We elaborate upon the Herty-Illner macroscopic traffic models which include special non-local forces. The first chapter presents these in relation to the traffic models of Aw-Rascle and Zhang, arguing that non-local forces are necessary for a realistic description of traffic. The second chapter considers travelling wave solutions for the Herty-Illner macroscopic models. The travelling wave ansatz for the braking scenario reveals a curiously implicit nonlinear functional differential equation, the jam equation, whose unknown is, at least to conventional tools, inextricably self-argumentative! Observing that analytic solution methods fail for the jam equation yet succeed for equations with similar coefficients raises a challenging problem of pure and applied mathematical interest. An unjam equation analogous to the jam equation explored by Illner and McGregor is derived. The third chapter outlines refinements for the Herty-Illner models. Numerics allow exploration of the refined model dynamics in a variety of realistic traffic situations, leading to a discussion of the broadened applicability conferred by the refinements: ultimately the prediction of stop-and-go waves. The conclusion asserts that all of the above contribute knowledge pertinent to traffic control for reduced congestion and ameliorated vehicular flow. / Graduate

The Aw-Rascle-Zhang (ARZ) model has become a favorable ``second order" macroscopic traffic model, which corrects several shortcomings of the Payne-Whitham (PW) model. The ARZ model possesses a family of flow rate versus density (FD) curves, rather than a single one as in the ``first order" Lighthill-Whitham-Richards (LWR) model. This is more realistic especially during congested traffic state, where the historic fundamental diagram data points are observed to be set-valued. However, the ARZ model also possesses some obvious shortcomings, e.g., it assumes multiple maximum traffic densities which should be a ``property" of road. Instead, we propose a Generalized ARZ (GARZ) model under the generic framework of ``second order" macroscopic models to overcome the drawbacks of the ARZ model. A systematic approach is presented to design generic ``second order" models from historic data, e.g., we construct a family of flow rate curves by fitting with data. Based on the GARZ model, we then propose a phase-transition-like model that allows the flow rate curves to coincide in the free flow regime. The resulting model is called Collapsed GARZ (CGARZ) model. The CGARZ model keeps the flavor of phase transition models in the sense that it assume a single FD function in the free-flow phase. However, one should note that there is no real phase transition in the CGARZ model. To investigate to which extent the new generic ``second order" models (GARZ, CGARZ) improve the prediction accuracy of macroscopic models, we perform a comparison of the proposed models with two types of LWR models and their ``second order" generalizations, given by the ARZ model, via a three-detector problem test. In this test framework, the initial and boundary conditions are derived from real traffic data. In terms of using historic traffic data, a statistical technique, the so-called kernel density estimation, is applied to obtain density and velocity distributions from trajectory data, and a cubic interpolation is employed to formulate boundary condition from single-loop sensor data. Moreover, a relaxation term is added to the momentum equation of selected ``second order" models to address further unrealistic aspects of homogeneous models. Using these inhomogeneous ``second order" models, we study which choices of the relaxation term &tau are realistic. / Mathematics