Stability of the Multiple-Access Channel Under Maximum Broadcast Loads

Abstract

We investigate deterministic broadcasting on multiple-access channels in the framework of adversarial queuing. A protocol is stable when the number of packets stays bounded, and it is fair when each packet is eventually broadcast. We address the question if stability and fairness can be achieved against the maximum injection rate of one packet per round. We study three natural classes of protocols: acknowledgment based, full sensing and fully adaptive. We show that no adaptive protocol can be both stable and fair for the system of at least two stations against leaky-bucket adversaries, while this is achievable against window adversaries. We study in detail small systems of exactly two and three stations attached to the channel. For two stations, we show that bounded latency can be achieved by a full-sensing protocol, while there is no stable acknowledgment-based protocol. For three stations, we show that bounded latency can be achieved by an adaptive protocol, while there is no stable full-sensing protocol. We develop an adaptive protocol that is stable for any number of stations against leaky-bucket adversaries. The protocol has ${\mathcal O}(n^2)$ packets queued simultaneously, which is proved to be best possible as an upper bound. We show that protocols that do not use queue sizes at stations in an effective way or are greedy by having stations with nonempty queues withhold the channel cannot be stable in systems of at least four stations.