  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 counter-intuitive asymmetric solutions to this signal coordination
problem. We further explore how these theoretical solutions degrade as traffic
conditions vary and automotive density increases.
