The influence of trip length distribution on urban traffic in network-level models

Modeling urban traffic on the level of network is a wide research area oriented to the development of ITS. In this thesis properties of models based on MFD (Macroscopic Fundamental Diagram) are studied. The idea behind MFD is to say that the state of the traffic inside an urban zone is fully determined by its accumulation (the number of traveling vehicles) and that the dynamics of accumulation is caused by the inflow of vehicles (flow of vehicles that enter the zone or start their trips from inside). Nowadays, two different philosophies of modeling dynamics of accumulation exist in the literature. The first one (outflow-MFD) postulates that the outflow of vehicles depends on accumulation. The second one (speed-MFD) postulates that the space-mean speed of vehicles depends on accumulation. The second philosophy already has strong empirical support based on observations of traffic inside many big (kilometer-scale) urban areas around the world. Thus, different speed-MFD models are of great scientific interest.

The thesis is mainly devoted to the comparison of so-called PL model (which assumes the existence of both speed-MFD and outflow-MFD) and TB model (which assumes the equality of speeds of vehicles and explicitly assumes the existence of trip length distribution). It was shown that PL model cannot accurately describe the dynamics of accumulation after the jump of inflow. In this case TB model is more preferable. Moreover, it was proven that PL model is a specific case of TB model for the exponential trip length distribution. This makes TB model more attractive than PL model for the practical usage.

TB model can be formulated mathematically either as integral equation or nonlocal PDE. Thus, the main drawback of TB model that can be an obstacle in practice is its computational complexity. In this thesis it was proposed to approximate TB model with a simpler model which does not require precise information about the trip length distribution. This so-called M model operates with only the mean and the standard deviation of distribution and has a form of ODE. The analytical comparison between PL, TB and M models proved that M model is much closer to TB model than PL model in the case of constant speed-MFD. The more realistic case of decreasing speed-MFD was studied through the numerical tests and also showed the same effect. Thus, the main conclusion of the study is that M model has practical potential as an elegant and computationally cheap approximation of TB model. Also, given that in the case of constant speed TB model is a type of LTI system, it can be expected that M model might be useful for a wide range of problems that are not related to transportation. However, in this thesis the conclusion about the small difference between M and TB models was made for the inflows that are typical for the transportation field. More precisely, only peak hour shaped (smooth and with small jumps) functions were studied.

Despite the simple form of M model it was found that there exists even more simple ODE approximation of TB model. This so-called $\alpha$ model works quite well for both smooth and jumping inflows except the case of a short period of time following the jump of inflow. Thus, it might be another good alternative to TB model. The main advantage of $\alpha$ model in the case of realistic speed-MFD function is its convex formulation which allows $\alpha$ model to be efficiently used inside optimization frameworks.