On the Stability of Dynamical Multi-Commodity Flow Networks

TL;DR

This paper analyzes the stability of dynamical multi-commodity flow networks in transportation, establishing conditions for stable free-flow states and exploring the emergence of non-free flow equilibria.

Contribution

It introduces a capacity region for stable free-flow equilibria and provides a basin of attraction estimate using contraction arguments.

Findings

Existence of a convex capacity region for stability.

Stable free-flow equilibrium points are locally asymptotically stable within this region.

Non-free flow equilibria can occur outside the stability region.

Abstract

We study a class of dynamical multi-commodity flow networks in transportation networks. These are modeled as dynamical systems describing the evolution of the densities of a number of different commodities across the cells of a transportation network. Each cell is characterized by commodity-specific increasing demand functions returning the maximum outflow of each commodity from the cell as a function of the current density of that commodity, as well as a decreasing supply function returning the total maximum inflow that is allowed in the cell as a function of the current aggregate density in the cell. Every commodity is characterized by a different routing matrix, whose entries describe the turning ratios between adjacent cells. We identify a (typically convex) capacity region: for exogenous inflow vectors belonging to that region, we prove the existence of a locally asymptotically stable free-flow equilibrium point. Building on a contraction argument, we also provide an estimate of the basin of attraction of such free-flow equilibrium point. Finally, we analyze a simple special case showing that, when the exogenous inflow vector does not belong to the region of stability, non-free flow equilibrium points might arise.