Solvable optimal velocity models and asymptotic trajectory

Abstract

In the Optimal Velocity Model proposed as a new version of Car Following\nModel, it has been found that a congested flow is generated spontaneously from\na homogeneous flow for a certain range of the traffic density. A\nwell-established congested flow obtained in a numerical simulation shows a\nremarkable repetitive property such that the velocity of a vehicle evolves\nexactly in the same way as that of its preceding one except a time delay $T$.\nThis leads to a global pattern formation in time development of vehicles'\nmotion, and gives rise to a closed trajectory on $\\Delta x$-$v$\n(headway-velocity) plane connecting congested and free flow points. To obtain\nthe closed trajectory analytically, we propose a new approach to the pattern\nformation, which makes it possible to reduce the coupled car following\nequations to a single difference-differential equation (Rondo equation). To\ndemonstrate our approach, we employ a class of linear models which are exactly\nsolvable. We also introduce the concept of ``asymptotic trajectory'' to\ndetermine $T$ and $v_B$ (the backward velocity of the pattern), the global\nparameters associated with vehicles' collective motion in a congested flow, in\nterms of parameters such as the sensitivity $a$, which appeared in the original\ncoupled equations.\n