A continuum theory of traffic dynamics for freeways with special lanes

This paper presents a generalized theory of kinematic waves for freeways with two vehicle types and a set of lanes reserved for one of the vehicle classes. The theory is not restricted to freeways on which the special lanes are clearly identified by signs and pavement markings; e.g. for high occupancy vehicles. It may also apply if the restrictions are self-imposed, such as would occur on a freeway segment upstream of a busy off-ramp where the existing traffic naturally avoids the 'far-side' lanes. Of particular interest are oversaturated time periods because the original theory of kinematic waves proposed by Lighthill and Whitham [Proceedings Royal Society, A 229, 281-345 (1955)] and Richards (1956) does not recognize that different traffic conditions (queues and speeds) may arise on the two sets of lanes, and that these may affect the two vehicle classes in different ways. It should be intuitive that whether a single coalesced queue forms on both sets of lanes or separate queues form on each set should depend on the traffic composition by vehicle class. In an attempt to furnish a reasonable depiction of these phenomena, a pair of conservation-type partial differential equations in the densities of the two vehicle types is used to model the freeway. In their simplest form, the equations only require the introduction of one additional parameter (representing the fraction of lanes allocated to each vehicle type) over the basic kinematic wave theory. The model is attractive because the nature of its solution can be described in complete physical detail by means of simple intuitive diagrams that show how the simple kinematic wave model is improved. The paper also introduces an exact solution method for problems with piece-wise constant initial data that can either be applied graphically or numerically. For general (piece-wise smooth) data, the numerical technique can be used to approximate the true solution as closely as desired in a way that does not deteriorate with time. A less precise finite-difference approach that always moves vehicles forward is also presented. (C) 1997 Elsevier Science Ltd.