Asymptotic stability region of slotted-Aloha

TL;DR

This paper derives an approximate stability condition for slotted-Aloha systems with many users, proving its accuracy through theory and simulations, and extends the results to CSMA systems.

Contribution

It introduces a new approximate stability criterion for large N, proven to be exact asymptotically, and demonstrates its accuracy for small systems and extends to CSMA.

Findings

The approximate stability condition is exact as N grows large.

Numerical experiments show high accuracy even for small N.

Results extend to more efficient CSMA protocols.

Abstract

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.