MATHS IA

Research question: How can the ideal distance headways of a traffic flow be determined?

INTRODUCTION
Considering the vast development in transportation technology in recent history, the
amount of vehicles and other transportation means are more extensive than ever. But this
development also comes with prevailing problems in practice. As a result, traffic has become one
of the main issues for today’s urban life.
The concept of traffic systems has been an issue of interest among many theoreticians, who
consist of people from physicists to mathematicians. The main aims of their studies was to
determine the laws of traffic flow, i.e. the interactions between travelers and the infrastructure,
analyze the fundamental properties of the traffic flow models, designate the main problems in
practice, and reveal the causes of these problems.
But considering the difficulty of conducting a precise real life experiment, these
theoreticians have developed models supported by real life data. The flow in these models bears
some characteristics important for evaluation. Firstly, the flow does not follow a certain sequence
because of the human factor and the its effects, thus the flow is highly random. So the models
produced are only able to make forecasts, but because they depend on practical data, evaluations
on the theories can be considered reliable. When the traffic flow data is processed, it shows
nonlinear characteristics.

THEORETICAL BACKGROUND
The main properties of a traffic flow is velocity, density and flow. Velocity(v) is the average
road the vehicles travel in a certain time. The first is time mean velocity, in which velocity is
calculated using fixed reference points in the road. The second is space mean velocity, which
calculates the data from snapshots during the whole road, and is usually considered more accurate.
Density(d) is the number of vehicles per unit roadway. Flow (q) is the number of vehicles that cross
a certain checkpoint in a certain amount of time.
These fundamental properties are utilized in order to establish a basis for any traffic flow
modelling and evaluation. They usually account for the main variables of a model and provide the
 quantitative basis for the evaluation of the traffic flow with respect to the model formulated, such
as in Kerner’s Three-Phase Traffic Theory, where Kerner uses the relationship between the density
and velocity in order to designate the congested areas and their propagative behavior (CITE
HERE).
The relationship between these are demonstrated by traffic theoreticians using the
fundamental diagrams of traffic flow. These diagrams include the density-velocity, density-flow,
and flow-velocity graphs. They are derived using empirical traffic data and have been proved
parallel to the many research made on the topic.

Diagram 1 to the left

demonstrates the relationship between the velocity and density of a traffic flow in a given time and
space. The relation between these two properties follow a nonlinear, inversely proportional
behavior. As in the diagram, as the density in the link, the portion of the road specified, increases,
the velocity decreases. This can also be established intuitively, one can imagine as the number of
vehicles increase by unit length, thus as the headway distance decreases, drivers will tend to slow
down due to the decreased area they have to accelerate and hesitation to crash. As denoted in the
graph, the exponential behavior will reveal how these properties give rise to the changes in the flow
properties. In the green area denoted as free flow, the density decreases increasingly with respect
to the velocity; whereas beyond the boundary between the free flow and bound flow, the behavior
shifts to decreasingly decreasing. This derivation is useful for analysts to determine the
infrastructural regulations.
Diagram 2 above demonstrates the relationship between the density and the flow rate, i.e
flux, of the traffic flow. As seen in the curve, the relationship between velocity and flux contains a
critical point which abruptly changes the trend of the curve, denoted as VC. A concept to be explained
later in the Diagram 3, VC stands for the critical velocity, which determines the maximum velocity
 that can be
experienced in free
flow. Moreover, the
peak also
corresponds to the
Qmax, which is used to
determine the
maximum flow rate,
the maximum amount
of vehicles that can enter a specific link at a given time period. On the right side of the critical
velocity, as the flux increases, the curve follows a decreasingly increasing behaviour, and after
peaking at VC, it follows a decreasingly decreasing behaviour, which implies a directly proportionate
relation in contrary with the inversely proportionate relationship. As denoted on the curve, the curve
is separated into the three phases of traffic flow corresponding to the areas in Diagram 2, and in
addition, the graph is also separated into stable and unstable conditions which imply that any
irregularity in the unstable area, where the flux begins decreasing due to decreasing velocity.

Diagram 3 to the left

demonstrates the relationship
between the density and the
flow rate of the traffic. As in
the
velocity-flow rate diagram, this
graph also contains a shift from
an directly proportionate
behaviour to an inversely
proportionate behaviour as the
density increases, but in contrast, as the graph moves rightward of the Qmax, there occurs a linear
relationship between the flow and the density. The slope of the
Diagram 3
density-flow rate graph equals to the velocity of the traffic flow (derived from the equation q=d.v).
 The relationship between these properties are formulated as below:
q=d.v
AIM
Among the many prevalent problems in traffic phenomenon, issues accounting for the
safety of the travel is the most important in my opinion. For this reason, I was intrigued to examine
how can an ideal behaviour for an safe traffic flow could be achieved. For the burden of safety is
considerably on the behaviour of the drivers, due to the role of random error in a vast majority of
car accidents, especially in the United Kingdom where is the subject of my investigation, my model
will attempt to provide a model for analysing traffic flow data considering the ideal stopping
distances, i.e the ideal distance headways, for drivers with respect to the average velocity of the
drivers and the density of the link.
As I created my model, I utilized several assumptions in order to be able to properly
calculate and evaluate on the limited data I could access, listed as below:
1. As I calculated the distance headways in the specific time period and space, I assumed that
the vehicles on the road all had the same distance headways, i.e the vehicles were
homogeneously spread throughout the road, in order to be able to calculate the ideal average
densities that corresponded velocity values from the ideal distance headways using the
equation h+l=1/d, where h is the distance headway, l is the vehicle length and d is the
density of the road.
2. As I calculated the average density from the empirical data I gathered using the equation
q=d.v, since the data is separated into fifteen minute intervals, to prevent any ambiguity
over the instantaneous densities, I assumed the calculated density value to be homogenous
for the whole interval.
3. I assumed that the average vehicle length, as calculated by the Department of
Transportation of United Kingdom, of four meters is the length of all individual vehicles.

First, going through the raw data I gathered from the Department of Transportation of the
UK, I only had the average velocity and the flow rate values accounting for the fundamental
properties of the traffic flow. Because the raw data was split into fifteen minute intervals, I
quadrupled the flow rate in order to negate the discretion in the units of the velocity (km/h) and
flow rate (n/15min). After that, I calculated the average densities for the fifteen minute intervals. I
used the equation q=d.v in the form of q/v=d to calculate the average density. An excerpt from the
data I gathered is below to demonstrate how I underwent the calculation in Table 1.
After the calculation, I determined the fundamental diagram of average density - velocity
diagram, in Diagram 4, in order to later use as I compare the ideal function of density and velocity
and the data.

Time Link Length Average Speed (v) Flow Flow(n/h) Average Density
(km) (km/h) (n/15min) (n/km)

12:00 2.76 104.13 25 100 0.960338

AM

12:15 2.76 91.14 22 88 0.965548

AM

12:30 2.76 104.78 19 76 0.725329

AM

12:45 2.76 110.05 14.75 59 0.53612

AM

1:00 2.76 104.35 14 56 0.536655

AM

Table 1
Diagram 4

DETERMINING THE IDEAL FUNCTION OF DENSITY vs VELOCITY

Initially, I analyzed the data determined by the Department of Transportation of United

Kingdom, given below in Diagram 5 for typical stopping distances, the average distance required
for a driver to stop when encountering a stationary object (vehicle, animal etc.) on its trajectory,
which I will from now on refer as the ideal distance headway (h). As the next step, I calculated
the density values in number of vehicles per kilometer, n/km, for each velocity and corresponding
ideal distance headway as shown below:

h+l=1/d

where h is the ideal headway distance, l is the vehicle length and d is density.

E.g: For v=32 km/h;

h=0.012 km, l=0.004 km.
Thus,
(10-3)*(12+4)=1/d
103*(1/16)=d
d=62.5 vehicles per km

Diagram 5
 After repeating the same procedure for the whole data, I derived Table 2 below.

Velocity (km/h) Density (n/km)

32 62.5

48 37.03704

64 25

80 17.54386

96 12.98701

112 10

Table 2

From this set of data, I derived the linear best fit line and the corresponding function by
using the least squares technique. The least squares technique may be defined as “a mathematical
procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the
squares of the offsets ("the residuals") of the points from the curve.” To specify, the offsets I used
to calculate the are defined as vertical offsets, demonstrated in Diagram 6, which are
mathematically denoted as yi-f(xi), the difference between the actual y value from the data and
the y value derived from the best fit function. The function is derived from the sums of squares of
the vertical offset (R) values and afterwards calculating the covariance and variance between the
values. Below the least squares fitting technique is explained step by step.

Diagram 6
(I will transform the equations I wrote on excel in the next draft.)
Step 1: First begin by defining the sum of squares of R (1) and the function aimed to be achieved
(2).

(1) (2)
Step 2: In order to determine the best fit data, equalize the derivatives of f(x) according to every
coefficient(3).

(3)
Step 3: After calculating the derivatives, the functions shown in (4) are determined.
 (4)

Step 4: Transform the equations into matrix form.

(5)
Step 5: Leaving the coefficients alone by having the inverse matrix of the 2x2 matrix.

(6)

(7)
Step 6: Transform the matrices back into algebraic form (8) in order to determine the sum of
squares (9) and consequently the variance of the coefficients (10).

(8)
 (9)

(10)
Step 7:

REFERENCES
https://en.wikipedia.org/wiki/Three-
phase_traffic_theory#Definitions_.5BJ.5D_and_.5BS.5D_of_the_phases_J_and_S_in_congeste
d_traffic
https://en.wikipedia.org/wiki/Traffic_flow#History
https://en.wikipedia.org/wiki/Three-phase_traffic_theory#Criticism_of_the_theory
http://victorknoop.eu/research/papers/chapter_vanwee.pdf
http://www.norbertwiener.umd.edu/Education/m3cdocs/Presentation2.pdf
http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_Fall2012/Files/Park.pdf
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0
ahUKEwjP6L_FurPTAhWFWBoKHSPdBBQQFghJMAM&url=https%3A%2F%2Fkluedo.ub.uni-
kl.de%2Ffiles%2F2899%2Fthesis2kluedo.pdf&usg=AFQjCNFO6W5zosQM3q-
AdmK8msn9udtGDQ&sig2=lAKR2n4ZObGg0YQmz87q1Q
http://www.traffic-simulation.de/uphill.html
http://www.kgm.gov.tr/SiteCollectionDocuments/KGMdocuments/Istatistikler/TrafikveUlasimBilgil
eri/12IlYollariTrafikUlasimBilgileri.pdf