Asymptotic Stability Region of Slotted Aloha

We analyze the stability of standard, buffered, slotted-Aloha systems. Specifically, we consider a set ofNusers, each equipped with an infinite buffer. Packets arrive into useri's buffer according to some stationary ergodic Markovian process of intensityλᵢ. At the beginning of each slot, if userihas packets in its buffer, it attempts to transmit a packet with fixed probabilitypᵢover a shared resource/channel. The transmission is successful only when no other user attempts to use the channel. The stability of such systems has been open since their very first analysis in 1979 by Tsybakov and Mikhailov. In this paper, we propose an approximate stability condition that is provably exact when the number of usersNgrows large. We provide theoretical evidence and numerical experiments to explain why the proposed approximate stability condition is extremely accurate even for systems with a restricted number of users (even two or three).