16-Civ-B10 Traffic Engineering · December 2013

Question 4 of 7: Deterministic Queueing at an Incident Bottleneck

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

Notes on this paper

Paper format. National Examinations, December 2013 — 98-Civ-B10 Traffic Engineering. Three-hour, open-book examination; any non-communicating calculator is permitted. Seven questions are printed and five complete solutions are required, all questions being of equal value. The printed grading scheme is Q1 (a) 6, (b) 6, (c) 8; Q2 (a) 10, (b) 10; Q3 (a) 10, (b) 10; Q4 20; Q5 20; Q6 (a)–(d) 5 each; Q7 (a)–(d) 5 each. The paper also states that if doubt exists as to the interpretation of a question the candidate should submit a clear statement of any assumptions made, and that any data required but not given can be assumed. All seven questions are worked below.

Reference texts.

Garber, N. J. and Hoel, L. A., Traffic and Highway Engineering, 5th ed. — Ch. 4 (traffic engineering studies: spot-speed statistics, volume and travel-time studies, the moving-vehicle method), Ch. 6 (fundamental principles of traffic flow: time-mean and space-mean speed, speed–density models, Poisson arrivals, deterministic queueing), Ch. 8 (intersection control and signalisation: progression and time–space diagrams), Ch. 9 (capacity and level of service for highway segments), Ch. 10 (capacity and level of service at signalised intersections). This is the principal reference for the subject.
Transportation Research Board, Highway Capacity Manual (HCM) — basic freeway segments and signalised-intersection saturation flow.
Transportation Association of Canada (TAC), Geometric Design Guide for Canadian Roads — Canadian lane-width, shoulder and clearance practice.
TAC, Manual of Uniform Traffic Control Devices for Canada (MUTCDC) — Canadian signal warrants, timing and coordination practice.
Institute of Transportation Engineers, Traffic Engineering Handbook — signal-system progression and alternate-system design.

Check: assumed reference-table values. Questions 2(a) and 2(b) are capacity problems whose input list — lane width, lateral obstruction, per-cent heavy vehicles, a specific grade, design speed and a target level of service — is exactly the argument list of the classical Highway Capacity Manual equations, but the paper does not reproduce the lookup tables. Consistent with the paper's own instruction that any data required but not given may be assumed, every table value used is stated explicitly at the point of use, drawn from one coherent edition family (HCM 1985/1994, ideal capacity 2,000 pc/h/ln for the freeway segment). Substituting another edition's tables rescales the final flow rate but changes neither the method nor the arithmetic chain; the sensitivity is discussed in the Question 2(a) concept note.

Question 4: Deterministic Queueing at an Incident Bottleneck (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 constant arrival stream meeting a service rate that changes twice.

Period	Clock time	Service capacity	Rate (veh/min)
Arrivals (throughout)	from 7:00	3,500 veh/h	58.333
Full closure	7:00 – 7:15	0	0
Partial opening	7:15 – 7:30	2,500 veh/h	41.667
Fully reopened	from 7:30	5,000 veh/h	83.333

Find. The queueing (cumulative arrival–departure) diagram, and from it: the time the queue dissipates, the longest queue length, the total delay, the average delay per vehicle, and the longest wait experienced by any single vehicle.

Approach. Plot cumulative arrivals and cumulative departures against time. The vertical gap between the curves is the queue length at that instant, the horizontal gap is the delay to an individual vehicle, and the area enclosed is the total delay. Work in vehicles per minute throughout so that areas come out in vehicle-minutes.

Convert every rate to vehicles per minute. Areas on the diagram are only meaningful if the time unit is consistent: $$\lambda=\frac{3500}{60}=58.333,\qquad \mu_{2}=\frac{2500}{60}=41.667,\qquad \mu_{3}=\frac{5000}{60}=83.333\ \text{veh/min}$$ Note the ordering $\mu_{2}<\lambda<\mu_{3}$: the queue must keep growing through the partial-opening period and can only start to shrink after 7:30.
Write the two cumulative curves. Cumulative arrivals are linear throughout, while cumulative departures are piecewise linear with a break at each capacity change. Measuring $t$ in minutes after 7:00, $$A(t)=58.333\,t,\qquad D(t)=\begin{cases} 0 & 0\le t\le 15\\ 41.667\,(t-15) & 15\le t\le 30\\ 625+83.333\,(t-30) & t\ge 30 \end{cases}$$ where the constant 625 is the number discharged during the partial opening, $41.667\times 15$.
Find the longest queue length. The queue is $Q(t)=A(t)-D(t)$, and since it grows whenever arrivals outrun departures it must peak at the last instant before full capacity returns, namely 7:30. Evaluating at both breakpoints, $$Q(15)=875-0=875\ \text{veh},\qquad Q(30)=1750-625$$ $$\boxed{Q_{max}=1{,}125\ \text{vehicles at }7{:}30\ \text{a.m.}}$$ At an average of about 7.5 m of queue per vehicle per lane, 1,125 vehicles spread over the freeway's lanes is a queue of the order of 2 to 4 km — the practical reason incident clearance is timed in minutes.
Find the time of queue dissipation. After 7:30 the queue drains at the difference between the restored capacity and the arrival rate: $$\text{drain rate}=\mu_{3}-\lambda=83.333-58.333=25.0\ \text{veh/min}$$ so clearing 1,125 vehicles takes $$\Delta t=\frac{1125}{25.0}=45.0\ \text{min after }7{:}30$$ $$\boxed{\text{the queue dissipates at }8{:}15\ \text{a.m.}\ (t=75\ \text{min})}$$ The arithmetic control is that the two cumulative curves must meet there: $A(75)=58.333\times 75=4375$ and $D(75)=625+83.333\times 45=4375$. They agree exactly.
Compute the total delay as the area between the curves. The enclosed region is made of three straight-sided pieces — a triangle while the freeway is shut, a trapezoid while it is partly open, and a triangle while the queue drains: $$\begin{aligned} A_{1}&=\tfrac{1}{2}(15)(875)=6{,}562.5\\ A_{2}&=\tfrac{1}{2}(875+1125)(15)=15{,}000\\ A_{3}&=\tfrac{1}{2}(45)(1125)=25{,}312.5 \end{aligned}$$ Summing the three, $$\boxed{\text{total delay}=46{,}875\ \text{veh}\cdot\text{min} =781.25\ \text{veh}\cdot\text{h}}$$ At a nominal value of travel time, this single 15-minute closure imposes something of the order of 780 vehicle-hours of delay — the number that justifies incident-response programmes.
Compute the average delay per vehicle. Every vehicle arriving before the queue clears is delayed, and that is $A(75)=4{,}375$ vehicles. Hence $$\bar{w}=\frac{46{,}875}{4{,}375}$$ $$\boxed{\bar{w}=10.71\ \text{min per vehicle}\ (\approx 10\ \text{min }43\ \text{s})}$$
Identify the longest wait of any vehicle. An individual vehicle's delay is the horizontal separation between the curves, so the worst-delayed vehicle is not the one that arrives when the queue is longest. Under first-in first-out service, the vehicle that waits longest is the last one discharged before full capacity is restored — vehicle number 625, which leaves at exactly 7:30. It arrived at $$t_{arr}=\frac{625}{58.333}=10.71\ \text{min}\quad(7{:}10{:}43\ \text{a.m.})$$ and departed at $t=30$ min, so its delay is $$w_{max}=30-10.71$$ $$\boxed{w_{max}=19.29\ \text{min}\ (\approx 19\ \text{min }17\ \text{s})}$$ This is confirmed by examining the wait as a function of arrival time: for vehicles arriving before 10.71 min the wait is $15+0.4t$ (rising), and for those arriving after it the wait is $22.5-0.3t$ (falling), so the maximum sits precisely at the changeover. Note that $w_{max}$ is nearly twice the average delay — a reminder that reporting only the average understates what the worst-affected drivers experience.

Queueing diagram: cumulative arrivals (blue) against cumulative departures (red). The vertical gap is the queue length — 875 vehicles at 7:15 and a maximum of 1,125 at 7:30; the horizontal gap is an individual vehicle's delay, longest (19.29 min) for the vehicle that arrives at 7:10:43 and is discharged as full capacity returns; the shaded area is the total delay of 46,875 vehicle-minutes. The curves meet at 8:15, when the queue clears.

Quantity	Result
Queue at 7:15 (end of full closure)	875 vehicles
Longest queue length	1,125 vehicles, at 7:30 a.m.
Time of queue dissipation	8:15 a.m. (75 min after the incident)
Total delay	46,875 veh·min = 781.25 veh·h
Vehicles affected	4,375
Average delay per vehicle	10.71 min (10 min 43 s)
Longest wait of any vehicle	19.29 min (19 min 17 s), for the vehicle arriving 7:10:43

Topic. Deterministic (D/D/1) queueing analysis of an incident bottleneck using cumulative arrival and departure curves.

Key relations. Queue length $Q(t)=A(t)-D(t)$ (vertical gap); individual delay $w(t)=D^{-1}(A(t))-t$ (horizontal gap); total delay $=\int [A(t)-D(t)]\,dt$ (enclosed area); average delay $=$ total delay divided by the number of vehicles affected; dissipation when $A(t)=D(t)$, reached after $Q_{max}/(\mu-\lambda)$ once capacity exceeds demand.

Why this works. The cumulative-curve construction works because both curves count the same vehicles in the same order, so the region between them can be read three different ways without any further modelling. Reading vertically gives the number of vehicles between the two curves at one instant — the queue. Reading horizontally gives the time between one vehicle's arrival and its departure — that vehicle's delay. Reading the area gives vehicle-minutes, the sum of all individual delays. The one modelling assumption is that service is first-in first-out and that arrivals and departures are deterministic, which is why the queue is a vertical stack at a point rather than a physical length along the road; converting to a physical queue needs a jam-density assumption, as done above. The key structural insight is that the maximum queue and the maximum individual delay occur at different times and to different vehicles: the queue peaks when the bottleneck is relieved, while the unluckiest driver is the one who joins just early enough to be served last under the restricted capacity.

Common pitfall. The classic mistake is to report the longest wait as the delay of the vehicle arriving when the queue is longest, or to equate the longest wait with the maximum queue divided by the discharge rate. Both confuse the vertical and horizontal readings of the same diagram. A second frequent error is arithmetic rather than conceptual: mixing vehicles per hour into an area computed over minutes, which corrupts the total delay by a factor of 60 — convert every rate to a common time unit first. A third is assuming the queue starts draining the moment the freeway is partially reopened; with 2,500 veh/h of capacity against 3,500 veh/h of demand it is still growing, just more slowly, and only the ordering $\mu_{2}<\lambda<\mu_{3}$ reveals this.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., Ch. 6 (Fundamental Principles of Traffic Flow) — deterministic queueing analysis and queueing diagrams; Ch. 4 for the travel-time and delay study methods that supply the input rates.

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