Traveling waves in an optimal velocity model of freeway traffic

Car-following models provide both a tool to describe traffic flow and algorithms for autonomous cruise control systems. Recently developed optimal velocity models contain a relaxation term that assigns a desirable speed to each headway and a response time over which drivers adjust to optimal velocity conditions. These models predict traffic breakdown phenomena analogous to real traffic instabilities. In order to deepen our understanding of these models, in this paper, we examine the transition from a linear stable stream of cars of one headway into a linear stable stream of a second headway. Numerical results of the governing equations identify a range of transition phenomena, including monotonic and oscillating travelling waves and a time- dependent dispersive adjustment wave. However, for certain conditions, we find that the adjustment takes the form of a nonlinear traveling wave from the upstream headway to a third, intermediate headway, followed by either another traveling wave or a dispersive wave further downstream matching the downstream headway. This intermediate value of the headway is selected such that the nonlinear traveling wave is the fastest stable traveling wave which is observed to develop in the numerical calculations. The development of these nonlinear waves, connecting linear stable flows of two different headways, is somewhat reminiscent of stop-start waves in congested flow on freeways. The different types of adjustments are classified in a phase diagram depending on the upstream and downstream headway and the response time of the model. The results have profound consequences for autonomous cruise control systems. For an autocade of both identical and different vehicles, the control system itself may trigger formations of nonlinear, steep wave transitions. Further information is available [Y. Sugiyama, Traffic and Granular Flow (World Scientific, Singapore, 1995), p. 137].