Day-to-day Flow Dynamics With User Learning Under Stochastic User Equilibrium

View from the angle of transportation management, it is important to understand the traffic flow evolution process toward user equilibrium (deterministic or stochastic). The day-to-day traffic modeling approach has great flexibility to describe the flow evolution process when it includes a wide range of assumed behavior rules into consideration. In the previous research of day-to-day models, travelers’ past experience have important impact on their cost prediction and consequently determines their route choice. Xiao et al (2016)constructed a second-order flow-based day-to-day dynamics from the combination of travelers’learning updating process and flow switching process. The second-order flow evolution was analyzed by analogizing to the spring damping system:the potential energy and kinetic energy of a traffic network were defined and the intrinsic parameters such as "travelers’sensitivity", "memory decay rate" were discussed. All of the study, however, has been restricted to the case of a system which converges to a deterministic user equilibrium (DUE). The purpose of the present paper is to extend the above model to the case of dynamical system converging to stochastic user equilibrium (SUE) in order to capture the drivers’ perceptual errors in evaluating of travel cost. The stochastic user equilibrium model introduces the randomness in the route-choice process assuming that drivers commit some estimation errors while choosing the shortest routes. In this paper, we provide a new day-to-day path flow dynamical system to describe traveler’s route choice behavior more precisely. We show that stationary points of this system are stochastic user equilibrium. A different energy analogy is introduced to understand the flow dynamics better. By regarding the total energy as the Lyapunov function, we assign an physical meaning to objective function of minimum programming problem of model presented in Fisk(1980). And we show that the dynamical system will approach the UE path flow pattern eventually with the defined total energy decreasing to its minimum. And then, by numerical experiments, the effects of the parameters in the model, such as the sensitivity parameter, the memory decay rate and the dispersion parameter, will be discussed. In reality, the link-based flow data are always easier to obtain than the path-based flow data. Due to this reason, we also formulate a link-based day-to-day model considering travelers’ learning behavior and explore some theoretical properties of this model. We prove that the model will finally evolve to UE point no matter what initial point is set. At last, we designed a virtual experiment to simulate the day-to-day route choice process with 268 participants in 26 days. Based on the realized day-to-day path flows and travel times, some well-established day-to-day models in the literature are calibrated and compared. The parameter fitting of the new model proposed in this paper is conducted, showing that incorporating stochastic factors in the second-order day-to-day model could improve the explanation power.