted ALOHA protocol

A version of the discrete-time slotted ALOHA protocol operating with finitely many buffered terminals is considered. The stability region is defined to be the set of vectors of arrival rates lambda =( lambda ₁,. . ., lambda _M) for which there exists a vector of transmission probabilities such that the system is stable. It is assumed that arrivals are independent from slot to slot, and the following model for the arrival distribution in a slot is assumed: the total number of arrivals in any slots is geometrically distributed, with the probability that such an arrival is at node i being lambda _i times ( Sigma _k lambda _k)^-1, independent of the others. With this arrival model, it is proven that the closure of the stability region of the protocol is the same as the closure of the Shannon capacity region of the collision channel without feedback, as determined by J.L. Massey and P. Mathys (1985). At present it is not clear if this result depends on the choice of arrival distribution. The basic probabilistic observation is that the stationary distribution and certain conditional distributions derived from it have positive correlations for bounded increasing functions.