2.3 Hydrodynamic Shockwave Analysis & Traffic Queuing Models

2.3 Hydrodynamic Shockwave Analysis & Traffic Queuing Models

PTOE Exam Focus: Shockwave analysis and queuing theory represent the core quantitative tools for evaluating bottleneck propagation, incident clearance times, and intersection delays. Master the calculation of shockwave speed ($w = \Delta q / \Delta k$), queue dissipation duration using Rankine-Hugoniot jump conditions, cumulative delay polygons under deterministic ($D/D/1$) queuing, and stochastic queue metrics ($M/M/1$ vs. $M/D/1$).

---

1. Hydrodynamic Continuum & LWR Kinematic Wave Theory

In 1955 and 1956, M.J. Lighthill, G.B. Whitham, and P.I. Richards developed the LWR Hydrodynamic Model, treating traffic flow on long, uninterrupted roadways as a continuous 1D compressible fluid governed by the Continuity Equation (Conservation of Vehicles):

$$\frac{\partial k}{\partial t} + \frac{\partial q}{\partial x} = 0$$

Where $k(x,t)$ is density and $q(x,t)$ is flow rate at location $x$ and time $t$.

Since macroscopic flow is a function of density ($q = q(k)$), applying the chain rule gives:

$$\frac{\partial k}{\partial t} + \left(\frac{dq}{dk}\right) \frac{\partial k}{\partial x} = 0$$

Characteristic Wave Velocity ($c_k$):

The quantity $c_k = \frac{dq}{dk}$ represents the speed at which an infinitesimal change in density propagates along the traffic stream:

1. Uncongested Regime ($k < k_c$): Slope $\frac{dq}{dk} > 0 \implies$ Small perturbations travel downstream along the roadway.
2. Capacity Point ($k = k_c$): Slope $\frac{dq}{dk} = 0 \implies$ Small perturbations remain stationary ($c_k = 0$).
3. Congested Regime ($k > k_c$): Slope $\frac{dq}{dk} < 0 \implies$ Small perturbations travel upstream against the direction of traffic flow ($c_k < 0$).

---

2. Macroscopic Shockwave Velocity (Rankine-Hugoniot Jump Condition)

When a finite, discontinuous change occurs between two distinct traffic states—State A ($q_A, k_A, u_A$) and State B ($q_B, k_B, u_B$)—a shockwave forms at the boundary. Applying the principle of vehicle conservation across the moving boundary interface yields the Rankine-Hugoniot shockwave jump condition:

$$w_{AB} = \frac{q_B - q_A}{k_B - k_A} = \frac{\Delta q}{\Delta k}$$

Where $w_{AB}$ is the propagation velocity of the shockwave boundary in mph or km/h.

 Traffic Flow --->  [ State A: q_A, k_A ]  |  [ State B: q_B, k_B ]
                                         <---
                                  Shockwave Boundary (w_AB)

Directional Rules for Shockwave Velocity:

• $w > 0$ (Forward / Downstream): The shockwave moves in the direction of traffic flow.
• $w < 0$ (Backward / Upstream): The shockwave moves backward against traffic flow (queue expands upstream).
• $w = 0$ (Stationary): The shockwave remains pinned at a fixed physical location (e.g., permanent geometric lane drop or peak crest).

Relative Speed of Vehicles Entering the Shockwave:

$$\nu_{rel, A} = u_A - w_{AB}$$ $$\nu_{rel, B} = u_B - w_{AB}$$

---

3. Typologies of Traffic Shockwaves

---

4. Bottleneck Queue Growth & Clearance Analysis

Consider an incident on a freeway where a lane is blocked for duration $T_0$, creating an upstream queue (State B), after which the blockage is cleared, allowing vehicles to discharge at capacity (State C):

1. Approach State A: Normal free-flow upstream traffic ($q_A, k_A$).
2. Queued State B: Congested standstill/metered traffic upstream of incident ($q_B, k_B$).
3. Discharge State C: Unconstrained capacity discharge after incident removal ($q_C = q_{\max}, k_C = k_c$).

Shockwave Velocities:

• Queue growth wave: $w_{AB} = \frac{q_B - q_A}{k_B - k_A} < 0$
• Queue clearing wave: $w_{BC} = \frac{q_C - q_B}{k_C - k_B} < 0$
• Downstream recovery wave: $w_{AC} = \frac{q_C - q_A}{k_C - k_A}$

Time to Full Queue Dissipation ($T_{\text{clear}}$):

Because $|w_{BC}| > |w_{AB}|$, the recovery wave catches the forming wave at time $T_{\text{clear}}$ measured from incident onset:

$$T_{\text{clear}} = T_0 \left(\frac{w_{BC} - w_{AC}}{w_{BC} - w_{AB}}\right) = T_0 \left(\frac{w_{BC}}{w_{BC} - w_{AB}}\right) \quad \text{(if } w_{AC} \approx 0\text{)}$$

Maximum Queue Length ($L_{\max}$):

$$L_{\max} = |w_{AB}| \times T_{\text{clear}}$$

---

5. Deterministic Queuing Theory (D/D/1 Models)

Deterministic queuing ($D/D/1$) analyzes bottlenecks where arrival rate $\lambda(t)$ and departure capacity $\mu(t)$ are known deterministic functions over time. Performance is evaluated using Cumulative Vehicle Diagrams ($A(t)$ and $D(t)$):

 Cumulative Vehicles
     ^
     |                     A(t) [Arrival Curve: slope = λ1]
     |                   *  /
     |                 *   /  A(t) [Demand Drops: slope = λ2]
     |               *    /---
     |      Q(t)   *     /   |
     |       |   *      /    | D(t) [Departure Curve: slope = μ]
     |       v *       /     |
     |       *        /      |
     |     *         /       |
     |   *          /        |
    0+--*----------*---------*-----> Time (t)
        0         T1        t_q

Geometric Properties of Cumulative Curves:

1. Queue Length ($Q(t)$): The vertical distance between arrival curve $A(t)$ and departure curve $D(t)$ at time $t$: $$Q(t) = A(t) - D(t)$$
2. Individual Vehicle Delay ($d(t)$): The horizontal distance from arrival time $t$ to departure time $t'$, representing the wait time of vehicle $N$: $$d(N) = t_{\text{depart}}(N) - t_{\text{arrive}}(N)$$
3. Total Aggregate Delay ($D_{\text{total}}$): The total area enclosed between $A(t)$ and $D(t)$ over the entire queuing interval $[0, t_q]$: $$D_{\text{total}} = \int_0^{t_q} [A(t) - D(t)]\, dt \quad (\text{vehicle-hours})$$
4. Average Delay per Vehicle ($\bar{d}$): $$\bar{d} = \frac{D_{\text{total}}}{N_{\text{total}}}$$

Multi-Period Surge Formulations:

If an initial demand surge $\lambda_1 > \mu$ lasts for duration $T_1$, after which demand drops to $\lambda_2 < \mu$:

• Maximum Queue: $Q_{\max} = (\lambda_1 - \mu) T_1$
• Time to Dissipate Queue ($t_{\text{diss}}$ after $T_1$): $t_{\text{diss}} = \frac{Q_{\max}}{\mu - \lambda_2}$
• Total Queue Duration: $t_q = T_1 + t_{\text{diss}} = T_1 \left(1 + \frac{\lambda_1 - \mu}{\mu - \lambda_2}\right) = T_1 \left(\frac{\lambda_1 - \lambda_2}{\mu - \lambda_2}\right)$
• Total Aggregate Delay (Area of Triangle): $D_{\text{total}} = \frac{1}{2} Q_{\max} t_q = \frac{1}{2} (\lambda_1 - \mu) T_1 \cdot t_q$

---

6. Stochastic Queuing Systems (M/M/1, M/D/1, and M/G/1)

When arrivals and service times exhibit random variations, deterministic models underestimate delay. Stochastic queuing models use Kendall's notation ($A/B/c/K/N/Z$):

• $A$ = Arrival distribution ($M$ = Markovian/Poisson, $D$ = Deterministic, $G$ = General)
• $B$ = Service distribution ($M$ = Exponential, $D$ = Constant, $G$ = General)
• $c$ = Number of parallel service channels/servers

A. M/M/1 Queuing System (Poisson Arrivals, Exponential Service, 1 Server)

• Traffic Intensity / Utilization: $\rho = \frac{\lambda}{\mu}$ (System is stable if and only if $\rho < 1$).
• Average Number of Vehicles in System (Waiting + Service): $$L = \frac{\rho}{1 - \rho} = \frac{\lambda}{\mu - \lambda}$$
• Average Number of Vehicles in Queue (Waiting Line): $$L_q = \frac{\rho^2}{1 - \rho} = \frac{\lambda^2}{\mu(\mu - \lambda)} = L - \rho$$
• Average Time Spent in System (Little's Law: $L = \lambda W$): $$W = \frac{L}{\lambda} = \frac{1}{\mu - \lambda}$$
• Average Waiting Time in Queue ($L_q = \lambda W_q$): $$W_q = \frac{L_q}{\lambda} = \frac{\rho}{\mu - \lambda} = \frac{\lambda}{\mu(\mu - \lambda)} = W - \frac{1}{\mu}$$
• Probability of $n$ Vehicles in System: $P_n = (1 - \rho)\rho^n$.

B. M/D/1 Queuing System (Deterministic Constant Service Time)

When service time is perfectly uniform (such as automated toll gates or fixed-cycle metered signals), service variance $\sigma_s^2 = 0$:

$$L_q = \frac{\rho^2}{2(1 - \rho)}$$ $$W_q = \frac{\rho}{2\mu(1 - \rho)}$$

Critical PTOE Takeaway: Under identical traffic intensity $\rho$, an $M/D/1$ queue yields exactly half the queue length and waiting delay of an $M/M/1$ queue ($L_{q, M/D/1} = \frac{1}{2} L_{q, M/M/1}$). Eliminating service variance cuts queuing delays by 50%!

C. M/G/1 Queuing System (Pollaczek-Khinchine Formula)

For general service time distribution with mean service time $1/\mu$ and variance $\sigma_s^2$:

$$L_q = \frac{\lambda^2 \sigma_s^2 + \rho^2}{2(1 - \rho)}$$ $$W_q = \frac{L_q}{\lambda} = \frac{\lambda \sigma_s^2 + \rho / \mu}{2(1 - \rho)}$$