A Kinematic Wave Traffic Flow Model for Mixed Traffic

In this paper we extend the Lighthill-Whitham-Richards kinematic wave traffic flow model to describe traffic with different types of vehicles, where all types of vehicles are completely mixed and travel at the same group velocity. A study of such a model with two vehicle classes (e.g., passenger cars and trucks) shows that, when both classes of traffic have identical free-flow speeds, the model 1) satisfies first-in-first-out rule, 2) is anisotropic, and 3) has the usual shock and expansion waves, and a family of contact waves. Different compositions of vehicle classes in this model propagate along contact waves. Such models can be used to study traffic evolution on long crowded highways where low performance vehicles entrap high performance ones. 1 BACKGROUND Vehicular traffic on highways often comprises different types of vehicles with varying driving performances. This heterogeneity affects traffic flow characteristics in significant ways, a fact that has long been recognized by the transportation engineering profession. For example, in the computation of flow capacity on a highway or at a signalized intersection, the Highway Capacity Manual recommends a series of adjustments to take account of the capacity reduction caused by heavy vehicles (i.e., trucks/buses/recreational vehicles). If one is interested in the effects of heavy vehicles on traffic flow over space and time, however, the Highway Capacity Manual procedures are not adequate. For this one needs a dynamic model for mixed traffic. Mixed traffic can be modeled at three different levels—microscopic, mesoscopic and macroscopic. It is perhaps most straightforward to model mixed traffic on a microscopic level—one simply endow, at one extreme, each individual vehicle with different performance and behavior characteristics. Many commercially available simulation packages, such as CORSIM, PARAMICS, and VISSIM allow the specification of multiple vehicle classes. Major challenges arise when one models mixed traffic on a mesoscopic level, mainly due to the correlation between various probability distributions of vehicular speeds. Nevertheless, a number of mesoscopic models of mixed traffic have been developed in recent years (1, 2). Aggregation of mesoscopic models of mixed traffic through expectation operations lead to multi-class traffic flow models in the macroscopic level. There is, however, another approach to develop macroscopic mixed traffic flow models. This is the approach of continuum modeling. It is this approach that we shall adopt in developing a traffic flow model for mixed traffic. In the continuum description of traffic flow, vehicular traffic is described as a special kind of fluid that are characterized by its concentration (density, ρ), mean velocity (v), and vehicle flux (flow rate q), all are functions of space (x) and time (t). The starting point of any continuum model of traffic flow is the conservation of vehicles ), , ( ) , ( ) , ( 2 1 2 1 t x q t x q dx t x t x