⚡ electrical engineering

Stochastic and Dynamic Fundamental Diagram for Mixed Traffic

This study proposes a stochastic and dynamic fundamental diagram framework for mixed traffic that demonstrates how both the penetration rate and the specific sequencing of automated and human-driven vehicles significantly influence the magnitude and variability of traffic hysteresis.

Original authors: Jiwan Jiang, Soyoung Ahn

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiwan Jiang, Soyoung Ahn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine traffic as a long line of people holding hands, trying to walk at the same speed. Sometimes, the person at the front stumbles or speeds up, and that little wobble ripples down the line. In traffic engineering, this ripple is called a "hysteresis loop." Think of it like a rubber band: when you stretch it (traffic gets crowded), it snaps back differently than how it stretched. The paper shows that if you just look at the average number of people in the line, you miss the whole story of how that rubber band snaps and stretches.

Here is what this paper does, broken down into simple concepts:

1. The Problem: The "Static" Map vs. The "Real" Ride

Traditionally, engineers use a "Fundamental Diagram" to predict traffic. Think of this like a static map of a city. It tells you, "If there are 50 cars per mile, the speed will be 30 mph." It assumes traffic is calm and steady.

But real traffic is messy. It's more like a live video of a rollercoaster. When a disturbance happens (like a sudden brake), the traffic doesn't just move along the map; it traces a messy, elliptical loop. The paper calls this "traffic hysteresis." The old maps can't explain why traffic sometimes gets stuck even after the road clears up.

2. The New Players: Humans vs. Robots

Now, we are mixing Human-Driven Vehicles (HDVs) with Automated Vehicles (AVs).

Humans are like a group of friends walking together. They react differently depending on how fast they are going, how tired they are, and how much space they have. Their reactions are non-linear and a bit chaotic. If the person in front moves a little, a human might overreact and slam the brakes.
AVs (Robots) are like a marching band. They follow strict rules. If the leader moves, the robot calculates exactly how to move. Their reactions are linear and predictable.

The paper asks: What happens when you mix the chaotic friends with the marching band?

3. The Secret Sauce: "Describing Function Analysis"

The math in the paper is heavy, but the idea is simple. The authors use a trick called Describing Function Analysis.
Imagine you are trying to understand how a wobbly, non-linear object (like a human driver) reacts to a steady rhythm (like a drumbeat). Instead of trying to solve the impossible equation for every single wobble, the authors say, "Let's just look at the main beat." They turn the messy human behavior into a "roughly linear" version that they can calculate. It's like turning a jagged, scribbled line into a smooth, straight line just enough to make the math work.

4. The Big Discovery: It's Not Just How Many, It's Where

The researchers ran thousands of computer simulations (like rolling dice millions of times) to see how traffic behaves with different mixes of humans and robots. They found two huge things:

A. The Order Matters (The Seating Chart)
It doesn't matter just how many robots are in the line; it matters where they sit.

Scenario 1: Robots First. If the robots are at the front, they start a very precise, rhythmic wobble. When the humans behind them feel this wobble, they get confused and overreact because humans get more jittery the bigger the wobble gets. This creates a huge, messy loop (big traffic jams).
Scenario 2: Humans First. If the humans are at the front, they create a messy wobble. But when the robots behind them feel it, they calmly smooth it out. This creates a smaller, tighter loop (less traffic trouble).
Analogy: Imagine a line of people passing a bucket of water. If the first person (robot) pours water perfectly, the second person (human) might panic and spill it. But if the first person (human) spills a little, the second person (robot) might catch it perfectly. The order changes the outcome.

B. More Robots Usually Helps, But Placement is Key
Generally, having more robots in the line makes the traffic smoother and reduces the size of those "hysteresis loops." The robots act like shock absorbers. However, the paper warns that if you put the robots in the wrong spots (like at the very front), they might actually make the human drivers more jittery, leading to bigger problems than if you had fewer robots.

5. The Conclusion

The paper concludes that to understand traffic, we can't just count the number of robots. We have to look at the sequence.

The Old Way: "We have 50% robots, so traffic will be 50% better."
The New Way: "We have 50% robots, but because they are sitting at the front, the traffic might actually be worse than if they were at the back."

In short, traffic is a dance. The paper shows that knowing the music (the number of robots) isn't enough; you have to know who is leading the dance (the sequence) to understand how the whole line will move.

1. Problem Statement

Traditional traffic flow theory relies on the Fundamental Diagram (FD), which depicts a static, deterministic relationship between flow, density, and speed under steady-state equilibrium. However, real-world traffic rarely remains in equilibrium; it exhibits traffic hysteresis, where the flow-density path traces an elliptical loop rather than a single curve.

Existing literature faces three critical gaps when addressing mixed traffic (comprising Automated Vehicles [AVs] and Human-Driven Vehicles [HDVs]):

Deterministic Assumptions: Most models treat car-following (CF) behavior as deterministic, ignoring the intrinsic randomness of human reaction times and the parametric uncertainty of AV controllers.
Neglect of Sequencing: While penetration rates (the percentage of AVs) are often studied, the spatial sequence (ordering) of AVs and HDVs within a platoon is frequently overlooked. Field observations suggest that clustering vs. random dispersion significantly impacts macroscopic performance.
Dynamic Complexity: There is a lack of a comprehensive framework that integrates high-order vehicle dynamics, stochastic behavior, and vehicle sequencing to explain how hysteresis loops evolve in mixed traffic.

2. Methodology

The authors propose a Sequence-Based Stochastic Dynamic FD framework that bridges microscopic driving behaviors with macroscopic traffic states. The methodology consists of three core components:

A. Describing Function Analysis (DFA)

To handle the nonlinearity of HDV car-following models (e.g., OVM, GFM, FVDM, IDM) while retaining the ability to perform frequency-domain analysis, the authors employ DFA.

Linearization: DFA approximates nonlinear systems by retaining only the fundamental harmonic of the output when driven by a sinusoidal input.
Transfer Functions: This allows the derivation of approximate, amplitude-dependent linear transfer functions ( $G(j\omega)$ ) for nonlinear CF laws. This bridges the gap between nonlinear HDV models and linear AV controllers.
Harmonic Balance: The method ensures dynamic consistency by solving for the amplitude and phase shift where the assumed sinusoidal response matches the system's output.

B. Probabilistic Framework for Mixed Platoons

The study models a platoon of $N$ vehicles led by a sinusoidal disturbance.

Recursive Amplitude Model: The oscillation amplitude of the $l$ -th vehicle ( $A_l$ ) is recursively calculated based on the amplitude of the predecessor ( $A_{l-1}$ ) and the vehicle's specific CF law and parameters.
Sources of Uncertainty: The framework incorporates:
1. Model Heterogeneity: Different CF laws (e.g., IDM vs. OVM for HDVs; Linear vs. Higher-order for AVs).
2. Parameter Randomness: CF parameters are treated as random variables drawn from prior distributions (calibrated via Approximate Bayesian Computation on NGSIM and MA datasets).
3. Sequence Variability: Explicit enumeration of vehicle ordering (e.g., AV-first, HDV-first, alternating, random).

C. Stochastic Dynamic FD Formulation

Macroscopic Mapping: Microscopic spacing and velocity oscillations are mapped to macroscopic density ( $\tilde{k}$ ) and flow ( $\tilde{q}$ ).
Monte Carlo Simulation: Since closed-form probability density functions (PDFs) for the joint distribution of density and flow are intractable due to the complexity of the recursive stochastic process, the authors use Monte Carlo simulations. They sample thousands of sequences and parameter sets to generate the joint PDF of the dynamic FD.

3. Key Contributions

Novel Framework: Development of the first stochastic, sequence-based dynamic FD that explicitly accounts for both vehicle class heterogeneity (AV vs. HDV) and spatial sequencing.
Analytical-Computational Hybrid: Successful integration of Describing Function Analysis (for analytical tractability of nonlinearities) with Monte Carlo simulation (to capture full stochastic distributions).
Quantification of Sequencing Effects: Demonstration that the order of vehicles is as critical as the penetration rate in determining traffic stability and hysteresis magnitude.
Calibration: Calibration of CF model probabilities and parameters using real-world high-resolution trajectory data (I-80 and I-24 datasets) and controlled AV experiments (MA dataset).

4. Key Results

The study conducted numerical experiments with a 21-vehicle platoon, varying AV penetration rates (0% to 100%) and sequencing scenarios.

A. Impact of Vehicle Sequencing

AV-First Scenario (AVs leading): Produced the largest hysteresis loops and highest flow variability.
- Mechanism: AVs initiate structured oscillations. As these propagate to trailing HDVs, the HDVs' nonlinear, amplitude-dependent behavior amplifies the oscillations, leading to erratic and wide hysteresis loops.
HDV-First Scenario (HDVs leading): Produced the smallest hysteresis loops and lowest variability.
- Mechanism: HDVs at the front generate initial disturbances, but the subsequent AVs (which are less sensitive to amplitude) dampen and stabilize the oscillations as they propagate downstream.
Alternating/Random: Resulted in intermediate outcomes, confirming that mixing strategies matter.

B. Impact of AV Penetration Rate

Hysteresis Damping: Generally, increasing AV penetration reduces the magnitude (area) and variability of the hysteresis loop.
- High-Density Convex Hull Area: Decreased from 861.82 (0% AV) to 694.34 (100% AV) $veh^2/(km \cdot h)$ .
- Flow Standard Deviation: Decreased from 104.29 to 72.19 $veh/h$.
Non-Monotonicity: The improvement is not strictly linear. In some intermediate penetration scenarios (e.g., 33.3% or 66.7%), the variability can temporarily increase depending on the specific sequence, highlighting that "more AVs" does not always equal "better stability" without considering placement.

C. Frequency Response Analysis

HDV models (OVM, GFM, FVDM, IDM) generally exhibit marginal stability (gain $|G| \approx 1$ ) at low frequencies, making them prone to wave amplification.
IDM showed the highest sensitivity to parameter choice, resulting in the widest spread in dynamic response.

5. Significance and Implications

Theoretical Advancement: The paper moves beyond static FDs and deterministic models, providing a rigorous mathematical basis for understanding the stochastic and dynamic nature of traffic in the era of automation.
Policy and Management:
- Platoon Formation: Traffic management strategies should consider vehicle sequencing. For instance, placing AVs at the rear of a mixed platoon (or ensuring HDVs lead) may be more effective at suppressing stop-and-go waves than placing AVs at the front.
- Transitional Era Management: During the partial adoption phase of AVs, simply increasing the number of AVs is insufficient; their spatial distribution must be optimized to minimize hysteresis and maximize capacity.
Future Directions: The framework suggests that future controller tuning for AVs should explicitly account for the stochastic nature of human drivers and the potential for sequence-dependent instability.

In conclusion, the study demonstrates that traffic hysteresis in mixed environments is governed not just by the composition of AVs and HDVs, but critically by how they interact through their specific sequencing, offering a new lens for designing stable mixed-traffic systems.