  We analyze the stability of standard, buffered, slotted-Aloha systems.
Specifically, we consider a set of $N$ users, each equipped with an infinite
buffer. Packets arrive into user $i$'s buffer according to some stationary
ergodic Markovian process of intensity $\lambda_i$. At the beginning of each
slot, if user $i$ has packets in its buffer, it attempts to transmit a packet
with fixed probability $p_i$ over 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 users $N$ grows 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). We finally extend the
results to the case of more efficient CSMA systems.
