Travel Time Reliability in Stochastic Kinematic Flow Models

This paper analyzes the time-dependent relationship between the mean and variance of travel time on a single corridor under rush hour like congestion patterns. To model this phenomenon, we apply the LWR ((Lighthill & Whitham, 1955), (Richards, 1956)) theory on a homogenous freeway with a discontinuous bottleneck at its downstream end, assuming a uni-modal demand profile with a stochastic peak. We establish conditions for typical counterclockwise hysteresis loops under these assumptions. It is demonstrated that shapes of the fundamental diagram which always produce a counterclockwise loop can be interpreted as an indication of aggressive driving behavior, while deviations may occur under defensive driving. This classification enables a detailed explanation of the qualitative physical mechanisms behind this pattern, as well as an analysis of the causes for quantitatively limited deviations. Some of the mathematical properties of the LWR model identified in our analysis have not yet been addressed in the literature and we critically examine the extent to which these reflect actual traffic flow behavior. Our considerations are supported by numerical experiments. The obtained results aim to improve the fundamental understanding of the physical causes of this hysteresis pattern and to facilitate its better estimation in traffic planning and control.