A multi-class vehicular flow model
for aggressive drivers.
R. M. Velasco
Aggressive multi-class model
A multi-class vehicular flow model
for aggressive drivers.
W. Marques Jr., R. M. Velasco, A. R.
Méndez
27 October, 2015
Universidade Federal do Paraná, Brazil.
Universidad Autónoma Metropolitana-Iztapalapa, México.
Universidad Autónoma Metropolitana-Cuajimalpa, México.
R. M. Velasco
Aggressive multi-class model
Abstract. The kinetic theory approaches to vehicular traffic modeling have
given very good results in the understanding of the dynamical phenomena
involved. In this work we deal with the kinetic approach modeling of a
traffic situation where there are many classes of aggressive drivers. Their
aggressivity is characterized through their relaxation times. The reduced
Paveri-Fontana equation is taken as a starting point to set the model.
The kinetic equation is taken to write the balance equations for the density
and the average speed in each drivers class. In this model we consider that
each class of drivers preserve the corresponding aggressivity, in such a way
that there will be no adaptation effects. It means that the number of drivers
in a class is conserved. Some characteristics of the model are explored with
the usual methods.
R. M. Velasco
Aggressive multi-class model
Outline
1
Introduction.
R. M. Velasco
Aggressive multi-class model
Outline
1
2
Introduction.
The kinetic model.
R. M. Velasco
Aggressive multi-class model
Outline
1
2
3
Introduction.
The kinetic model.
Interaction integrals.
R. M. Velasco
Aggressive multi-class model
Outline
1
2
3
4
Introduction.
The kinetic model.
Interaction integrals.
Two classes of drivers.
R. M. Velasco
Aggressive multi-class model
Outline
1
2
3
4
5
Introduction.
The kinetic model.
Interaction integrals.
Two classes of drivers.
The equilibrium state.
R. M. Velasco
Aggressive multi-class model
Outline
1
2
3
4
5
6
Introduction.
The kinetic model.
Interaction integrals.
Two classes of drivers.
The equilibrium state.
Stability analysis.
R. M. Velasco
Aggressive multi-class model
Introduction
In the literature, traffic flow in highways is described through different approaches going from the microscopic to the macroscopic points of view
[1, 2, 3].
R. M. Velasco
Aggressive multi-class model
Introduction
In the literature, traffic flow in highways is described through different approaches going from the microscopic to the macroscopic points of view
[1, 2, 3]. Our goal in this work is the construction of a macroscopic model
coming from the reduced Paveri-Fontana equation for aggressive two-user
classes of drivers, which do not loose their aggressivity.
R. M. Velasco
Aggressive multi-class model
Introduction
In the literature, traffic flow in highways is described through different approaches going from the microscopic to the macroscopic points of view
[1, 2, 3]. Our goal in this work is the construction of a macroscopic model
coming from the reduced Paveri-Fontana equation for aggressive two-user
classes of drivers, which do not loose their aggressivity. We characterize
each class by means of the time of response, which is shortly called as the
relaxation time τ1 , τ2 , τ1 6= τ2 . In our treatment the distribution function
fi (c, x, t) is calculated with a model for the averaged desired speed, which
introduces the aggressivity.
R. M. Velasco
Aggressive multi-class model
Introduction
In the literature, traffic flow in highways is described through different approaches going from the microscopic to the macroscopic points of view
[1, 2, 3]. Our goal in this work is the construction of a macroscopic model
coming from the reduced Paveri-Fontana equation for aggressive two-user
classes of drivers, which do not loose their aggressivity. We characterize
each class by means of the time of response, which is shortly called as the
relaxation time τ1 , τ2 , τ1 6= τ2 . In our treatment the distribution function
fi (c, x, t) is calculated with a model for the averaged desired speed, which
introduces the aggressivity. Then the distribution function is written in terms
of local variables such as the densities and average speeds, leading to a closure relation.
R. M. Velasco
Aggressive multi-class model
Introduction
In the literature, traffic flow in highways is described through different approaches going from the microscopic to the macroscopic points of view
[1, 2, 3]. Our goal in this work is the construction of a macroscopic model
coming from the reduced Paveri-Fontana equation for aggressive two-user
classes of drivers, which do not loose their aggressivity. We characterize
each class by means of the time of response, which is shortly called as the
relaxation time τ1 , τ2 , τ1 6= τ2 . In our treatment the distribution function
fi (c, x, t) is calculated with a model for the averaged desired speed, which
introduces the aggressivity. Then the distribution function is written in terms
of local variables such as the densities and average speeds, leading to a closure relation. The conditions to obtain an equilibrium state are obtained and
a linear stability analysis is done.
R. M. Velasco
Aggressive multi-class model
The kinetic model
The reduced Paveri-Fontana equation comes from an integration over the
instantaneous desired speed giving place to an average called c0 (c, x, t),
where c is the actual speed of vehicles.
R. M. Velasco
Aggressive multi-class model
The kinetic model
The reduced Paveri-Fontana equation comes from an integration over the
instantaneous desired speed giving place to an average called c0 (c, x, t),
where c is the actual speed of vehicles.
∂fi (c)
∂fi (c) ∂
+c
+
∂t
∂x
∂c
c0 (c) − c
fi (c)
τi
!
X
=q
ρi fj (c)ξi (c)+ρj fi (c)ψj (c) ,
j
(1)
R. M. Velasco
Aggressive multi-class model
The kinetic model
The reduced Paveri-Fontana equation comes from an integration over the
instantaneous desired speed giving place to an average called c0 (c, x, t),
where c is the actual speed of vehicles.
∂fi (c)
∂fi (c) ∂
+c
+
∂t
∂x
∂c
c0 (c) − c
fi (c)
τi
!
X
=q
ρi fj (c)ξi (c)+ρj fi (c)ψj (c) ,
j
(1)
here q = 1 − p is the probability of overpassing and it can be a function of
density.
R. M. Velasco
Aggressive multi-class model
The kinetic model
The reduced Paveri-Fontana equation comes from an integration over the
instantaneous desired speed giving place to an average called c0 (c, x, t),
where c is the actual speed of vehicles.
∂fi (c)
∂fi (c) ∂
+c
+
∂t
∂x
∂c
c0 (c) − c
fi (c)
τi
!
X
=q
ρi fj (c)ξi (c)+ρj fi (c)ψj (c) ,
j
(1)
here q = 1 − p is the probability of overpassing and it can be a function of
density.
The interaction is separated according to active ψi (c) or passive ξi (c) interaction terms.
R. M. Velasco
Aggressive multi-class model
They are defined as
R. M. Velasco
Aggressive multi-class model
They are defined as
Z
(w − c)
ψi (c) =
w <c
fi (w )
dw ,
ρi
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i = 1, 2
active,
Aggressive multi-class model
(2)
They are defined as
Z
(w − c)
ψi (c) =
w <c
fi (w )
dw ,
ρi
Z
(w − c)
ξi (c) = fi (c)
w >c
fj (w )
dw ,
ρj
R. M. Velasco
i = 1, 2
active,
i = 1, 2,
Aggressive multi-class model
passive.
(2)
(3)
They are defined as
Z
(w − c)
ψi (c) =
w <c
fi (w )
dw ,
ρi
Z
(w − c)
ξi (c) = fi (c)
w >c
fj (w )
dw ,
ρj
i = 1, 2
active,
i = 1, 2,
passive.
(2)
(3)
The quantities fi (c), c0 (c), ψi (c), ξi (c) also depend on (x, t) but their
writing has been avoided to shorten the notation.
R. M. Velasco
Aggressive multi-class model
They are defined as
Z
(w − c)
ψi (c) =
w <c
fi (w )
dw ,
ρi
Z
(w − c)
ξi (c) = fi (c)
w >c
fj (w )
dw ,
ρj
i = 1, 2
active,
i = 1, 2,
passive.
(2)
(3)
The quantities fi (c), c0 (c), ψi (c), ξi (c) also depend on (x, t) but their
writing has been avoided to shorten the notation.
The active interaction comes from faster vehicles interacting with the slow
ones and cause a decrease of the distribution function.
R. M. Velasco
Aggressive multi-class model
They are defined as
Z
(w − c)
ψi (c) =
w <c
fi (w )
dw ,
ρi
Z
(w − c)
ξi (c) = fi (c)
w >c
fj (w )
dw ,
ρj
i = 1, 2
active,
i = 1, 2,
passive.
(2)
(3)
The quantities fi (c), c0 (c), ψi (c), ξi (c) also depend on (x, t) but their
writing has been avoided to shorten the notation.
The active interaction comes from faster vehicles interacting with the slow
ones and cause a decrease of the distribution function.
On the other hand, the passive term refers to fast vehicles impeded by
vehicles with speed c, causing an increase in the distribution function.
R. M. Velasco
Aggressive multi-class model
They are defined as
Z
(w − c)
ψi (c) =
w <c
fi (w )
dw ,
ρi
Z
(w − c)
ξi (c) = fi (c)
w >c
fj (w )
dw ,
ρj
i = 1, 2
active,
i = 1, 2,
passive.
(2)
(3)
The quantities fi (c), c0 (c), ψi (c), ξi (c) also depend on (x, t) but their
writing has been avoided to shorten the notation.
The active interaction comes from faster vehicles interacting with the slow
ones and cause a decrease of the distribution function.
On the other hand, the passive term refers to fast vehicles impeded by
vehicles with speed c, causing an increase in the distribution function.
Both give a contribution to the balance in the kinetic equation [7].
R. M. Velasco
Aggressive multi-class model
The macroscopic variables
We define the densities and average speeds as
R. M. Velasco
Aggressive multi-class model
The macroscopic variables
We define the densities and average speeds as
Z
Z
fi (c)
ρi (x, t) = fi (c)dc,
vi (x, t) = c
dc = hcii ,
ρi
R. M. Velasco
Aggressive multi-class model
(4)
The macroscopic variables
We define the densities and average speeds as
Z
Z
fi (c)
ρi (x, t) = fi (c)dc,
vi (x, t) = c
dc = hcii ,
ρi
(4)
Their dynamic equations are obtained from the reduced Paveri-Fontana
equation when it is multiplied by functions of the instantaneous speed c
and integrated.
R. M. Velasco
Aggressive multi-class model
The macroscopic variables
We define the densities and average speeds as
Z
Z
fi (c)
ρi (x, t) = fi (c)dc,
vi (x, t) = c
dc = hcii ,
ρi
(4)
Their dynamic equations are obtained from the reduced Paveri-Fontana
equation when it is multiplied by functions of the instantaneous speed c
and integrated.
X
∂ρi
∂ρi vi
+
= −q
ρi ρj vj − vi + hψi ij − hξj ii ,
∂t
∂x
j
R. M. Velasco
Aggressive multi-class model
i = 1, 2
(5)
The macroscopic variables
We define the densities and average speeds as
Z
Z
fi (c)
ρi (x, t) = fi (c)dc,
vi (x, t) = c
dc = hcii ,
ρi
(4)
Their dynamic equations are obtained from the reduced Paveri-Fontana
equation when it is multiplied by functions of the instantaneous speed c
and integrated.
X
∂ρi
∂ρi vi
+
= −q
ρi ρj vj − vi + hψi ij − hξj ii ,
∂t
∂x
i = 1, 2
j
where h...ii means the average over the fi distribution function.
R. M. Velasco
Aggressive multi-class model
(5)
The macroscopic variables
We define the densities and average speeds as
Z
Z
fi (c)
ρi (x, t) = fi (c)dc,
vi (x, t) = c
dc = hcii ,
ρi
(4)
Their dynamic equations are obtained from the reduced Paveri-Fontana
equation when it is multiplied by functions of the instantaneous speed c
and integrated.
X
∂ρi
∂ρi vi
+
= −q
ρi ρj vj − vi + hψi ij − hξj ii ,
∂t
∂x
i = 1, 2
(5)
j
where h...ii means the average over the fi distribution function. It should
be noticed that hξj ii − hψi ij = vj − vi is satisfied.
R. M. Velasco
Aggressive multi-class model
As a consequence we obtain that both densities satisfy conservation equations as it should be due to their lack of adaptation between classes of
drivers.
R. M. Velasco
Aggressive multi-class model
As a consequence we obtain that both densities satisfy conservation equations as it should be due to their lack of adaptation between classes of
drivers.
In a similar way the equations of motion for fluxes read as
R. M. Velasco
Aggressive multi-class model
As a consequence we obtain that both densities satisfy conservation equations as it should be due to their lack of adaptation between classes of
drivers.
In a similar way the equations of motion for fluxes read as
X
∂ρi vi
∂
ρi
+
(ρi vi2 + Pi ) = (Vi0 − vi ) + q
ρi ρj hcξi ij + hcψj ii , (6)
∂t
∂x
τi
j
R. M. Velasco
Aggressive multi-class model
As a consequence we obtain that both densities satisfy conservation equations as it should be due to their lack of adaptation between classes of
drivers.
In a similar way the equations of motion for fluxes read as
X
∂ρi vi
∂
ρi
+
(ρi vi2 + Pi ) = (Vi0 − vi ) + q
ρi ρj hcξi ij + hcψj ii , (6)
∂t
∂x
τi
j
where Vi0 (x, t) comes from the average
R of the desired speed now taken over
the instantaneous speed c, and Pi = (c −vi )2 fi (c)dc = ρi Θi is the i−class
traffic pressure and Θi the speed variance.
R. M. Velasco
Aggressive multi-class model
The distribution function
First of all we consider aggressive drivers characterized by the parameter
ω in such a way that the average desired speed is assumed as Vi0 (c) =
ωvi , ω > 1.
R. M. Velasco
Aggressive multi-class model
The distribution function
First of all we consider aggressive drivers characterized by the parameter
ω in such a way that the average desired speed is assumed as Vi0 (c) =
ωvi , ω > 1.
According to this model we have found [8] an exact solution, which can also
be written in terms of the local variables ρi , vi .
R. M. Velasco
Aggressive multi-class model
The distribution function
First of all we consider aggressive drivers characterized by the parameter
ω in such a way that the average desired speed is assumed as Vi0 (c) =
ωvi , ω > 1.
According to this model we have found [8] an exact solution, which can also
be written in terms of the local variables ρi , vi .
α ρi
fi (c) =
Γ(α) vi
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αc
vi
α−1
αc
exp −
vi
Aggressive multi-class model
(7)
The distribution function
First of all we consider aggressive drivers characterized by the parameter
ω in such a way that the average desired speed is assumed as Vi0 (c) =
ωvi , ω > 1.
According to this model we have found [8] an exact solution, which can also
be written in terms of the local variables ρi , vi .
α ρi
fi (c) =
Γ(α) vi
where the parameter α =
τi the relaxation time.
αc
vi
τi (1−pi )ρei vie
ω−1
R. M. Velasco
α−1
αc
exp −
vi
(7)
, ωi is the aggresivity parameter and
Aggressive multi-class model
The distribution function
First of all we consider aggressive drivers characterized by the parameter
ω in such a way that the average desired speed is assumed as Vi0 (c) =
ωvi , ω > 1.
According to this model we have found [8] an exact solution, which can also
be written in terms of the local variables ρi , vi .
α ρi
fi (c) =
Γ(α) vi
αc
vi
α−1
αc
exp −
vi
τ (1−p )ρe v e
(7)
i i i
where the parameter α = i ω−1
, ωi is the aggresivity parameter and
τi the relaxation time. The quantity α depends on the model parameters
and the traffic situation measured by (ρei , vie ), which correspond to the
equilibrium state.
R. M. Velasco
Aggressive multi-class model
The distribution function
First of all we consider aggressive drivers characterized by the parameter
ω in such a way that the average desired speed is assumed as Vi0 (c) =
ωvi , ω > 1.
According to this model we have found [8] an exact solution, which can also
be written in terms of the local variables ρi , vi .
α ρi
fi (c) =
Γ(α) vi
αc
vi
α−1
αc
exp −
vi
τ (1−p )ρe v e
(7)
i i i
where the parameter α = i ω−1
, ωi is the aggresivity parameter and
τi the relaxation time. The quantity α depends on the model parameters
and the traffic situation measured by (ρei , vie ), which correspond to the
equilibrium state.
It should be noticed that ρi (x, t), vi (x, t) in Eq. (7) are local variables.
R. M. Velasco
Aggressive multi-class model
Now the interaction terms
R. M. Velasco
Aggressive multi-class model
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
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Aggressive multi-class model
(8)
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
(8)
can be calculated with the distribution function given in Eq. (7), where it
is assumed that there are two classes of drivers with the same α parameter,
but different relaxation times.
R. M. Velasco
Aggressive multi-class model
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
(8)
can be calculated with the distribution function given in Eq. (7), where it
is assumed that there are two classes of drivers with the same α parameter,
but different relaxation times.
• Within user-class interactions.
R. M. Velasco
Aggressive multi-class model
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
(8)
can be calculated with the distribution function given in Eq. (7), where it
is assumed that there are two classes of drivers with the same α parameter,
but different relaxation times.
• Within user-class interactions.
These interactions correspond to the case i = j, then
R. M. Velasco
Aggressive multi-class model
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
(8)
can be calculated with the distribution function given in Eq. (7), where it
is assumed that there are two classes of drivers with the same α parameter,
but different relaxation times.
• Within user-class interactions.
These interactions correspond to the case i = j, then
hcξi ii + hcψi ii = vi2 − hc 2 ii = −Θi ,
R. M. Velasco
i = 1, 2.
Aggressive multi-class model
(9)
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
(8)
can be calculated with the distribution function given in Eq. (7), where it
is assumed that there are two classes of drivers with the same α parameter,
but different relaxation times.
• Within user-class interactions.
These interactions correspond to the case i = j, then
hcξi ii + hcψi ii = vi2 − hc 2 ii = −Θi ,
i = 1, 2.
• Between user-class interactions.
R. M. Velasco
Aggressive multi-class model
(9)
Now the interaction terms
Iij (c) = hcξi ij + hcψj ii ,
(8)
can be calculated with the distribution function given in Eq. (7), where it
is assumed that there are two classes of drivers with the same α parameter,
but different relaxation times.
• Within user-class interactions.
These interactions correspond to the case i = j, then
hcξi ii + hcψi ii = vi2 − hc 2 ii = −Θi ,
i = 1, 2.
(9)
• Between user-class interactions.
The active and passive interactions between different classes for (i 6= j) can
be written in a closed form, however we have found that a simpler and very
good approximation for them
R. M. Velasco
Aggressive multi-class model
(
Iij = vi vj 1 −
α + 1 vj
α vi
!
α + 1 vi
H(vi − vj ) + 1 −
α vj
R. M. Velasco
!
Aggressive multi-class model
)
H(vi − vj ) , (10)
(
Iij = vi vj 1 −
α + 1 vj
α vi
!
α + 1 vi
H(vi − vj ) + 1 −
α vj
!
where H(vi − vj ) is the step function.
R. M. Velasco
Aggressive multi-class model
)
H(vi − vj ) , (10)
(
Iij = vi vj 1 −
α + 1 vj
α vi
!
α + 1 vi
H(vi − vj ) + 1 −
α vj
!
)
H(vi − vj ) , (10)
where H(vi − vj ) is the step function.
In order to begin the analysis we will consider that v2 > v1 and in such a
case the interaction terms simplify so that
R. M. Velasco
Aggressive multi-class model
(
Iij = vi vj 1 −
α + 1 vj
α vi
!
α + 1 vi
H(vi − vj ) + 1 −
α vj
!
)
H(vi − vj ) , (10)
where H(vi − vj ) is the step function.
In order to begin the analysis we will consider that v2 > v1 and in such a
case the interaction terms simplify so that
I11 = −Θ1 , I22 = −Θ2 , I12 = 0, I21 ' −(v2 − v1 )2 .
R. M. Velasco
Aggressive multi-class model
(11)
(
Iij = vi vj 1 −
α + 1 vj
α vi
!
α + 1 vi
H(vi − vj ) + 1 −
α vj
!
)
H(vi − vj ) , (10)
where H(vi − vj ) is the step function.
In order to begin the analysis we will consider that v2 > v1 and in such a
case the interaction terms simplify so that
I11 = −Θ1 , I22 = −Θ2 , I12 = 0, I21 ' −(v2 − v1 )2 .
(11)
The speed variance is calculated with the distribution function given in Eq.
(7), hence
R. M. Velasco
Aggressive multi-class model
(
Iij = vi vj 1 −
α + 1 vj
α vi
!
α + 1 vi
H(vi − vj ) + 1 −
α vj
!
)
H(vi − vj ) , (10)
where H(vi − vj ) is the step function.
In order to begin the analysis we will consider that v2 > v1 and in such a
case the interaction terms simplify so that
I11 = −Θ1 , I22 = −Θ2 , I12 = 0, I21 ' −(v2 − v1 )2 .
(11)
The speed variance is calculated with the distribution function given in Eq.
(7), hence
Θi (x, t) =
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vi2
,
α
Aggressive multi-class model
(12)
a result which allows us to identify 1/α as the variance prefactor, which
according to empirical data at low vehicle densities is constant, here we
have taken α = 100 [5].
R. M. Velasco
Aggressive multi-class model
a result which allows us to identify 1/α as the variance prefactor, which
according to empirical data at low vehicle densities is constant, here we
have taken α = 100 [5].
The last ingredient to close the dynamic equations is given through the
traffic pressure, it contains the speed variance and an anticipation term
measured with the spatial derivative of the average speed, then
R. M. Velasco
Aggressive multi-class model
a result which allows us to identify 1/α as the variance prefactor, which
according to empirical data at low vehicle densities is constant, here we
have taken α = 100 [5].
The last ingredient to close the dynamic equations is given through the
traffic pressure, it contains the speed variance and an anticipation term
measured with the spatial derivative of the average speed, then
Pi =
ρi vi2
∂vi
− µi
,
α
∂x
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i = 1, 2.
Aggressive multi-class model
(13)
a result which allows us to identify 1/α as the variance prefactor, which
according to empirical data at low vehicle densities is constant, here we
have taken α = 100 [5].
The last ingredient to close the dynamic equations is given through the
traffic pressure, it contains the speed variance and an anticipation term
measured with the spatial derivative of the average speed, then
Pi =
ρi vi2
∂vi
− µi
,
α
∂x
i = 1, 2.
(13)
where µi is a constant analogous of a viscosity and it is different for each
class of drivers. It should be said that this hypothesis can be justified in
terms of the kinetic model [6, 8].
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Aggressive multi-class model
The closed set of dynamical equations
R. M. Velasco
Aggressive multi-class model
The closed set of dynamical equations
∂ρ1 ∂ρ1 v1
+
= 0,
∂t
∂x
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Aggressive multi-class model
(14)
The closed set of dynamical equations
∂ρ1 ∂ρ1 v1
+
= 0,
∂t
∂x
(14)
∂ρ2 ∂ρ2 v2
+
= 0,
∂t
∂x
(15)
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Aggressive multi-class model
The closed set of dynamical equations
∂ρ1 ∂ρ1 v1
+
= 0,
∂t
∂x
(14)
∂ρ2 ∂ρ2 v2
+
= 0,
∂t
∂x
(15)
∂v1
∂v1
1 ∂P1 v1∗ − v1
+ v1
=−
+
,
∂t
∂x
ρ1 ∂x
τ1
(16)
R. M. Velasco
Aggressive multi-class model
The closed set of dynamical equations
∂ρ1 ∂ρ1 v1
+
= 0,
∂t
∂x
(14)
∂ρ2 ∂ρ2 v2
+
= 0,
∂t
∂x
(15)
∂v1
∂v1
1 ∂P1 v1∗ − v1
+ v1
=−
+
,
∂t
∂x
ρ1 ∂x
τ1
(16)
∂v2
∂v2
1 ∂P2 v2∗ − v2
+ v2
=−
+
∂t
∂x
ρ2 ∂x
τ2
(17)
R. M. Velasco
Aggressive multi-class model
The closed set of dynamical equations
∂ρ1 ∂ρ1 v1
+
= 0,
∂t
∂x
(14)
∂ρ2 ∂ρ2 v2
+
= 0,
∂t
∂x
(15)
∂v1
∂v1
1 ∂P1 v1∗ − v1
+ v1
=−
+
,
∂t
∂x
ρ1 ∂x
τ1
(16)
∂v2
∂v2
1 ∂P2 v2∗ − v2
+ v2
=−
+
∂t
∂x
ρ2 ∂x
τ2
(17)
v1∗ = wv1 + τ1 (1 − p)ρ1 I11 + τ1 (1 − p)ρ2 I12 ,
(18)
where
R. M. Velasco
Aggressive multi-class model
The closed set of dynamical equations
∂ρ1 ∂ρ1 v1
+
= 0,
∂t
∂x
(14)
∂ρ2 ∂ρ2 v2
+
= 0,
∂t
∂x
(15)
∂v1
∂v1
1 ∂P1 v1∗ − v1
+ v1
=−
+
,
∂t
∂x
ρ1 ∂x
τ1
(16)
∂v2
∂v2
1 ∂P2 v2∗ − v2
+ v2
=−
+
∂t
∂x
ρ2 ∂x
τ2
(17)
v1∗ = wv1 + τ1 (1 − p)ρ1 I11 + τ1 (1 − p)ρ2 I12 ,
(18)
v2∗ = wv1 + τ2 (1 − p)ρ1 I21 + τ2 (1 − p)ρ2 I22 ,
(19)
where
R. M. Velasco
Aggressive multi-class model
The equilibrium state.
First of all we define some dimensionless quantities
R. M. Velasco
Aggressive multi-class model
The equilibrium state.
First of all we define some dimensionless quantities
δ=
ρe1
,
ρe2
β=
R. M. Velasco
τ1
τ2
χ=
µ1
.
µ2
Aggressive multi-class model
(20)
The equilibrium state.
First of all we define some dimensionless quantities
δ=
ρe1
,
ρe2
β=
τ1
τ2
χ=
µ1
.
µ2
(20)
The equilibrium state is obtained when we assume that the traffic state
corresponds to densities and average speed independent of time and the
spatial coordinate.
R. M. Velasco
Aggressive multi-class model
The equilibrium state.
First of all we define some dimensionless quantities
δ=
ρe1
,
ρe2
β=
τ1
τ2
χ=
µ1
.
µ2
(20)
The equilibrium state is obtained when we assume that the traffic state
corresponds to densities and average speed independent of time and the
spatial coordinate.
Let us call the variables in equilibrium as (ρe1 , ρe2 , v1e , v2e ), then the set of
equations is solved and as a result we have that
R. M. Velasco
Aggressive multi-class model
The equilibrium state.
First of all we define some dimensionless quantities
δ=
ρe1
,
ρe2
β=
τ1
τ2
χ=
µ1
.
µ2
(20)
The equilibrium state is obtained when we assume that the traffic state
corresponds to densities and average speed independent of time and the
spatial coordinate.
Let us call the variables in equilibrium as (ρe1 , ρe2 , v1e , v2e ), then the set of
equations is solved and as a result we have that
ρe1 v1e =
α(ω − 1)
,
τ1 (1 − p e )
R. M. Velasco
Aggressive multi-class model
The equilibrium state.
First of all we define some dimensionless quantities
δ=
ρe1
,
ρe2
β=
τ1
τ2
χ=
µ1
.
µ2
(20)
The equilibrium state is obtained when we assume that the traffic state
corresponds to densities and average speed independent of time and the
spatial coordinate.
Let us call the variables in equilibrium as (ρe1 , ρe2 , v1e , v2e ), then the set of
equations is solved and as a result we have that
ρe1 v1e =
α(ω − 1)
,
τ1 (1 − p e )
v2e =
v1e ±
β + 2α ±
δ
q
(β + 2α)2 − 4(1 + αδ) αδ
2(1 + αβ)
.
(21)
R. M. Velasco
Aggressive multi-class model
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
R. M. Velasco
Aggressive multi-class model
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
Also, the overpasing probability must be specified and we have taken 1−p =
ρeff , where ρeff = ρ/ρmax is the effective density seen by each class of
drivers.
R. M. Velasco
Aggressive multi-class model
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
Also, the overpasing probability must be specified and we have taken 1−p =
ρeff , where ρeff = ρ/ρmax is the effective density seen by each class of
drivers.
We consider the effective density for the slow class as the corresponding
density meaning that (ρ1 )eff = ρ1 /ρmax .
R. M. Velasco
Aggressive multi-class model
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
Also, the overpasing probability must be specified and we have taken 1−p =
ρeff , where ρeff = ρ/ρmax is the effective density seen by each class of
drivers.
We consider the effective density for the slow class as the corresponding
density meaning that (ρ1 )eff = ρ1 /ρmax . On the other hand for the fast
class (ρ2 )eff = (ρ1 + ρ2 )/ρmax [7, 9].
R. M. Velasco
Aggressive multi-class model
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
Also, the overpasing probability must be specified and we have taken 1−p =
ρeff , where ρeff = ρ/ρmax is the effective density seen by each class of
drivers.
We consider the effective density for the slow class as the corresponding
density meaning that (ρ1 )eff = ρ1 /ρmax . On the other hand for the fast
class (ρ2 )eff = (ρ1 + ρ2 )/ρmax [7, 9].
To perform the linear stability analysis we consider a small perturbation of
variables around their equilibrium values
R. M. Velasco
Aggressive multi-class model
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
Also, the overpasing probability must be specified and we have taken 1−p =
ρeff , where ρeff = ρ/ρmax is the effective density seen by each class of
drivers.
We consider the effective density for the slow class as the corresponding
density meaning that (ρ1 )eff = ρ1 /ρmax . On the other hand for the fast
class (ρ2 )eff = (ρ1 + ρ2 )/ρmax [7, 9].
To perform the linear stability analysis we consider a small perturbation of
variables around their equilibrium values
ρi = ρei + ρ̂i exp(ikx − σt),
R. M. Velasco
vi = vie + v̂i exp(ikx − σt),
Aggressive multi-class model
(22)
The stability analysis.
It should be noticed that v1∗ = v1e and an empirical fundamental diagram is
e = (dv /dρ )e a quantity
introduced, so v1e = v1e (ρ1 ), besides we define γ11
1
1
which is always negative.
Also, the overpasing probability must be specified and we have taken 1−p =
ρeff , where ρeff = ρ/ρmax is the effective density seen by each class of
drivers.
We consider the effective density for the slow class as the corresponding
density meaning that (ρ1 )eff = ρ1 /ρmax . On the other hand for the fast
class (ρ2 )eff = (ρ1 + ρ2 )/ρmax [7, 9].
To perform the linear stability analysis we consider a small perturbation of
variables around their equilibrium values
ρi = ρei + ρ̂i exp(ikx − σt),
vi = vie + v̂i exp(ikx − σt),
(22)
where the perturbation has been expanded in modes with a wave vector k
and a complex frequency called σ in such a way that the stability condition
for the equilibrium state is given by Reσ > 0.
R. M. Velasco
Aggressive multi-class model
The hypotheses we have introduced simplify the set of equations and a
dipersion relation can be obtained from the following matrix,
R. M. Velasco
Aggressive multi-class model
The hypotheses we have introduced simplify the set of equations and a
dipersion relation can be obtained from the following matrix,
 −σ + ikv e
1
γe
 (v1e )2 ik
 αρe − τ11

1
1

0


a
− τ1
2
0
0
−σ + ikv2e
ik(v2e )2
a
− τ2
αρe
2
2
ikρe1
µ
−σ + α+2
ikv1e + ρe1 k 2 + τ1
α
1
1
0
a
− τ3
2
−σ +

0



,


ikρe2
2
α+2 ikv e + µ2 k
2
α
ρe
2
where its determinant must vanish.
R. M. Velasco
0
Aggressive multi-class model
+
1−a4
τ2
(23)
The hypotheses we have introduced simplify the set of equations and a
dipersion relation can be obtained from the following matrix,
 −σ + ikv e
1
γe
 (v1e )2 ik
 αρe − τ11

1
1

0


a
− τ1
2
0
0
−σ + ikv2e
ik(v2e )2
a
− τ2
αρe
2
2
ikρe1
µ
−σ + α+2
ikv1e + ρe1 k 2 + τ1
α
1
1
0
a
− τ3
2
−σ +
0

0



,


ikρe2
2
α+2 ikv e + µ2 k
2
α
ρe
2
+
1−a4
τ2
(23)
where its determinant must vanish.
Due to the fact that the macroscopic equations are valid in a kind of hydrodynamical limit (k → 0), we will expand the roots in the dispersion
relation around k = 0 and take terms up to order k 2 ,
R. M. Velasco
Aggressive multi-class model
The hypotheses we have introduced simplify the set of equations and a
dipersion relation can be obtained from the following matrix,
 −σ + ikv e
1
γe
 (v1e )2 ik
 αρe − τ11

1
1

0


a
− τ1
2
ikρe1
0
0
−σ + ikv2e
ik(v2e )2
a
− τ2
αρe
2
2
µ
−σ + α+2
ikv1e + ρe1 k 2 + τ1
α
1
1
0
a
− τ3
2
−σ +
0

0



,


ikρe2
2
α+2 ikv e + µ2 k
2
α
ρe
2
+
1−a4
τ2
(23)
where its determinant must vanish.
Due to the fact that the macroscopic equations are valid in a kind of hydrodynamical limit (k → 0), we will expand the roots in the dispersion
relation around k = 0 and take terms up to order k 2 ,
σ = σ0 + kσ1 + k 2 σ2 + O(k 3 ),
and there will be four different roots, which will be called as Σi .
R. M. Velasco
Aggressive multi-class model
(24)
Σ1 = ikc1 +
Σ2 =
1
τ1
+
ik
α
e
e
e
e
(2v1 + αv1 − αγ11 ρ1 ) +
Σ3 =
ik
a4 − 1
Σ32 = −
1 − a4
k 2 τ1 2
e e 2
c1 − (α + 1)(γ11 ρ1 )
αρe1
(25)
i
k2 h
e
e 2
e e e
e e 2
αµ1 − τ1 ρ1 (v1 ) + 2ρ1 v1 γ11 − α(ρ1 γ11 )
+ ...
αρe1
(26)
e
e
2
(a4 − 1)v2 − a2 ρ2 + k Σ32
"
τ2
α(a4 −
1)3
#
2 e 2
e
e
e
(a4 − 1) (v2 ) − a2 ρ2 2v2 (a4 − 1) + αa2 ρ2 ,
i
h
ik
2
e
e
(a4 − 1)v2 (2 + α) + αa2 ρ2 + k Σ42
α(a4 − 1)
i
h
h
1
e
3
e
e 2
e
e e
Σ42 =
(a4 − 1)ρ2 v2 τ2 (a4 − 1)v2 − 2a2 ρ2 + α (a4 − 1) µ2 − ρ2 τ2 (a2 ρ2 )
(a4 − 1)3 αρe2
Σ4 =
τ2
(27)
+
R. M. Velasco
Aggressive multi-class model
(28)
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
R. M. Velasco
Aggressive multi-class model
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
First of all the root Σ1 shows us that the mode associated with the density
in class 1 propagates with speed
R. M. Velasco
Aggressive multi-class model
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
First of all the root Σ1 shows us that the mode associated with the density
in class 1 propagates with speed
e e
c1 = v1e + γ11
ρ1 ,
R. M. Velasco
Aggressive multi-class model
(29)
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
First of all the root Σ1 shows us that the mode associated with the density
in class 1 propagates with speed
e e
c1 = v1e + γ11
ρ1 ,
and when its real part is positive, it determines stability in this mode
R. M. Velasco
Aggressive multi-class model
(29)
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
First of all the root Σ1 shows us that the mode associated with the density
in class 1 propagates with speed
e e
c1 = v1e + γ11
ρ1 ,
(29)
and when its real part is positive, it determines stability in this mode
e ρ2 i
Σ1
k 2 h c1
γ11
1
Re
=
−
(α
+
1)
.
(30)
τ1 (v1e )2
αρe1 v1e
v1e
R. M. Velasco
Aggressive multi-class model
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
First of all the root Σ1 shows us that the mode associated with the density
in class 1 propagates with speed
e e
c1 = v1e + γ11
ρ1 ,
(29)
and when its real part is positive, it determines stability in this mode
e ρ2 i
Σ1
k 2 h c1
γ11
1
Re
=
−
(α
+
1)
.
(30)
τ1 (v1e )2
αρe1 v1e
v1e
In this case the fundamental diagram chosen for v1e determines its behavior
and the corresponding stability property.
R. M. Velasco
Aggressive multi-class model
The results given show four time scales in the problem (red terms), all of
them must be positive in order to obtain a stable solution for the equilibrium
state.
First of all the root Σ1 shows us that the mode associated with the density
in class 1 propagates with speed
e e
c1 = v1e + γ11
ρ1 ,
(29)
and when its real part is positive, it determines stability in this mode
e ρ2 i
Σ1
k 2 h c1
γ11
1
Re
=
−
(α
+
1)
.
(30)
τ1 (v1e )2
αρe1 v1e
v1e
In this case the fundamental diagram chosen for v1e determines its behavior
and the corresponding stability property. If we take the Greenshields diagram
we obtain a stability region for ρe1 < 43 veh/km, which means that we are
at low density.
R. M. Velasco
Aggressive multi-class model
On the other hand, the time scale in the leading term of root Σ2 is 1/τ1
which is obviously positive.
R. M. Velasco
Aggressive multi-class model
On the other hand, the time scale in the leading term of root Σ2 is 1/τ1
which is obviously positive.
It means that modes in class 1 are stable up to certain density fixed by the
fundamental diagram chosen.
R. M. Velasco
Aggressive multi-class model
On the other hand, the time scale in the leading term of root Σ2 is 1/τ1
which is obviously positive.
It means that modes in class 1 are stable up to certain density fixed by the
fundamental diagram chosen.
4
Concerning class 2 we have two time scales 1−a
τ2 which is positive according
to 1 − a4 .
R. M. Velasco
Aggressive multi-class model
On the other hand, the time scale in the leading term of root Σ2 is 1/τ1
which is obviously positive.
It means that modes in class 1 are stable up to certain density fixed by the
fundamental diagram chosen.
4
Concerning class 2 we have two time scales 1−a
τ2 which is positive according
to 1 − a4 .
2α 1 − a4
=
ζ + δ(ζ − 1/δ) − 1,
(31)
ω−1
δζ
R. M. Velasco
Aggressive multi-class model
On the other hand, the time scale in the leading term of root Σ2 is 1/τ1
which is obviously positive.
It means that modes in class 1 are stable up to certain density fixed by the
fundamental diagram chosen.
4
Concerning class 2 we have two time scales 1−a
τ2 which is positive according
to 1 − a4 .
2α 1 − a4
=
ζ + δ(ζ − 1/δ) − 1,
(31)
ω−1
δζ
R. M. Velasco
Aggressive multi-class model
Lastly, the mode corresponding to the class 2 density is also a propagating
mode, its speed being
i
1 h
c2
2 2
2
=
1
+
(δ
+
2)ζ
δ
+
α(ζδ
−
1)
,
(32)
v2e
βδ 2 ζ
R. M. Velasco
Aggressive multi-class model
Lastly, the mode corresponding to the class 2 density is also a propagating
mode, its speed being
i
1 h
c2
2 2
2
=
1
+
(δ
+
2)ζ
δ
+
α(ζδ
−
1)
,
(32)
v2e
βδ 2 ζ
And the real part in Σ32 must be positive to obtain stability in the mode
corresponding to the density in class 2.
R. M. Velasco
Aggressive multi-class model
2
References
[1]
Helbing, D.; Traffic and related self-driven many-particle systems Rev. Mod. Phys. 1067, (2001)
[2]
Kerner, B. S.; Introduction to Modern Traffic Flow Theory and Control, Springer (2009).
[3]
Treiber, M. and Kesting, A.; Traffic Flow Dynamics Springer (2013).
[4]
Marques Jr., W. and Méndez, A. R.; On the kinetic theory of vehicular traffic flow: Chapman-Enskog expansion versus
Grad’s moment method Physica A 392 3430-3440 (2013).
[5]
Shvetsov, V., Helbing, D.; Macroscopic dynamics of multiline traffic Phys. Rev. E 59, 6328 (1999).
[6]
Méndez, A. R. and Velasco, R. M.; Kerner’s free-synchronized phase transition in a macroscopic traffic flow model with
two classes of drivers J. Phys. A: Math. Theor. (FTC) 46, 462001 (2013).
[7]
Hoogendoorn, S., Bovy, P.; A macroscopic model for multiple user class traffic operations: derivation, analysis and
numerical results, Report VK 2205.328 (1998).
[8]
Velasco, R. M., Marques-Jr., W.; Navier-Stokes-like equations for traffic flow, Phys. Rev. E 72, 046102, (2005).
[9]
Van Wageningen-Kessels , F.; Multi-class continuum traffic models: Analysis and simulation methods, PhD thesis
TU-Delft (2013).
[10]
Van Wageningen-Kessels, F.; Anisotropy in generic multi-class traffic models, Transportmetrica A, 9, 451 (2013).
R. M. Velasco
Aggressive multi-class model
R. M. Velasco
Aggressive multi-class model