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

TL;DR

This paper analyzes how monotone distributed routing policies affect the throughput and overload behavior of dynamical flow networks, proving conditions for optimal throughput and characterizing overload scenarios.

Contribution

It establishes the existence of globally stable equilibria for maximal throughput under monotone routing and characterizes overload dynamics when capacity constraints are violated.

Findings

Networks achieve maximal throughput under certain conditions.

Overload leads to linear growth in flow densities on network links.

Results are derived using an $l_1$-contraction principle for monotone systems.

Abstract

The paper investigates the throughput behavior of single-commodity dynamical flow networks governed by monotone distributed routing policies. The networks are modeled as systems of ODEs based on mass conversation laws on directed graphs with limited flow capacities on the links and constant external inflows at certain origin nodes. Under monotonicity assumptions on the routing policies, it is proven that a globally asymptotically stable equilibrium exists so that the network achieves maximal throughput, provided that no cut capacity constraint is violated by the external inflows. On the contrary, should such a constraint be violated, the network overload behavior is characterized. In particular, it is established that there exists a cut with respect to which the flow densities on every link grow linearly over time (resp. reach their respective limits simultaneously) in the case where the buffer capacities are infinite (resp. finite). The results employ an $l_1$-contraction principle for monotone dynamical systems.