16-Civ-B10 Traffic Engineering · December 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 98-Civ-B10 Traffic Engineering, National Examinations, December 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. 3 (driver characteristics and the PIEV process), Ch. 4 (traffic-engineering studies: spot-speed studies, time-mean and space-mean speed), Ch. 6 (fundamental principles of traffic flow, deterministic and stochastic queueing) and Ch. 8 (intersection control: saturation flow, change and clearance intervals, Webster’s green split and delay). Transportation Research Board, Highway Capacity Manual — signalised-intersection capacity, degree of saturation, control delay and level of service. AASHTO, A Policy on Geometric Design of Highways and Streets, and Transportation Association of Canada, Geometric Design Guide for Canadian Roads — stopping and passing sight distance, crest vertical-curve design and K-values. FHWA, Manual on Uniform Traffic Control Devices, and Transportation Association of Canada, Manual of Uniform Traffic Control Devices for Canada — traffic-signal warrants and no-passing-zone marking.
Assumptions declared under the paper’s NOTE 1 and NOTE 2 (“any data required, but not given, can be assumed”). These are used throughout and are not repeated in every question: bus passenger-car equivalent \(E_B = 2.0\); pedestrian walking speed \(S_p = 1.2\ \text{m/s}\); driver perception–reaction time 2.5 s and deceleration \(a = 3.4\ \text{m/s}^2\) for stopping sight distance; AASHTO/TAC metric sight-distance eye and object heights; first-in–first-out discipline in every queueing calculation.
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.
The signal actually displays a green interval \(G\), an amber (yellow) interval \(A\) and an all-red interval \(R\); the sum \(I = A + R\) is the intergreen or change-and-clearance period. Real discharge, however, does not match those displayed intervals. When the signal turns green the first few drivers must react and accelerate, so the departure headways of the first four or five vehicles are longer than the saturation headway; this shortfall is the start-up lost time \(l_1\), typically 2–3 s. At the other end of the phase, vehicles continue to enter the intersection during part of the amber, so some of the amber is used productively; the unusable remainder is the clearance lost time \(l_2\).
Effective green is the idealised interval during which the approach is assumed to discharge at exactly the saturation flow rate \(s\), defined so that the rectangle \(s \times g\) contains the same number of vehicles as the real, ramped discharge profile:
\[ g = G + A - l_1 - l_2 \qquad\text{or, equivalently,}\qquad g = G + I - \ell \]
where \(\ell = l_1 + l_2\) is the total lost time for that phase. Effective red is simply the rest of the cycle for that movement, \(r = C - g\): the interval during which no vehicles are assumed to depart at all, even though a few are physically clearing on amber. The two idealisations are complementary and they are what make the rectangular queueing model of Questions 5 and 7 work — the arrival curve is continuous, the departure curve is flat over \(r\) and rises at slope \(s\) over \(g\). The intersection lost time per cycle is \(L = \sum \ell_i\), and the capacity of a lane group follows immediately as \(c = s\,(g/C)\). A common source of confusion is that the effective green of a phase can be longer than its displayed green (because part of the amber is used), while the effective red is always longer than the displayed red.
Time-mean speed (TMS) is the arithmetic average of the spot speeds of vehicles passing a fixed point during a stated period:
\[ \bar{u}_t = \frac{1}{n}\sum_{i=1}^{n} u_i \]
It is what a radar gun, a loop-pair or a laser meter produces, because those instruments sample vehicles as they arrive at one cross-section. Space-mean speed (SMS) is the average speed of the vehicles occupying a length of road at one instant, and it is obtained as the harmonic mean of the individual speeds, or equivalently by dividing a known length by the average travel time over it:
\[ \bar{u}_s = \frac{n}{\sum_{i=1}^{n} \left( 1/u_i \right)} = \frac{L}{\bar{t}} \]
The two are linked by Wardrop’s relation \( \bar{u}_t = \bar{u}_s + \sigma_s^{2}/\bar{u}_s \), where \(\sigma_s^2\) is the variance of the space-mean-speed distribution. Because a variance is never negative, TMS is always greater than or equal to SMS, with equality only when every vehicle travels at the same speed; the gap widens as the speed distribution spreads. The practical point is that only the space-mean speed satisfies the fundamental relation \(q = k\,\bar{u}_s\), so density, capacity and travel-time studies must use SMS. Time-mean speed is nevertheless the correct statistic for enforcement, spot-speed studies, 85th-percentile speed-limit setting and sight-distance design, because those applications care about what individual drivers do at a location.
Both are demand-responsive alternatives to pretimed (fixed-time) control, and both rely on vehicle detectors, a minimum (initial) green, a unit extension or passage time, and a maximum green.
Semi-actuated control places detectors only on the minor approaches. The major street is the resting phase: it holds green indefinitely and is interrupted only when a detector on the side street (or a pedestrian push-button) registers a call. The side-street green then runs for at least its minimum, extends while vehicles keep arriving within the passage time, and terminates on gap-out or on max-out, after which the controller returns to the arterial. This is the standard treatment where a low-volume road meets a busy arterial, and it has the important advantage that the arterial phase can still be coordinated with adjacent signals, because the background cycle length is preserved.
Fully-actuated control places detectors on every approach. No phase is a resting phase; each one is called by demand, is skipped entirely when no call exists, and is extended vehicle by vehicle between its minimum and maximum green. The cycle length and the splits therefore vary from cycle to cycle in response to the actual arrival pattern. Fully-actuated control gives the lowest delay at an isolated intersection with unpredictable or strongly fluctuating demand, and it handles unbalanced or intermittent movements gracefully. Its drawback is the mirror image of the semi-actuated advantage: because the cycle length floats, a fully-actuated intersection cannot readily be coordinated into a progressive system, and it costs more to install and maintain because of the additional detection.
PIEV is the four-stage model of the driver’s response to a stimulus, and the interval it describes is the perception–reaction time \(t\) used in every sight-distance and change-interval formula.
P — Perception: the driver senses the stimulus (a stopped vehicle, a signal changing to amber, a pedestrian stepping off the kerb). I — Identification or Intellection: the driver interprets what has been sensed and understands its significance. E — Emotion: the driver decides on the appropriate response — brake, steer, accelerate, sound the horn — a judgement stage strongly affected by experience, age, fatigue, alcohol and distraction. V — Volition: the driver executes the chosen response, moving the foot to the brake pedal or turning the wheel; the vehicle only begins to decelerate at the end of this stage.
Design values for the total PIEV time depend on the task. AASHTO and the TAC Geometric Design Guide adopt 2.5 s for stopping sight distance, which is conservative enough to cover roughly the 90th percentile of drivers in a simple braking situation. About 1.0 s is used in the amber-interval (dilemma-zone) formula \(A = t + v/(2a + 2gG)\), because the driver is already alert and attending to the signal. Values of 2.0 s or more are used for complex urban decision points and for decision sight distance, where the driver must select among several manoeuvres. PIEV time enters stopping sight distance as the brake-reaction distance \(0.278\,V t\), which at 100 km/h is about 70 m — roughly a third of the total SSD — so it is far from a second-order term.
The Manual on Uniform Traffic Control Devices sets out eight warrants, at least one of which must be satisfied before a traffic control signal is installed. Meeting a warrant is a necessary but not a sufficient condition: an engineering study must still show that the signal will improve the overall safety and operation of the intersection. The eight are:
Four of them, discussed:
Warrant 1 — Eight-Hour Vehicular Volume. The basic volume warrant, applied when the sheer weight of intersecting traffic is the problem. It has two independent conditions. Condition A (minimum vehicular volume) is met when, for each of any eight hours of an average day, both the major-street total volume and the higher-volume minor-street approach exceed tabulated thresholds — for a one-lane-each intersection, 500 veh/h on the major street and 150 veh/h on the minor approach. Condition B (interruption of continuous traffic) recognises the opposite problem, a major street so heavy that minor-street drivers cannot find gaps; its major-street threshold is higher (750 veh/h) and its minor-street threshold lower (75 veh/h). The thresholds are reduced to 70 % where the 85th-percentile major-street speed exceeds 70 km/h or the community has fewer than 10 000 people.
Warrant 3 — Peak Hour. Intended for the intersection that operates acceptably almost all day but breaks down for one hour — the classic case being an industrial plant, office park or arena that discharges a large volume over a short period. It is satisfied when, for one hour of an average day, the minor-street delay exceeds four vehicle-hours for a one-lane approach (five for two lanes), the minor-street volume exceeds 100 veh/h for one lane, and the total entering volume exceeds 650 veh/h. Because this warrant addresses a single hour, the MUTCD cautions that it should be applied only where the peaking is genuinely unusual, since a signal installed for one hour penalises the other twenty-three.
Warrant 4 — Pedestrian Volume. Applied where vehicular volume alone would not justify a signal but pedestrians cannot cross safely. It is satisfied when the pedestrian volume crossing the major street is at least 100 per hour for each of any four hours, or 190 in any one hour, and there are fewer than 60 adequate gaps per hour in the vehicle stream, and no other signalised crossing lies within 90 m. The warrant is the formal recognition that a road can be functionally impassable on foot while remaining free-flowing for vehicles, and it is normally implemented with pedestrian signal heads and a push-button.
Warrant 7 — Crash Experience. The reactive, safety-driven warrant. It requires that five or more reported crashes of types susceptible to correction by a signal (predominantly right-angle collisions) have occurred within a twelve-month period, each involving injury or property damage above the reporting threshold; that adequate trial of less restrictive remedies — improved signing, sight-line clearance, turn restrictions, stop control — has failed to reduce the frequency; and that volumes are not less than 80 % of the Warrant 1 thresholds. The 80 % qualifier prevents a signal being installed at an intersection so lightly travelled that a signal would create more rear-end collisions than the right-angle crashes it prevents.
Check — Canadian practice. The question names the U.S. MUTCD explicitly, so the eight warrants above are the correct answer. In Canada the equivalent document is the TAC Manual of Uniform Traffic Control Devices for Canada, which replaces the pass/fail warrant list with a point-scoring traffic signal warrant matrix: points are accumulated for vehicle volume, delay, collision experience and pedestrian activity, and a total of 100 points indicates that a signal is justified. The underlying considerations are the same; only the decision instrument differs.
