Day-to-day flow dynamics with user learning

Xiao et al. (2013) constructed a second-order flow-based day-to-day dynamics derived from a combined model of flow and perceived cost. The second-order dynamics was analyzed by analogizing to a damped oscillation system: “mass” and “energy” of the day-to-day dynamic traffic network was defined, the intrinsic properties such as “angular frequency”, “damping ratio” and system stability were discussed. However, discussion on the possible negative path flows was excluded in their paper: the nonnegativity of path flows was assured by a strong assumption but not the flow evolution process itself. This paper relaxes the assumption of positive path flows in the day-to-day model proposed by Xiao et al. (2013) which simultaneously considered travelers’ learning and route switching behavior. With nonnegative-path-flow constraints, evolution trajectories of the day-to-day dynamics become discontinuous. As a result, the flow dynamics with regard to flows and perceived costs cannot be converted to an equivalent second-order ordinary equation set anymore, and the analogy in Xiao et al. (2013) cannot be conducted. With separable link travel time functions, we show that the dynamic path flows still converge to Wardrop’s user equilibrium path flow set. Rigorous proof is established based on a generalized invariance theorem with a piecewise continuous Lyapunov function. Stability and some other system properties are examined by numerical examples.