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

some traffic patterns. Typically the existence of such waves in traffic flow can be deduced from both kinematic (continuous fluid-like) models and discrete (“particle”) car-following models. Most are too detailed for inclusion here, but we can gain some insight into these phenomena by examining some simple “steady-state” models. In this context, the steady state represents an equilibrium or perhaps a neutrally stable solution, much like a ball on a flat table—if disturbed, it will not move away indefinitely (instability) or return to its starting point (stability) but will remain where it is placed. This is neutral stability. But we start with an introduction to the probabilistic approach to traffic flow. And as pedestrians, we should be particularly interested in the
gaps
between the vehicles! We shall think about the gaps in what follows.
X
=
Pr
: PROBABILITY IN THE CITY
     
    Probabilistic (or stochastic) models incorporate, by definition, an element of randomness. This word is not to be understood in the common sense as haphazard; a more precise definition is given below. In this context it can mean that there is a probability distribution for, say, the size of gaps in a line of traffic. We can view traffic or the gaps in traffic as a distribution in space or time. In space, a length of single lane road (for simplicity) will have a distribution of vehicles along it at any given moment in time—a snapshot view. Alternatively we may identify a fixed location on the road with vehicles passing this positionas time goes on. The first situation is a distribution of intervals in space, and the second is a distribution in time. Such distributions (or series of events) are termed
random
provided that [ 19 ]
(i) each event (e.g., vehicle arrival time) is independent of any other, and
 (ii) equal intervals of time (or space) are equally likely to contain equal numbers of events (e.g., vehicles).
    There are several important distributions of interest in traffic flow studies; we shall briefly examine two of them—the
Poisson
and (displaced)
negative exponential distributions
. The latter is a simple generalization of a negative exponential distribution. The former gives the probability of a specified number of vehicles along a section of road at a given time, or passing a given point in a certain time interval. The latter provides the probability of a time or distance “gap” of a specified length in a specified time or distance. More precisely, it describes the time between events in a
Poisson process
, that is, a process in which events occur continuously and independently at a constant average rate. A derivation of the Poisson distribution is given in Appendix 4 .
X
=
P
(
t
): TRAFFIC GAPS IN THE CITY?
     
    This is an important question that has direct relevance to a topic mentioned above: pedestrians crossing roads. When I walk to work I have several roads to cross, and not all of them have crosswalks. There have also been quite a few occasions when I am in the middle of a crosswalk and cars go right by me (once it was a police cruiser that nearly knocked me down). There is a flavor of probability theory in this chapter, but not to worry, the applications we’ll be making are very straightforward.
    We’ll call
P
(
t
) the probability that no vehicle passes a certain point in a time interval
t
. Suppose that, over another period of time
T
a large number
N
of cars pass that same point. What do we mean by large in this context? Let’s take
N
> 100. The average number of cars in a time interval
t
is then
n
=
Nt
/
T
. For events that are equally likely to occur at any time, the distribution of times between the events is well described
exponential distribution
. For example, it is often used for modeling the behavior of items with a constant failure rate.It also has the advantage of taking a simple mathematical form. We define the exponential distribution

     
    P
(
t
) also describes the probability that there is a gap