2.1 Macroscopic Traffic Stream Models & Fundamental Diagram (q = k × u)

2.1 Macroscopic Traffic Stream Models & Fundamental Diagram (q = k × u)

PTOE Exam Focus: The macroscopic fundamental relationship $q = k \times \bar{u}s$ forms the bedrock of traffic operations engineering. Expect quantitative exam problems testing the mathematical derivation of Space Mean Speed from spot speed datasets, the application of Wardrop's speed variance identity, and the calculation of critical capacity parameters ($u_f, k_j, k_c, u_c, q{\max}$) across Greenshields, Greenberg, and Underwood speed-density formulations.

---

1. Fundamental Traffic Stream Parameters

Macroscopic traffic flow theory treats vehicular traffic as a continuous compressible fluid, aggregating individual vehicle trajectories into three primary stream parameters:

1. Flow Rate ($q$ or $V$): The equivalent hourly rate at which vehicles pass a fixed point on a lane or roadway during a specified time interval less than one hour (typically 15 minutes): $q = \frac{N}{T}$ Where $N$ is the observed vehicle count over observation period $T$ (hours). On the PTOE exam, hourly volume ($V$) is expanded to peak 15-minute flow rate ($v$) using the Peak Hour Factor ($PHF$): $v = \frac{V}{PHF} = \frac{V}{4 \times V_{15}}$

2. Density ($k$ or $K$): The number of vehicles occupying a given length of lane or roadway at a single instant in time (spatial snapshot), expressed in vehicles per mile per lane (veh/mi/ln) or vehicles per kilometer per lane (veh/km/ln): $k = \frac{N}{L}$ Where $N$ is the number of vehicles concurrently present along roadway segment length $L$. Density can also be estimated from electronic presence loop detector occupancy ($Occ$, the percentage of time the detection zone is occupied): $k = \frac{5280 \times Occ}{100 \times (L_v + L_d)}$ Where $L_v$ is average vehicle length (typically 18–20 ft) and $L_d$ is the detector loop length (typically 6 ft).

3. Speed ($u$ or $v$): The rate of vehicle motion, expressed in miles per hour (mph) or kilometers per hour (km/h). Because speed varies among individual drivers, macroscopic stream analysis requires precise statistical averaging.

---

2. Space Mean Speed vs. Time Mean Speed (Wardrop's Variance Identity)

A frequent trap on the PTOE exam is confusing Time Mean Speed (TMS) with Space Mean Speed (SMS). The fundamental equation of traffic flow ($q = k \times u$) holds strictly and exclusively for Space Mean Speed.

Mathematical Definitions

• Time Mean Speed ($\bar{u}_t$): The arithmetic mean of spot speeds of all vehicles passing a specific roadway cross-section during a given time interval: $\bar{u}_t = \frac{1}{N} \sum_{i=1}^{N} u_i$ TMS reflects what a roadside radar gun, lidar, or single inductive loop detector records over time.

• Space Mean Speed ($\bar{u}_s$): The harmonic mean of spot speeds, representing the average speed of all vehicles occupying a given roadway segment over a spatial length $L$: $\bar{u}_s = \frac{L}{\bar{t}} = \frac{L}{\frac{1}{N}\sum_{i=1}^N t_i} = \frac{N}{\sum_{i=1}^N \frac{1}{u_i}}$ Where $t_i = L / u_i$ is the individual travel time of vehicle $i$ across segment length $L$.

Wardrop's Variance Identity

In 1952, John Glen Wardrop formulated the mathematical proof establishing the exact structural relationship between TMS and SMS: $\bar{u}_t = \bar{u}_s + \frac{\sigma_s^2}{\bar{u}_s}$ Where $\sigma_s^2$ is the sample variance of vehicle speeds about the Space Mean Speed: $\sigma_s^2 = \frac{1}{N} \sum_{i=1}^N (u_i - \bar{u}_s)^2$

Engineering Implications:

• Because $\sigma_s^2 \ge 0$, Time Mean Speed is always greater than or equal to Space Mean Speed ($\bar{u}_t \ge \bar{u}_s$).
• $\bar{u}_t = \bar{u}_s$ if and only if all vehicles travel at identical speeds (zero speed variance, $\sigma_s^2 = 0$).
• Faster vehicles cross a point detector more frequently than slower vehicles relative to their spatial presence, causing TMS to place disproportionate weight on high-speed vehicles. SMS weights vehicles inversely by travel time, correctly capturing spatial density.
• Substituting TMS into $q = k \times u$ will systematically overestimate actual roadway flow and capacity.

---

3. The Fundamental Diagram of Traffic Flow

The macroscopic behavior of a traffic stream is governed by the continuous interrelationship between $q$, $k$, and $u$. Plotting these pairwise produces the three curves of the Fundamental Diagram:

   Flow (q)                 Speed (u)                 Flow (q)
     ^                        ^                         ^
q_max|      *  (Capacity)  u_f|*                     q_max|      *  (Capacity)
     |    *   *               | *                         |    *   *
     |  *       *             |   *                       |  *       *
     | *         *            |     *                     | *         *
    0+-------------*->k      0+-------*---->k            0+-------------*->u
     0     k_c    k_j         0  k_c  k_j                 0     u_c    u_f
     (Flow-Density)           (Speed-Density)             (Flow-Speed)

Key Boundary Parameters:

• Free-Flow Speed ($u_f$): The theoretical speed of vehicles as density approaches zero ($k \to 0$). Drivers travel unhindered by other traffic.
• Jam Density ($k_j$): The maximum physical density of a roadway segment where vehicles are bumper-to-bumper in complete gridlock ($u = 0$, $q = 0$). Typical passenger car jam density is $180\text{--}240\text{ veh/mi/ln}$ ($40\text{--}50\text{ ft/veh}$).
• Critical Density ($k_c$ or $k_m$): The density at which the roadway delivers its absolute maximum flow rate (capacity, $q_{\max}$).
• Critical Speed ($u_c$ or $u_m$): The Space Mean Speed corresponding to maximum capacity flow.
• Capacity ($q_{\max}$ or $C$): Maximum sustainable stream throughput: $q_{\max} = k_c \times u_c$.

Operational Regimes:

1. Uncongested / Stable / Undersaturated Regime ($k < k_c$, $u > u_c$):
   • Flow increases as density increases (positive slope: $\frac{dq}{dk} > 0$).
   • Individual drivers maintain speed autonomy; perturbations dampen out.
2. Congested / Forced Flow / Oversaturated Regime ($k > k_c$, $u < u_c$):
   • Flow decreases as density increases (negative slope: $\frac{dq}{dk} < 0$).
   • Operating under stop-and-go queue conditions; disturbances propagate upstream as shockwaves.

---

4. Single-Regime Macroscopic Stream Formulations

Traffic engineers utilize mathematical models relating speed and density to predict flow performance. The three foundational single-regime models are Greenshields, Greenberg, and Underwood.

A. Greenshields Model (Linear Speed-Density)

Bruce Greenshields (1935) proposed a direct linear decline in speed with increasing density: $u(k) = u_f \left(1 - \frac{k}{k_j}\right)$

Multiplying by density $k$ yields the parabolic flow-density relationship: $q(k) = k \cdot u(k) = u_f \left(k - \frac{k^2}{k_j}\right)$

Differentiating with respect to $k$ and setting $\frac{dq}{dk} = 0$ identifies the critical capacity parameters: $\frac{dq}{dk} = u_f \left(1 - \frac{2k}{k_j}\right) = 0 \implies k_c = \frac{k_j}{2}$ $u_c = u(k_c) = u_f \left(1 - \frac{k_j/2}{k_j}\right) = \frac{u_f}{2}$ $q_{\max} = k_c \cdot u_c = \left(\frac{k_j}{2}\right)\left(\frac{u_f}{2}\right) = \frac{u_f k_j}{4}$

• Properties: Perfectly symmetric parabola centered at $k_c = k_j/2$. Parabolic flow-speed relationship: $q(u) = k_j(u - u^2/u_f)$. Simple and widely used for planning, but underestimates speeds at low densities and overestimates speeds near jam density.

B. Greenberg Model (Logarithmic Speed-Density)

Harold Greenberg (1959) derived a logarithmic model based on 1D fluid hydrodynamic continuity: $u(k) = u_c \ln\left(\frac{k_j}{k}\right)$

Corresponding flow-density equation: $q(k) = u_c \cdot k \ln\left(\frac{k_j}{k}\right)$

Setting $\frac{dq}{dk} = 0$: $\frac{dq}{dk} = u_c \ln\left(\frac{k_j}{k}\right) - u_c = 0 \implies \ln\left(\frac{k_j}{k_c}\right) = 1 \implies k_c = \frac{k_j}{e} \approx 0.368 k_j$ $q_{\max} = u_c k_c = \frac{u_c k_j}{e}$

• Properties: Excellent fit for dense, congested urban freeway and tunnel traffic (such as the Lincoln Tunnel data from which it was calibrated). However, as density approaches zero ($k \to 0$), speed mathematically approaches infinity ($u \to \infty$), meaning it cannot model free-flow speed.

C. Underwood Model (Exponential Speed-Density)

Robin Underwood (1961) addressed Greenberg's low-density limitation with an exponential formulation: $u(k) = u_f \exp\left(-\frac{k}{k_c}\right)$

Corresponding flow-density equation: $q(k) = u_f \cdot k \exp\left(-\frac{k}{k_c}\right)$

Setting $\frac{dq}{dk} = 0$: $\frac{dq}{dk} = u_f \exp\left(-\frac{k}{k_c}\right) \left(1 - \frac{k}{k_c}\right) = 0 \implies k = k_c$ $u_c = u(k_c) = u_f e^{-1} = \frac{u_f}{e} \approx 0.368 u_f$ $q_{\max} = k_c u_c = \frac{u_f k_c}{e}$

• Properties: Models free-flow conditions with high accuracy ($u = u_f$ at $k = 0$). However, speed approaches zero only asymptotically as $k \to \infty$, meaning the model lacks a finite jam density ($k_j = \infty$).

---

5. Macroscopic Model Comparison Matrix

---

6. Worked PTOE Calculation Example

Problem Statement:

A macroscopic traffic study on a 3-lane suburban freeway segment calibrates a linear Greenshields speed-density model defined as: $u = 64 - 0.40 k$ Where speed $u$ is in mph and density $k$ is in veh/mi/ln.

1. Determine the free-flow speed ($u_f$) and jam density ($k_j$).
2. Calculate the critical density ($k_c$), critical speed ($u_c$), and per-lane capacity ($q_{\max}$).
3. If a loop detector currently records a lane flow of $1800\text{ veh/hr/ln}$ at a speed of $30\text{ mph}$, determine the operating regime (congested vs. uncongested).

Solution:

1. Boundary Parameters:

   • At $k = 0$, $u_f = 64\text{ mph}$.
   • At $u = 0$, $0 = 64 - 0.40 k_j \implies k_j = \frac{64}{0.40} = 160\text{ veh/mi/ln}$.

2. Critical Values:

   • Critical density: $k_c = \frac{k_j}{2} = \frac{160}{2} = 80\text{ veh/mi/ln}$.
   • Critical speed: $u_c = \frac{u_f}{2} = \frac{64}{2} = 32\text{ mph}$.
   • Maximum lane capacity: $q_{\max} = k_c \times u_c = 80 \times 32 = 2560\text{ veh/hr/ln}$.

3. Operating Regime Analysis:

   • At operating speed $u = 30\text{ mph}$, operating density is: $k = \frac{q}{u} = \frac{1800\text{ veh/hr/ln}}{30\text{ mph}} = 60\text{ veh/mi/ln}$
   • Comparing to critical values:
     • $k = 60\text{ veh/mi/ln} < k_c = 80\text{ veh/mi/ln}$
     • Operating speed $u = 30\text{ mph}$ is less than $u_c = 32\text{ mph}$, but checking consistency with the model: $u(60) = 64 - 0.40(60) = 40\text{ mph}$.
     • Since the observed speed ($30\text{ mph}$) is below the critical speed ($32\text{ mph}$) for the measured flow, the traffic stream is operating in the congested (forced flow / oversaturated) regime.