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proposition article anglais .pdf



Nom original: proposition article anglais.pdf
Titre: urban mobility
Auteur: cyril enault

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Gravitation dynamics for the urban mobility ; a modelling
of transportation system, urban growth and automobile usage.
Introduction
The phenomenon of urban growth is not specific in the XXth century because the city exists
since the Antiquity. Nevertheless, we attend since the 50s, everywhere in the western
countries, a renewal of growth to be put in the credit, at least in Europe and in Asia in the
recovery according to war with its Baby boom. The urban sprawl or the extension of suburbs
in fine layers is good what characterizes most the current city (Ewing 1997).
In front of this phenomenon, new by its scale, it was advisable first of all to encircle it, to
estimate hit and then to understand mechanisms (as the studies of Mills 1992, Margo 1992,
Leroy, Sonstelie 1983). Diverse questions then appeared putting actually the real question:
how to control and to dike an extension to the anarchy of the urban suburbs? It is in this spirit
that amenagers threw tracks for a more rational and scientific study of the urban territories.
Many studies have discuses about the link of urban form and transportation (Newmann,
Kenworthy 1989, 1991, 1996, Muller 1995, Kenworthy and Laube 1996, 1999, Camagni,
Gibelli and Rigamonti 2002, Cervero 2002, Wheaton 2003, Badoe and Miller 2004, Massot,
Armoogum, Bonnel, Caubel, 2006, Milakis, Vlastos, Barbopoulos, 2008, Lin, Yang 2009).
Certainly the road is very long to succeed in putting clearly in link the phenomena without
reducing them to simplistic logics. Well before this questioning, it is clearly turned out that
the knowledge of the forms and the demonstrations of the urban sprawl stay another stake in
size for the geographers and the economists. To a certain extent, we are still not enough far
from the city completely feigned and analyzed in all its complexity, as suggest it the
geographers least to the fact of the question of the urban modelling.
And nevertheless, the quantitative analysis of the urban sprawl evolved well since the 50s.
The first studies on the urban shape date 1951 (Clark). In an illuminating article, the author
shows then the important recurrence of the negative exponential mathematical logic for the
densities of population according to the distance in the center of cities. The important sample
of big cities all over the world is not nevertheless that a statistical oddity. Indeed, years later
Clark, Alonso (1964), confirms a certain regularity at the level of the pensions and of the
urban organization which will be resumed by Muth (1969) and connected directly with the
density. Mills (1969) will confirm then the recurrence of the densities including for
developing countries (Mills, Tan 1980). Other studies will be also led later this way for the
Latin American countries (Ingram, Alan 1981). Finally Bussière (1972) puts an end to the
debate by showing by the maximization of entropy that the negative exponential shape would
be a law of balance of the urban. This assumption is too the argument for Wilson (1970).
More difficult is the approach of the polycentric city. Many papers analyses this specific
form, very classic for the actual city (Batty 2001, Cervero 1989, Ogawa Fujita 1982, Gordon
Richardson 1996, Heikkila Gordon Kim Peiser, Richardson 1989 and too the density studies :
Griffith 1981, Anas Arnott Small 1998, Mahmassani Bajj Tong 1988, Mc Donald Prather
1994, Mc Donald 1987, Small Song 1994, Whang Zhou 1999, Zheng 1991).
Always on this question of the forms, it is also difficult to pass next to the works concerning
the fractals in which the authors show still a recurrence of the forms and it to establish it
typologies (Batty Longley 1986). Descriptive aspect, certainly for these works but of the very
big other interest as far as we do not conceive any more the territory by means of the
Euclidian but good classic geometry on the basis of another shape of metrics; the study of the

city thus advances strongly on the question of the description of the forms but also in a
measure on the explanation.
On this last point, this type of studies is also economists' fact and to a lesser extent of
geographers. These works present a big interest but remain very often very theoretical in them
principles and thus at once difficult to access but also rather with a difficult integration and a
complex managing within the framework of local and purely territorial analysis.
More recently, the works of the geographers more widely turned to the modelling of system
of dynamic nature via equations and different formalisms (Batty Xie 1994, Batty Xie, Sun
1999, Xie 1996). The cellular automata with the multi-multi agent systems the biggest corpus
of works on this question today. Interesting above all for their results, they remain
nevertheless rather binding in the measure or the logics cannot be easily modified.
Other works bottom the choice of the purely mathematical analysis and so opt for a more
flexible approach by dynamic systems. This is the case of the studies of Pumain (1982), and
Pumain, Sanders Saint Julien (1989)
This article positions in this second trend. It takes support on a series of hypotheses which it
discusses and fixes:
The gravitation models
The models of fluid mechanics (law of the traffic)
The general mechanics of Newton
The usage of the automobile in the space and in the time.
These elements serve then as support for a completely mathematical demonstration ending in
the end in a dynamic system in three differential equations.
In this theoretical corpus, the main question remains the shape taken by the relation urban
speed/density of population/ usage of the car. The urban sprawl, as far as it follows this type
of logic does it always lead to similar results? And to a certain extent it reports the reality of
the phenomenon?
To answer these questions, we suggest defining first of all the working hypotheses, then
detailing the demonstration of our system to finish finally by its general study.

1. Quick review of traffic and formalism gravity models
1.1. State of the premises on gravity models
Historically, Ravenstein (1885) was the first, in the 19th century, to explore the analogy
between the laws of physics and the Social Science. Since then, the first exercises were
widely amended (Carrothers (1956), awesome and Bramhall (1960), Batten and Boyce
(1987)), which is that there are now several models in competition.
The first and simplest expression assumes the existence of a proportional relationship between
the flow and the mass of population.
Thus, the stream emitted from a place i is expressed as a simple broadcast from the Mi
population is:
Qi = GM i [1]
With Qi road flow emitted from the place i
G constant
The population of the place i, Mi.

Model [1] is by definition a space and then plans a mass population Mi which it does not
define the distribution in space.
A first development is to introduce the notion of distance in model [1], this suggests that the
flow varies with distance instead of diffusion i.
M
Qi = G i [2]
xi
Where xi is the radial distance i instead.
By staying on the strict analogy with the laws of physics, it is also possible to formalize the
interaction between two points i and j of space. It is then assumed that the exchanged global
flux exceeds the simple addition and thus the emission of i multiplied by the emission of j,
then include the famous law of gravitation:
MiM j
M Mj
Qi = G i
= G
[3]
xij xij
xij2
Still today widely employed, this model has however a clear limit. Indeed, the human flow
differs from the electrostatic field because it is bounded by i or j, hence different likely
formalism. A new modelling assumptions provide, inter alia, that the flow emitted from a
place i that is distributed by the masses of people. There are then several variants of
formalization for population densities. The maximization of entropy (Wilson 1970) from the
most common model is the model of Clark (1951). Departing from [1] and by substituting Mi
by a negative exponential, it gets:
Qi = GMe −αxi [4]
This expression is very close to the more classical type gravity formulation for an amendment
on the distance (xij exponent n).
[4] Model can be generalized as the gravity of base, from a place of unique to i issue[1]place of
issue. Global flow observed is then the sum is:
Qi = G ∑i M i e −α i xi
[5]
Note that form [5] is close to the expression of the polycentric densities of traditional
geographical economy through a G-factor.
Model [5] has been refined in a formalization of 1970 differentiating origin and destination [2].
Note that in the event of confusion between origins and destinations; the model comes down
to [5].
1.2. Macroscopic models of traffic
If the gravity model gives the aggregate measure of the traffic between two masses of
population or the general expression of the flow in a place of space, traffic models introduce
[3]
, on the other hand the dynamics. They are constructed from the analogy with the laws of
fluid flow.
In the literature, there are two major families of traffic model: microscopic so-called models
that track the movement across the vehicle (in terms of speed, of accélération…) and on
which we do we'll not and the so-called models macroscopic [4] in fact, systems based on three
expressions. The best known and simplest of them is the LWR model [5].
As common core, these macroscopic models are based on three physical quantities.
- the density of vehicles

Ki

=

∂N i
∂xi

[6]

Or even Ki vehicle concentration is equivalent to the variation in the amount of vehicles
present in a place i or by variation of xi distance.
-

speed

It simply plans by the quotient of the distance on the duration
∂xi
Vi =
[7]
∂t
- Finally the flow or stream road

∂N i
[8]
∂t
It is determined by the amount of vehicles quotient Ni by duration instead.
Expressions [6], [7] and [8] allow to obtain the characteristic relationship, first equation
common to all macroscopic models of traffic.
Qi = K iVi [9]
For the dynamic equation of the model (second expression of the system), it will distinguish
the first order of higher order models.
In the case of the first order (LWR), considering that the flow is constant in the section
balanced while in the second case, there may be an imbalance in the whole. It employs the
latter in the case of high congestion causing the dissemination of the plugs in dynamics on
other sectors of the network. It is characterized by stopping and restarting vehicles
movements.
When considering the principle of balance (first-order model), it is possible to express a
relationship to translate conservation of flows between upstream and downstream of a mass
urban i[6] is:
Qi

x2

∫ [K1 (x1 , t )

x1

=

− K 2 ( x 2 , t )]dx =

t2

∫ [Q (x, t )
2

2

− Q1 ( x, t1 )]dt

[10]

t1

They deduct that in a first-order model, there is therefore a strict balance between the
incoming and the outgoing flows [7] is:
∂Qi
∂K i
+
= 0 [11]
∂xi
∂t
Higher-order models, they are far more complex. A common formalization substitute
equilibrium model [11] by two equations partial. A term of Celerity discern however between
the different exercises. The main variants are models of Payne (1971), Ross (1988),
Papageorgiou et al (1990), Del Castillo (1993) or Zhang (1998-2000).
For the third and last equation the simplest system, there are a wide variety of expression in
competition. Is defined by "fundamental diagram" the General relationship that connects the
flow concentration
The first formalization of this statistical relationship has been proposed in the 1930s[8].
Qi = − aK i2 + bK i [12]
With a and b both parameters evaluate statistically.
Since then, other formalisms, in competition, could be proposed.
Drew model (1968)
Qi = aK i K i + bK i

The expression of Greenberg
Qi = aK i log(K i ) + bK i
The relationship of Underwood
Qi = aK i e − bK i
Finally the relationship of May (1990), the most current
Qi = aK i e −bK i
Other more general forms can also exist as
2

1 K
−  i
b  Kr





b

Qi = aK i e
The overall profile of the fundamental diagram is presented in figure 1 (with as fitting the
model of Greenshields).
It will be noted that the cloud stretches along the curve with two clearly defined situations.
1/ a part where the speed increases with the flow and concentration, talking about condition of
fluidity.
2/ a part where the speed decreases with flow increase[9] concentration, it then sets this
configuration of saturated or congestion.

Figure 1. Point cloud of the flow rate (every 15 minutes)
for a station of South of Paris: Villeneuve Saint Georges (April 2009).
Models to simulate traffic are now numerous and widely used by economists, and
geographers traffic engineers. In this theoretical bloom, it is often difficult to make a choice
for application as each model has its interests and its limits. The purpose of the next part is to
first find the most relevant assumptions for our problem in this theoretical corpus and then
combine them to achieve a dynamic formalization for the density of population-speed road

2. Definition of hypotheses and research of the system dynamics speed roaddensity of population
2.1. Definition of our model assumptions
2.1.1. Assumptions on the space
A) To begin with, we define the space as a Euclidean smooth plan fully accessible, which
implies a perfect capillary network road.
B) Each portion of the territory i is characterized by a mass of population Mi (that is
consider by the density). geographic position-independent
C) It will also be assumed that the city consists of i urban centers: polycentric city
D) Proposed Vi road speed is average in point i, which for an empty space leads to a plan
2.1.2. A gravity of population attraction
Our hypothesis is that the population in a place i space is originally a stream of Qi. The global
space of flows, in a polycentric perspective is therefore being a sum of gravitational
interactions, in what we had envisaged in the first part as:
Qi = G ∑i M i e −α i xi
The justification of the exponential model rather than the hyperbolic is due to the difficulty of
the last to make account flows to the centre i. Indeed, modeling provides an infinity while it is
obvious that the stream emitted by the people Mi..
The sum is also to discuss; We assume as well as in the polycentric configuration, each space
i is interacting with its neighbor i + 1, but also that there is within spaces not competition.
Urban space is rather the sum of the forces of each place i.
2.1.3. Operation of the stream according to the laws of fluid flow
Following this assumption, we assume that the stream can be expressed also by the traffic
models.
We're first choice of the model of Greenshieds as fundamental diagram, which is irrelevant to
an urban scale.
The characteristic relationship is naturally taken as hypothesis
Finally, it remains the second relationship which should discuss the principle.
In absolute terms, the flows are in equilibrium in a given network. Off, we consider the
dissemination of a flow in porous space which implies gains or losses in the road network.
This function of gain or loss must be carefully established.
We emit the idea that the entry of additional vehicles on the road is the result of a program of
mass but that should be considered primarily as a variation. Furthermore, it should be to
associate a parameter Ri, relating to the place i, a function in reality which will be the rate of
use of vehicles.
It will be assumed that the flow entering a network corresponds to the outflow through the
loss or gain function multiplied by a factor of gi.
x2

∫ [K1 (x1 , t )

x1

− K 2 (x 2 , t )]dx =

t2

∫ [Q (x, t )
2

2

− Q1 (x, t1 )]dt − Ri ∫ [g i ( x1 , t ) −

t1

With gi(x,t) density of population in the xi space and time t.
Ri service of rates of use of the car. xi distance
The result of fact is the following simplified model:

g i (x 2 , t )]dx

[13]

∂Qi
∂K i
∂g
+
= Ri i [14]
∂xi
∂t
∂t
Accordingly, the following system will be assumed.
Qi = K iVi
∂Qi
∂K i
∂g
+
= Ri i
∂t
∂xi
∂t
Qi

= − aK i2

+ bK i

2.1.4. Forces and the forces travel exercise
It is assumed in our space, that the movement is the result of the distribution of the masses of
people. The result is indeed, an organization founded on the principle of Clark (1951), as we
had suggested but also logic of complex movements is governed by an amount n of forces.
They have their origin in population density. It therefore deducts that speed or even flows are
in interaction with densities according to the usual laws of motion in a space that will initially
set as empty. Newton's second law gives the key or the link between on the one hand speed
and gravity on the other hand.
Either:

dV i
[15]
dt
Pi represents the forces of gravity acting at a time given t and place i on the object in motion
in our space and τi a parameter i instead.

∑P

i

= τi

2.1.5. Use of the automobile rate and link to the space and time
Ri is here to define. It can be in direct relation to congestion (in sum concentration) or Ki.
According to what can be observed in reality, the rate of use of the car is often regarded as a
probability. We therefore here assume that you can associate with a logistic function for the
link to the concentration of vehicle.
Following the occupation of the road, can be assumed that the choice will perform on a type
of mode of travel. It is assumed that the traffic congestion leads to massive use of the public
transport a decrease in the rate of use of the car.
Therefore the inference is the following function:
dRi
= − ηRi (1 − Ri ) [16]
dK i
With η parameter to calibrate. Ki concentration of vehicle and Ri automobile motorization
rates
2.2.

The General model
2.2.1. Express the dynamics of speed

The purpose of this paragraph is from the assumptions of §2.1 to search a dynamic expression
of speed density.
To do this, is part of the flow-concentration relationship. By introducing the speed, it is shown
from [9] and [12] that:
Vi = − aK i + b [17]

Where Vi is the average speed, Ki vehicle concentration and a and b of the parameters to be
determined.
Then look for the dynamics of the speed by the derivative is:
dVi
dK
= −a i
dt
dt
Let's derivative of Ki.
To do this, should be introduced [14]. It is established that:

dVi
dt

= − αGg i

− aRi

dg i
dt

[18]

2.2.2. Express the dynamics of density and return on the dynamics of speed
For each point i space, it will set a benchmark of Frenet curvilinear coordinates [10] giving an
opportunity to highlight the link density speed through Newton's second law.
In logic of attraction gravity, any flow is considered as converging or diverging towards all of
the places i (following a polycentric principle). The path then followed by a mobile will be the
distribution function gravity of the masses of population for all of the places i.
In fact, the observed movement oscillates between different trajectories points i. The
following schema explicitly the terms and conditions of the trajectory of the mobile following
the Frenet [11] marker is:

Figure 2. Movement and gravity attraction
In this model the radial component is by definition equal to Vi²/xi[12]. It is also possible that
the movements are in contrary sense to the people. In this case, then, part Vi²/xi becomes
negative because contrary to the orientation of the coordinate system of Frenet.

A random function will then propose to model the movement of the centre to the periphery or
periphery to the centre[13] or more specifically, this allows formalize centrifugal movement or
centripetal (it then chose a piped relation random function to promote the dispersion or
concentration).
Either
Entier ( random () +1+ f )
aléa = (− 1)
(f) is defined as a parameter of randomization for "piper" probability. It is included between 0
and 0.5. This setting is close to 0.5, more likely to get movements of concentration is high.
The sum of Pi forces expressed the polycentric gravity model defined by [5]
Hence according to [15] and [18]:
dg i
2τ i
dV
=
AléaVi i
dt
Gxi
dt
Either:
2τ i
dg i
dg


=
AléaVi  aRi i + aGαg i 
dt
Gxi
dt


from which (with 2αa/xi = ξ and 2a/Gxi = Φ ))
2τ iαaAléaVi g i
dg i
xi
Aléaτ i ξVi g i
=
=
[19]
2aτ i Aléa
dt
1 − Aléaτ i ΦRiVi
1 −
RiVi
Gxi
We search then to express the derivative of velocity with the derivative of the density is:
dVi
Aléaτ i aξVi g i Ri
= − αaGg i −
dt
1 − Aléaτ i ΦRiVi
By asking αaG = λ and has aξ = Γ, you get:

dVi
dt

= − λg i



Aléaτ i ΓVi g i Ri
1 − Aléaτ i ΦRiVi

[20]

2.2.3. Expression of the dynamics of the rate of use of the automobile and final model
By definition can be found the dynamics of the rate of use of the automobile by the following
formula:
dRi
dRi dK i
=
dt
dK i dt
Gold from [16], [18] and [19], it is possible to write:


dRi
Vi g i Ri
= − ηRi (1 − Ri ) Aléaτ i Γ
+ λg i  [21]
dt
1 − AléaΦτ i RiVi


The final system can therefore be written:

[22]

dVi
dt
dg i
dt

Aléaτ i ΓVi g i Ri
1 − Aléaτ i ΦRiVi
Aléaτ i ξVi g i
=
1 − Aléaτ i ΦRiVi
= − λg i





dRi
Vi g i Ri
= − ηRi (1 − Ri ) Aléaτ i Γ
+ λg i 
1 − AléaΦτ i RiVi
dt


The result we get here is a dynamic system involving direct with road speed Vi population
density as well as the rate of use of the car laughed for adjustment variables. In practice this
means that the evolution of the speed is on the evolution of the density, the evolution of the
rate and this interactively.
The system has no solution for the three functions Vi, Ri and gi are linked in time by their
derivatives.
By its construction, it also contains a random part alea.
The parameter τi represents the local rate of growth of population density, it will calculate the
following formula:
∆g i
τi =
∆t
Finally, the model follows the evolution of the trio speed density rate and it to a place i
indefinite space. Therefore, it allows in an application to multiple geographic entities to
follow trajectories differentiated spaces.
In this, can be said that, mathematically, our model is similar to the field of vectors of a
geographic area.
Beyond formalism itself, it is also important to understand the spatial logics support in our
model. To do this, we propose in our suite to analyze the different configurations that the
model may take based on the values of parameters, speed and density.

3. Differentiated spatial analysis of urban dynamic system
In the system [22], spatial patterns may be considered by varying the values of gi, laughed and
Vi. The value of the discriminant is gi density. Will be subsequently decompose our urban
space, from a theoretical point of view in 4 zones with distance to the centre of the city. The
example of Paris chose to set distance values.
Later in this article we will present the results of density projections on fixed values (gi,
laughed and Vi) and also by setting the parameters λ, ξ, Φ, Γand η.

Figure 3. Typology of the developments in urban areas and urban fringe.

3.1. The evolution of the centre: crater or growth?
The first case we encounter is that of the city centre. We must then differentiate types of
developments.
1 / saturated growth
2 / the reversal of growth or the crater density
In the first case, gi density is evolving favorably in a constant manner until the speed is
positive (whether as long as there is no congestion). In parallel, the speed decreases in a
constant manner in exponential. The rate of use of the car also decreased from in our example
of 0.75 to 0.4.
Then, the situation changed fundamentally since the speed becomes negative. The latter
stabilizes at a value. Density, growth is more soft until that believes are drying up and until it
reaches saturation (in our example average 5000 inhabitants/km²).
The rate of use of the car also decreased and also stabilizes at a threshold value of 0.4
(asymptote to 0.4).
This means that for this range of value, the system [22] becomes stationary.
In this second case, the growth of density is also important in the early but quickly reached an
inversion point (black point on the chart) where the density undergoes a reversal. Evolution
becomes negative and density decreases slowly at first and then accelerates then to stabilize.
This phenomenon is explained by the speed that also decreases and then becomes negative
term (congestion). The use of the automobile rate decreases also tends to a minimum (in the
period when Vi tends to 0) and then re - increased very slightly to stabilize at a low level to
0.45.
.

Figure 4. Evolution of the density for the two scenarios from the city centre

3.2. The evolution of the suburbs and urban fringe
In these two configurations, we are somewhat in the same logic as for the city centre but with
developments staggered in time. Thus, density starts to levels much lower than for the centre
and feels significantly faster than for the centre to reach a threshold at the end of the
simulation to 4000 inhabitants/km².

Figure 4. Evolution of the density in the urban suburbs
Vi speed decreased steadily from the beginning of the simulation and then tends to 0. Rate, it
decreases sufficiently modest manner and tends quickly to a threshold to 0.9.
The trend of the system is at steady state.
The difference between the two scenarios comes from the values currently in play. For the
suburbs, thresholds are important and suburban tends to evolve in the city centre (with a
difference at the level of public transport). For the urban fringe, on hand with thresholds of
transit to 0 or nearly, gradually, is introduced lines. The part of the public transport increased
slightly but this has important implications for the dynamics of the population (x by 3 or 4 of
the density). In parallel, the speed decreases also on a regular basis with a basic level well
more than in the suburbs. In short, the urban fringe tends to integrate into the suburbs.
3.3. The evolution of rural areas
In this scenario, density is set at a much lower level than in the rest of the space, the rate of
transport in common is 0, the speed is close to 80, 90 km/h. Finally, local growth rate τi is
negative (originality of this scenario: negative migration rate, negative population growth
rate)).
The evolution in the long term is a stability of speed to the same value, a rate of public transit
to 0 and finally a very slight decrease in the density, but regular. Rural areas therefore empty
of their population.

Figure 5. Evolution of the density of rural in abandonment
In our three situations, we well see that in the end, the logics are similar and that the overall
principle of dynamics is the expansion of the growth of the area of growth of the city. The
processes are therefore clearly a broadcast.
City centre can evolve in two ways, but in the end, it tends toward a crater, suburban
integrates with the centre, the urban fringe becomes suburban and finally what is rural isolated
decreases until it reversed the logic of rates (demographic and migration).
Conclusion
A brief literature review on traffic models and the gravity model to identify the contours of
the key assumptions of a coupled approaches density-speed-rate of use of the car.
Amended, different basic laws have been progressively integrated in an analytical and formal
approach.
What then is the logic at work? This is a more or less complex system dynamically linking
population density, the average aggregate speed road as well as the use of the car. There is an
indivisibility which is the fact that the three equations are related by their respective
derivatives in this link. Accordingly, and as often in these cases, there is no analytical
solution.
The main results of this study are theoretical identification of a dynamic complex and
differentiated from urban spaces. It is mainly logic in concentric rings. Also the urban center
is subject to a trend changing either logistics or crater. As urban fringe, suburbs respond to the
same base but with developments that are staggered in time (logistic model against
exponential). Finally, isolated rural space is characterized by a trend decline in some cases
could evolve into a dynamic of fringe. Seen well, the city and its outskirts are advancing
surface by progressively converting spaces adjacent to it.
A general point of view, the main interest of this model comes from its relative simplicity of
implementation in a process of simulation (in a geographic information system), that we have
not undertaken here. Then, and most important is the fact that there are today still few models
that couple collective transport, urban form and the type of operation of the networks.

The limits of such an approach are the reduction of urban logic to only three functions. It is
obvious that the dynamic, in all its complexity is much more difficult to identify.
Furthermore, one could also argue the too theoretical aspect of such an approach but this is
not the case where in previous papers, we had explored similar logic, certainly does not
necessarily involving three equations, but this does not fundamentally change the problem.
You could easily propose an application in a geographic information system that we have
coupled with a programming example in Visual Basic in Excel [14]sheet. Improvements are to
consider, in the first place on the space itself (introduction of space perception and not more
plan) or on the consequences of the evolution of the rack densities (a probabilistic approach
might be the key).
[1]

polycentrism, the most common cases
Model of Wilson 1970
[3]
Lebacque J.P. 1993 for a preview
[4]
Models with an interest in the General conditions of the traffic on the segments covered by the three sizes
flow, speed and density.
[5]
Lightill MH, Witham GB and Richard PI 1955
[6]
For macroscopic model the simplest cases of order 1.
[7]
Non-broadcast plugs or congestion in space and time
[8]
Greenshields 1935
[9]
Or occupancy rate is defined as follows with TOi occupancy of the roadway, Ki concentration, 2 constants: L
length of the segment and l what the trafician call electrical length (distance from minimum extent for the
collection of data traffic for electromagnetic loops)
[10]
The Frenet marker is defined as a mobile perspective coordinate system centered on the object in motion. He
is described by two coordinates: the tangential coordinate which is tangent to the trajectory of the object and the
radial coordinate which is perpendicular to the latter. The tangential coordinate is formalized by the dV/dt
acceleration and the radial coordinate by the square of the velocity by the distance to the centre of curvature is
V²/xi. It may in certain cases, linearize the two coordinates by writing M = dt/dt + V²/xi
[11]
In which it does keep not tangential part, which would lead to a spiral movement.
[12]
According to the laws of gravity acceleration
[13]
According to a process of dice with 2 faces either 1 or -1 with a part to increase the chances of obtaining
either a 1 or a - 1.
[14]
Enault 2011
[2]

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