Probability Distributions of Travel Times on Arterial Networks: Traffic Flow and Horizontal Queuing Theory Approach

In arterial networks, traffic flow dynamics are driven by the presence of traffic signals, for which precise signal timing is difficult to obtain in arbitrary networks or might change over time. A comprehensive model of arterial traffic flow dynamics is necessary to capture its specific features in order to provide accurate traffic estimation approaches. From hydrodynamic theory, arterial traffic dynamics are modeled under specific assumptions standard in transportation engineering. This flow model is used to develop a statistical model of arterial traffic. The statistical approach is essential to capture the variability of travel times among vehicles: (1) the delay experienced by a vehicle depends on the time when it enters the link (in relation to the signal green/red phases) and this entrance time can occur at any random time during the cycle and (2) the free flow speed of a vehicle depends both on the driver and on external factors (jaywalking, double parking, etc.) and is another source of uncertainty. These two sources of uncertainty are captured by deriving the probability distribution of delays (from hydrodynamic theory) and modeling the nominal free flow travel time as a random variable (which encodes variability in driving behavior). An analytical expression is derived for the probability distribution of travel times between any two locations on an arterial link, parameterized by traffic parameters (cycle time, red time, free flow speed distribution, queue length and queue length at saturation). The model is validated using probe vehicle data collected during a field test in San Francisco, as part of the Mobile Millennium system. The numerical results show that the new distribution derived in this article more accurately represents the actual distribution of travel times than other distributions that are commonly used to represent travel times (normal, log-normal and Gamma distributions). It is also shown that the model performs particularly well when the amount of data available is small. This is very promising as the volume of probe vehicle data available in real time to most traffic information systems today remains sparse.