On the Stability of Freeway Traffic
Traffic flow theory 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. It is shown that 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, including any recovery phase of a rush hour where the freeway's average density declines, the distribution of flow and density quickly becomes uneven. 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. All these effects are quantified with formulas and verified with simulations.