16-Civ-B10 Traffic Engineering · December 2013

Question 1 of 7: Poisson Arrivals and Negative Exponential Headways

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 1: Poisson Arrivals and Negative Exponential Headways (20 marks: (a) 6, (b) 6, (c) 8)

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 counted sample of arrivals in 120 successive 30-second intervals, tabulated below. The third column is the product of the first two, so the table already supplies everything needed to form the sample mean.

Cars per 30 s interval, x	Number of intervals, f	Total cars observed, xf
0	10	0
1	15	15
2	30	60
3	20	60
4	20	80
5	10	50
6	5	30
7	5	35
8	3	24
9	2	18
Totals	120	372

Find. (a) the Poisson probability of exactly x arrivals per 30-second interval for x = 0 to 9; (b) the corresponding expected number of intervals out of 120; and (c) the probability that a randomly chosen headway is at least 12 seconds, treating headways as negative exponential.

Observed interval counts against the fitted Poisson frequencies (λ = 3.1 vehicles per 30 s). The fitted distribution reproduces the shape of the sample, under-predicting the 0-car intervals and over-predicting the 2- and 3-car intervals — the signature of arrivals that are slightly more clustered than pure randomness.

Approach. A Poisson process has a single parameter, so estimate the mean arrival rate from the sample, substitute it into the Poisson mass function for part (a), scale by the number of intervals for part (b), and exploit the fact that the headways of a Poisson arrival process are necessarily negative exponential for part (c).

Part (a) — estimate the mean arrival rate from the sample. The Poisson parameter is the mean number of arrivals per counting interval, which for grouped data is the weighted mean $$\lambda=\frac{\sum x f}{\sum f}=\frac{372}{120}$$ so that $$\boxed{\lambda = 3.1\ \text{vehicles per 30-second interval}}$$ Because the totals column of Table 1 was already provided, this is simply 372 vehicles spread over 120 intervals. Equivalently the street carries $3.1\times 120 = 372$ vehicles in the 60 minutes of observation, or 372 vehicles per hour.
Apply the Poisson mass function. With the rate now fixed, the probability of exactly $x$ arrivals in one interval is $$P(x)=\frac{\lambda^{x}e^{-\lambda}}{x!}=\frac{3.1^{x}e^{-3.1}}{x!}$$ Taking $x=2$ as the worked instance, $$P(2)=\frac{3.1^{2}\,e^{-3.1}}{2!}=\frac{9.61\times 0.045049}{2}=0.2165$$ The remaining nine values follow by the same substitution and are collected in the results table below. Note $e^{-3.1}=0.045049$, which is itself $P(0)$.
Part (b) — convert probabilities to expected frequencies. The theoretical frequency is the probability multiplied by the number of observations, since each of the 120 intervals is an independent trial: $$F(x)=N\,P(x)=120\,P(x)$$ For $x=2$ this gives $F(2)=120\times 0.2165=25.98$ intervals, against 30 observed. The largest single discrepancy is at $x=0$, where the model expects only 5.41 intervals with no arrivals but 10 were counted.
Check the fit by summation. A useful arithmetic control is that the ten theoretical frequencies must sum to slightly less than the sample size, the shortfall being the probability of ten or more arrivals: $$\sum_{x=0}^{9}F(x)=119.83\quad\text{against}\quad N=120$$ so the truncated tail accounts for $120-119.83=0.17$ of an interval, i.e. $P(x\ge 10)=0.0014$. That the two totals agree to two parts in a thousand confirms that both the rate and the individual probabilities were evaluated correctly.
Part (c) — express the arrival rate per second. Headway is a continuous time measurement, so the rate must be carried in vehicles per second rather than per 30-second interval: $$\lambda_{s}=\frac{3.1\ \text{veh}}{30\ \text{s}}=0.10333\ \text{veh/s} \qquad\Rightarrow\qquad \bar{h}=\frac{1}{\lambda_{s}}=9.68\ \text{s}$$ The mean headway of 9.68 seconds is the reciprocal of the arrival rate; it is the scale parameter of the exponential distribution that follows.
Apply the negative exponential headway distribution. If arrivals are Poisson then the interval between successive arrivals is negative exponential, and the probability of a headway at least $t$ is the probability of no arrival during $t$: $$P(h\ge t)=e^{-\lambda_{s}t}$$ Substituting $t=12$ s, $$P(h\ge 12)=e^{-0.10333\times 12}=e^{-1.24}$$ $$\boxed{P(h\ge 12\ \text{s}) = 0.2894 \approx 28.9\,\%}$$ So roughly two headways in every seven exceed 12 seconds — about 107 of the 372 vehicles observed in the hour follow their predecessor by 12 seconds or more. This is the quantity a gap-acceptance or unsignalised-crossing study needs, since a 12-second gap is comfortably usable by a turning or crossing driver.

Cars per 30 s, x	Intervals observed	(a) Theoretical probability P(x)	(b) Theoretical frequency 120 P(x)
0	10	0.0450	5.41
1	15	0.1397	16.76
2	30	0.2165	25.98
3	20	0.2237	26.84
4	20	0.1733	20.80
5	10	0.1075	12.90
6	5	0.0555	6.66
7	5	0.0246	2.95
8	3	0.0095	1.14
9	2	0.0033	0.39
Totals	120	0.9986	119.83

Part	Quantity	Result
(a), (b)	Mean arrival rate λ	3.1 veh per 30 s (372 veh/h)
(c)	Arrival rate λ_s	0.10333 veh/s
(c)	Mean headway	9.68 s
(c)	P(headway ≥ 12 s)	0.2894 (28.9 %)

Topic. Fitting a Poisson arrival model to counted traffic, and the exponential headway distribution that is its continuous-time twin.

Key relations. Poisson mass function $P(x)=\lambda^{x}e^{-\lambda}/x!$; expected frequency $F(x)=N\,P(x)$; negative exponential headway $P(h\ge t)=e^{-\lambda_{s}t}$ with mean headway $\bar h = 1/\lambda_{s}$; and the shifted form $P(h\ge t)=e^{-(t-\tau)/(\bar h-\tau)}$ when a minimum headway $\tau$ is imposed.

Why this works. The Poisson distribution arises whenever arrivals are independent, the probability of an arrival in a vanishing interval is proportional to its length, and simultaneous arrivals are impossible. Low-to-moderate flow on an urban street away from an upstream signal satisfies these conditions reasonably well, which is why one parameter suffices — and why the Poisson mean and variance are equal. That last property is the fitting diagnostic: here the sample mean is 3.1 and the sample variance is about 3.9, so the counts are mildly over-dispersed (more clustered than random), which is exactly what the figure shows as an excess of empty intervals. Part (c) needs no new assumption because the exponential headway law is a mathematical consequence of Poisson counting, not a second model: the event "headway exceeds $t$" and the event "zero arrivals in $t$" are the same event, so $P(h\ge t)=P(0\ \text{in}\ t)=e^{-\lambda_{s}t}$.

Common pitfall. Two errors dominate. First, forgetting to convert $\lambda$ from vehicles-per-interval to vehicles-per-second before entering the exponential: using $e^{-3.1\times 12}$ instead of $e^{-0.10333\times 12}$ returns essentially zero and is off by eleven orders of magnitude. Second, computing $P(h\ge 12)$ as $1-P(h\le 12)$ but then mis-stating the complement — note that $P(h\ge t)$ takes the plain exponential while $P(h< t)=1-e^{-\lambda_s t}$; mixing the two is the most common sign error in gap studies. A third, subtler slip is expecting the ten theoretical frequencies to sum to exactly 120; they cannot, because the tabulation truncates the tail at nine.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., Ch. 6 (Fundamental Principles of Traffic Flow) — Poisson arrival and headway distributions; Ch. 4 for the underlying volume-study method.

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