Towards Stability Criteria For Multidimensional Distributed Systems: Buffered Aloha Case

A distributed system can be viewed as a multidimensional, not necessarily Markovian, stochastic process over a large (typically infinite) state space. Assessing stability of such multidimensional systems is notoriously difficult. In this paper we consider the standard discrete-time slotted ALOHA system with a finite number of buffered users. The stability region for t1us system is known only for two users and for the symmetric system. We propose a new method of studying the stability of distributed systems including ALOHA by means of a simple concept of isolating single users, applying Loynes' stability criteria for an isolated queue, and using stochastic dominance to verify required stationarity assumptions in the Loynes' criterion. As a result, we derive sufficient conditions and necessary conditions for stability of the ALOHA system, and we also indicate that these conditions are sufficient and necessary. In fact, our method allows to assess stability of a subset of users in the ALOHA system. Such a stability we name partial stability, and it is of considerable jnterest to engineers. Finally, we generalize our approach to assess stability of a class of distributed systems using a more sophisticated extension of the ALOHA system. This generalized approach is next illustrated on coupled-processors systems. ·This research WiLS supported in part by llle NSF grants CCR-8900305 and NCR-8702115, by AFOSR gra.nl90-0107, and by grant ROl LM05118 from the Nalional Library of Medicine.