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The paper investigates the throughput behavior of single-commodity dynamicalflow networks governed by monotone distributed routing policies. The networksare modeled as systems of ODEs based on mass conversation laws on directedgraphs with limited flow capacities on the links and constant external inflowsat certain origin nodes. Under monotonicity assumptions on the routingpolicies, it is proven that a globally asymptotically stable equilibrium existsso that the network achieves maximal throughput, provided that no cut capacityconstraint is violated by the external inflows. On the contrary, should such aconstraint be violated, the network overload behavior is characterized. Inparticular, it is established that there exists a cut with respect to which theflow densities on every link grow linearly over time (resp. reach theirrespective limits simultaneously) in the case where the buffer capacities areinfinite (resp. finite). The results employ an $l_1$-contraction principle formonotone dynamical systems.

Throughput Optimality and Overload Behavior of Dynamical Flow Networks under Monotone Distributed Routing