Moving-Vehicle Method for Volume and Travel Time: 16-Civ-B10 Traffic Engineering, Undated paper, Q6

Question 6: Moving-Vehicle Method for Volume and Travel Time (5 × 4 = 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. Eight test-vehicle runs in each direction over the same section, each logging the run's travel time, the number of vehicles met travelling in the opposite direction, the number that overtook the test vehicle, and the number the test vehicle overtook.

Find. The averages of every logged column in both directions, then the traffic volume and the mean traffic travel time in each direction.

The moving-vehicle (Wardrop) test section. Eight runs are made in each direction; each run logs the travel time, the vehicles met head-on, and the vehicles that overtook or were overtaken by the test vehicle.

Approach. Average each column over the eight runs, then apply Wardrop's moving-observer relations: the volume in one direction is estimated from the vehicles met while running against that stream, corrected for overtaking during the run with it; the mean travel time of the stream is the test vehicle's own travel time corrected by the net overtaking rate.

Part (a) — Average every logged column. With eight runs per direction the arithmetic means are:

Direction	Mean travel time (min)	Mean opposing count	Mean overtaking the test vehicle	Mean overtaken by the test vehicle
Westbound	$\bar t_w=4.17125$	$M_w=111.75$	$O_w=3.000$	$P_w=2.500$
Eastbound	$\bar t_e=3.75750$	$M_e=101.75$	$O_e=2.500$	$P_e=2.250$

For example, the westbound travel times sum to 33.37 min over eight runs, giving 4.17125 min, and the westbound opposing counts sum to 894 vehicles, giving 111.75. Note that $M_w$ counts eastbound vehicles (met while running west) and $M_e$ counts westbound vehicles.

Part (b) — Eastbound traffic volume. Wardrop's volume relation for a direction uses the vehicles met while running against it, adjusted for net overtaking while running with it, divided by the sum of the two mean run times: $$q_e=\frac{M_w+O_e-P_e}{\bar t_w+\bar t_e}$$ $$q_e=\frac{111.75+2.50-2.25}{4.17125+3.75750}=\frac{112.00}{7.92875}=14.126\ \text{veh/min}$$ $$\boxed{q_e=847.6\ \text{veh/h}}$$
Part (c) — Westbound traffic volume. The mirror relation, using the westbound vehicles met on the eastbound runs: $$q_w=\frac{M_e+O_w-P_w}{\bar t_w+\bar t_e}=\frac{101.75+3.00-2.50}{7.92875}=\frac{102.25}{7.92875}=12.896\ \text{veh/min}$$ $$\boxed{q_w=773.8\ \text{veh/h}}$$ The eastbound stream is about 10% heavier, which is consistent with its shorter mean run time only because the section is uncongested; the direction with more traffic is not automatically the slower one.
Part (d) — Average travel time of eastbound traffic. The test vehicle is not a random member of the stream: if more vehicles overtook it than it overtook, it was travelling slower than average, and its own run time must be corrected downward by the net overtaking count divided by the volume: $$\bar t_{\text{traffic},e}=\bar t_e-\frac{O_e-P_e}{q_e}=3.75750-\frac{2.50-2.25}{14.126}=3.75750-0.01770$$ $$\boxed{\bar t_{\text{traffic},e}=3.740\ \text{min}}$$
Part (e) — Average travel time of westbound traffic. $$\bar t_{\text{traffic},w}=\bar t_w-\frac{O_w-P_w}{q_w}=4.17125-\frac{3.00-2.50}{12.896}=4.17125-0.03877$$ $$\boxed{\bar t_{\text{traffic},w}=4.132\ \text{min}}$$ The correction is small in both directions (0.5% and 0.9%) because the test driver was instructed to "float" with the stream, which is exactly the intent of the method — a large correction would be a warning that the driver was not floating.
Sanity-check the result against the section length. The section length is not stated, but the results are internally consistent regardless: taking a nominal 4.0 km section, the eastbound space mean speed would be $4.0/(3.740/60) = 64.2$ km/h and the westbound $4.0/(4.132/60) = 58.1$ km/h, giving densities of $847.6/64.2 = 13.2$ and $773.8/58.1 = 13.3$ veh/km. The near-equality of the two densities on a road of the same cross-section in both directions is a strong independent check that the volumes and travel times hang together.

Check — column interpretation and section length. The two overtaking columns are read as printed: "no. of vehicles that overtook the test vehicle" is $O$ and "no. of vehicles overtaken by the test vehicle" is $P$, so the net correction $O-P$ is positive when the test vehicle ran slower than the stream. Reversing the two columns would change each travel time by about 0.02–0.04 min and each volume by under 0.1%, so the conclusion is insensitive to that reading. The section length is not given, so speeds and densities in Step 6 are illustrative only, computed on a nominal 4.0 km; the answers to parts (a)–(e) require no length.

Part	Quantity	Symbol	Result
(a)	Mean travel time, westbound / eastbound runs	$\bar t_w,\bar t_e$	4.171 min / 3.758 min
(a)	Mean opposing count, westbound / eastbound runs	$M_w,M_e$	111.75 / 101.75 veh
(a)	Mean overtaking the test vehicle, W / E	$O_w,O_e$	3.000 / 2.500 veh
(a)	Mean overtaken by the test vehicle, W / E	$P_w,P_e$	2.500 / 2.250 veh
(b)	Eastbound traffic volume	$q_e$	848 veh/h (847.6)
(c)	Westbound traffic volume	$q_w$	774 veh/h (773.8)
(d)	Average travel time, eastbound traffic	$\bar t_{\text{traffic},e}$	3.740 min
(e)	Average travel time, westbound traffic	$\bar t_{\text{traffic},w}$	4.132 min

Topic. The moving-observer (Wardrop) method — simultaneous estimation of directional volume and mean travel time from a single instrumented vehicle making paired runs.

Key relations. $q = (M_{\text{opposing run}} + O - P)/(\bar t_w + \bar t_e)$; $\bar t_{\text{traffic}} = \bar t_{\text{test}} - (O-P)/q$; and the fundamental relation $q = k\,u_s$ used for the density cross-check.

Why this works. The count of vehicles met while driving against a stream is the number that crossed a moving "gate", and over the combined run time $\bar t_w + \bar t_e$ that gate sweeps the whole section exactly once relative to the stream — which is why the sum of the two run times, not either one alone, is the denominator. The overtaking terms then correct for the test vehicle not being an average member of the stream: each vehicle that passes it represents one vehicle the moving gate failed to count, and each vehicle it passes represents one counted twice. The same net overtaking count converts the driver's own travel time to the stream's mean, which is why the method delivers volume and travel time from one data set instead of requiring separate volume counts and floating-car runs.

Common pitfall. Dividing by only one direction's run time — the denominator is $\bar t_w + \bar t_e$ for both directions, and using $\bar t_e$ alone roughly doubles the eastbound volume. The second is pairing the wrong opposing count with a direction: the eastbound volume comes from the counts made on the westbound runs. The third is signing the overtaking correction backwards, which moves the travel time the wrong way.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., §4.3 (travel-time and delay studies; the moving-vehicle technique).

Direction	Mean travel time (min)	Mean opposing count	Mean overtaking the test vehicle	Mean overtaken by the test vehicle
Westbound	\(\bar t_w=4.17125\)	\(M_w=111.75\)	\(O_w=3.000\)	\(P_w=2.500\)
Eastbound	\(\bar t_e=3.75750\)	\(M_e=101.75\)	\(O_e=2.500\)	\(P_e=2.250\)

Part	Quantity	Symbol	Result
(a)	Mean travel time, westbound / eastbound runs	\(\bar t_w,\bar t_e\)	4.171 min / 3.758 min
(a)	Mean opposing count, westbound / eastbound runs	\(M_w,M_e\)	111.75 / 101.75 veh
(a)	Mean overtaking the test vehicle, W / E	\(O_w,O_e\)	3.000 / 2.500 veh
(a)	Mean overtaken by the test vehicle, W / E	\(P_w,P_e\)	2.500 / 2.250 veh
(b)	Eastbound traffic volume	\(q_e\)	848 veh/h (847.6)
(c)	Westbound traffic volume	\(q_w\)	774 veh/h (773.8)
(d)	Average travel time, eastbound traffic	\(\bar t_{\text{traffic},e}\)	3.740 min
(e)	Average travel time, westbound traffic	\(\bar t_{\text{traffic},w}\)	4.132 min

Question 6 of 7: Moving-Vehicle Method for Volume and Travel Time

Question 6: Moving-Vehicle Method for Volume and Travel Time (5 × 4 = 20 marks)