Effect of the Trip-Length Distribution on Network-Level Traffic Dynamics: Exact and Statistical Results

This paper presents additional results of the generalized bathtub model for urban networks, including a simpler derivation and exact solutions for uniformly distributed trip lengths. It is shown that in steady state this trip-based model is equivalent to the more parsimonious accumulation-based model, and that the trip-length distribution has merely a transient effect on traffic dynamics, which converge to the same point in the macroscopic fundamental diagram (MFD). To understand the statistical properties of the system, a queueing approximation method is proposed to compute the network accumulation variance. It is found that (i) the accumulation variance is much larger than predicted by traditional queueing models, due to the nonlinear dynamics imposed by the MFD, (ii) the trip-length distribution has no effect on the accumulation variance, indicating that the proposed formula for the variance might be universal, (iii) the system exhibits critical behavior near the capacity state where the variance diverges to infinity. This indicates that the tools from critical phenomena and phase transitions might be useful to understand congestion in cities.