A multi-commodity dynamical model for traffic networks
A dynamical model for traffic networks is proposed and analyzed. In the traffic network, the transportation demands are considered as multi-commodity flows where each commodity has a unique destination. The network is modeled by a multigraph where at each node each commodity splits among the outgoing links in a way such that the drivers are more likely to avoid a road when the density on it increases. It will be shown that if the graph has no cycles, the density of each commodity on each link will converge to a unique limit that does not depend on the initial state.
Network resilience, namely structural robustness of the network with respect to perturbations, is also studied. In particular, it is shown that if all commodities have access to all outgoing links, the network can manage perturbations whose magnitude is less than a quantity which plays the natural role of residual capacity of an equilibrium. If instead not all commodities have access to all links, overreaction of the network to perturbations implies that even small perturbations might be amplified
and start a cascade.
Finally, the idea of back-pressure is employed to provide a simple distributed control strategy. Analogously to the single commodity case, such actual strategy is able to back-propagate the information that congestion is happening ahead, thus allowing the drivers to reroute even if their decision is based on local information only.
Master's Thesis ISRN LUTFD2/TFRT--5925--SE, Department of Automatic Control, Lund University, Sweden, 2013.