Stability of N interacting queues in random-access systems

We revisit the stability problem of systems consisting of N buffered terminals accessing a common receiver over the collision channel by means of the standard ALOHA protocol. We find that in the slotted ALOHA system queues have "instability rank" based on their individual average arrival rates and transmission probabilities. If a queue is stable, then the queue with lower instability rank is stable as well. The instability rank is used to intelligently set up the dominant systems. And the stability inner and outer bounds can be found by bounding the idle probability of some queues in the dominant system. Through analyzing those dominant systems one by one, we are able to obtain inner and outer bounds for stability. These bounds are tighter than the known ones although they still fail to identify the exact stability region for cases of N>2. The methodology used is new and holds promise for successfully addressing other similar stability problems.