Throughput optimal distributed routing in dynamical flow networks

A class of distributed routing policies is shown to be throughput optimal for single-commodity dynamical flow networks. The latter are modeled as systems of ODEs based on mass conservation laws on directed graphs with maximum flow capacities on links and constant external inflow at some origin nodes. Distributed routing regulates the flow splitting at each node, as a function of information on the densities of the local links around the nodes. Under monotonicity properties of routing, it is proven that, if no cut capacity constraint is violated by the external inflow, then a globally asymptotically stable equilibrium exists and the network achieves maximal throughput. This holds for finite or infinite buffer capacities for the densities. The overload behavior, if any cut capacity constraint is violated, is also characterized: there exists a cut on which the link densities grow linearly in time for infinite buffer capacities, while they simultaneously reach their respective buffer capacities, when these are finite. Numerical simulations illustrate and confirm the theoretical contributions.