On the macroscopic stability of freeway traffic

A simple model of traffic flow is used to analyze the spatio-temporal distribution of flow and density on closed-loop homogeneous freeways with many ramps, which produce inflows and allow outflows. As we would expect, if the on-ramp demand is space-independent then this distribution tends toward uniformity in space if the freeway is either: (i) uncongested; or (ii) congested with queues on its on-ramps and enough inflow to cause the average freeway density to increase with time. In all other cases, however, including any recovery phase of a rush hour where the freeway's average density declines, the distribution of flow and density quickly becomes uneven. This happens even under conditions of perfect symmetry, where the percentage of vehicles exiting at every off ramp is the same. The flow-density deviations from the average are shown to grow exponentially in time and propagate backwards in space with a fixed wave speed. A consequence of this type of instability is that, during recovery, gaps of uncongested traffic will quickly appear in the unevenly congested stream, reducing average flow. This extends the duration of recovery and invariably creates clockwise hysteresis loops on scatter-plots of average system flow vs. density during any rush hour that oversaturates the freeway. All these effects are quantified with formulas and verified with simulations. Some have been observed in real networks. In a more practical vein, it is also shown that the negative effects of instability diminish (i.e., freeway flows increase) if (a) some drivers choose to exit the freeway prematurely when it is too congested and/or (b) freeway access is regulated in a certain traffic-responsive way. These two findings could be used to improve the algorithms behind VMS displays for driver guidance (finding a), and on-ramp metering rates (finding b).

[1]  G. F. Newell.  A simplified theory of kinematic waves in highway traffic, part I: General theory , 1993 .

[2]  Piotr Olszewski,et al.  Area-wide traffic speed-flow model for the Singapore CBD , 1995 .

[3]  Carlos F. Daganzo,et al.  Lane-changing in traffic streams , 2006 .

[4]  R. Bertini,et al.  Empirical Study of Traffic Features at a Freeway Lane Drop , 2005 .

[5]  Siamak Ardekani,et al.  Urban Network-Wide Traffic Variables and Their Relations , 1987, Transp. Sci..

[6]  M. Lighthill,et al.  On kinematic waves I. Flood movement in long rivers , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[7]  G. F. Newell.  A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks , 1993 .

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

[9]  Carlos F. Daganzo,et al.  THE CELL TRANSMISSION MODEL, PART II: NETWORK TRAFFIC , 1995 .

[10]  Christine Buisson,et al.  Exploring the Impact of Homogeneity of Traffic Measurements on the Existence of Macroscopic Fundamental Diagrams , 2009 .

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

[12]  Hani S. Mahmassani,et al.  Network performance under system optimal and user equilibrium dynamic assignments: Implications for , 1993 .

[13]  Carlos F. Daganzo,et al.  Urban Gridlock: Macroscopic Modeling and Mitigation Approaches , 2007 .

[14]  C. Daganzo,et al.  Macroscopic Fundamental Diagrams for Freeway Networks: Theory and Observation , 2011 .

[15]  Carlos F. Daganzo,et al.  THE NATURE OF FREEWAY GRIDLOCK AND HOW TO PREVENT IT. , 1995 .

[16]  Nikolaos Geroliminis,et al.  Hysteresis Phenomena of a Macroscopic Fundamental Diagram in Freeway Networks , 2011 .

[17]  M. Cassidy,et al.  Some traffic features at freeway bottlenecks , 1999 .

[18]  Siamak Ardekani,et al.  Characterizing Traffic Conditions in Urban Areas , 1984, Transp. Sci..

[19]  N. Geroliminis,et al.  Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings - eScholarship , 2007 .

[20]  C. Daganzo,et al.  Exploring the Effect of Turning Maneuvers and Route Choice on a Simple Network , 2010 .

[21]  C. Daganzo,et al.  Macroscopic relations of urban traffic variables: Bifurcations, multivaluedness and instability , 2011 .

[22]  Jorge A. Laval,et al.  Effects of Merging and Diverging on Freeway Traffic Oscillations , 2010 .