Ergodicity of M-dimensional random walks and random access systems

We apply a recent set of results of Mensikov and Malysev that concern necessary or sufficient conditions for ergodicity of constrained M-dimensional random walks to the problem of stability of M coupled queueing systems that describe a system of M buffered terminals accessing a common channel by means of a discrete-time ALOHA protocol. We obtain a necessary and sufficient condition for the stability of such a system. Although the condition does not yield a descriptive characterization of the stability region (because it is stated in terms of joined queue size distribution), it allows a reduction of the stability problem of a M-user system to the determination of the steady state distribution of a (M - 1)-user system.