X and the City: Modeling Aspects of Urban Life by John A. Adam Page A

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, whereis the average speed.may be interpreted as a measure of the mean change in speed per unit distance of the journey.
Question:
jerk
is defined as the time derivative of the acceleration, that is,
j
(
t
) =
da
/
dt
. Do you think there is any usefulness to defining a quantity similar acceleration noise (as in equation (8.10), namely
jerk noise
?
     

     
Figure 8.7. Headway geometry for a two-lane highway.
     
X
=
T h
: OVERTAKING IN THE CITY
     
    We conclude this chapter with a simple mathematical model of overtaking and passing on a straight section of road. In Figure 8.7 a vehicle in the right lane moves into the left lane at speed
V
1 >
V
0 to overtake and pass a slower-moving vehicle. The equations, if not the diagram, are independent of which side of the Atlantic the vehicle resides! The
headway T H
is the time gap between adjacent oncoming vehicles moving at speed
V
2 .
    Clearly,

     
    Suppose that a reasonable time in which to complete a passing maneuver by a car traveling at 55 mph is 8 seconds, the headway required for an oncoming vehicle traveling at 65 mph in the opposing lane is
T H
≈ 15 seconds. For interstates or other multilane highways, the needed gaps are smaller; we just change the sign of
V
2 (obviously we require
V
1 <
V
2 ). In this case for the same speeds
T H
≈ 1.2 seconds!

Chapter 9
PROBABILITY IN THE CITY

     
     
     
    There are many topics under the umbrella “road traffic” that are amenable to mathematical investigation. Examples include traffic flow on the open road and at intersections, parking problems, accident rates, design of road systems for new towns and expanding cities, traffic lights and other control systems, transportation and scheduling problems, to name but a few. And as in so many aspects of mathematical modeling, there are two basic choices: to define the problem in a
deterministic
or a
probabilistic
context. The former can subdivide farther into continuum or discrete (or equivalently, macroscopic or microscopic) approaches, each with its own advantages and disadvantages. In the continuum approach the flow of traffic is treated as a fluid; properties of individual vehicles (“fluid particles”) are not considered. Such models are often referred to as being
kinematic
—the motion (as opposed to the “forces” behindthat motion) is of prime consideration. By contrast, car-following models are
dynamic
in the sense that accelerations and decelerations (and by implication, the forces) are implicit in their description. Although it is outside the scope of this book, it is perhaps worth mentioning that the discrete approach is often used to model flocking in starlings, shoaling in fish, and other swarming behavior (in ant, bee, and locust populations, for example).
    One very interesting feature common to both natural and man-made phenomena is the topic of stability vs. instability. Will those waves on the lake grow or die out? Will that crack in the windshield continue to grow or stop? (Almost certainly the former!) Will that traffic bottleneck disappear by the time I get there, or will it get worse? A typical traffic “instability” arises from time lags in the response of a driver to the accelerations and decelerations of the vehicle in front. A small lag can grow as it passes, in a wavelike manner from car to car. Indeed, it is accurate to say that terms like “shock wave” and “expanding wave” are quite appropriate to describe