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Macroscopic Modeling Framework Unifying Kinematic Wave Modeling and Three-Phase Traffic Theory

Modeling breakdown probabilities or phase-transition probabilities is an important issue when assessing and predicting the reliability of traffic flow operations. Looking at empirical spatiotemporal patterns, these probabilities clearly are a function not only of the local prevailing traffic conditions (density, speed) but also of time and space. For instance, the probability that a start-stop wave occurs generally increases when moving upstream away from the bottleneck location. A simple partial differential equation is presented that can be used to model the dynamics of breakdown probabilities, in conjunction with the well-known kinematic wave model. The main assumption is that the breakdown probability dynamics satisfy the way information propagates in a traffic flow, that is, they move along with the characteristics. The main result is that the main characteristics of the breakdown probabilities can be reproduced. This is illustrated through two examples: free flow to synchronized flow (F-S transition) and synchronized to jam (S-J transition). It is shown that the probability of an F-S transition increases away from the on ramp in the direction of the flow; the probability of an S-J transition increases as one moves upstream in the synchronized flow area.