Second-order fluid dynamics models for travel times in dynamic transportation networks

Concerns traffic congestion, especially in intelligent vehicle highway systems (IVHS); addressing such questions as how traffic patterns form and how congestion can be alleviated. Understanding drivers' travel times is key to this problem. We present macroscopic models for determining analytical forms for travel times. We take a fluid dynamics approach. We propose two second-order nonseparable macroscopic models for analytically estimating travel time functions : the polynomial travel time (PTT) model and the exponential travel time (ETT) model. These models incorporate second-order effects such as reaction of drivers to upstream and downstream congestion. Based on piecewise linear and piecewise quadratic approximations of the departure flow rates, we propose different classes of travel time functions for the first-order separable PTT and ETT models, and present the relationship between these functions. We show how the analysis of the first-order separable PTT model extends to the second-order model with nonseparable velocity functions for acyclic networks. We analyze the second-order separable ETT model where the queue propagation term is not neglected. We thus reduce the analysis to a Burgers equation and then to the heat equation.