• 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.

• 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 ($\rho = \lambda / \mu$).

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)

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 ($h$).

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 $q_{max} = u_o \times k_o$:

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

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.

• 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.

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.

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.

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.