g. puppo, a 2-populations kinetic model for …...a 2-populations kinetic model for vehicular tra c...

A2-populationskinetic modelfor vehicular

traffic

G. Puppo,M. Semplice,G. Visconti

Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

A 2-populations kinetic modelfor vehicular traffic

G. Puppo1 M. Semplice2 G. Visconti1

1Dipartimento di Scienza ed Alta TecnologiaUniversita degli Studi dell’Insubria

2Dipartimento di MatematicaUniversita degli Studi di Torino

SIMAI 2014Taormina, 7-10 July 2014

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Fundamental DiagramFundamental diagram as a basic tool to study traffic flow:

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 330

q

ρ

Figure: Fundamental diagram based on experimental data referring to one-weektraffic flow in viale del Muro Torto, Rome, Italy.

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuummathematical models. In Encyclopedia of Complexity and Systems Science,volume 22, pages 9727-9749. Springer, New York, 2009.

Phase transition

Free phase

the flow of vehiclesincreases linearly

Congested phase

the flow values arescattered

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Our aim

stochastic behaviour of drivers

⇓ ?

scattered experimental measures

⇑ ?

interactions between vehicles with heterogeneous features

⇒ traffic as a mixture of two classes of vehicles, e.g.:

class of cars: faster and shorterclass of trucks: slower and longer

Why kinetic?

X microscopic scale: not computationally competitive;√mesoscopic scale: a simple approach to recover themacroscopic quantities and to model the microscopicinteractions;

X macroscopic scale: requires a closure law.

3 / 19


traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Our aim

stochastic behaviour of drivers

⇓ ?

scattered experimental measures

⇑ ?

interactions between vehicles with heterogeneous features

⇒ traffic as a mixture of two classes of vehicles, e.g.:

class of cars: faster and shorterclass of trucks: slower and longer

Why kinetic?

X microscopic scale: not computationally competitive;√mesoscopic scale: a simple approach to recover themacroscopic quantities and to model the microscopicinteractions;

X macroscopic scale: requires a closure law.3 / 19


traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

1-population kinetic modeldiscrete velocity kinetic framework: V = {v1, . . . , vn},where v1 = 0 and vn is the maximum speed;

spatially homogeneous problem: the experimentaldiagrams are constructed by assuming that the traffic flowis stationary and homogeneous in space;

fj = fj(t) is the distribution function of vehicles travelingat speed vj;

Ajh,k is the probability that a vehicle changes its speed vhin a new speed vj after an interaction with a vehicle withspeed vk.

f ′j(t) =

n∑h,k=1

fh(t)fk(t)Ajh,k − fj(t)

n∑k=1

fk(t), j = 1, . . . , n

L. Fermo and A. Tosin, A fully-discrete-state theory approach tomodeling vehicular traffic. SIAM J. Appl. Math., 73(4):1533-1556,2013. 4 / 19


traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Critical considerations

The model provides a single-valued curve, thus it does notjustify the large dispersion of the flow in the congested phase.

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

2-populations kinetic modelWe generalize the model to two classes of vehicles, thus wedefine:

Vc = {v1, . . . , vn} ⊃ Vt = {v1, . . . , vm}, velocity space of carsand trucks, with m < n;

fj = fj(t), gj = gj(t) distribution functions of cars andtrucks with j-th speed class;

f(t, v) =∑nj=1 fj(t)δvj (v), g(t, v) =

∑mj=1 gj(t)δvj (v) kinetic

distribution functions of cars and trucks;

lc, lt average length of cars and trucks, L length of road;

Nc = L∑nj=1 fj , Nt = L

∑nj=1 gj vehicle number on the

road;

Therefore NclcL ∈ [0, 1], Ntlt

L ∈ [0, 1] and the total occupiedspace is:

0 ≤ s =Nclc +Ntlt

L≤ 1

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

2-populations kinetic model

dfjdt

= Qj(f, f)︸︷︷︸cars-cars

+Qj(f, g)︸︷︷︸cars-trucks

, j = 1, . . . , n (1)

dgjdt

= Qj(g, g)︸︷︷︸trucks-trucks

+Qj(f, g)︸︷︷︸trucks-cars

, j = 1, . . . ,m (2)

where:

Qj(f, f) =∑nh,k=1 fhfkA

jh,k − fj

∑nk=1 fk, j = 1, . . . , n

Qj(f, g) =∑nh=1

∑mk=1 fhgkB

jh,k − fj

∑mk=1 gk, j = 1, . . . , n

Qj(g, g) =∑mh,k=1 ghgkC

jh,k − gj

∑mk=1 gk, j = 1, . . . , m

Qj(g, f) =∑mh=1

∑nk=1 ghfkD

jh,k − gj

∑nk=1 fk, j = 1, . . . , m

and for any fixed j, Ajh,k ∈ Rn×n, Bj

h,k ∈ Rn×m, Cjh,k ∈ Rm×m,

Djh,k ∈ Rm×n, give the probability transition between two microscopic

states (i.e. speed classes).

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Rules of interactionvh = speed class of the candidate vehicle;vk = speed class of the field vehicle;

vh ≤ vk

vh vkvh+1

1− s

s

vh > vk

vhvk s 1− s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions


vh ≤ vk

vh vk

vh+1

1− s

s

vh > vk

vhvk s 1− s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions


vh ≤ vk

vh vkvh+1

1− s

s

vh > vk

vhvk s 1− s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions


vh ≤ vk

vh vkvh+1

1− s

s

vh > vk

vhvk

s 1− s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions


vh ≤ vk

vh vkvh+1

1− s

s

vh > vk

vhvk s

1− s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions


vh ≤ vk

vh vkvh+1

1− s

s

vh > vk

vhvk s 1− s8 / 19


traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Table of games - Aj

Ajh,k = P(vh → vj/vk), with vh, vj , vk ∈ Vc:

vh < vk :{

Ahh,k = 1− α(1− s)

Ah+1h,k = α(1− s)

vh > vk :{

Akh,k = 1− α(1− s)

Ahh,k = α(1− s)

vh = vk :

h = k = 1

A11,1 = 1− α(1− s)

A21,1 = α(1− s)

1 < h = k < n

A

h−1h,h = (1− α)s

Ahh,h = 1− α− (1− 2α)s

Ah+1h,h = α(1− s)

h = k = n

An−1n,n = (1− α)s

Ann,n = 1− (1− α)s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Table of games - Cj

Cjh,k = P(vh → vj/vk), with vh, vj , vk ∈ Vt:

vh < vk :{

Chh,k = 1− α(1− s)

Ch+1h,k = α(1− s)

vh > vk :{

Ckh,k = 1− α(1− s)

Chh,k = α(1− s)

vh = vk :

h = k = 1

C11,1 = 1− α(1− s)

C21,1 = α(1− s)

1 < h = k < m

C

h−1h,h = (1− α)s

Chh,h = 1− α− (1− 2α)s

Ch+1h,h = α(1− s)

h = k = m

Cm−1m,m = (1− α)s

Cmm,m = 1− (1− α)s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Table of games - Bj

Bjh,k = P(vh → vj/vk), with vh, vj ∈ Vc and vk ∈ Vt:

vh < vk :{

Bhh,k = 1− α(1− s)

Bh+1h,k = α(1− s)

vh > vk :{

Bkh,k = 1− α(1− s)

Bhh,k = α(1− s)

vh = vk :

h = k = 1

B11,1 = 1− α(1− s)

B21,1 = α(1− s)

1 < h = k ≤ m < n

B

h−1h,h = (1− α)s

Bhh,h = 1− α− (1− 2α)s

Bh+1h,h = α(1− s)

h = k = m = n

Bn−1n,n = (1− α)s

Bnn,n = 1− (1− α)s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Table of games - Dj

Djh,k = P(vh → vj/vk), with vh, vj ∈ Vt and vk ∈ Vc:

vh < vk :

h = m

{D

hh,k = 1

h 6= m

Dhh,k = 1− α(1− s)

Dh+1h,k = α(1− s)

vh > vk :{

Dkh,k = 1− α(1− s)

Dhh,k = α(1− s)

vh = vk :

h = k = 1

D11,1 = 1− α(1− s)

D21,1 = α(1− s)

1 < h = k ≤ m

D

h−1h,h = (1− α)s

Dhh,h = 1− α− (1− 2α)s

Dh+1h,h = α(1− s)

h = k = m

Dm−1m,m = (1− α)s

Dmm,m = 1− (1− α)s

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Macroscopic variablesMacroscopic quantities resulting from 2-populations model:

Density:

ρc(t) =∑nj=1 fj(t) ρt(t) =

∑mj=1 gj(t)

Flow:

qc(t) =∑nj=1 vjfj(t) qt(t) =

∑mj=1 vjgj(t)

Speed:

uc(t) = qc(t)ρc(t)

ut(t) = qt(t)ρt(t)

Construction of fundamental diagrams

1 fix s ∈ [0, 1];2 choose ρc(0), ρt(0) such that ρc(0)lc + ρt(0)lt = s;

3 look at asymptotic distributions fej , j = 1, . . . , n and

gej , j = 1, . . . ,m;

4 define the total flow as q = qc + qt.

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Properties∑nj=1A

jh,k =

∑nj=1B

jh,k =

∑mj=1 C

jh,k =

∑mj=1D

jh,k = 1∀h, k;

conservation of mass: dρc

dt = 0 and dρt

dt = 0;

the equilibria do not depend on the initial conditionsfj(0), gj(0);

the flow rate depends on ρc, ρt and thus for any given s ∈ [0, 1]there are different values of qc + qt;

indifferentiability principle if lc = lt and Vc = Vt: the1-population model and 2-populations one are consistent withFj = fj + gj .

P. Andries and K. Aoki and B. Perthame, A consistent BGK-typemodel for gas mixtures. Technical report, Institut National DeRecherche En Informatique Et En Automatique, 2001. Rapportde recherche n◦4230

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Fundamental diagrams - density

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Other fundamental diagrams

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Achievements and applications

the scattered data in the congested phase:

1 depend on the interactions between vehicles whichgenerate the heterogeneous composition of traffic;

2 do not depend on the stochastic behavior of drivers.

in order to study traffic phenomena, one can investigatedifferent quantities related to the flow:

1 q = Pc∑nk=1 vkfk + Pb

∑mk=1 vkgk

where Pc, Pb are the number of transportable people oncars and buses;

2 q = Mv(lv, hv)∑nk=1 vkfk +Mc(lc, hc)

∑mk=1 vkgk

where Mv,Mc are the transportable mass on vans andtrucks.

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Kinetic is essential

2-populations macroscopic model:

S. Benzoni-Gavage and R.M. Colombo. An n-populations model fortraffic flow. European Journal of Applied Mathematics,14(05):587-612, 2003.

The diagram is conditioned by the closure law of the hyperbolicsystem:

qi(ρ) = Vi(1− ρ1 − ρ2)ρi, i = 1, 2

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traffic


Introduction

1-population

2-populations

Fundamentaldiagrams

Conclusions

Perspectives

Further analysis can be developed to analyze the followingaspects:

1 two vehicle classes in other kinetic traffic model;

Wegener, R. and Klar, A.. A kinetic model for vehicular trafficderived from a stochastic microscopic model. Transport Theoryand Statistical Physics, 25:785-798, 1996.

2 a 2-populations kinetic model for multilane road;

Bonzani, I. and Gramani Cumin, L.M.. Modelling andsimulations of multilane traffic flow by kinetic theory methods.Computers and Mathematics with Applications, 56:2418-2428,2009.

3 a 2-populations kinetic model with continuous velocityspace: spatially homogeneous and inhomogeneous case.

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g. puppo, a 2-populations kinetic model for …...a 2-populations kinetic model for vehicular tra c...

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