A stochastic queueing model for travel time reliability with spatial density heterogeneity

This paper constructs a theoretical model connecting day-to-day travel time reliability (TTR) and vehicle density, testing it on freeway stretches in Minneapolis - St. Paul and San Diego. We establish the presence of a counter-clockwise hysteresis loop between average vehicle density and experienced travel time standard deviation. We also observe that vehicle density heterogeneity exerts a more profound effect on TTR during moderate congestion. We find that a time-homogeneous Poisson process accurately characterizes the vehicle arrival process on segments of freeway stretches, both with and without on-ramps, during the morning peak period. The travel times for these segments align well with an exponential distribution. Through an analysis of the travel time correlations among different segments of the freeway stretch, we find that this correlation is higher during congestion offset periods, resulting in more severe travel time fluctuations due to increased density heterogeneity. Using the M/M/1 queueing model, we derive a model encapsulating the variance of experienced travel time, average vehicle density, and density heterogeneity for freeway stretches consisting of multiple segments. We use the ‘birth-death process’ to model the relationship between average vehicle density and density heterogeneity. By integrating these two models, we derive a novel model that accurately approximates freeway stretch TTR based on average vehicle density and traffic flow. We apply our models to TTR changes on the two selected freeway stretches to validate across different time periods and select two subnetworks, each containing three additional freeway stretches, to validate across spatial dimensions. The results from these experiments corroborate the robustness and generality of our model across both temporal and spatial contexts.