Macroscopic network-level traffic models: bridging fifty years of development toward the next era

Network macroscopic fundamental diagrams (NMFD) and related network-level traffic dynamics models have received both theoretical support and empirical validation with the emergence of new data collection technologies. However, the extent to which network-level macroscopic traffic models may be ready for practical implementation remains to be ascertained. This paper aims to shed light on this matter by reviewing the 50-year history of macroscopic modeling of urban networks and assessing remaining gaps and opportunities for further development of both theory and applications. To this end, the existing literature's chronology is mapped onto three eras of development, and the corresponding theories, assumptions, and limitations are outlined and discussed in two streams, equilibrium relations and traffic dynamics. Among the topics pertaining to equilibrium relations, the highlighted gaps include the lack of empirical studies on the hysteresis and bifurcation phenomena, the existence of multi-modal NMFD (3D-NMFD) in different traffic conditions, the factors that might affect the (3D-) NMFD shape, the accuracy of speed-NMFDs in particular different bus speed NMFDs, and the passenger-oriented NMFDs. Research gaps pertaining to traffic dynamics include the analytical solution of trip-based models, the FIFO violation in the delay-based models, the definition of outflow and entrance functions, the notion of active network length, the trip length distribution, and the path flow distribution. Future research directions targeting topics that might shape the potential next era include the practical implementation of NMFD-based control strategies, the application of NMFD in quality of service assessments, and NMFD in the presence of new technologies such as connected and autonomous vehicles (CAVs).