Global asymptotic stability of max-min congestion control schemes

A number of congestion control schemes which adopt the max-min fairness criterion and do not require maintenance of per flow states within the network have been proposed in the literature. The establishment of global asymptotic stability remains an open challenging research topic. In this paper, we show global asymptotic stability of two decentralized max-min congestion controllers in the absence and presence of queueing dynamics when the propagation delays are assumed to be zero. In particular, we show that these max-min congestion control schemes applied to a network of arbitrary topology, guarantee that the user sending rates converge asymptotically to the max-min allocation values for any arbitrary feasible initial condition. The second algorithm which accounts for queueing dynamics, typical in store and forward networks, does not only guarantee convergence of the sending rates to their max-min allocation values but also guarantees that the queue sizes within the network all converge to zero when there is only one bottleneck link at each path.