An analytical framework to model uncertainty in urban network dynamics using Macroscopic Fundamental Diagrams

Recent studies have proposed using well-defined relationships between network productivity and accumulation—otherwise known as Network or Macroscopic Fundamental Diagrams (MFDs)—to model the dynamics of large-scale urban traffic networks. This provides a computationally efficient way to study these complex systems and facilitates the design and control of novel large-scale traffic management strategies. However, empirical and simulation evidence suggests that MFDs are rarely well-defined. Instead, they exhibit large amounts of scatter and uncertainty, which suggests a range of network productivities may be observed for any given accumulation. This paper examines the impact of this MFD uncertainty and uncertainty in aggregate-level vehicle demands (i.e., vehicle exit and entry rates) on large-scale network behavior. It is shown that these uncertainties can cause fundamentally different aggregate network behaviors than would be expected if they were ignored, including unexpected congestion or gridlock. An analytically derived Markov Chain framework is proposed that can be used to model aggregate network dynamics while explicitly accounting for these types of uncertainties, which are very likely to arise on realistic urban networks. Comparison between the analytical predictions and numerical simulations suggest that the Markov Chain framework can accurately predict traffic dynamics under uncertainty for both single- and multi-region networks. Since this framework relies on the careful discretization of both time and accumulation within individual regions of a network, guidance is also provided on how to best select these discretization parameters the most accurate results.