Day-to-Day Road Pricing and Network Performance Analysis via Output Stability

In this paper, we devise three day-to-day (DTD) road pricing schemes from their static counterparts to achieve several traffic management objectives such as regulating the traffic dynamics to the desired equilibrium, traffic restraint control, and balancing the fairness and efficiency of traffic networks, respectively. Qualitative analyses involved in the stability and convergence analysis as well as the DTD evolution of network performance admit challenges ranging from partial state stability, multiple equilibria, non-measurable system states to complex nonlinear network performance functionals. To handle these hurdles, we establish a unified stability analysis framework for qualitative analysis and control synthesis of continuous-time DTD traffic dynamics via the notion of output stability. Under this framework, by devising a proper measurable output function for the DTD dynamics in question, we can perform the relevant qualitative analyses for several continuous-time DTD traffic dynamics under different traffic control measures. For example, we investigate the DTD evolution of network performance indexes such as unfairness and inefficiency (e.g., price of anarchy (PoA)) to show how the proposed flow-dependent link toll scheme can balance the fairness and efficiency of traffic systems. We demonstrate that the proposed output stability framework can offer much more flexibility to perform qualitative analysis for these complex processes by devising suitable output functions. Finally, we show the merits of the road pricing schemes in conjunction with the output stability via numerical simulations.