On Generalized Proportional Allocation Policies for Traffic Signal Control

The fast-increasing demand and relatively slow growth of infrastructure capacity are providing a strong motivation for research in real-time urban traffic controls that make the best use of novel sensing in order to increase efficiency and resilience of the transportation system. In our contribution, we focus on a class of dynamic feedback traffic signal control policies that are based on a generalized proportional allocation rule. The proposed traffic signal controls are decentralized (they make use of local information only), scalable (they are independent of the network size and topology), and universal (they do not rely on any information about external inflows or turning ratios). In spite of their fully distributed nature, we prove that such control policies achieve a global objective, maximum throughput, in that they stabilize the urban traffic network whenever possible under the given capacity constraints. The traffic model we consider consists in a network of interconnected vertical queues with deterministic dynamics driven by physical laws (conservation of mass and preservation of non-negativity of the traffic volumes) as well as scheduling constraints (described as a set of phases, each phase consisting in a subset of lanes that can be be given green light simultaneously). This results in a differential inclusion for which we prove existence and, in the special case of orthogonal phases, uniqueness of continuous solutions via a generalization of the reflection principle. Stability is then proved by interpreting the generalized proportional allocation controllers as minimizers of a certain entropy-like function that is then used as a Lyapunov function for the closed-loop system.