TrafficJAms: Modelling Traffic Jams in Aberdeen

Animations

IDM: Phantom Jam on a Circular Road

35 vehicles following the Intelligent Driver Model on a ring road. A single braking perturbation amplifies through the platoon, creating a backward-propagating phantom traffic jam. Cars coloured green (free flow) through red (stopped).

Nagel-Schreckenberg: Cars on a Highway

30 vehicles on a 120-cell periodic highway governed by the stochastic cellular automaton rules (accelerate, brake, random slowdown, move). Top panel shows real-time car positions with speed colouring; bottom panel builds up the space-time diagram revealing backward-propagating jam waves.

LWR: Shockwave Propagation on a Corridor

Godunov scheme solving the LWR hyperbolic PDE with Greenshields fundamental diagram. An initial high-density block (jam) disperses: backward shock propagates upstream while a rarefaction fan spreads downstream. Road strip colour indicates local congestion.

Bando OVM: Phantom Jam Emergence

25 vehicles on a circular road following the Optimal Velocity Model. Initially in near-uniform free flow, a small position perturbation triggers a Hopf bifurcation — the speed bar chart on the right shows the instability growing into a full stop-and-go wave rotating around the ring.

Aberdeen City Centre: Multi-Agent Simulation

200 vehicles navigating Aberdeen's real street network (fetched via OSMnx, 800m radius around Marischal College). Each vehicle follows IDM car-following on actual road segments, re-routing on arrival. Colour indicates speed: green = free flow, red = congested.

More Animations

Animation	Description
	Headlights & Taillights — braking cars glow red, accelerating cars show white headlights
	Speed Sparkline — rolling mean speed overlay inside the ring road
	Near-Miss Detection — red triangles flash when gap < 3m
	IDM vs Bando — side-by-side comparison of jam dynamics
	NaSch Parameter Sweep — 4 randomisation values in a grid
	3D Highway View — rotating 3D perspective of NaSch
	Dynamic Assignment — flow pulses through Aberdeen network
	Waveform Strip — audio-visualiser style speed range display
	Vehicle-Hours Lost — cumulative delay counter
	Phase Portrait — gap vs speed limit cycle in phase space

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Why Aberdeen?

Aberdeen presents distinctive traffic modelling challenges:

• Limited road network: geography constrains the city between the Rivers Don and Dee, the North Sea coast, and surrounding hills, creating natural bottlenecks.
• Bridge dependence: key pinch points at the Bridge of Don, Bridge of Dee, and King Street/Market Street corridors.
• The AWPR effect: the Aberdeen Western Peripheral Route (opened 2018) rerouted strategic traffic but shifted congestion patterns.
• Tidal commuter flows: strong directional bias from residential north/south into the city centre and energy-sector business parks.
• Port and industrial traffic: heavy goods vehicles serving Aberdeen Harbour and the energy industry mix with commuter traffic.

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Mathematical Modelling Approaches

1. Macroscopic (Continuum) Models

Treat traffic as a fluid and model aggregate density, flow, and velocity.

LWR Model (Lighthill-Whitham-Richards)

The foundational first-order hyperbolic PDE:

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}\bigl(Q(\rho)\bigr) = 0$$

where $\rho(x,t)$ is vehicle density and $Q(\rho) = \rho , v(\rho)$ is the flow-density relation (fundamental diagram). Shockwave formation corresponds to jam onset.

Aberdeen application: model jam propagation along single-corridor routes like King Street (A956) or the A90 approach to the Bridge of Don.

Payne-Whitham (second-order) Model

Adds a momentum equation with an anticipation/pressure term:

$$\frac{\partial v}{\partial t} + v\frac{\partial v}{\partial x} = \frac{V_e(\rho) - v}{\tau} - \frac{c_0^2}{\rho}\frac{\partial \rho}{\partial x}$$

where $V_e(\rho)$ is the equilibrium speed-density relation, $\tau$ is a relaxation time, and $c_0$ controls driver anticipation. Better captures stop-and-go waves.

Aberdeen application: model the oscillatory congestion on the A90/A96 merge zones where drivers react to downstream queues.

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2. Microscopic (Car-Following) Models

Simulate individual vehicle dynamics; useful for detailed intersection and merge studies.

Intelligent Driver Model (IDM)

$$\dot{v}_\alpha = a\left[1 - \left(\frac{v_\alpha}{v_0}\right)^\delta - \left(\frac{s^*(v_\alpha, \Delta v_\alpha)}{s_\alpha}\right)^2\right]$$

$$s^*(v, \Delta v) = s_0 + vT + \frac{v,\Delta v}{2\sqrt{ab}}$$

where $v_0$ is desired speed, $s_0$ is minimum gap, $T$ is safe time headway, $a$ and $b$ are acceleration/braking parameters.

Aberdeen application: simulate individual vehicle behaviour at the Anderson Drive / A90 junction or roundabouts like the Haudagain.

Optimal Velocity Model (Bando)

$$\dot{v}_\alpha = \kappa\bigl[V_{\mathrm{opt}}(s_\alpha) - v_\alpha\bigr]$$

Simple but captures spontaneous jam formation via Hopf bifurcation when sensitivity $\kappa$ is in a critical range.

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3. Network / Graph-Theoretic Models

Aberdeen's road network as a directed graph $G = (V, E)$ with nodes (junctions) and edges (road segments).

Static Traffic Assignment (Wardrop Equilibrium)

Find link flows $\mathbf{x}$ such that all used routes between each OD pair have equal (and minimal) cost:

$$c_p = c_q \quad \forall \text{ used paths } p,q \text{ between the same OD pair}$$

Solved via convex optimisation (Beckmann formulation):

$$\min_{\mathbf{x}} \sum_{e \in E} \int_0^{x_e} t_e(\omega),d\omega$$

subject to flow conservation and non-negativity. Link cost functions typically follow the BPR formula:

$$t_e(x_e) = t_e^0 \left[1 + \alpha\left(\frac{x_e}{C_e}\right)^\beta\right]$$

Aberdeen application: evaluate how closing or upgrading a single link (e.g., Union Street bus gates) redistributes flow across the entire network. Quantify the Braess paradox potential in Aberdeen's constrained topology.

Dynamic Traffic Assignment (DTA)

Extends the above to time-varying demands and link travel times; models rush-hour wave propagation across the network.

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4. Cellular Automaton Models

Nagel-Schreckenberg Model

Discrete space-time-velocity model on a 1D lattice. At each time step:

1. Acceleration: $v \leftarrow \min(v+1,, v_{\max})$
2. Braking: $v \leftarrow \min(v,, d-1)$ where $d$ is gap to next car
3. Randomisation: with probability $p$, $v \leftarrow \max(v-1,, 0)$
4. Movement: advance $v$ cells

Aberdeen application: fast simulation of phantom jam emergence on long stretches (A90 dual carriageway). Easy to extend to multi-lane with lane-changing rules. The stochastic element captures realistic driver variability.

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5. Queueing-Theoretic Models

Model intersections and bottlenecks as queueing systems.

M/D/1 or M/G/1 queues for signalised junctions: vehicles arrive as a Poisson process (rate $\lambda$), service time is the green phase allocation, and the queue length $L_q$ follows:

$$L_q = \frac{\rho^2}{2(1-\rho)} \quad \text{(M/D/1)}$$

where $\rho = \lambda / \mu$ is the utilisation.

Tandem / network queues: chain multiple intersections (e.g., the sequence of traffic lights along Union Street or King Street) as a Jackson network to analyse spillback and blocking.

Aberdeen application: optimise signal timings along the Market Street / Guild Street corridor; quantify delays at the Beach Boulevard / Esplanade signals.

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6. Stochastic and Data-Driven Approaches

Mean-field game (MFG) models: each driver minimises their own cost functional coupled to the population density:

$$\partial_t u + H(x, \nabla u) = \epsilon \Delta u + F(\rho)$$ $$\partial_t \rho - \nabla \cdot (\rho , \nabla_p H) = \epsilon \Delta \rho$$

Useful for modelling route-choice equilibria with many interacting agents.

Bayesian traffic state estimation: fuse sparse sensor data (e.g., SCOOT loop detectors on Aberdeen's signal network, Google/Waze travel times) with model predictions via Kalman filtering or particle filters to estimate real-time density fields.

Machine learning surrogates: train neural networks on simulation outputs or historical data (e.g., Traffic Scotland / Aberdeen City Council open data) to provide fast traffic state prediction for scenario testing.

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7. Suggested Starting Points for Aberdeen

Approach	Best suited for	Data needs
LWR / Payne-Whitham	Corridor-level jam propagation (A90, A96)	Flow-density measurements at a few points
IDM simulation	Junction/roundabout design (Haudagain, Bridge of Don)	Turning counts, signal plans
Nagel-Schreckenberg CA	Quick scenario testing, phantom jam statistics	Approximate flow rates
Network assignment	City-wide route redistribution, policy evaluation	OD demand matrix, link capacities
Queueing models	Signal optimisation on arterial corridors	Arrival rates, signal timing plans
MFG / data-driven	Real-time prediction, route guidance	Sensor/GPS trace data

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Key Aberdeen Data Sources

• Aberdeen City Council traffic count and signal data
• Traffic Scotland real-time and historical journey time data
• Transport Scotland STATS19 accident records and traffic surveys
• OpenStreetMap road network geometry for graph extraction
• Uber Movement / Google Environmental Insights travel time datasets

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Simulation Results

All models are implemented in the trafficjams/ package. Run with:

pip install -r requirements.txt
python -m simulations.run_all

LWR Model — Jam Propagation on A90-like Corridor

The Godunov scheme resolves shockwave propagation from an initial high-density block. The upstream shock travels backward (visible as the leftward-moving density front) while a rarefaction wave fans out downstream — classic LWR behaviour matching the Greenshields fundamental diagram.

Payne-Whitham — Stop-and-Go Waves at Merge Zone

The second-order model with anticipation term captures stop-and-go oscillations near the merge zone. Unlike the first-order LWR, the momentum equation allows velocity to deviate from equilibrium, producing richer wave structures.

Intelligent Driver Model — Spontaneous Jam on Circular Road

50 vehicles on a 1km circular road. A small initial speed perturbation on one vehicle amplifies through the platoon, producing a persistent traffic jam that propagates backward — the classic "phantom jam" phenomenon. Speed profiles show oscillatory convergence.

Bando OVM — Phantom Jams via Hopf Bifurcation

The Optimal Velocity Model demonstrates how uniform flow becomes unstable when the sensitivity parameter κ lies in a critical range. The mean speed drops and speed variance increases as the system evolves into stop-and-go waves.

Nagel-Schreckenberg — Cellular Automaton Space-Time Diagram

The stochastic CA model produces realistic jam patterns on a 500-cell lattice with 100 vehicles. Backward-propagating jam waves (dark diagonal bands) emerge spontaneously from the random braking rule — a hallmark of the NaSch model.

Network Assignment — Wardrop Equilibrium on Aberdeen Network

Beckmann formulation solved with BPR cost functions on an 8-node Aberdeen-inspired network. At equilibrium, used paths have approximately equal costs (Wardrop's first principle). Path 2 (Anderson Drive) carries the most flow due to higher capacity.

M/D/1 Queueing — Signalised Corridor Delays

Queue length and delay grow nonlinearly with utilisation, approaching infinity as ρ → 1. A 4-intersection corridor (modelling Union Street or King Street) shows how delays compound through multiple signals. The ρ = 0.85 threshold marks typical onset of severe congestion.

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References

• Lighthill, M.J. & Whitham, G.B. (1955). On kinematic waves II: A theory of traffic flow on long crowded roads. Proc. Royal Society A, 229(1178).
• Treiber, M. & Kesting, A. (2013). Traffic Flow Dynamics. Springer.
• Nagel, K. & Schreckenberg, M. (1992). A cellular automaton model for freeway traffic. J. Phys. I France, 2(12).
• Wardrop, J.G. (1952). Some theoretical aspects of road traffic research. Proc. ICE, 1(3).