Stability of the Max-Weight Routing Protocol in Dynamic Networks and and Scheduling at Critical Loads

We study the stability of the MAX-WEIGHT protocol for combined routing and scheduling in communication networks Previous work has shown that this protocol is stable or a versarial multicommodity traffic in subcritically loaded static networks and for single-commodity traffic in critically loaded dynamic networks. We show: The MAX-WEIGHT protocol is stable for adversarial multicommodity traffic in adversarial dynamic networks whenever the network is subcritically loaded. The MAX-WEIGHT protocol is stable for fixed multicommodity traffic in fixed networks even if the network is critically loaded. The latter result has implications for the running time of the MAX-WEIGHT protocol when it is used to solve multicommodity flow problems. In particular, for a fixed problem instance we show that if the value of the optimum solution is known, the MAX-WEIGHT protocol finds a flow that is within a (1 - epsilon)-factor of optimal in time O(1/epsilon) (improving the previous bound of O(1/epsilon(2))). If the value of the optimum solution is not known, we show how to apply the MAX-WEIGHT algorithm in a binary search procedure that runs in O(1/epsilon) time.