THEORETICAL FOUNDATION OF MACROSCOPIC TRAFFIC MODELS

On the basis of the elementary laws of individual driver behavior concerning the acceleration and interaction of vehicles a gas-kinetic traffic model is constructed. This yields theoretical relations for the fundamental diagram and the equilibrium variance-density relation, but above all it allows the derivation of macroscopic traffic equations. These build a hierarchy of non-closed equations which can be closed by different zeroth-order approximations. In this way one obtains the traffic equations of Lighthill and Whitham or those of Phillips. Alternatively one can derive Euler-like traffic equations which are proposed here since they include additional dynamical equations for the velocity variance. The new variance equation is able to describe the empirically observed increase of variance directly before a traffic jam develops.

[1]  Wilbur S. Smith,et al.  Transportation planning and technology , 1972 .

[2]  M Cremer,et al.  A fast simulation model for traffic flow on the basis of Boolean operations , 1986 .

[3]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : notes added in 1951 , 1951 .

[4]  D. Helbing.  High-fidelity macroscopic traffic equations , 1995 .

[5]  Warren F. Phillips,et al.  A kinetic model for traffic flow with continuum implications , 1979 .

[6]  E B Lieberman,et al.  DEVELOPMENT AND TESTING OF INTRAS, A MICROSCOPIC FREEWAY SIMULATION MODEL. VOLUME I. PROGRAM DESIGN AND PARAMETER CALIBRATION , 1976 .

[7]  Kerner,et al.  Structure and parameters of clusters in traffic flow. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  W. R. Buckland.  Theory of Traffic Flow , 1961 .

[9]  Joel Keizer,et al.  Statistical Thermodynamics of Nonequilibrium Processes , 1987 .

[10]  Helbing.  Improved fluid-dynamic model for vehicular traffic. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Robert Herman,et al.  Kinetic theory of vehicular traffic , 1971 .

[12]  Nagatani.  Jamming transition in the traffic-flow model with two-level crossings. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  S. L. Paveri-Fontana,et al.  On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis , 1975 .

[14]  Robert Herman,et al.  Traffic Dynamics: Analysis of Stability in Car Following , 1959 .

[15]  Carlos F. Daganzo,et al.  TRANSPORTATION AND TRAFFIC THEORY , 1993 .

[16]  A J Santiago,et al.  IDENTICAL TRAFFIC STREAMS IN THE TRAF-NETSIM SIMULATION PROGRAM , 1990 .

[17]  Ajay K. Rathi,et al.  THE NEW NETSIM SIMULATION PROGRAM , 1990 .

[18]  D. Gazis,et al.  Nonlinear Follow-the-Leader Models of Traffic Flow , 1961 .

[19]  Peter M. A. Sloot,et al.  Modelling and Simulation , 1988, Systems Analysis and Simulation 1988, I: Theory and Foundations. Proceedings of the International Symposium held in Berlin (GDR), September 12–16, 1988.

[20]  D. Haar,et al.  Statistical Physics , 1971, Nature.

[21]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[22]  E. Alberti,et al.  Contributions to the Boltzmann-like approach for traffic flow—A model for concentration dependent driving programs , 1978 .

[23]  Martin Hilliges,et al.  A phenomenological model for dynamic traffic flow in networks , 1995 .

[24]  Leonard Rach,et al.  Research Directions in Computer Control of Urban Traffic Systems , 1979 .

[25]  Reinhart D Kuhne,et al.  KNOWLEDGE-BASED OPTIMIZATION OF LINE CONTROL SYSTEMS FOR FREEWAYS , 1992 .

[26]  L. Goddard,et al.  Operations Research (OR) , 2007 .

[27]  C. Daganzo.  Requiem for second-order fluid approximations of traffic flow , 1995 .

[28]  津田 剛,et al.  道路投資の経済分析(Transportation Research Record,No.490,1974) , 1975 .

[29]  P. Nelson.  A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions , 1995 .

[30]  Markos Papageorgiou,et al.  Applications of Automatic Control Concepts to Traffic Flow Modeling and Control , 1983 .

[31]  G. F. Newell.  Nonlinear Effects in the Dynamics of Car Following , 1961 .

[32]  The National Academy of Sciences , 1928, Science.

[33]  E. Montroll,et al.  Traffic Dynamics: Studies in Car Following , 1958 .

[34]  M Cremer,et al.  AN EXTENDED TRAFFIC MODEL FOR FREEWAY CONTROL , 1985 .

[35]  George A. Bekey,et al.  Mathematical models of public systems , 1971 .

[36]  P. K. Munjal,et al.  PROPAGATION OF ON-RAMP DENSITY PERTURBATIONS ON UNIDIRECTIONAL TWO- AND THREE-LANE FREEWAYS , 1971 .

[37]  I. Prigogine,et al.  A Boltzmann-Like Approach for Traffic Flow , 1960 .

[38]  J. Holm,et al.  交通量調査デ-タを用いた交通推計モデル(Traffic Engineering and Control,Vol.17No.4,1976) , 1977 .

[39]  R. B. Potts,et al.  Car-Following Theory of Steady-State Traffic Flow , 1959 .

[40]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .