A Synopsis of Simulation and Mobility Modeling in Vehicular Ad
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![Page 1: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/1.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
ON THE MODELING OF VEHICULAR
TRAFFIC
Ph.D. CourseComplex Systems in Engineering Sciences
Luisa FermoDipartimento di Matematica, Politecnico di Torino
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ON THE MODELING OF VEHICULAR TRAFFIC
The system to be modeled: Vehicular Traffic along a one-way road. A first
description.
“The width of a traffic shock only encompasses a few vehicles ”
“ A fluid particle responds to stimuli from the front and frombehind, but a car is an anisotropic particle that mostly responds tofrontal stimuli ”
“ Unlike molecules, vehicles have personalities (e.g., aggressive andtimid) that remain unchanged by motion”
C.F. Daganzo
Requiem for second-order fluid approximation of traffic flow ,
Transp. Res., 29B (1995), 277/286
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ON THE MODELING OF VEHICULAR TRAFFIC
The first step towards modeling: the choice of the scale of representation.
◮ Microscopic scale: Each vehicle is individually followed.The model writes as
xi = ai [t, {xk}Nk=1, {xk}
Nk=1], i = 1, ...,N
where xi is the scalar position of the i-th vehicle,t is the time,xi is the velocity,ai is a function describing the acceleration.
These models are not competitive from a computational pointof view.
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ON THE MODELING OF VEHICULAR TRAFFIC
The first step towards modeling: the choice of the scale of representation.
◮ Macroscopic scale: Each vehicle is not individually followed.An example is given by
∂ρ
∂t+
∂q
∂x= 0
where ρ is the density,q is the flux.
Some of these models recover missing information fromexperimental observation in steady flow conditions.
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ON THE MODELING OF VEHICULAR TRAFFIC
The first step towards modeling: the choice of the scale of representation.
◮ Kinetic Scale
1. Each vehicle is identified at time t∗ by◮ its position x
∗
∈ Dx∗ = [0, L] with L > 0 the length of theroad;
◮ its velocity v∗
∈ Dv∗ = [0, Vmax ] with Vmax > 0 the maximumspeed allowed along the road.
Remark: In the following we will consider the dimensionlessquantities:
x =x∗
L∈ Dx ≡ [0, 1], v =
v∗
Vmax
∈ Dv ≡ [0, 1],L
Vmax
t∗ = t.
The set {x , v} defines the microscopic state of the vehicle.
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ON THE MODELING OF VEHICULAR TRAFFIC
The first step towards modeling: the choice of the scale representation.
2. In order to describe the overall system a distribution function
f = f (t, x , v) : [0, Tmax ] × Dx × Dv → R+
is introduced.
f (t, x , v)dxdv represents the infinitesimal number of vehiclesthat at time t are located in [x , x + dx ] and travel with aspeed belonging to [v , v + dv ].
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ON THE MODELING OF VEHICULAR TRAFFIC
The first step towards modeling: the choice of the scale representation.
3. The model writes as:
∂f
∂t+ v
∂f
∂x= J[f ]
where J describes the interactions among vehicles.
4. One can compute macroscopic quantities like the density
ρ(t, x) =
∫
Dv
f (t, x , v)dv ,
and the flux
q(t, x) =
∫
Dv
v f (t, x , v)dv .
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ON THE MODELING OF VEHICULAR TRAFFIC
Modeling the vehicular traffic at the Kinetic Scale
Remark on the Kinetic Approach
Problem:Like the macroscopic models, the kinetic approach is based on acontinuum hyphothesis.
⇓
This assumption is not phisically satisfied by cars along a road.
⇓
A Possible Solution: Discrete velocity kinetic models.
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ON THE MODELING OF VEHICULAR TRAFFIC
Discrete velocity kinetic models: the basic idea
1. Introduce in Dv = [0, 1] a grid
Iv = {v1, v2, ..., vn−1, vn}
with v1 ≡ 0,vn ≡ 1,
vi < vi+1, ∀i = 1, ..., n − 1.
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ON THE MODELING OF VEHICULAR TRAFFIC
Discrete velocity kinetic models: the basic idea
2. The overall system is now described by
f (t, x , v) =n
∑
i=1
fi (t, x)δvi(v)
where the n functions fi (t, x) = [0, Tmax ] × Dx → R+ are the
distribution functions of the speed classes.
fi (t, x)dx denotes the infinitesimal number of vehicles havingspeed vi that at time t are in [x , x + dx ].
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ON THE MODELING OF VEHICULAR TRAFFIC
Discrete velocity kinetic models: the basic idea
3. The model writes as:
∂fi∂t
+ vi∂fi∂x
= Ji [f], ∀i = 1, ..., n
where f = (f1, ...fn) and Ji describes the interactions amongvehicles.
4. One can compute macroscopic quantities like the density
ρ(t, x) =n
∑
i=1
fi (t, x)
and the flux
q(t, x) =n
∑
i=1
vi fi (t, x).
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ON THE MODELING OF VEHICULAR TRAFFIC
Discrete velocity kinetic model: the interaction term Ji
The kinetic models describe the interactions by appealing to thefollowing guidelines:
1. Cars are regarded as points, their dimensions are negligible;
2. Interactions are binary. More precisely we will call
A. Candidate vehicle: the vehicle that change its state;B. Field vehicle: the vehicle that causes such a change;C. Test vehicle: an ideal vehicle of the system whose microscopic
state is targeted by a hypothetical observer;
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ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity model: the interaction term Ji
3. Interactions modify by themselves only the velocity of thevehicles, not their positions;
4. Vehicles are anisotropic particles;
5. Interactions are conservative, in the sense that they preservethe total number of vehicles of the system. Thus the operatorJi [f] is required to satisfy
n∑
i=1
Ji [f] = 0.
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ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity model: the interaction term Ji
If the interactions are such that 1 − 5 are fulfilled then
Ji [f] = Gi [f, f] − fiLi [f],
where
◮ Gi [f, f] is the i-th gain operator giving the amount per unittime that get the velocity vi ;
◮ Li [f] is the i-th loss operator giving the amount of vehicles perunit time that lose the velocity vi .
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ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity model: the interaction term Ji
The interactions among vehicles are described in a stochastic way.Then,
◮ A table of games Aihk is introduced such that
Aihk ≥ 0,
n∑
i=1
Aihk = 1, ∀i = 1, ..., n.
◮ Moreover, an encounter rate ηhk is defined.
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ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity model: a look at the literature
V. Coscia, M. Delitala and P. FrascaOn the mathematical theory of vehicular traffic flow II. Discrete velocity kinetic
models ,Int. J. Non-Linear Mech., 42(3) (2007), 411-421.
M. Delitala and A. TosinMathematical modeling of vehicular traffic: a discrete kinetic theory approach ,Math. Models Methods Appl. Sci., 17 (2007), 901-932.
C. Bianca and E. CosciaOn the coupling of steady and adaptive velocity grids in vehicular traffic
modelling,
24(2) (2011), 149-155.
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear
![Page 17: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/17.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity models with Local Interactions
The models write as:
∂fi∂t
+ vi∂fi∂x
= Ji [f], ∀i = 1, ..., n
where
Ji [f] =n
∑
h,k=1
ηhkAihk fhfk − fi
n∑
k=1
ηik fk
V. Coscia, M. Delitala and P. FrascaOn the mathematical theory of vehicular traffic flow II. Discrete velocity kinetic
models ,Int. J. Non-Linear Mech., 42(3) (2007), 411-421.
C. Bianca and E. CosciaOn the coupling of steady and adaptive velocity grids in vehicular traffic
modelling,
24(2) (2011), 149-155.
![Page 18: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/18.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity models with non local interactions
◮ One define Interaction length ξ the distance between theinteracting field and the candidate vehicle.
◮ If x is the position of the candidate vehicle and ξ is the lengthinteraction then one can define the interaction interval orvisibility zone as Iξ(x) = [x , x + ξ].
◮ One introduce a weight function w(x , y) weigthing theinteraction over the visibility zone.
M. Delitala and A. TosinMathematical modeling of vehicular traffic: a discrete kinetic theory approach ,Math. Models Methods Appl. Sci., 17 (2007), 901-932.
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear
![Page 19: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/19.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
Kinetic discrete velocity models with non local interactions
The models write as
∂fi∂t
+ vi∂fi∂x
= Ji [f], ∀i = 1, ..., n (1)
where
Ji [f] =n
∑
h,k=1
∫
Iξ
ηhk(t, y)Aihk(t, y)fh(t, x)fk(t, y)w(x , y)dy
− fi (t, x)
n∑
k=1
∫
Iξ
ηik(t, y)fk(t, y)w(x , y)dy .
M. Delitala and A. Tosin
Mathematical modeling of vehicular traffic: a discrete kinetic theory approach ,Math. Models Methods Appl. Sci., 17 (2007), 901-932.
A. Bellouquid, E. De Angelis and L. Fermo
Towards the modeling of vehicular traffic as a complex system: a kinetic approach,Math. Models Methods Appl. Sci., (2012), to appear.
![Page 20: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/20.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
All the terms appearing in the equations are modelled taking intoaccount the Traffic Phase, namely, traffic states having specificempirical spatiotemporal features. These features are specific onlyto a single traffic phase. It is characterized by a certain set ofstatistical properties of traffic variables (density, mean velocity,flux).
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
![Page 21: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/21.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
Classically there are two traffic phase:
◮ Free flow where vehicles are able to change a lane and topass. The maximum density achievable under free flow iscalled critical density ρc .
◮ Congested flow occurs when the vehicle density is highenough. Here the speed is lower than the lowest speed in freeflow.
![Page 22: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/22.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
Kerner identify three traffic phase:
◮ Free flow (F) where vehicles are able to change a lane and topass. The maximum density achievable under free flow iscalled critical density ρc .
◮ Congested flow occurs when the vehicle density is highenough. Here the speed is lower than the lowest speed in freeflow.
◮ Syncronized flow (S) Kerner describes synchronized flow asthe phase at which vehicles are accelerating to meet free flowtraffic.
◮ Wide moving jam (J) When vehicles move up the highwaythrough bottlenecks. The minimum density achieved undercongestion is called jam density ρj .
B. S. KernerThe Physics of Traffic,Springer 2004.
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ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
Now, we come back to equation (1) and model terms ηhk and Aihk .
◮ At first, we introduce a parameter α identifying theenvironmental conditions:
α = α0 +ρc
ρcmax(1 − α0),
where α0 is the minimum value of α identified by experiments,ρc is the critical density,ρcmax is the maximum critical density.
◮ Then we define the encounter rate η = ηhk = 1 + αρ2, ∀h, k .
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
![Page 24: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/24.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
◮ We give a table of games according to the three traffic phase.In FREE FLOW (0 ≤ ρ ≤ ρc)
Aihk =
1, i = n;
0, otherwise.(2)
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
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ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
◮ In WIDE MOVING JAM (ρj ≤ ρ ≤ 1)
Aihk =
1, i = 1;
0, otherwise.(3)
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
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ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
◮ In SYNCRONIZED FLOW (ρc < ρ < ρj)
A. Interaction with faster vehicles.
Aihk =
1 − α (ρj + ρc − ρ), i = h;
α(i − h)
1k
∑
i=h+1
1
i − h
(ρj + ρc − ρ), i = h + 1, ..., k ;
0, otherwise.
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
![Page 27: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/27.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
B. Interaction with slower vehicles.
Aihk =
α (ρj + ρc − ρ), i = h;
[1 − α (ρj + ρc − ρ)](h − i)
h−1∑
i=k
(h − i)
, i = k , ..., h − 1;
0, otherwise.
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
![Page 28: ON THE MODELING OF VEHICULAR TRAFFICcalvino.polito.it/fismat/poli/pdf/PhD_Courses/Traffic_Fermo.pdfON THE MODELING OF VEHICULAR TRAFFIC The first step towards modeling: the choice](https://reader034.fdocuments.net/reader034/viewer/2022050715/5d4f442a88c993c2058b5cdf/html5/thumbnails/28.jpg)
ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
C. Interaction with equally fast vehicles.
Aihk =
(1 − α)(h − i)
h−1∑
i=1
(h − i)
(1 − ρj − ρc + ρ), i = 1, ...h − 1;
α + (1 − 2α)(ρs + ρc − ρ), i = h;
α 1(i − h)
1n
∑
i=h+1
1
i − h
(ρj + ρc − ρ), i = h + 1, ..., n.
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
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ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
C. Interaction with equally fast vehicles.
Ai11 =
1 − α(ρj + ρc − ρ), i = 1;
α(ρj + ρc − ρ), i = 2;
0, otherwise.
Ainn =
α (1 − ρj − ρc + ρ), i = n − 1;
1 − α (1 − ρj − ρc + ρ), i = n;
0, otherwise.
A. Bellouquid, E. De Angelis and L. Fermo
Towards the modeling of vehicular traffic as a complex system: a kinetic approach,Math. Models Methods Appl. Sci., (2012), to appear.
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ON THE MODELING OF VEHICULAR TRAFFIC
A Kinetic discrete velocity model
Only for the sake of simplicity here we have consideredthe microscopic state {x , v}.
In the following paper
A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic
approach,Math. Models Methods Appl. Sci., (2012), to appear.
the microscopic state {x , v , u} is considered where u is anadditional variable denoting the quality of the micro-systemdriver-vehicle.
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ON THE MODELING OF VEHICULAR TRAFFIC
The non stationary problem
◮ We study the evolution of the system.The solution will depend on the time and mathematically themodel will be described by
1. Integro-differential equations;2. Initial conditions;3. Boundary conditions.
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ON THE MODELING OF VEHICULAR TRAFFIC
Example: Formation of Clustering
The model:
∂fi∂t
+ vi∂fi∂x
= Ji [f]
fn(0, x) = 100 sin2 (10π(x − 0.2)(x − 0.3)), x ∈ [0.2, 0.3],
fn−1(0, x) = 50 sin2 (10π(x − 0.5)(x − 0.6)), x ∈ [0, 5, 0.6]
fi (0, x) = 0, ∀i = 1, ..., n − 2, ∀x ∈ Dx
fi (t, 0) = fi (t, 1) ∀i = 1, ..., n
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ON THE MODELING OF VEHICULAR TRAFFIC
Example: Formation of Clustering in bad road conditions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=1.14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρt=1.40
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ON THE MODELING OF VEHICULAR TRAFFIC
Example: Formation of Clustering in optimal road conditions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=1.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
ρ
t=2.19
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ON THE MODELING OF VEHICULAR TRAFFIC
The spatially homogeneous problem
The model writes as
dfijdt
= η
n∑
h,k=1
Aihk fh(t) fk(t) − fi (t)
n∑
k=1
fk(t)
,
fi (0) = f 0i ,
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ON THE MODELING OF VEHICULAR TRAFFIC
Velocity Diagram
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ
ξ
α=0.95
α=0.7
α=0.5
α=0.3
Figure: Velocity diagram: mean velocity ξ versus density ρ
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ON THE MODELING OF VEHICULAR TRAFFIC
Fundamental Diagram
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
ρ
q
α=0.95
α=0.7
α=0.5
α=0.3
Figure: Fundamental diagram: flux q versus density ρ