This dissertation proposes two new models and discusses two applications of day-to-day dynamic models. A generalized invariance theorem is adopted to investigate the stability issues of these models when the widely-used Lyapunov’s second theorem and Lasalle’s invariance principle are not applicable. Flow dynamics incorporating tradable credit scheme and congestion pricing scheme are investigated. The effectiveness of both schemes can be assured under mild conditions.

The first model combines traveler’s perceiving, learning and route switching behavior while assuring nonnegative flows. With separable link travel time functions, the dynamic path flows converge to the Wardrop’s user equilibrium path flow set. In the second model, the rational behavior adjustment process in Yang and Zhang...[ Read more ]

This dissertation proposes two new models and discusses two applications of day-to-day dynamic models. A generalized invariance theorem is adopted to investigate the stability issues of these models when the widely-used Lyapunov’s second theorem and Lasalle’s invariance principle are not applicable. Flow dynamics incorporating tradable credit scheme and congestion pricing scheme are investigated. The effectiveness of both schemes can be assured under mild conditions.

The first model combines traveler’s perceiving, learning and route switching behavior while assuring nonnegative flows. With separable link travel time functions, the dynamic path flows converge to the Wardrop’s user equilibrium path flow set. In the second model, the rational behavior adjustment process in Yang and Zhang (2009) is extended to incorporate boundedly rational user equilibrium. Specific models are constructed and numerical examples are conducted for demonstration.

The first application of day-to-day dynamics is to investigate the price and flow dynamics of a tradable credit scheme. A continuous dynamic model in a finite time horizon is proposed to describe travelers’ learning behavior and the evolution of network flows and credit price. Existence and uniqueness of the equilibria are established. The conditions for stability and convergence of the dynamic system as the time horizon extends to infinity and the impact of limited implementation time horizon on the system behavior are investigated.

The second application examines the availability of a trial-and-error toll scheme with day-to-day flow dynamics. The trial-and-error method was proposed in Yang et al. (2004) for discovering system optimal flow pattern and toll scheme with the absence of demand functions: the links tolls were firstly calculated based on some target flow pattern and then imposed on the network, after which the user equilibrium flow pattern was instantly reached and utilized to update the target flows. In this dissertation, the assumption on the instantaneous realization of user equilibrium in Yang et al. (2004) is relaxed and their iterative scheme is extended for implementing the road pricing in a traffic network with day-to-day flow dynamics and evolution. The path flows are assumed to evolve following the “excess travel cost dynamics” during each inter-trial period and the user equilibrium state may not be achieved at the end of each inter-trial period. With mild assumptions on the flow evolution process, the trial-and-error method is still applicable to identify the system optimal link tolls and decentralize the system optimal flows, while neither the demand functions nor the mechanism of the flow evolution are explicitly required. A methodology is developed for both updating the toll charges and choosing the inter-trial periods to assure the iterative approach converging towards the system optimum. Some numerical examples are conducted to support the theoretical findings.