Existence of equilibrium in a continuous dynamic queueing model for traffic networks with responsive signal control

Many real-life traffic systems incorporate responsive traffic signal control, i.e. where the green time assigned to a stage at a junction depends on the queue lengths on the various approaches. By making signals responsive one might expect the queueing pattern to approach equilibrium, i.e. a queueing pattern for which the responsive policy tells us to leave the signal settings unchanged. However, changing junction signal settings changes the costs of traversing the approaches to the junction and traffic may change route in the light of this. Hence, a responsive signal system is really at equilibrium only if it is at equilibrium with respect to its own rules and also with respect to the re-routing of traffic. The paper gives a framework for responsive signal control in the dynamic queueing model in terms of stage pressures. Three responsive signal policies are considered: delay-minimisation, equisaturation and P-0. A dynamical system is specified that describes both changes to signals due to the responsive signal policy and changes to route inflows due to the re-routing of traffic. An implicit function theorem is utilised in showing that the swap vector for the dynamical system is a continuous function of the route flow vector and green time vector. Then by Schauder's fixed point theorem, there exists equilibrium of the dynamical system. Finally, the responsive policies are compared with fixed signals in network simulations.