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 literature. The stability of some of these schemes in networks of arbitrary topology, has been established analytically only for small deviations about the equilibrium point. The establishment of global asymptotic stability remained an open challenging research topic. In this paper we show global asymptotic stability for a decentralized max-min congestion controller which does not maintain per flow states within the network. In particular, we show that a max-min congestion control scheme applied to a network of arbitrary topology, guarantees that the user sending rates converge asymptotically to the max-min allocation values for any arbitrary feasible initial condition. The paper resolves a long standing research problem in this field.