The stability region of the finite-user slotted ALOHA protocol

Consideration is given to a version of the discrete-time slotted ALOHA protocol operating with finitely many buffered terminals. The stability region is defined to be the set of vectors of arrival rates λ=(λ₁, . . ., λ_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 that the total number or arrivals in any slot is geometrically distributed, with the probability that such an arrival is at node i being λ_i/Σλ_k, over all k, independent of others. With this arrival model it is proved that the closure of the stability region of the protocol is the same as the Shannon capacity region of the collision channel without feedback. The basic probabilistic observation is that the stationary distribution and certain conditional distributions derived from it have positive correlations for bounded increasing functions. Similar techniques may be of use in studying other interacting system of queues. At present it is not clear if the result depends on the choice of arrival distribution