A hydrodynamic theory based statistical model of arterial traffic

In arterial networks, the dynamics of traffic flows are driven by the presence of traffic signals. A comprehensive model of the dynamics of arterial traffic flow is necessary to capture the specifics of arterial traffic and provide accurate traffic estimation. From hydrodynamic theory of traffic flow, we model the dynamics of arterial traffic under specific assumptions which are standard in transportation engineering. We use this flow model to develop a statistical model of arterial traffic. First, we derive an analytical expression for the spatial distribution of vehicles. This encompasses the fact that the average density of vehicles is higher close to the traffic signals because of the delays experienced by the vehicles. Second, we derive the probability distribution of total and measured delay (to be defined specifically in the document). The delays experienced by vehicles traveling on a link of the network depend on the time (from the beginning of the cycle) when they enter the link. We model the probability of delay for a path between two arbitrary points on the link. The probability distribution of measured delay takes into account the sampling scheme to derive the probability of the observed delay from probe vehicles sampled uniformly in time. Finally, we use the probability distribution of delays and a model of driving behavior to derive the probability distribution of travel times between any two arbitrary points on a link. The analytical derivations are parameterized by traffic variables (cycle time, red time, model of free flow speed, queue length and queue length at saturation). The models estimates queue length (and thus congestion levels), signal parameters and variability of driving behavior. We show that the probability distributions of travel times on an arterial links are quasi-concave. The probability distributions of travel times between any arbitrary location on the link are finite mixture distributions where each component represents a class of vehicles depending on the characteristics of its delay. We prove that each component of the mixture distribution is log-concave, which enables the use of specific optimization algorithm. The distributions derived in this report are used as fundamental building block for arterial traffic estimation using sparse travel time measurements from probe vehicles used in subsequent work. ∗Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, [email protected], http://eecs.berkeley.edu/~aude †Department of Industrial Engineering and Operations Research, University of California, Berkeley ‡Department of Civil and Environmental Engineering, Systems Engineering, University of California, Berkeley