X and the City: Modeling Aspects of Urban Life (25 page)

Figure 8.6. Two vehicles approaching on an initially perpendicular path.

A compromise of sorts between the straight stop of length
D
and the circular turn of radius 2
D
is the “spiral stop.” This is accomplished by turning and braking simultaneously, and the vehicle will trace out a spiral path. The straight stop is achieved by applying (in the simple case) a constant force at 180° to the direction of motion. The circular path arises when that force is applied at 90° to the direction of motion. When this force is applied at some other constant angle
γ
to the direction of motion, the result is an equiangular spiral trajectory. In polar coordinates the equation of the path takes the form
r
₁
(
θ
) =
r
₀
e
^{(2cot
γ
)
θ}
, where
r
₀
is a constant (see
Appendix 6
for details).

What factors are important in studying what might be called “traffic engineering”? Clearly, weather and road conditions and the driver’s response to a changing traffic environment are all significant and indeed, interrelated. A twisty two-lane road poses different problems from a six-lane Interstate highway or “main drag” in a city. The effects of increased traffic volume, road repairs, or adjacent building projects on congestion are generally difficult to answer quantitatively, but the concept of acceleration noise—the root-mean-square of the acceleration of a vehicle—is a useful one in determining some answers to questions of this type.

If
v
(
t
) and
a
(
t
) are respectively the speed and acceleration of a vehicle at time
t
, assumed continuous, then the average acceleration over a trip lasting time
T
is

It is interesting to note that if the initial and final speeds are the same, then
ā
= 0. This will certainly be the case if the vehicle starts from rest and stops at the end of the trip!

The
acceleration noise σ
is the RMS (root-mean-square) of
a
(
t
) −
ā
, that is,

Exercise:
Show that

Clearly, when
ā
= 0, then

Exercise:
Calculate this quantity for several simple analytic choices of
a
(
t
).

Why might this concept be a useful one for traffic engineering? If we think about a car that is driven fairly smoothly (i.e., with no violent acceleration or braking), we would expect the quantity
σ
to be small (in a sense to be discussed later). If the vehicle is driven with such accelerations and decelerations,
σ
will be large. Recall that the slope of a speed-time graph at a given point is the acceleration at that point. In effect, the acceleration noise is a measure of the smoothness of the speed-time graph—the smaller
σ
, the smoother the journey. A narrow, crowded road with sharp turns will give rise, other things being equal, to a higher value of
σ
. Those reckless drivers (never ourselves, of course) we see so frequently weaving in and out of traffic will engender high
σ
-values. Instead of wishing to shake a fist at them, or inwardly raging at them, perhaps we should just content ourselves with this fact. At this point, an amusing image comes to mind: after passengers alight from a car, they each in turn raise a card above their heads with an estimate of the
σ
-value for the just-completed trip!

In a very interesting article by Jones and Potts (1962), several possible answers to this question are provided, based on experimental data. After extensive investigations in Adelaide and its environs, they were able to draw several conclusions, some of them perhaps not surprising:

1. Given, say, a two-lane road and a four-lane road through hilly countryside,
σ
is much greater for the former than the latter.

2. For roads in hilly countryside,
σ
is smaller for an uphill journey than for a downhill one.

3. If two drivers drive at different speeds below the “design speed” of a highway,
σ
is about the same.

4. If one or both drivers exceeds the design speed of the highway,
σ
is higher for the faster driver.

5. An increase in the volume of traffic increases
σ
.

6. Similarly, an increase in traffic congestion resulting from parked cars, frequently stopping buses, cross traffic, pedestrians, etc., increases
σ
.

7. A suitably calibrated value of
σ
may provide a better measure of traffic congestion than travel or stopped times.

8. High values of
σ
indicate a potentially dangerous situation; they
may
be a measure of higher accident rates.

Naturally we must ask, what are typically high and low values for
σ
? The authors found that
σ
= 1.5 ft/s
²
is a high value, and
σ
= 0.7 ft/s
²
is a low one.

Another factor influenced by the size of
σ
is an economic one: fuel efficiency. This is not just important for trucking companies or individual truckers, of course: it is increasingly important for the average driver as well. Trucks are fitted with tachographs to record the driving behavior of the truckers, and presumably have the effect of providing an incentive to (in effect) lower the value of
σ
.

Generally, then, the smoothness of a journey can be measured by the acceleration noise—the standard deviation of the accelerations; furthermore, it is known that the acceleration distribution is essentially normal. It could well be a useful measure of the danger posed by erratic drivers, for whom
σ
is high. It also increases in tunnels, the reasons probably being narrow lanes, artificial lighting, and confined conditions. However, it is not always sensitive to changing traffic conditions, especially in city centers or major highways leading into them, where traffic congestion is common and the average speeds are low. It is also the case that a given value of
σ
may correspond to more than one traffic situation, for example, journeys at low speeds in dense traffic or faster journeys interspersed with traffic lights. One possibility to avoid this ambiguity is to use a modification of
σ
, namely
, where
is the average speed.
may be interpreted as a measure of the mean change in speed per unit distance of the journey.