Traffic Flow Theory

Traffic Flow Theory

Traffic flow theory involves the development of mathematical relationships among the primary elements of a traffic stream: flow, density, and speed. These macroscopic relationships help engineers understand traffic behavior, predict congestion, and design efficient transportation facilities.

Time-Space Diagrams

A fundamental graphical tool used to analyze traffic streams.

Time-Space Diagram

A two-dimensional plot with time (usually in seconds) on the x-axis and distance (usually in meters or feet) on the y-axis. The trajectory of a single vehicle is plotted as a line or curve on this graph.

Interpreting Time-Space Diagrams

Vehicle Trajectories: The path of an individual vehicle over time and space.
Speed ( $u$ ): The slope of the trajectory line ( $\Delta x / \Delta t$ ). A steeper slope indicates a higher speed. A horizontal line (zero slope) indicates a stopped vehicle.
Headway ( $h$ ): The horizontal distance between two parallel trajectories at a constant distance (e.g., passing a specific point).
Spacing ( $s$ ): The vertical distance between two parallel trajectories at a constant time (e.g., a snapshot of the roadway).
Shockwaves: When traffic states change (e.g., from free-flow to congested), the boundary between these states forms a shockwave, visually identifiable as the intersection points of different trajectory slopes.

Queueing Theory Applications

Mathematical modeling of waiting lines in transportation facilities.

Queueing Models in Traffic

Deterministic Queueing (D/D/1): Used for predictable situations like toll booths or signalized intersections where arrivals and departures are constant. It uses cumulative input and output curves (bottleneck models) to calculate total delay and maximum queue length.
Stochastic Queueing (M/M/1, M/G/1): Used when arrivals are random (Poisson distribution) and service times are variable (exponential or general). Essential for analyzing uncontrolled intersections, parking lots, or airport runways where the exact timing of arrivals cannot be known.
Key Metrics: Average waiting time, average queue length, maximum queue length, and system utilization factor ( $\rho = \lambda / \mu$ ).

Fundamental Equation of Traffic Flow

The core relationship linking the three main macroscopic parameters—flow (

q

), density (

k

), and speed (

u

)—is given by:

q = k \times u

Where:

$q$ = Flow rate: Number of vehicles passing a point per hour (veh/hr or vph)
$k$ = Density: Number of vehicles occupying a given length of a lane or roadway at a given instant (veh/km or veh/mi)
$u$ = Space mean speed: The average speed of vehicles over a specific length of roadway (km/hr or mph)

Fundamental Traffic Variables

Traffic Flow Theory (Greenshields Model)

Model Parameters

Free-Flow Speed (u_f)80 km/h

Jam Density (k_j)100 veh/km

Current Traffic State

Current Density (k)25 veh/km

Speed (u)60.0 km/h

Flow (q)1500 veh/h

Capacity (q_max)2000 veh/h

Optimum Speed40.0 km/h

Flow vs. Density (q-k)

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Speed vs. Density (u-k)

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To understand the macroscopic variables (

q, k, u

), we must first understand their microscopic counterparts.

q

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Flow ( $q$ ): The macroscopic rate at which vehicles pass a fixed point.
Headway ( $h$ ): The microscopic time interval between the passage of the front bumpers of consecutive vehicles.

They are inversely related:

q = \frac{3600}{h_{avg}}

(Where $h_{avg}$ is in seconds, yielding flow in vehicles per hour)

k

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Density ( $k$ ): The macroscopic concentration of vehicles over a length of road.
Spacing ( $s$ ): The microscopic distance between the front bumpers of consecutive vehicles.

They are inversely related:

k = \frac{1000}{s_{avg}}

(Where $s_{avg}$ is in meters, yielding density in vehicles per kilometer)

u

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Time Mean Speed ( $u_t$ ): The arithmetic mean of the spot speeds of vehicles passing a fixed point over a period of time. It is measured using radar guns or loop detectors.
Space Mean Speed ( $u_s$ ): The harmonic mean of spot speeds, representing the average speed of vehicles present on a segment of roadway at a given instant. This is the speed ( $u$ ) used in the fundamental equation $q=ku$ .

Key Takeaways

Flow ( $q$ ), Density ( $k$ ), and Speed ( $u$ ) form the three macroscopic variables linked by $q = k \times u$ .
Space Mean Speed ( $u_s$ ) is the harmonic mean of individual speeds and properly links macroscopic flow to density.

Time-Space Diagrams (Trajectories)

Visualizing vehicle movement over time and space.

A fundamental tool in traffic flow theory is the Time-Space Diagram. It plots the position of a vehicle (

x

) on the vertical axis against time (

t

) on the horizontal axis.

The line representing a vehicle's path is called its trajectory.
The slope of the trajectory ( $dx/dt$ ) at any point represents the vehicle's speed.
A steeper slope indicates a faster speed; a horizontal line indicates a stopped vehicle.
The vertical distance between two parallel trajectories is the spacing ( $s$ ).
The horizontal distance between two parallel trajectories is the time headway ( $h$ ).

Measurement Methods

The Moving Observer Method

A practical field technique for simultaneously estimating traffic flow (

q

) and average speed (

u

) on a roadway segment is the Moving Observer Method. A test vehicle travels along the segment in both directions, recording:

The number of vehicles it overtakes.
The number of vehicles that overtake the test vehicle.
The number of opposing vehicles met (when traveling in the opposite direction).
The travel time of the test vehicle. Through algebraic relationships, these counts and times can accurately estimate the stream flow and speed without installing stationary sensors.

Traffic Stream Models

These models attempt to describe the empirical relationships between speed, density, and flow under uninterrupted conditions.

Greenshields Model (Linear)

The simplest and most widely taught macroscopic model. It assumes a linear, inverse relationship between speed and density. As the road gets more crowded, drivers slow down linearly.

u = u_f \left( 1 - \frac{k}{k_j} \right)

Where:

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$u_f$ = Free-flow speed: The theoretical speed when density is zero (an empty road).
$k_j$ = Jam density: The theoretical maximum density when speed is zero (complete gridlock).

Substituting this into the fundamental equation (

q = uk

), we get a parabolic flow-density relationship:

q = u_f k - \frac{u_f}{k_j} k^2

q_{max}

Understanding why capacity occurs at exactly half of the free-flow speed and jam density.

To find the mathematical peak of the parabolic flow-density curve (which represents the absolute maximum capacity of the roadway segment), we apply differential calculus.

We start with the parabolic flow equation derived from Greenshields:

q = u_f k - \frac{u_f}{k_j} k^2

To find the maximum flow (

q_{max}

), we take the derivative of flow (

q

) with respect to density (

k

) and set it to zero:

\frac{dq}{dk} = u_f - 2 \left( \frac{u_f}{k_j} \right) k = 0

Solving for

k

, we find the optimal density (

k_o

2 \left( \frac{u_f}{k_j} \right) k_o = u_f \implies k_o = \frac{k_j}{2}

Substituting

k_o = k_j / 2

back into the original linear speed equation yields the optimal speed (

u_o

u_o = u_f \left( 1 - \frac{k_j / 2}{k_j} \right) = u_f \left( 1 - 0.5 \right) = \frac{u_f}{2}

Finally, applying the fundamental equation

q_{max} = u_o \times k_o

q_{max} = \left( \frac{u_f}{2} \right) \times \left( \frac{k_j}{2} \right) = \frac{u_f \times k_j}{4}

Note

Maximum Flow (Capacity)
In the Greenshields model, maximum flow (

q_{max}

, also called capacity) occurs exactly when the traffic stream is at half the jam density and half the free-flow speed.

Optimum Density ( $k_o$ ): $k_j / 2$
Optimum Speed ( $u_o$ ): $u_f / 2$

q_{max} = \frac{u_f \times k_j}{4}

Greenberg's Logarithmic Model

Greenshields breaks down at very low densities (predicting finite speed instead of matching free-flow limits realistically) and very high densities. Greenberg proposed a logarithmic model that fits dense traffic conditions much better, though it fails at low densities (predicting infinite speed at zero density).

Greenberg's Logarithmic Model

Calculates the speed of traffic based on a logarithmic relationship with density, particularly effective for dense traffic conditions.

$$
u = u_o \ln\left( \frac{k_j}{k} \right)
$$

Key Takeaways

The Greenshields Model forms a linear inverse relationship between speed and density, causing a parabolic flow-density curve.
Optimum flow (capacity) occurs halfway to the jam density ( $k_j/2$ ) and halfway to free-flow speed ( $u_f/2$ ).

Macroscopic Fundamental Diagram (MFD)

Extending flow theory from a single roadway segment to an entire urban network.

While traditional models analyze a single link, the Macroscopic Fundamental Diagram (MFD) (or Network Fundamental Diagram) plots the relationship between the average network flow (production) and the average network density (accumulation) for an entire city region or downtown core.

MFD Characteristics

It proves that an entire urban network behaves similarly to a single road: as the number of vehicles in the downtown area increases, the total network throughput increases up to a critical capacity point.
If accumulation exceeds that critical point, gridlock sets in, and total network throughput drops drastically.
Engineers use the MFD to implement perimeter metering (e.g., holding cars at suburban traffic signals) to ensure the downtown core never exceeds its critical accumulation, thus maintaining maximum city-wide throughput.

Shockwave Theory

Traffic conditions on a roadway are rarely uniform. When there is a sudden change in capacity (due to a lane closure, a crash, or a red light) or a sudden change in demand (like a stadium emptying out), the boundary between the two different traffic states propagates along the highway. This moving boundary is called a shockwave.

Understanding shockwaves is critical for safety analysis, as they represent areas where drivers must suddenly decelerate, leading to high risks of rear-end collisions. Traffic engineers use shockwave theory to calculate how fast a queue will grow, how long it will take to dissipate, and where to place advanced warning signs for stopped traffic.

A shockwave in traffic represents a boundary between two distinct states of traffic flow (e.g., free-flowing traffic hitting a sudden queue). It is the propagation of a change in density and flow along the roadway.

The speed of the shockwave (

u_{sw}

) is calculated as the slope of the chord connecting the two traffic states (Points 1 and 2) on the flow-density (

q-k

) curve.

u_{sw} = \frac{q_2 - q_1}{k_2 - k_1}

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Forward Moving Shockwave ( $u_{sw} \gt 0$ ): The boundary moves in the direction of traffic. (e.g., A queue dissipating after a green light turns on).
Backward Moving Shockwave ( $u_{sw} \lt 0$ ): The boundary moves against the direction of traffic. (e.g., A queue forming and growing backward from a red light or an accident).
Stationary Shockwave ( $u_{sw} = 0$ ): The boundary stays in one place. (e.g., The front of a queue at a steady bottleneck where arrival rate exactly equals departure rate).

Key Takeaways

Shockwaves visualize moving boundaries between two distinct traffic states with differing flows and densities.
Speed of a shockwave represents the rate of queue growth or dissipation, aiding in bottleneck delays calculation.

Microscopic Car-Following Models

Modeling the behavior of individual drivers reacting to the car in front of them.

While macroscopic models look at the stream as a whole, car-following models mathematically describe how a single driver (the follower) adjusts their acceleration based on the actions of the vehicle immediately ahead (the leader). These form the basis of modern microscopic traffic simulation software (like VISSIM).

The General Motors (GM) Model

The most famous family of car-following models posits that a driver's response (acceleration/deceleration) is proportional to the stimulus (the relative speed between the two cars) and inversely proportional to their spacing.

\text{Response} = \text{Sensitivity} \times \text{Stimulus}

a_{n+1}(t+\Delta t) = \left[ \frac{\alpha_l \cdot u_{n+1}^m(t+\Delta t)}{[x_n(t) - x_{n+1}(t)]^l} \right] \times [u_n(t) - u_{n+1}(t)]

Where:

$a_{n+1}$ is the acceleration of the following vehicle after a reaction time $\Delta t$ .
$u_n - u_{n+1}$ is the relative speed (stimulus).
$x_n - x_{n+1}$ is the spacing between the vehicles.
$\alpha, l, m$ are calibration parameters.

Gap Acceptance Theory

Crucial for modeling unsignalized intersections, roundabouts, and lane changing. A driver on a minor road will only pull into the major traffic stream if the time headway (gap) between approaching vehicles is greater than their personal Critical Gap ( $t_c$ ). If the gap is rejected, they wait for the next one. This theory uses probability distributions (like the negative exponential distribution) to predict how long a driver will be delayed before finding a safe gap.

Queuing Theory

Queuing theory is used to analyze delays, queue lengths, and waiting times at bottlenecks (like toll booths or intersections).

Deterministic Queuing (D/D/1)

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Assumptions: Arrival rate ( $\lambda$ ) and service rate ( $\mu$ ) are constant and deterministic. There is 1 server.
Mechanism: A queue forms whenever the arrival rate exceeds the service rate ( $\lambda \gt \mu$ ).
Analysis: Often visualized using cumulative arrival and departure curves over time. Total Delay is the area between these two curves.

Stochastic Queuing (M/M/1)

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Assumptions: Arrivals are random (following a Poisson distribution, hence Markovian 'M'), service times are exponentially distributed (also 'M'), and there is 1 server.
Condition: The system only reaches a steady state if the utilization ratio $\rho = \lambda / \mu \lt 1$ . If $\rho \ge 1$ , the queue grows infinitely.
Key Metric: Average queue length ( $L_q$ ) is given by: $L_q = \frac{\rho^2}{1-\rho}$

M/M/1 Delay Analysis

Calculates the average time a vehicle spends waiting in the queue and the total time spent in the system.

$$
W_q = \frac{\lambda}{\mu(\mu - \lambda)}
$$

Total Time in System

Calculates the total average time a vehicle spends in the M/M/1 queuing system (wait time + service time).

$$
W = W_q + \frac{1}{\mu} = \frac{1}{\mu - \lambda}
$$

Key Takeaways

Queuing theory assesses bottlenecks by analyzing arrival rates ( $\lambda$ ) vs. service rates ( $\mu$ ).
When $\lambda \gt \mu$ , queues grow infinitely; analyzing these helps predict delays and needed capacity expansion.

Mathematical Formulations in Flow Theory

The Greenshields Model Equation

The Greenshields Model assumes a linear relationship between speed and density:

u = v_f \left( 1 - \frac{k}{k_j} \right)

Where

v_f

is free-flow speed and

k_j

is jam density. Substituting this into the fundamental equation (

q = u \times k

) yields a parabolic relationship for flow:

q = v_f k - \left( \frac{v_f}{k_j} \right) k^2

Maximum capacity (

q_{max}

) occurs when density is exactly half of the jam density (

k_m = k_j / 2

) and speed is half of the free-flow speed (

u_m = v_f / 2

q_{max} = \frac{v_f \times k_j}{4}

Shockwave Velocity

The velocity of a shockwave (

u_w

) propagating between two different traffic states (state 1 and state 2) is defined as the change in flow divided by the change in density:

u_w = \frac{q_2 - q_1}{k_2 - k_1}

Key Takeaways

The fundamental equation of macroscopic traffic flow is $q = k \times u$ (Flow = Density × Speed).
Time Mean Speed is the arithmetic average of spot speeds; Space Mean Speed is the harmonic average and must be used in the fundamental equation.
The Greenshields Model assumes a linear relationship between speed and density, leading to a parabolic relationship between flow and density where capacity ( $q_{max}$ ) occurs at $u_f/2$ and $k_j/2$ .
Shockwave Theory describes how changes in traffic states (like a queue forming or dissipating) propagate along a roadway.
Queuing Theory is used to calculate delays and queue lengths at bottlenecks, analyzing the balance between arrival rates and service rates.

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Time-Space Diagrams

Time-Space Diagram

Interpreting Time-Space Diagrams

Queueing Theory Applications

Queueing Models in Traffic

Fundamental Equation of Traffic Flow

Fundamental Traffic Variables

Traffic Flow Theory (Greenshields Model)

Model Parameters

Current Traffic State

Flow vs. Density (q-k)

Speed vs. Density (u-k)

Flow (qqq) and Headway (hhh)

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Density (kkk) and Spacing (sss)

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Speed (uuu)

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Time-Space Diagrams (Trajectories)

Measurement Methods

The Moving Observer Method

Traffic Stream Models

Greenshields Model (Linear)

Checklist

The Derivation of Maximum Capacity (qmaxq_{max}qmax​)

Note

Greenberg's Logarithmic Model

Greenberg's Logarithmic Model

Macroscopic Fundamental Diagram (MFD)

MFD Characteristics

Shockwave Theory

Checklist

Microscopic Car-Following Models

The General Motors (GM) Model

Gap Acceptance Theory

Queuing Theory

Deterministic Queuing (D/D/1)

Checklist

Stochastic Queuing (M/M/1)

Checklist

M/M/1 Delay Analysis

Total Time in System

Mathematical Formulations in Flow Theory

The Greenshields Model Equation

Shockwave Velocity

Flow ( $q$ ) and Headway ( $h$ )

Density ( $k$ ) and Spacing ( $s$ )

Speed ( $u$ )

The Derivation of Maximum Capacity ( $q_{max}$ )