Traffic Flow for the Morning Commute

We imagine that drivers in the morning commute must traverse some homogeneous section of highway of length l > 0 and finite capacity q_m to reach their workplaces. Each driver has a specified latest time at which he can exit the road and still be at work on time, but, because of the capacity and other traffic flow restrictions, it may be necessary that some drivers be at work earlier than their scheduled times if no one is to be late. Traffic is treated as a continuous fluid in the manner proposed by Lighthill and Whitham. This represents a generalization to l > 0 of a theory described by Hendrickson and Kocur for point bottlenecks (l = 0) and a revision of a theory presented by Mahmassani and Herman. That waves can travel with only finite speed causes the schedule delay to be larger for l > 0 than for l = 0 and possibly be positive even if the demand rate is always less than q_m. The conclusions, however, are quite different from those given by Mahmassani and Herman. If there is a cost per unit of trip time and another (lower) cost per unit of schedule delay, there is a well-defined system optimal assignment. There is some question, however, regarding the stability of the user optimal assignment.