Macroscopic Fundamental Diagram (MFD)

• MFD is a network-level relationship between aggregated traffic density and flow that exhibits a rise–peak–fall structure with a critical density and maximum throughput.
• It uses various formulations—including reservoir, multi-region, and speed-based models—to capture overall traffic dynamics and inform control strategies.
• The MFD framework is pivotal for real-time state estimation, perimeter control, and planning while addressing challenges like heterogeneity, hysteresis, and uncertainty.

Searching arXiv for recent and foundational papers on Macroscopic Fundamental Diagram to ground the article. The Macroscopic Fundamental Diagram (MFD) is a network-level relation between aggregated traffic state variables in an urban region, most commonly average density or accumulation and average flow, production, or speed. In its canonical form, the diagram exhibits a rise–peak–fall structure: network flow increases with density at low occupancy, reaches a maximum $q^*$ at a critical density $k^*$, and then decreases as congestion dominates. Equivalent formulations use accumulation $n$ and output $o(n)$, or production $\mathcal{P}$ and heterogeneity $\gamma$, reflecting the fact that the MFD is both a compact traffic law and a reduced-order state description for monitoring, control, and planning. The same literature also stresses that an MFD is neither universal nor automatically single-valued: its existence and shape depend on network homogeneity, topology, control, and transient disturbances such as hysteresis (Aghamohammadi et al., 2018, Taillanter et al., 2023, Oh et al., 2020, Hammerl et al., 2024).

1. Core definition and state variables

In the aggregate network interpretation, the MFD is often written as

$Q = Q(n),$

where $Q$ is average network flow and $n$ is the number of vehicles inside the network. An associated outflow relation is

$o(n)=Q(n)\frac{L}{\ell},$

with $k^*$0 the total network length and $k^*$1 the average trip length. This converts average flow into the number of trip completions per unit time and yields the basic reservoir conservation law

$k^*$2

in which $k^*$3 is inflow demand (Aghamohammadi et al., 2018).

A closely related formulation uses network accumulation and production. In a heterogeneity-aware representation,

$k^*$4

where $k^*$5 is accumulation and $k^*$6 captures spatial heterogeneity in density. For vehicular MFDs, accumulation and production can be computed from link densities $k^*$7, link flows $k^*$8, and segment lengths $k^*$9, while $n$0 is defined as the standard deviation of segment densities. This formulation makes explicit that identical accumulation levels need not imply identical realized throughput when congestion is distributed differently across space (Oh et al., 2020).

Urban arterial studies also write the MFD in terms of network-averaged density and flow,

$n$1

with corresponding heterogeneity measures

$n$2

These definitions are central in work showing that MFD performance depends not only on mean density but also on how unevenly density and flow are distributed across links (Zhang et al., 2011).

At a larger urban scale, the MFD is typically understood as a relation between average car density $n$3 and average car flow $n$4, with critical density $n$5 and maximum network flow $n$6 defining the best operating point of the network. Those quantities are treated as practically important for perimeter control, congestion pricing, traffic signal optimization, congestion propagation analysis, and road-network planning (Taillanter et al., 2023).

2. Dynamical formulations and model classes

The simplest MFD-based dynamical model is the reservoir or bathtub model, in which the city is represented as a single homogeneous reservoir governed by accumulation conservation. This representation underlies a large share of MFD-based dynamic traffic assignment (DTA) because it replaces link-by-link dynamics with aggregate loading and completion processes. The same review notes that the MFD is arguably independent of origins, destinations, and route choice, which is precisely what makes it useful for macroscopic modeling (Aghamohammadi et al., 2018).

Multi-region formulations partition the network into interacting homogeneous reservoirs. In two-region perimeter control, each region has regional accumulation $n$7, regional trip completion rate $n$8, and jam accumulation $n$9, while control acts through inter-region metering rates $o(n)$0 and $o(n)$1. The aggregate state can be written with OD-specific components,

$o(n)$2

and the nonlinear dynamics may be expressed as

$o(n)$3

In this setting, the MFD identifies the critical accumulation $o(n)$4 where throughput is maximized and provides the macroscopic law used to regulate cross-boundary flows (Chen et al., 27 May 2025).

A more detailed aggregate model is the trip-based MFD, also called the speed-MFD or generalized bathtub formulation. Rather than making travel time a function of instantaneous accumulation alone, it computes group-specific travel time through a virtual-traveler construction: $o(n)$5 where $o(n)$6 is trip length and $o(n)$7 is network speed as a function of accumulation. This retains the city as a single reservoir while allowing heterogeneous trip lengths and departure times to affect experienced travel times (Balzer et al., 2021).

Another extension embeds the MFD in a multi-region state-space model for ride-sourcing and ridesplitting. For each region $o(n)$8, the speed-MFD is

$o(n)$9

with production $\mathcal{P}$0. The model tracks vehicle states $\mathcal{P}$1, vehicle counts $\mathcal{P}$2, and total remaining distance $\mathcal{P}$3. The additional remaining-distance state is introduced through the so-called M-model, motivated by the fact that accumulation-based MFD models can incur errors when trip lengths vary over time, for example when vehicles cruise for passengers or shared rides are interrupted (Beojone et al., 2022).

The review literature also identifies a continuum-space line of work in which the speed–density relation $\mathcal{P}$4 acts as the continuum analogue of the MFD. The governing conservation law is

$\mathcal{P}$5

and in Hughes-type models,

$\mathcal{P}$6

This interpretation links MFD-based urban traffic theory to two-dimensional conservation-law models and to predictive or reactive DTA formulations (Aghamohammadi et al., 2018).

3. Heterogeneity, hysteresis, and uncertainty

A recurring result in the literature is that the MFD is sensitive to spatial heterogeneity. In the Singapore AMOD study, spatial density variability is represented by

$\mathcal{P}$7

and the vehicle MFD is written as

$\mathcal{P}$8

There, greater density heterogeneity is associated with lower production, and hysteresis at a given accumulation is quantified by

$\mathcal{P}$9

with total hysteresis

$\gamma$0

The same work interprets clockwise hysteresis as a signature of less efficient and less stable recovery at equal accumulation (Oh et al., 2020).

In arterial networks governed by adaptive traffic signals, higher density heterogeneity correlates with lower flow, and flow heterogeneity is strongly anticorrelated with performance. Under time-dependent boundary conditions, intricate hysteresis loops appear in the MFDs and are strongly correlated with density heterogeneity. The reported mechanism is that loading and recovery can traverse different spatial distributions even when the mean density is the same, yielding different aggregate flows (Zhang et al., 2011).

Freeway-corridor theory generalizes the usual single clockwise-loop picture. For a corridor with a location-dependent downstream bottleneck governed by the LWR conservation law,

$\gamma$1

the MFD $\gamma$2 versus $\gamma$3 can form a figure-eight hysteresis loop when the fundamental diagram is nonlinear and either $\gamma$4 with full queue clearance or $\gamma$5. In that analysis, the upper branch is clockwise and the lower branch counter-clockwise. The same paper reports empirical evidence from two California bottlenecks that a continuous bottleneck causes less hysteresis than a discontinuous one under otherwise comparable conditions: the normalized hysteresis area is $\gamma$6 for I-880 N and $\gamma$7 for SR-41 N, a $\gamma$8 reduction with $\gamma$9 (Hammerl et al., 2024).

Recent calibration work treats the MFD as a capacity envelope with physically meaningful uncertainty rather than a single deterministic curve. Using a $Q = Q(n),$0-trapezoidal MFD,

$Q = Q(n),$1

the upper and lower bounds are interpreted as loading and recovery capacities, and a phase-aware coefficient $Q = Q(n),$2 is used to separate them. In that framework, the gap between the bounds is interpreted as capacity drop, estimated at about $Q = Q(n),$3 (Ma et al., 27 Aug 2025).

A stochastic formulation pushes this further by modeling exit-flow variation as a Wiener-driven process. In that work, the stochastic MFD preserves the accumulation-based structure but treats exit flow as $Q = Q(n),$4, with $Q = Q(n),$5 bounded through a nonlinear transformation of Brownian motion. The resulting stochastic differential equation and the forward Fokker–Planck equation generate time-evolving state distributions rather than a single trajectory and reproduce both hysteresis and gridlock (Qi, 2022).

4. Estimation, calibration, and data-driven reconstruction

Empirical MFD estimation is difficult when full network observability is unavailable. One approach constructs an MFD from floating car data (FCD) and loop detector counts using Edie’s generalized definitions,

$Q = Q(n),$6

then estimates the probe penetration rate from detector crossings and uses the calibrated MFD operationally to infer network density from probe speeds even when real-time penetration is unknown. In the reported microsimulation study, the mean error in estimated average density is about $Q = Q(n),$7 at $Q = Q(n),$8 FCD penetration and about $Q = Q(n),$9 at $Q$0 penetration (Knoop et al., 2020).

A second line of work replaces traditional loop or probe inputs with license plate cameras (LPCs) and road congestion indices (RCIs). The resulting MFD with volume-delay relationship (MFD-VD) begins from

$Q$1

uses $Q$2 for network trip time, and derives the implicit ODE

$Q$3

On a $Q$4 network in Chengdu, the fitted accumulation-based MFD achieves $Q$5 on 33 workdays and $Q$6 on the daily-averaged profile. Its central theoretical result is an observability-invariant saturation metric: $Q$7 which allows estimation of the ratio-to-critical-value even when only a finite set of LPCs is available and the true detection proportion is unknown (Han et al., 2023).

Sparse sensing has also motivated learning-based MFD reconstruction. A recent framework combines a custom Multi-Task Physics-Informed Neural Network (MTPINN) with Model-Agnostic Meta-Learning (MAML) across multiple cities. The learner predicts flow, critical occupancy, and maximum flow jointly while penalizing violations of a bi-parabolic MFD structure. On the UTD19 dataset, the meta-learned model reports mean MSE $Q$8 with 75 detectors, $Q$9 with 50, $n$0 with 25, and $n$1 with 10, compared with $n$2, $n$3, $n$4, and $n$5 for a standalone MTPINN. The same work summarizes the improvement as an average MSE reduction of roughly $n$6 to $n$7, depending on detector subset size (Roark et al., 19 Aug 2025).

The network-level nature of the MFD has also been used as the backbone of link-level reconstruction. A recent hybrid global-local framework defines a Local Correction Factor (LCF) so that

$n$8

or more generally,

$n$9

The correction factor is learned with a Graph Attention Network (GAT) for spatial dependency and a Gated Recurrent Unit (GRU) for temporal dependency, together with a strategic network partitioning method. In the reported experiments, average error in path travel time relative to MFD-based results is reduced to approximately $o(n)=Q(n)\frac{L}{\ell},$0 (2405.14257).

5. Control, operations, and planning applications

Perimeter control is one of the classical uses of the MFD because the diagram identifies the critical accumulation region where throughput is maximized. In a recent two-region formulation, the conventional set-point perimeter control problem is extended to an optimal tracking perimeter control problem (OTPCP), where the goal is to track a time-varying desired trajectory rather than stabilize to a fixed point. The control is solved with adaptive dynamic programming and integral reinforcement learning, without a well-calibrated dynamics model. In the reported numerical experiments, the proposed tracking perimeter control achieves a $o(n)=Q(n)\frac{L}{\ell},$1 reduction in total travel time and a $o(n)=Q(n)\frac{L}{\ell},$2 improvement in cumulative trip completion relative to a set-point controller (Chen et al., 27 May 2025).

The MFD has also become a system-level diagnostic for emerging mobility services. In the Singapore AMOD study, introducing automated mobility-on-demand increases maximum vehicle accumulation by about $o(n)=Q(n)\frac{L}{\ell},$3–$o(n)=Q(n)\frac{L}{\ell},$4, increases total 24-hour VKT by $o(n)=Q(n)\frac{L}{\ell},$5–$o(n)=Q(n)\frac{L}{\ell},$6, and increases total hysteresis over AM and PM peaks by roughly $o(n)=Q(n)\frac{L}{\ell},$7–$o(n)=Q(n)\frac{L}{\ell},$8, while passenger production remains relatively unchanged. The same analysis reports that total network energy rises by $o(n)=Q(n)\frac{L}{\ell},$9–$k^*$00, whereas total $k^*$01 and PM emissions decrease because the AMOD fleet is assumed to be battery electric (Oh et al., 2020).

Demand management formulations use the MFD to couple network congestion and behavioral response. In a trip-based MFD model with a tradable credit scheme (TCS), users choose between car and public transport through a logit model while car travel times are generated endogenously by the network state. Under an optimized TCS that minimizes total travel time, the reported optimum is $k^*$02 credits with total travel time reduced by about $k^*$03. Under a mixed objective combining travel time and carbon emissions, the optimum is $k^*$04 credits, with total travel time reduced by about $k^*$05, carbon emissions reduced by about $k^*$06, and public transport share increased by about $k^*$07 points (Balzer et al., 2021).

Regional control has recently been extended to electric-vehicle charging demand. In a 16-region framework, regional traffic dynamics are represented by a triangular MFD,

$k^*$08

and the joint optimization coordinates route guidance, external demand admission, and charging-station assignment. In the reported case study, under heavy traffic the proposed method reduces average total time by about $k^*$09 and travel time by about $k^*$10 relative to a no-demand-management benchmark, while outperforming nearest-charging and shortest-path baselines across all tested traffic conditions (Wen et al., 1 May 2026).

At the planning scale, the MFD has been coupled with the Cell Transmission Model (CTM) for the expressway network design problem in multiple subregions. In that hybrid model, subregions follow MFD dynamics and candidate expressways are modeled by CTM cells, while route choice is determined by stochastic user equilibrium. The results indicate that new expressways can significantly alleviate traffic congestion in the initial stages of planning, but that marginal benefits diminish as the expressway network expands if demand does not continue to increase. Variations in traffic demand between subregions also produce different construction schemes, emphasizing the role of demand distribution in budget allocation (Di et al., 2024).

6. Structural determinants, related diagrams, and open issues

A major research direction concerns how MFD shape depends on network structure. Numerical modeling and dimensional arguments have yielded scaling laws for the critical density $k^*$11 and maximum flow $k^*$12 in terms of road density $k^*$13, intersection density $k^*$14, car length $k^*$15, and maximum velocity. The key structural variable is

$k^*$16

interpreted as the average number of cars that can fit on a road segment between two intersections. The reported scaling for the critical density is

$k^*$17

which implies that $k^*$18 depends strongly on road density and only weakly on intersection density. Capacity is governed by a crossover relation

$k^*$19

so short effective link lengths and intersection spillbacks become decisive when $k^*$20 is small (Taillanter et al., 2023).

Signal control provides a related but distinct macroscopic perspective. For signalized urban road segments, a parsimonious FD of the form

$k^*$21

has been parameterized by average green split $k^*$22 through

$k^*$23

Empirically, higher green split shifts the FD upward, meaning higher speeds at the same flow. This is explicitly presented as being closely related to MFD thinking, but at the scale of signalized segments rather than classical whole-network MFDs (Zhang et al., 10 Jan 2025).

Another adjacent development is fundamental-diagram-constrained dynamic optimal transport. There, the traffic-theory flux relation is imposed locally as

$k^*$24

inside a Benamou–Brenier optimal transport problem. The paper explicitly states that this is not the classical MFD used in network traffic analysis; it is a local, pointwise admissibility condition inspired by the same saturation logic (Dong et al., 28 Jul 2025).

Several limitations and controversies are therefore well established. The review literature emphasizes that MFD validity depends on network topology, block length, turn-only lanes, and traffic signal timing, and that the assumptions required for a single-valued MFD are often not explicitly verified. It also notes that much continuum-space pedestrian and vehicular literature is mathematically MFD-like but historically developed without direct awareness of MFD theory. Open problems identified there include departure time choice, improved numerical methods, system optimum properties, real-time applicability of continuum-space models, and anisotropic formulations for intersecting flows (Aghamohammadi et al., 2018).

Taken together, these developments portray the MFD not as a single fixed curve but as a family of macroscopic closures for urban traffic systems. In the narrowest sense it is the aggregate network relation between accumulation and throughput; in broader usage it is the organizing principle behind reservoir models, multi-region control, state estimation under sparse sensing, uncertainty quantification, and several hybrid global-local or regional-link formulations. A plausible implication is that future MFD research will remain centered on three persistent issues already visible in the current literature: the conditions under which a well-defined macroscopic law exists, the treatment of hysteresis and uncertainty as intrinsic rather than residual phenomena, and the integration of the MFD with richer sensing, learning, and control architectures.