Deterministic queueing at a signalized approach: 16-Civ-A6 Highway Design, Construction, and Maintenance, December 2013, Q2

Question 2: Deterministic queueing at a signalized approach (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 signalized approach observed over two 60‑s cycles, each with a 30‑s red followed by a 30‑s green, analysed as a deterministic (D/D/1) queue.

Given data
Quantity	Value	Rate (veh/s)
Arrival rate, cycle 1	1,080 veh/h	0.30
Arrival rate, cycle 2	720 veh/h	0.20
Saturation (green) discharge	1,800 veh/h	0.50
Red / green interval	30 s / 30 s	—

Find. (a) the cumulative arrival/departure diagram; (b) the maximum queue; (c) the total and average vehicle delay over the two cycles.

Cumulative arrival curve A(t) (solid) and departure curve D(t) (dashed). Departures are zero on red and rise at the saturation rate on green while a queue exists. The vertical gap between the curves is the queue length; it peaks at 9 vehicles at the end of each red. The two curves meet at t = 120 s, so the queue clears exactly as the second green ends.

Approach. Build cumulative arrivals A(t) and departures D(t) in vehicles; the horizontal distance between them is a vehicle’s delay, the vertical distance is the queue, and the area between them is the total delay.

Convert rates to veh/s. Arrivals are $q_1=1080/3600=0.30$ veh/s in cycle 1 and $q_2=720/3600=0.20$ veh/s in cycle 2; the green discharges at $s=1800/3600=0.50$ veh/s while a queue remains.
Cumulative arrivals. $A(t)=0.30\,t$ for $0\le t\le 60$ and $A(t)=18+0.20\,(t-60)$ thereafter, giving $A(60)=18$ and $A(120)=30$ vehicles.
Red 1 (0–30 s). No departures, so the queue grows to $Q(30)=0.30\times30=9$ vehicles.
Green 1 (30–60 s). Discharge at $0.50$ against arrivals at $0.30$ gives a net clearance of $0.20$ veh/s. Starting from 9, the residual at the end of green is $Q(60)=9-0.20\times30=3$ vehicles — the queue does not fully clear in cycle 1.
Red 2 (60–90 s). No departures; arrivals now at $0.20$, so $Q(90)=3+0.20\times30=9$ vehicles.
Green 2 (90–120 s). Net clearance $0.50-0.20=0.30$ veh/s. The 9‑vehicle queue clears in $9/0.30=30$ s, i.e. exactly at $t=120$ s. Peak queue: $\boxed{Q_{\max}=9\text{ vehicles}}$ (reached at the end of each red).
Total delay = area between the curves. Summing the four trapezoidal strips of the queue‑versus‑time profile: $$D_{tot}=\tfrac12(30)(0{+}9)+\tfrac12(30)(9{+}3)+\tfrac12(30)(3{+}9)+\tfrac12(30)(9{+}0)=135+180+180+135.$$ Hence $\boxed{D_{tot}=630\ \text{veh}\cdot\text{s}=10.5\ \text{veh}\cdot\text{min}}$.
Average delay. All $A(120)=30$ vehicles that arrive are served, so $$\bar d=\frac{D_{tot}}{N}=\frac{630}{30}=\boxed{21\ \text{s/veh}}.$$

Final results — Question 2
Quantity	Result
Maximum queue length	9 vehicles
Queue clears at	t = 120 s (end of 2nd green)
Total vehicle delay	630 veh·s (10.5 veh·min)
Average delay per vehicle	21 s/veh

Topic. Deterministic (D/D/1) queueing analysis of an under‑saturated signal approach.

Key relations. Queue $Q(t)=A(t)-D(t)$; total delay $=\int[A(t)-D(t)]\,dt$ = area between curves; average delay $=\,$total delay$/N$; maximum queue occurs where the arrival and departure slopes cross (here, the end of each red).

Why this works. With deterministic, piece‑wise‑constant arrival and service rates the cumulative curves are straight‑line segments, so queue, delay and clearance times all follow from simple geometry. The horizontal gap is an individual vehicle’s wait (FIFO) and the enclosed area sums those waits over all vehicles.

Common pitfall. Assuming the queue clears within each green — here a 3‑vehicle residual is carried into cycle 2. Also forgetting that departures are zero during red and only reach the saturation rate while a queue is present.

Source. Garber & Hoel, Traffic and Highway Engineering, ch. 5 (queueing) and ch. 8 (signal timing).

Question 2 of 7: Deterministic queueing at a signalized approach

Question 2: Deterministic queueing at a signalized approach (20 marks)