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Two key aggregated traffic models are the relationship between average network flow and density (known as the network or flow macroscopic fundamental diagram [flow-MFD]) and the relationship between trip completion and density (known as network exit function or the outflow-MFD [o-FMD]). The flow- and o-MFDs have been shown to be related by average network length and average trip distance under steady-state conditions. However, recent studies have demonstrated that these two relationships might have different patterns when traffic conditions are allowed to vary: the flow-MFD exhibits a clockwise hysteresis loop, while the o-MFD exhibits a counter-clockwise loop. One recent study attributes this behavior to the presence of bottlenecks within the network. The present paper demonstrates that this phenomenon may arise even without bottlenecks present and offers an alternative, but more general, explanation for these findings: a vehicle’s entire trip contributes to a network’s average flow, while only its end contributes to the trip completion rate. This lag can also be exaggerated by trips with different lengths, and it can lead to other patterns in the o-MFD such as figure-eight patterns. A simple arterial example is used to demonstrate this explanation and reveal the expected patterns, and they are also identified in real networks using empirical data. Then, simulations of a congestible ring network are used to unveil features that might increase or diminish the differences between the flow- and o-MFDs. Finally, more realistic simulations are used to confirm that these behaviors arise in real networks. 

Relationships between average network productivity and accumulation or density aggregated 2 across spatially compact regions of urban networks—so called network Macroscopic Fundamental 3 Diagrams (MFDs)—have recently been shown to exist. Various analytical methods have been put 4 forward to estimate a network’s MFD as a function of network properties, such as average block 5 lengths, signal timings, and traffic flow characteristics on links. However, real street networks are 6 not homogeneous—they generally have a hierarchical structure where some streets (e.g., arterials) 7 promote higher mobility than others (e.g., local roads). This paper provides an analytical method 8 to estimate the MFDs of hierarchical street networks by considering features that are specific to 9 hierarchical network structures. Since the performance of hierarchical networks is driven by how 10 vehicles are routed across the different street types, two routing conditions— user equilibrium and 11 system optimal routing—are considered in the analytical model. The proposed method is first 12 implemented to describe the MFD of a hierarchical one-way limited access linear corridor and 13 then extended to a more realistic hierarchical two-dimensional grid network. For both cases, it is 14 shown that the MFD of a hierarchical network may no longer be unimodal or concave as 15 traditionally assumed in most MFD-based modeling frameworks. These findings are verified using 16 simulations of hierarchical corridors. Finally, the proposed methodology is applied to demonstrate 17 how it can be used to make decisions related to the design of hierarchical street network structures. 

Network macroscopic fundamental diagrams (MFDs) have recently been shown to exist in real-world urban traffic networks. The existence of an MFD facilitates the modeling of urban traffic network dynamics at a regional level, which can be used to identify and refine large-scale network-wide control strategies. To be useful, MFD-based modeling frameworks require an estimate of the functional form of a network’s MFD. Analytical methods have been proposed to estimate a network’s MFD by abstracting the network as a single ring-road or corridor and modeling the flow–density relationship on that simplified element. However, these existing methods cannot account for the impact of turning traffic, as only a single corridor is considered. This paper proposes a method to estimate a network’s MFD when vehicles are allowed to turn into or out of a corridor. A two-ring abstraction is first used to analyze how turning will affect vehicle travel in a more general network, and then the model is further approximated using a single ring-road or corridor. This approximation is useful as it facilitates the application of existing variational theory-based methods (the stochastic method of cuts) to estimate the flow–density relationship on the corridor, while accounting for the stochastic nature of turning. Results of the approximation compared with a more realistic simulation that includes features that cannot be captured using variational theory—such as internal origins and destinations—suggest that this approximation works to estimate a network’s MFD when turning traffic is present.