A formulation of unifiable multi-commodity kinematic wave model with relative speed ratios

Abstract

Observations suggest that the First-In-First-Out (FIFO) principle is usually violated on multi-lane roads. Jin (2017) formulated and solved unifiable multi-commodity kinematic wave models, when different commodities have the same contributions to overall traffic congestion (unifiable) but may travel at different speeds (non-FIFO). However, the construction of unifiable multi-commodity fundamental diagrams and the assumption of concave or convex commodity flow proportion functions are purely mathematical and lack behavioral explanations. Thus the existing formulation is only of pure mathematical and theoretical interests and too complicated for real-world calibration, validation, or applications.In this study, we present a new formulation of unifiable multi-commodity kinematic wave models to address the aforementioned limitations. We first introduce a new variable for the relative speed ratios of different commodities and particularly discuss constant relative speed ratios. The relative speed ratios are physically, behaviorally, and economically meaningful since they characterize drivers’ relative aggressiveness, values of times, and other features. We then present unifiable multi-commodity fundamental diagrams based on the relative speed ratios, which can be used to derive the mathematical generating functions in Jin (2017). Then we show that non-FIFO multi-commodity kinematic wave model is a system of non-strictly hyperbolic conservation laws and solve the Riemann problem for a two-commodity system with constant relative speed ratios, in which the commodity flow proportion function is either concave or convex. We also present an empirical evidence for the existence of unifiable multi-commodity fundamental diagrams with constant relative speed ratios, which help to demonstrate the advantage of the new formulation. In summary, the new formulation is as general as the original formulation, but physically more meaningful, theoretically easier to solve, and empirically simpler to calibrate. Finally we conclude the study with discussions on potential future applications.