16-Civ-A6 Highway Design, Construction, and Maintenance · December 2016

Question 4 of 7: Greenshields' Model and Shock-Wave Analysis

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examination, December 2016 — 98-Civ-A6, Transportation Planning & Engineering. Three hours, closed book (one two-sided aid sheet, approved calculator). Seven questions of equal value (20 marks each); any five constitute a complete examination. All seven are solved here, because the set is a study resource rather than a sat examination.

Reference texts. Mannering, Washburn & Kilareski, Principles of Highway Engineering and Traffic Analysis (Wiley) — traffic-stream models, deterministic queueing and shock waves; Papacostas & Prevedouros, Transportation Engineering and Planning (Prentice Hall) — the land-use/transport system and the four-step demand model; Ortuzar & Willumsen, Modelling Transport (Wiley) — trip generation, distribution, mode choice and traffic assignment; Sheffi, Urban Transportation Networks (Prentice Hall) — user-equilibrium assignment; Transportation Association of Canada, Geometric Design Guide for Canadian Roads — Canadian planning and design practice.

Question 4: Greenshields' Model and Shock-Wave Analysis (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A single-lane highway obeying Greenshields' linear speed–density relation, temporarily obstructed by a slow truck.

Given data — traffic stream and obstruction
Free-flow speed, $u_f$	100 km/h
Jam density, $k_j$	100 veh/km
Normal (state A) speed, $u_A$	80 km/h
Truck speed, $u_B$	15 km/h, over a distance of 1.0 km
Platoon (state B) density and flow	$k_B=85$ veh/km, $q_B=1275$ veh/h
Downstream condition after the exit	uncongested (discharge at capacity)

Find. Capacity and the density at capacity; the platoon length at the instant the truck leaves; and the further time the platoon takes to disappear.

Time–space diagram of the moving bottleneck. The truck advances at +15 km/h while the rear shock retreats at −5 km/h, so the platoon is 1.333 km long when the truck exits; the −35 km/h recovery wave then overtakes the shock 2.67 min later, 0.556 km upstream of the entry point.

Approach. Fix the three traffic states A (free flow), B (platoon) and C (capacity discharge) on the Greenshields fundamental diagram, then obtain each shock-wave speed as the chord slope $\omega=\Delta q/\Delta k$ between two states and track the wave trajectories on the time–space plane.

Capacity and the density at capacity. Greenshields assumes speed falls linearly with density, $$u=u_f\!\left(1-\frac{k}{k_j}\right)\;\Rightarrow\;q=uk=u_f\!\left(k-\frac{k^{2}}{k_j}\right)$$ Setting $dq/dk=0$ gives $k_m=k_j/2$ and hence $u_m=u_f/2$, so $$k_m=\frac{100}{2}=50\ \text{veh/km},\qquad u_m=\frac{100}{2}=50\ \text{km/h}$$ $$\boxed{q_{\max}=\frac{u_f k_j}{4}=\frac{(100)(100)}{4}=2500\ \text{veh/h at }k_m=50\ \text{veh/km}}$$
Fix the two traffic states. For the normal stream at 80 km/h, $$k_A=k_j\!\left(1-\frac{u_A}{u_f}\right)=100\!\left(1-\frac{80}{100}\right)=20\ \text{veh/km},\qquad q_A=u_Ak_A=1600\ \text{veh/h}$$ The platoon values quoted in the question are consistent with the same model, which is worth confirming before using them: $k_B=100(1-15/100)=85$ veh/km and $q_B=(15)(85)=1275$ veh/h, both as given. The stream is therefore operating well below capacity upstream ($1600\lt2500$ veh/h) and well below it inside the platoon as well — the platoon is congested-side, not capacity, flow.
Speed of the shock wave behind the truck. A shock wave is the boundary between two states, and its speed is the chord slope between them on the $q$–$k$ diagram: $$\omega_{AB}=\frac{q_B-q_A}{k_B-k_A}=\frac{1275-1600}{85-20}=\frac{-325}{65}$$ $$\boxed{\omega_{AB}=-5\ \text{km/h (moving upstream)}}$$ The negative sign means the tail of the platoon travels backwards along the highway at 5 km/h even though every vehicle in it moves forward.
Length of the platoon when the truck exits. The truck occupies the highway for $$t_1=\frac{1.0\ \text{km}}{15\ \text{km/h}}=0.0667\ \text{h}=4.0\ \text{min}$$ During that period the head of the platoon (the truck itself) advances at $+15$ km/h while the tail retreats at $-5$ km/h, so the platoon stretches at their relative speed: $$L=\big(u_B-\omega_{AB}\big)t_1=\big(15-(-5)\big)(0.0667)$$ $$\boxed{L=1.333\ \text{km, containing } k_B L = 85(1.333)\approx113\ \text{vehicles}}$$ The vehicle count checks independently against the rate at which vehicles cross the rear shock, $k_A(u_A-\omega_{AB})t_1=20(85)(0.0667)=113$ vehicles — the same answer from the upstream side.
Speed of the recovery wave. Once the truck leaves, the platoon leader accelerates and, with no congestion downstream, the queue discharges at capacity: state C is $k_C=50$ veh/km, $q_C=2500$ veh/h. The starting (recovery) wave between B and C therefore travels at $$\omega_{BC}=\frac{q_C-q_B}{k_C-k_B}=\frac{2500-1275}{50-85}=\frac{1225}{-35}$$ $$\boxed{\omega_{BC}=-35\ \text{km/h (moving upstream, seven times faster than the tail)}}$$
Time for the platoon to dissipate. The platoon disappears when the recovery wave, launched from the exit point, overtakes the rear shock. Both move upstream, so they close at the difference of their speeds: $$t_2=\frac{L}{|\omega_{BC}|-|\omega_{AB}|}=\frac{1.333}{35-5}=0.0444\ \text{h}$$ $$\boxed{t_2=2.67\ \text{min}\;(2\ \text{min }40\ \text{s})\ \text{after the truck exits}}$$ Tracking the two trajectories from the entry point gives the same instant, $t=6.67$ min after the truck entered, at a point 0.556 km upstream of where it joined — which is where the last queued vehicle finally regains free-flow speed.

Final results — Question 4
Quantity	Value
Capacity, $q_{\max}$	2,500 veh/h
Density and speed at capacity	50 veh/km, 50 km/h
Normal stream (state A)	$k_A=20$ veh/km, $q_A=1600$ veh/h
Shock-wave speed A→B	−5 km/h (upstream)
Platoon length at truck exit	1.333 km (≈ 113 vehicles)
Recovery-wave speed B→C	−35 km/h (upstream)
Dissipation time after exit	2.67 min (2 min 40 s)
Location where the platoon vanishes	0.556 km upstream of the entry point

Topic. Greenshields' traffic-stream model and shock-wave (kinematic-wave) analysis of a moving bottleneck.

Key relations. $u=u_f(1-k/k_j)$; $q=uk$; $k_m=k_j/2$, $u_m=u_f/2$, $q_{\max}=u_fk_j/4$; shock-wave speed $\omega_{12}=(q_2-q_1)/(k_2-k_1)$; platoon growth rate $=u_{\text{obstruction}}-\omega_{\text{tail}}$; dissipation time $=L/(|\omega_{\text{recovery}}|-|\omega_{\text{tail}}|)$.

Why this works. Conservation of vehicles across a moving discontinuity gives the Rankine–Hugoniot result that a shock travels at the chord slope joining the two states on the fundamental diagram. Once each state is a point on the $q$–$k$ curve, every wave in the problem is just a chord, and the whole event becomes a set of straight lines on the time–space plane whose intersections answer the question. That is why a slow truck can produce a queue that persists after it leaves: the recovery wave has to travel back along the whole platoon before the last vehicle is released, and it only gains on the tail at the difference of the two upstream speeds.

Common pitfall. Computing the dissipation time as $L/|\omega_{BC}|$, which ignores that the tail is still retreating and understates the time by 14 % (2.29 min instead of 2.67 min). Two further traps: assuming the platoon disappears the moment the truck exits (it does not — the queue is still 1.3 km long at that instant), and using the truck's speed as the platoon's wave speed. Also check that the given platoon density and flow satisfy the stated Greenshields parameters before relying on them; if they do not, the inconsistency must be flagged.

Source. Mannering, Washburn & Kilareski, Principles of Highway Engineering and Traffic Analysis, Ch. 5 (traffic-stream models and shock waves); Papacostas & Prevedouros, Transportation Engineering and Planning, Ch. 4; Roess, Prassas & McShane, Traffic Engineering, Ch. 5.

← Question 3 Next: Question 5 →