16-Civ-B10 Traffic Engineering · May 2015

Question 5 of 7: Moving-Vehicle Method for Volume and Travel Time

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

Notes on this paper

Paper format. 98-Civ-B10 Traffic Engineering, National Examinations, May 2015. Three-hour duration, OPEN BOOK, any non-communicating calculator permitted. Seven questions, all of equal value (20 marks each) with the mark split printed in the grading scheme; the paper requires a total of five solutions. All seven are worked below, because this document is a study resource rather than an exam script.

Reference texts. Garber, N. J. and Hoel, L. A., Traffic and Highway Engineering — Ch. 4 (traffic-engineering studies: volume counts, peak-hour factor, spot-speed studies), Ch. 6 (fundamental principles of traffic flow: the Greenshields model, deterministic and stochastic queueing, the moving-observer method) and Ch. 8 (intersection control: saturation flow, change and clearance intervals). Transportation Research Board, Highway Capacity Manual — basic-freeway-segment and two-lane-highway level-of-service criteria and the lane-width, lateral-clearance and heavy-vehicle adjustment factors. Transportation Association of Canada, Geometric Design Guide for Canadian Roads and the Manual of Uniform Traffic Control Devices for Canada — Canadian lane widths, crosswalk practice and clearance-interval policy. Institute of Transportation Engineers, Traffic Engineering Handbook — queueing applied to parking and toll facilities.

Check — declared assumptions, invoked under the paper’s own NOTE 1 and NOTE 2 (“any data required, but not given, can be assumed”, with a clear statement of the assumption). This paper is open-book and deliberately withholds several standard table values. The following are used consistently throughout and each is restated where it is first applied:

Bus (heavy-vehicle) passenger-car equivalent on the level intersection approaches of Question 1: E_B = 2.0.
Two-phase signal operation at the Question 1 intersection (one phase for the north–south pair, one for the east–west pair), with no start-up lost time given, so the whole intergreen is taken as lost time.
Maximum service flow rates and volume-to-capacity ratios for basic freeway segments at a 100 km/h design speed: 700/1000/1450/1800/2000 pc/h/lane is the 110 km/h family, so the 100 km/h family 600/1000/1450/1800/2000 pc/h/lane (v/c = 0.30/0.50/0.72/0.90/1.00) is used, with an ideal capacity of 2000 pc/h/lane.
Two-lane-highway ideal capacity 2800 pc/h both directions, with level-terrain v/c ratios 0.15/0.27/0.43/0.64/1.00 at 0 % no-passing zones, and a 50/50 directional split (f_d = 1.00).
Passenger-car equivalents on specific grades: E_T = 5 for trucks on both the 4 % / 1.0 km grade and the 3 % / 1.5 km grade, E_R = 3 for recreational vehicles on the 4 % grade, E_B = 1.6 for intercity buses on grades of 4 % or flatter, and E_T = 2 for trucks in general level terrain.

Different editions of the capacity manual tabulate these cells slightly differently. Where an alternative reading changes an answer materially, the alternative is computed and stated alongside; the method and the arithmetic are what carry the marks.

Question 5: Moving-Vehicle Method for Volume and Travel Time (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.

Observation	North-bound run	South-bound run
Test-car travel time, t (min)	2.80	2.90
Opposing vehicles met, M	78	95
Vehicles overtaking the test car, O	2.0	1.5
Vehicles passed by the test car, P	1.0	1.0

Find. The traffic volume in each direction, and the average travel time of the traffic stream (as distinct from that of the test car) in each direction.

Question 5 — the two moving-observer runs. The count of opposing vehicles met on each run sizes the flow in the other direction, while the overtaking and passing counts belong to the run made with the stream of interest.

Approach. Apply Wardrop’s moving-observer relations, taking care that the “met” count for a given direction comes from the run made against that direction, while the overtaking and passing counts come from the run made with it. Then correct the test car’s own travel time for the net number of vehicles that overtook it.

Part (a) — state the moving-observer volume relation and identify which counts belong to which direction. For a direction of interest, Wardrop’s relation is $$q = \frac{M_a + O_w - P_w}{t_a + t_w}$$ where $M_a$ is the number of vehicles of the direction of interest met while driving against it, and $O_w$ and $P_w$ are the vehicles overtaking and overtaken while driving with it. This crossing of subscripts is the single easiest place to lose the question. For the north-bound stream, the vehicles “met” are counted on the south-bound run, so $M_a = 95$, while $O_w = 2.0$ and $P_w = 1.0$ come from the north-bound run. The denominator is the sum of both run times, $$t_a + t_w = 2.90 + 2.80 = 5.70 \text{ min}$$
Compute the north-bound volume. Substituting, $$q_N = \frac{95 + 2.0 - 1.0}{5.70} = \frac{96.0}{5.70} = 16.842 \text{ veh/min}$$ and converting to an hourly rate, $$\boxed{\;q_N = 16.842 \times 60 = 1010.5 \approx 1011 \text{ veh/h}\;}$$
Part (b) — compute the south-bound volume by the mirror-image substitution. Now the direction of interest is south-bound, so the “met” count comes from the north-bound run ($M_a = 78$) and the overtaking and passing counts from the south-bound run ($O_w = 1.5$, $P_w = 1.0$): $$q_S = \frac{78 + 1.5 - 1.0}{5.70} = \frac{78.5}{5.70} = 13.772 \text{ veh/min}$$ $$\boxed{\;q_S = 13.772 \times 60 = 826.3 \approx 826 \text{ veh/h}\;}$$ The two volumes differ by about 22 %, and the larger belongs to the direction that was met more often — which is the sanity check on having crossed the subscripts correctly. Using the same-direction met count instead would return about 832 veh/h for the north-bound stream: a plausible number that nothing else in the question contradicts, which is exactly what makes the error dangerous.
Part (c) — correct the test car’s travel time to the stream’s travel time. The test car is not an average vehicle: if more vehicles overtake it than it overtakes, it was travelling slower than the mean and its own time overstates the stream’s. The correction is $$\bar{t} = t_w - \frac{O_w - P_w}{q}$$ For the north-bound stream, with $O_w - P_w = 2.0 - 1.0 = 1.0$ net vehicle and $q_N = 16.842$ veh/min, $$\bar{t}_N = 2.80 - \frac{1.0}{16.842} = 2.80 - 0.0594 = 2.741 \text{ min}$$ $$\boxed{\;\bar{t}_N = 2.74 \text{ min} = 164.4 \text{ s}\;}$$ The correction is small — 3.6 s — because the net overtaking count is only one vehicle against a flow of nearly 17 per minute.
Part (d) — repeat for the south-bound stream. Here $O_w - P_w = 1.5 - 1.0 = 0.5$ net vehicles and $q_S = 13.772$ veh/min, so $$\bar{t}_S = 2.90 - \frac{0.5}{13.772} = 2.90 - 0.0363 = 2.864 \text{ min}$$ $$\boxed{\;\bar{t}_S = 2.86 \text{ min} = 171.8 \text{ s}\;}$$ Both corrections are downward, as they must be when the test car is overtaken more often than it overtakes: the driver was slightly conservative relative to the stream in both directions. The south-bound stream is both lighter and slower, which is the pattern of a section on the off-peak side of a tidal corridor.

Question 5 — results
Quantity	Result
Sum of run times, t_a + t_w	5.70 min
(a) North-bound volume	16.84 veh/min = 1010.5 ≈ 1011 veh/h
(b) South-bound volume	13.77 veh/min = 826.3 ≈ 826 veh/h
(c) Average north-bound travel time	2.74 min (164.4 s)
(d) Average south-bound travel time	2.86 min (171.8 s)
Total two-way volume	1837 veh/h

Topic. The moving-observer (Wardrop–Charlesworth) method for simultaneous estimation of volume and travel time on a two-way section.

Key relations. $q = (M_a + O_w - P_w)/(t_a + t_w)$, where $M_a$ is the opposing count from the run made against the direction of interest and $O_w$, $P_w$ come from the run made with it; $\bar{t} = t_w - (O_w - P_w)/q$.

Why this works. The volume relation is a conservation argument, and seeing it makes the crossed subscripts obvious rather than arbitrary. While the test car drives against the stream of interest for a time $t_a$, it meets every vehicle of that stream that was anywhere in a stretch swept out by their combined closing speed — the count $M_a$ is therefore proportional to $q(t_a + t_w)$, because the relative displacement in that time equals the section length travelled by both. The overtaking correction handles the fact that the observer is embedded in the stream on the return run: if $O_w$ vehicles passed it and it passed $P_w$, the net $O_w - P_w$ vehicles crossed its position, and adding them recovers the count a stationary observer would have made. The travel-time correction is the same net count re-expressed as a time: $(O_w-P_w)/q$ is the time by which the test car lagged the mean, so subtracting it converts the driver’s own experience into the stream’s. The elegance is that a single pair of runs yields four numbers — two volumes and two travel times — with no roadside equipment at all.

Common pitfall. Overwhelmingly the dominant error is using the “met” count from the same-direction run, giving 831.6 veh/h for the north-bound volume here instead of 1010.5 — wrong by 18 %, yet entirely plausible on its face and contradicted by nothing else in the question. Guard against it by remembering that a stationary observer cannot count a stream by riding in it; the counting only happens on the opposing run. Two lesser slips: dividing by only one run time instead of the sum, which doubles both volumes, and adding rather than subtracting the net overtaking term in the travel-time correction, which pushes the stream’s travel time above the test car’s and implies the stream is slower than a vehicle it keeps overtaking.

Source. Garber & Hoel, Traffic and Highway Engineering, Ch. 4 (traffic-volume studies: the moving-observer method); Wardrop, J. G. and Charlesworth, G., “A method of estimating speed and flow of traffic from a moving vehicle”, Proceedings of the Institution of Civil Engineers.

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