Dynamic First-Order Modeling of Phase-Transition Probabilities

Summary Modeling breakdown probabilities or phase transition probabilities is an important issue when assessing and predicting the reliability of traffic flow operations. Looking at empirical spatio-temporal patterns, these probabilities clearly are not only a function of the local prevailing traffic conditions (density, speed), but also of time and space. For instance, the probability that start-stop wave occurs generally increases when moving upstream away from the bottleneck location. The dynamics of the breakdown probabilities are the topic of this paper. We propose a simple partial differential equation that can be used to model the dynamics of breakdown probabilities, in conjunction with a first-order model. The main assumption is that the breakdown probability dynamics satisfy the way information propagates in a traffic flow, i.e. they move along with the characteristics. The main result is that we can reproduce the main characteristics of the breakdown probabilities, such as observed by Kerner. This is illustrated by means of two examples: free flow to synchronized flow (F-S transition) and synchronized to jam (S-J transition). We show 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 we move upstream in the synchronized flow area. Note that all the examples shown in the paper are deterministic.