Symmetry breaking in optimal timing of traffic signals on an idealized two-way street

Simple physical models based on fluid mechanics have long been used to understand the flow of vehicular traffic on freeways; analytically tractable models of flow on an urban grid, however, have not been as extensively explored. In an ideal world, traffic signals would be timed such that consecutive lights turned green just as vehicles arrived, eliminating the need to stop at each block. Unfortunately, this “green-wave” scenario is generally unworkable due to frustration imposed by competing demands of traffic moving in different directions. Until now this has typically been resolved by numerical simulation and optimization. Here, we develop a theory for the flow in an idealized system consisting of a long two-way road with periodic intersections. We show that optimal signal timing can be understood analytically and that there are counterintuitive asymmetric solutions to this signal coordination problem. We further explore how these theoretical solutions degrade as traffic conditions vary and automotive density increases.

Figure 1

Space-time diagram for traffic flow. The red (dark gray) bars indicate locations and times for red lights and the green (light gray) bars indicate green lights. (a) illustrates the definitions of various variables and parameters in our model. (b) displays various vehicle trajectories. The dashed line (blue) indicates a car traveling slower than the green-wave speed $v_g$, the dash-dotted line (magenta) indicates a car traveling faster than $v_g$, and solid lines (black) indicate cars traveling exactly at $v_g$ both eastbound (starting on the left) and westbound (starting on the right). (c) displays vehicle trajectories near the green- and red-wave speeds. The solid line (black) indicates a car traveling at the green-wave speed. The dashed line (red) indicates a vehicle traveling slightly faster than the red-wave speed. The dash-dotted line (red) indicates a vehicle traveling slightly more slowly than the red-wave speed.

Figure 2

Theoretical efficiency versus $r_{\Delta }$ for (a) $r_C=0.34$, (b) $r_C=0.26$, (c) $r_C=0.25$, and (d) $r_C=0.13$. Green (light gray) indicates $E_{\mathrm{east}}$; red (dark gray) indicates $E_{\mathrm{west}}$; and black indicates $E_{\mathrm{tot}}$ assuming equal demand in both directions.

Figure 3

Optimality of the green wave. Intervals of $r_C$ for which the green wave optimizes the bidirectional efficiency are indicated by green (light gray) rectangles. Intervals of $r_C$ for which other timings are optimal are indicated by red (dark gray) rectangles.

Figure 4

Efficiency from simulation versus $r_{\Delta }$ for $r_C=0.34$. (a) shows the efficiency for increasing vehicle density: the solid curve (black) indicates the theoretical efficiency $\left(E_{\mathrm{tot}}\right)$; the dotted (green), dash-dotted (blue), and dashed (red) curves represent simulation results for vehicle densities of 10%, 50%, and 90% density, respectively. The bottom panels display the effects of variation in the traffic signal spacing (b) and vehicle speed (c) on the efficiency with 10% traffic density. Solid (black) indicates the theoretical efficiency; the dotted (green), dash-dotted (blue), and dashed (red) curves represent simulation results for 0%, 1%, and 5% standard deviation, respectively.

Figure 5

Predictions and simulations of various jamming transitions. (a) displays the eastbound efficiency for various values of $r_{\Delta }$ with $r_C=0.34$. Of particular interest are the green curve (dashed), which represents an eastbound green wave, and the red curve (solid), which is near an eastbound red wave. The markers correspond to the critical densities for platoon segmentation (circle) and platoon coalescence (square). (b) and (c) show the bidirectional (weighted average of east- and westbound) efficiency for $r_C=0.34$ and $r_C=0.26$, respectively, with curves representing green-wave peaks (green dashed curve), discontinuous peaks (red solid curve), and suboptimal timings (blue dash-dotted curve).