Traditional DTA models of large cities suffer from prohibitive computation times and calibration/validation can become major challenges faced by practitioners. The empirical evidence in 2008 in support of the existence of a Macroscopic Fundamental Diagram (MFD) on urban networks led to the formulation of discrete-space models, where the city is divided into a collection of reservoirs. Prior to 2008, a large body of DTA models based on pedestrian flow models had been formulated in continuum space as 2-dimensional conservation laws where the speed-density relationship can now be interpreted as the MFD. Perhaps surprisingly, we found that this continuum-space literature has been mostly unaware of MFD theory, and no attempts exist to verify the assumptions of MFD theory. This has the potential to create significant inconsistencies, and research is needed to analyze their extent and ways to resolve them. We also find that further research is needed to (i) incorporate departure time choice, (ii) improve existing numerical methods, possibly extending recent advances on the one-dimensional kinematic wave (LWR) model, (iii) study the properties of system optimum solutions, (iv) examine the real-time applicability of current continuum-space models compared to traditional DTA methods, and (v) formulate anisotropic models for the interaction of intersecting flows.

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This paper presents a formulation of the reactive dynamic user equilibrium problem in continuum form using a network-level Macroscopic Fundamental Diagram (MFD). Compared to existing continuum models for cities -all based in Hughes' pedestrian model in 2002 -the proposed formulation (i) is consistent with reservoir-type models of the MFD literature, shedding some light into the connection between these two modeling approaches, (ii) can have destinations continuously distributed on the region, and (iii) can incorporate multi-commodity flows without additional numerical error. The proposed multi-reservoir numerical solution method treats the multi-commodity component of the model in Lagrangian coordinates, which is the natural representation to propagate origin-destination information (and any vehicle-specific characteristic) through the traffic stream. Fluxes between reservoir boundaries are computed in the Eulerian representation, and are used to calculate the speed of vehicles crossing the boundary. Simple examples are included that show the convergence of the model and its agreements with the available analytical solutions. We find that (i) when origins and destinations are uniformly distributed in a region, the distribution of the travel times can be approximated analytically, (ii) the magnitude of the detours from the optimal free-flow route due to congestion increase linearly with the inflow and decreases with the square of the speed, and (iii) the total delay of vehicles in the network converges to the analytical approximation when the size of reservoirs tends to zero.

This paper extends the Stochastic Method of Cuts (SMoC) to approximate of the Macroscopic Fundamental Diagram (MFD) of urban networks and uses Maximum Likelihood Estimation (MLE) method to estimate the model parameters based on empirical data from a corridor and 30 cities around the world. For the corridor case, the estimated values are in good agreement with the measured values of the parameters. For the network datasets, the results indicate that the method yields satisfactory parameter estimates and graphical fits for roughly 50\% of the studied networks, where estimations fall within the expected range of the parameter values. The satisfactory estimates are mostly for the datasets which (i) cover a relatively wider range of densities and (ii) the average flow values at different densities are approximately normally distributed similar to the probability density function of the SMoC. The estimated parameter values are compared to the real or expected values and any discrepancies and their potential causes are discussed in depth to identify the challenges in the MFD estimation both analytically and empirically. In particular, we find that the most important issues needing further investigation are: (i) the distribution of loop detectors within the links, (ii) the distribution of loop detectors across the network, and (iii) the treatment of unsignalized intersections and their impact on the block length.