TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 11, Novembe
r
2014, pp. 77
3
8
~ 775
7
DOI: 10.115
9
1
/telkomni
ka.
v
12i11.61
28
7738
Re
cei
v
ed Ap
ril 24, 2014; Revi
sed
Jul
y
1, 2014; Accept
ed Jul
y
25, 2
014
Review of the Urban Traffic Modeling
Zhu Song*
1
, Zhiguang Qi
n
2
Univers
i
t
y
of Electronic Sci
enc
e and T
e
chno
l
o
g
y
of Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: toni11
0@1
6
3
.
com
1
, zgqin@
uestc.edu.c
n
2
A
b
st
r
a
ct
Now
adays, th
e urba
n traffic mod
e
li
ng, w
h
ich is
hel
pful i
n
pla
nni
ng a
n
d
control
lin
g the traffic
system
, has becoming
a r
e
searc
h
hots
p
ot of tra
ffic engineer
ing.
After dec
ades
of researc
h
and
deve
l
op
ment, there now
exist
s
hundre
d
s of mo
de
ls c
hoos
i
ng differe
nt mode
lin
g metho
d
s to simul
a
te the
traffic flow
. It is importa
nt for us to un
derst
and th
ese
mo
dels
by class
i
fying th
e
m
an
d
ana
ly
z
i
n
g
the
i
r
features. The features of traffi
c
mod
e
ls, inc
l
u
d
in
g the scal
a
b
ility
,
accuracy a
nd co
mp
utab
ilit
y
,
are beco
m
i
n
g
importa
nt indic
a
tors to meas
ure their perf
o
rmanc
e.
In this pa
per
, w
e
introduc
e an
d compar
e so
me
grou
nde
d
mo
d
e
ls. In p
a
rticul
ar
, w
e
analy
z
e
the a
d
v
anta
g
e
s an
d d
i
sadv
antag
es
of
ex
i
s
ting mo
dels, an
d
classify them
into three
cate
gori
e
s acc
o
rdi
ng the
i
r gr
an
ul
arity:
macrosc
opic, mes
o
sco
pic
a
nd micros
copi
c
mo
de
ls
.
Ke
y
w
ords
:
macroscopic models, mesoscopi
c models, m
i
croscopic m
odels,
surv
ey, car-following theory,
cellu
lar a
u
to
ma
ton mode
ls, ga
s-kinetic the
o
ry
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
With the increasi
ng num
b
e
r of vehicle
s
, t
he urban traffic system face
s many p
r
oble
m
s
one of
which
is traffic
co
nge
stion be
coming m
o
re
seri
ou
s day
after day. Th
e urb
an traffic
modeling is
helpful for mitigating traffic
c
o
nges
ti
on, b
e
ca
use it allo
ws u
s
to b
e
tter un
derstan
d,
plan, d
e
si
gn
and
optimize
the traffic sy
stem. The
r
e
e
x
ist hun
dre
d
s of urban
traff
i
c mo
del
s, wh
ich
cho
o
se different kind
s of method
s, su
ch as
probabili
ty and statistics, differe
ntia
l equation
s
a
n
d
nume
r
ical method
s. It is nece
s
sary
to classi
fy these traffic model
s for
comp
ari
ng their
advantag
es a
nd disadvant
age
s.
Acco
rdi
ng to the model granula
r
ity, which is
the leve
l of detail con
s
ide
r
ed in the
model,
we cla
s
sify traffic model
s into three
catego
rie
s
, macroscopi
c, meso
scop
ic (hybri
d) a
n
d
microsco
pic (sub
-mi
c
ro
sco
p
ic) m
odel
s.
Macro
s
copi
c model
s
view all
vehicl
es a
s
a w
hol
e, a
nd stu
d
y the
cha
r
a
c
teri
stics of the
entire traffic
flow. In parti
cula
r, they measure
the
variation of
traffic
flow parameters
, whic
h
inclu
de flo
w
rate, velocity
and d
e
n
s
ity, and a
nalyze
the rel
a
tion
sh
ip amo
ng the
s
e p
a
ramete
rs.
Although th
e
s
e m
odel
s
can de
scri
be t
he vari
ation
of som
e
traffi
c ph
eno
men
a
(e.g., the
stop
-
and-go wave
), they canno
t explain the
formati
on of these phen
o
m
ena du
e to the ignoran
ce o
f
the individ
ual
vehicl
e’s be
havior.
We
d
i
vide
ma
croscopi
c m
odel
s into two
cat
egori
e
s, tim
e
-
indep
ende
nt static mo
del
s and time-rel
ated dynami
c
mod
e
ls, a
c
cording to th
e co
rrel
a
tion
of
time. The tim
e
-related
dyn
a
mic mod
e
ls
con
s
id
er
th
e effect
of spa
c
e and
time correl
ation,
a
n
d
thus, they
co
uld be
mo
re
reali
s
tic th
an
time
-ind
epe
ndent static one
s
in som
e
ca
se
s.
Typ
i
cal
time-rel
a
ted dynamic
mo
dels (e.g.,
L
W
R mod
e
)
a
pply the fluid
dynami
cs,
whi
c
h i
s
a th
eory
of
fluid mech
anics
that de
als
with the
n
a
tural
scie
n
c
e of fluids
(li
q
uids a
nd ga
ses)
in
motio
n
,
to
cha
r
a
c
teri
ze t
he variation o
f
the traffic flow.
Meso
scopi
c
model
s a
s
su
me a set of
nearby vehicl
es a
s
a unit,
a so-call
ed “platoon
”,
and d
e
scribe
the inflo
w
a
nd outflo
w
o
f
each pl
ato
on. Spe
c
ifica
lly, these m
o
dels study th
e
comm
on be
h
a
vior of vehi
cle
s
in a
sa
me platoo
n. We g
r
ou
p m
e
so
scopi
c m
odel
s into two
categ
o
rie
s
: g
a
s-kin
e
tic m
o
dels
and
hybrid m
odel
s.
The first ga
s-kineti
c
mo
d
e
l pro
p
o
s
ed
by
Prigogi
ne an
d He
rman
ap
plies the
ga
s-dynamics, wh
ic
h is
a law t
hat explain
s
the beh
avior
of a
hypothetical ideal ga
s, to descr
ib
e the platoon. Hybri
d
models u
s
ua
lly combine di
fferent model
s
(e.g., a micro
s
copi
c model
mixed with a meso
scopi
c model
) to combine thei
r advantag
es a
nd
reme
dy their disa
dvantag
e
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
vie
w
of the Urba
n Traffi
c Modeling
(Zh
u
Song)
7739
Microsco
pic
model
s fo
cu
s on the
be
ha
vior
of in
dividual vehi
cle
s
, and
study
how one
vehicle dyn
a
m
ically interacts
with an
other.
Th
ese
model
s atte
mpt to describe the
overall
cha
r
a
c
teri
stics of th
e
system by i
n
te
grating
t
he
cha
r
a
c
teri
stic of ea
ch
in
dividual vehi
cle.
Microsco
pic
model
s have
three ca
te
g
o
rie
s
: ca
r-foll
o
win
g
model
s, cellula
r au
tomaton mod
e
ls
(pa
r
ticle h
opi
ng mod
e
ls) a
nd sub-mi
cro
s
copi
c mo
de
ls
. C
a
r
-
f
o
llow
i
n
g
mo
de
ls
ana
lyz
e
th
e
ve
hic
l
e
followin
g
beh
avior in on
e l
ane. Cell
ula
r
automaton
m
odel
s view in
dividual vehi
cles a
s
self-d
ri
ven
particl
es, whi
c
h is a coll
ect
i
on of particle
s
re
spo
nd to a rand
om perturbati
on by the motion of the
other
nea
rby
parti
cle
s
. Compa
r
ed
wit
h
othe
r
two kind
s of
mo
dels, su
b-mi
crosco
pic
mo
dels
descri
be
more detail
s
,
su
ch a
s
d
r
iver’
s
psycholo
g
ic
al
re
action
s,
re
spo
n
se to th
e traffic an
d
ca
r
lights, etc.
This
pape
r
cl
assifies t
r
affic mo
dels ma
inly in
model
gra
nula
r
ity. And we
re
sp
ectively
introdu
ce
so
me impo
rtant
model
s of each type
a
nd su
mma
rize the ch
ara
c
teristic
of these
model
s in
se
ction ma
cro
s
copi
c mo
del
s, meso
sc
opi
c model
s an
d
microsco
pic model
s. At th
e
end of ea
ch
sectio
n we
list the co
mpari
s
o
n
of the advantage, di
sadva
n
tage, appli
c
able
environ
ment
and modeli
ng method
s of the most
importa
nt models. B
e
sid
e
s, there’s a
con
c
lu
sio
n
ab
out cha
r
a
c
teri
stics of exis
ting model
s at the end of this arti
cle.
2. Macros
copic
Models
Macro
s
copi
c model
s co
nsi
der traffic flo
w
as
an e
n
tirety and they
do not ca
re a
bout the
behavio
r of i
ndividual ve
h
i
cle
s
. The
s
e
model
s
cont
a
i
n stati
c
ma
croscopi
c m
o
d
e
ls
and
dyna
mic
macro
s
copi
c model
s. The stand
ard
static model
s
incl
ude the re
cu
rsive mod
e
l, the start-arrive
model an
d the start-de
stin
ation model.
The dynam
i
c
models
cont
ain the first-o
r
de
r co
ntinuu
m
model (e.g., the LWR mo
d
e
l), and
se
co
nd-o
r
d
e
r
cont
inuum mo
del
s su
ch li
ke th
e Payne mod
e
l
and the Papa
georgiou m
o
d
e
l. The cla
ssif
i
cation of
ma
cro
s
copi
c mo
dels i
s
sh
own
in Figure 1.
Figure 1. The
Classificatio
n
Figure of Macrosco
pic M
odel
s
2.1.
Static Ma
cro
scopic Mod
e
ls
Static macro
s
copi
c mod
e
l
s
re
se
arch o
n
the time-in
depe
nden
ce
relation
shi
p
among
traffic param
eters
su
ch a
s
traffic
flow
veloc
i
ty
()
vx
, flo
w
rate
()
qx
and den
sity
()
x
at th
e
traffic
flow loc
a
tion
x
. There are three i
m
porta
nt Static
ma
crosco
pic m
odel
s: t
he recursive
model, the st
art-a
rrival mo
del and the
start-d
e
stin
atio
n model [1].
The re
cursi
ve model
:
The
re
cursiv
e mod
e
l aim
s
to fi
gure
out the traffic
flow rate
throug
h the
calcul
ation of t
he traffic fl
o
w
rate at e
a
ch
se
ction
re
cu
rsively.
The model
u
n
iformly
divides
the traffic
flow into
N
se
ction
s
a
n
d
let ea
ch
se
ct
ion i o
w
n
s
at most o
ne
ent
ran
c
e
i
r
and
one exit
i
s
. Let
1
i
q
and
i
q
re
spe
c
tively be the traffic inflow
ra
te and the
ou
tflow rate
of sectio
n
i
. By Equation (1
), we ca
n co
mpute
i
q
according
to
paramete
r
s
1
i
q
,
i
r
and
i
s
. As
the
model’
s
nam
e implie
s, we
can
cal
c
ulat
e the traffic fl
ow rate of an
y section
re
cursively fro
m
the
flow rate of previous o
n
e
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
38 – 775
7
7740
1
;1
,
2
,
,
ii
i
i
qq
r
s
i
N
(
1
)
The
schemati
c
dia
g
ram of
the re
cu
rsive mod
e
l
is
shown in Fi
gu
re 2. Se
ction
1 is th
e
begin
n
ing
se
ction of the traffic flow, an
d
0
q
is the i
n
itial flow rate. Ea
ch
se
ction i
s
see
n
unifo
rml
y
whi
c
h
contai
n
s
at m
o
st o
n
e
entra
nce an
d one
exit. The outp
u
t
i
q
of se
ction
i
is th
e input
of its
forwa
r
d se
ction
1
i
.
0
q
1
q
1
i
q
i
q
1
N
q
N
q
1
r
N
s
N
r
i
r
i
s
1
s
Figure 2. Sch
e
matic Di
ag
ram of the Re
cursive Mo
de
l
The Star
t-a
rr
ive model:
The start
-
arriv
e
model atte
mpts to cou
n
t
the traffic flow rate
throug
h the
a
rrival flo
w
rate of ea
ch
se
ction.
Th
e schematic diag
ram of this
m
odel i
s
the
sa
me
as th
e
re
cu
rsive m
odel.
The
differen
c
e is that
th
e
sta
r
t-a
rrive
model
define
a
pro
p
o
r
tional
variable
ij
a
a
s
the traffic flo
w
rate of
se
ction j which e
n
tered
from
entran
c
e
i
r
. Then we can
cal
c
ulate e
a
ch se
ction’
s flow rate a
c
cord
ing to form (2
):
1
;1
,
2
,
j
ji
i
j
i
qr
a
j
…,
N
,
,,
1
,
1
,
i
N
i
N
ii
ii
aa
a
a
(0
1
)
(2)
To cal
c
ulate t
he traffic flow rate at each
se
ction, we i
m
port the
NN
order sta
r
t-a
rriv
e
matrix of
ij
a
:
1,
1
1
,
2
1,
2,2
2
,
,
0
00
N
N
NN
aa
a
aa
a
A=
(
3
)
Then
we
set
the traffic fl
ow ve
ctor
12
,,
N
qq
q
q
，
and the
entrance traffic fl
ow
vec
t
or
12
,,
,
N
rr
r
r
.
And we s
i
mplify the matrix
form as
:
qr
A
(
4
)
That is to say
the traffic flow rate at ea
ch se
ction can
be figure
d
o
u
t from matri
x
A and
the entran
c
e f
l
ow vecto
r
r
.
The s
t
ar
t-de
stina
t
ion m
odel:
T
he
start-de
stinatio
n mod
e
l i
s
use
d
o
n
a
spe
c
ific
con
d
ition th
at the
road
is closin
g at th
e
end. Thi
s
mo
del d
e
fine
s a
pro
p
o
r
tional
variable
ij
b
the
rate of the exit traffic
flow at
i
s
which e
n
t
ered fro
m
e
n
tran
ce
i
r
. Then we can
cal
c
ulate e
a
ch
s
e
c
t
ion’s
exit flow rate in from (5):
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
vie
w
of the Urba
n Traffi
c Modeling
(Zh
u
Song)
7741
1
;1
,
2
,
,
j
ji
i
j
i
s
rb
j
N
(
5
)
And all vehicl
es exit at the end of se
ctio
n
N
, so we hav
e:
1;
1
,
2
,
N
ij
ji
bj
N
(
6
)
Then
we al
so impo
rt the
NN
orde
r
start-d
e
stinatio
n Ma
trix of
ij
b
in the form a
s
follow
：
Sr
B
(
7
)
We can al
so
expre
ss
ij
a
in Formul
a (8
):
1
;1
,
2
,
N
ij
ik
kj
ab
i
N
(
8
)
Static ma
croscopi
c m
odel
s are
con
s
tant
co
efficient
m
odel
s. The
u
s
age
of the
s
e
model
s
is limited
p
r
a
c
tically
be
cau
s
e th
ey can
n
o
t make
p
r
e
d
i
c
tion
s for a
c
cidental
event
s. Neverthel
e
ss,
studying on
static macroscopic models i
s
still me
aningful due to the un
stable m
easurement i
n
dynamic mo
d
e
l. In that case, only the mean val
ue
of a short pe
riod is valua
b
le which usu
a
lly
fluctuate st
ro
ngly.
2.2.
D
y
namic Ma
crosco
pic M
odels
Dynami
c
ma
crosco
pic
mod
e
ls mai
n
ly de
scribe
th
e
sp
atio-temp
o
ral
asso
ciation rules of
the traffic flow features, in
cludi
ng traffic flow ra
te, velocity and de
n
s
ity. The theoretical ba
sis of
dynamic
ma
cro
s
copi
c m
odel
s is th
e
fluid dy
nam
ics
model,
whi
c
h i
s
al
so kn
own a
s
the
contin
uum m
odel of traffic flow. Such m
odel
s con
s
id
e
r
traffic flow a
s
a com
p
re
ssible fluid form
ed
by a la
rge
nu
mber of vehi
cles
and
do
no
t menti
on th
e
individual
be
havior
of the
s
e vehi
cle
s
. We
can divid
e
the dynami
c
micro
s
copi
c m
odel
s into
two categ
o
rie
s
[2]. One cate
gory is the first-
orde
r
continu
u
m mod
e
ls
contain relatio
n
s b
e
twe
en
the traffic
flow veloc
i
ty
-de
n
s
ity or flow
rate-
den
sity. The
other
cate
gory is the
se
co
nd-o
r
d
e
r
cont
inuum m
odel
s contain
add
itional rel
a
xation
time to a
dapt
the velo
city
of vehicl
es wi
th
the
su
rro
u
nding
on
es.
The m
a
jo
r dif
f
eren
ce
bet
ween
these
2 categ
o
rie
s
is th
at wheth
e
r the
model
c
ontai
ns in
ertia term. It make th
ese t
w
o
cate
gorie
s
no differen
c
e
if the time consta
nt of th
e ine
r
tia term
is
set to
zero, whi
c
h m
e
a
n
s vehi
cle
s
can
instanta
neo
u
s
ch
ang
e thei
r velocity.
2.2.1.
The First-or
der Con
t
inu
u
m Model
The
rep
r
e
s
en
tative model
of the first-o
r
der
co
ntinuu
m model
introdu
ced i
n
thi
s
p
ape
r is
the LWR model proposed by
Lighthill and Whitham.
LWR model
:
Lighthill a
n
d
Whitha
m establi
s
h
ed their traffic
model
with the one
-
dimen
s
ion
a
l kineti
c
theo
ry of traffic flow in
19
5
5
. They cho
o
se the
prin
ciple of ma
ss
con
s
e
r
vation
of fluid dyna
mics in traffic flow an
d then form their f
i
rs
t-
ord
e
r ma
cro
s
copi
c
t
r
af
f
i
c
flow m
odel.
T
hey let
as the traffic flow dens
ity,
q
as the traffic flow rate,
t
a
s
tim
e
variable,
x
as th
e
spati
a
l displa
cem
ent of traffic
flow. Thu
s
,
(,
)
sx
t
is
the t
r
affic
flow generation rate.
There are three cases: the
first is
(,
)
0
sx
t
, whi
c
h indi
cate
s th
e co
nservatio
n
of the flow
rate;
the se
co
nd i
s
(,
)
0
sx
t
, whic
h means
the inlet tr
affic
flow; the las
t
is
(,
)
0
sx
t
, which means
the outlet traffic flow. The
continuity equ
ation
[3] of th
e traffic
flow is
s
h
own in (9):
(,
)
q
s
xt
tx
(
9
)
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LWR mo
del
a
nd ma
ny mo
d
e
ls
ba
sed
on
LWR a
s
sume
a
relation
shi
p
bet
wee
n
th
e traffic
flow velo
city and d
e
n
s
ity under
equili
bri
u
m state in
(10).
e
v
is the d
y
namic e
quili
brium vel
o
cit
y
of the traffic
flow:
(,
)
(
(,
)
)
e
vx
t
v
x
t
(
1
0
)
So the equati
on ca
n be tra
n
sformed into
:
+(
v
+
)
(
,
)
t
e
e
v
s
xt
x
(
1
1
)
LWR mo
del
can corre
c
tly d
e
scrib
e
the fo
rmat
ion
of th
e sh
ock
wave
s an
d the
dispersing
of traffic
congestion, but i
t
c
an not
describe
none-quilibri
um traffic fl
ow phenomena li
ke
the
gho
st traffic. In orde
r to describe the
s
e
phen
om
en
a, se
con
d
-o
rd
er continuo
us
model
s ba
se
d on
LWR model
wa
s pro
p
o
s
e
d
later.
2.2.2.
Second
-orde
r
Continu
u
m Models
Secon
d
-ord
er co
ntinuo
us mod
e
ls in
cl
ud
e mo
del
s li
ke th
e
Payne m
o
del a
nd
Papage
orgio
u
s mod
e
l whi
c
h ad
d a rela
xation time to LWR m
odel,
etc.
Payne mode
l:
To de
scrib
e
the no
ne-q
u
ilibriu
m
traffic ph
enom
en
on like gh
ost
traffic,
schola
r
s a
dd
vary moment
um equatio
n to LWR m
odel
and forme
d
fluid dynami
c
model such a
s
Payne model.
Acco
rdi
ng to
the id
ea
of t
he
ca
r-follo
wi
ng the
o
ry, P
a
yne p
r
op
oses th
e
co
rre
spondi
ng
dynamic e
q
u
a
tions [4] in Formul
a (12
)
. The mod
e
l de
fines
as the pressu
re ind
e
x;
x
as
the pre
s
su
re
term which
indicates th
e drive
r
’s
re
action
pro
c
e
ss to
stimul
a
t
ions;
as th
e
relaxation time;
1
()
e
vv
as the relaxation term whi
c
h indi
cat
e
s the rel
a
x pro
c
e
ss that
driver
adapt
s to the equilib
rium speed.
1
()
e
vv
vv
v
tx
x
(
1
2
)
The Payne
model i
s
abl
e to sim
u
late
the pr
opag
a
t
ion of nonli
n
ear
wave in
real ro
ad.
And it is the basi
s
model of
the ext
ensively used
simu
lation software FREFL
O
.
Payne mo
del
’s m
a
in
cont
ri
bution li
es in
the
relation
ship formula
o
f
the dyna
mi
c traffic
flow velocity-den
sity in Equation (13
)
.
is the rela
xation time and
x
is the flow’s
spatial
displ
a
cement
during
relaxa
tion time
.
(,
)
[
(
,
)
]
vx
t
v
x
x
t
(
1
3
)
The m
odel
can
simulate
prelimi
nary t
he ba
ckward
-sp
r
e
ad of t
r
affic cong
esti
on. The
r
e
are p
r
obl
ems on the ada
p
t
ive process
and nu
meri
cal cal
c
ulatio
n
.
Yet the main pro
b
lem is
the
relation
shi
p
a
s
sumption b
e
t
ween the traf
fic flow veloci
ty and densit
y.
Payne also summari
ze
s th
e se
co
nd-ord
e
r fluid dyn
a
m
ic mo
del’s
gene
ral form. Most of
the fluid dyn
a
mic m
odel
s can
be
written in vel
o
city
equatio
n fro
m
their
co
ntinuity equatio
n in
Formul
a (14
)
.
V
V
x
is
the transport term,
P
is the traffic
pressure,
is
the relaxation time,
e
V
is the dynamic equilibrium
traffic vel
o
city determined
by local
vehicl
es’
densi
ty,
1
P
x
is
the pre
s
sure term, and
1
()
e
VV
is
the relaxation term.
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Re
Pr
11
()
e
T
r
a
n
s
p
o
rt
T
e
rm
la
x
a
tio
n
T
e
r
m
es
s
u
re
T
e
r
m
VV
P
VV
V
tx
x
(
1
4
)
The main
differen
c
e
betwe
en these mo
dels
[5]
is the
traffic pressu
re
P
, relaxation time
and the dyna
mic equili
briu
m traffic velocity
e
V
.
The
relaxatio
n
time in
L
W
R mo
del i
s
set to 0. Payn
e mod
e
l a
nd
Papage
orgio
u
s m
odel
assume
0
()
()
/
(
2
)
e
PV
V
[6], and the avera
ge free/exp
e
ct speed
0
(0
)
e
VV
. Phillips
model a
s
sum
e
s
P
, and
is
the varian
ce
of the velocity [7].
Kühne model [8]
，
Ke
r
n
er
and Kon
häu
ser mod
e
l [9] define
0
V
P
x
，
whil
e
0
is a po
sitive con
s
tant,
is the
coeffici
ent of visco
sity. The term
V
x
in Kerner an
d Kon
häu
ser m
ode
l means the
visco
sity
term s
i
milar with
the
term
2
2
V
x
pro
p
o
s
ed by
Whith
a
m
b
e
fore, whi
c
h
i
s
importa
nt
to
filter
the
sho
c
k front.
Michalo
poul
os mod
e
l d
e
fine the re
laxation time
as a varia
b
le inversely
prop
ortio
nal to the traffic flow de
nsity [10
]. Wu model
introdu
ce
s a
momentum
equatio
n of the
one-dime
nsio
nal pip
e
flow into traffic fl
ow m
odel
i
n
the co
ndition
of the hyb
r
i
d
and
lo
w-sp
eed
traffic in Chi
na [11]. Bellouqui
d re
sea
r
ch
es
on th
e
hyperb
o
lic a
s
ymptotic lim
it of the discret
e
kineti
c
theo
ry
model
of veh
i
cula
r traffic [12]. Dag
a
n
z
o
re
se
arche
s
o
n
the a
nalysi
s
of the
sta
b
il
ity
of macro
s
co
pic traffic fl
ow [13].
Ng
oduy thin
ks
widely
scat
tered t
r
affic
flow rate-d
e
n
sity
relation
shi
p
i
s
cau
s
ed
by the rando
m variation
s
in
driving beh
avio
r. And
he sol
v
es
this probl
em
by adopting
a multi-cl
ass first-o
r
de
r m
odel with
a
stocha
stic
sett
ing in his m
o
del pa
ramete
rs
[14].
The mo
st p
r
ominent fe
ature
of micro
s
copi
c m
odel
s is that the
s
e mod
e
ls
do
not take
vehicle’
s in
di
vidual be
havi
o
r into
a
c
cou
n
t. We
sum
m
a
rize the
feat
ure
s
of th
ese
model
s i
n
Ta
ble
1.
Table 1. Feat
ure
s
of Macro
s
copi
c Model
s
Model
Feat
ures
Static
macr
oscopic
models
Being a constant model, static model
is lim
ited used in real cases.
D
y
namic
macr
oscopic
models
Being different fr
om common fluid, the
velocity
inversel
y proportional to t
he densit
y
in traffic flow
is a
n
inexplicable phenomenon
in the d
y
namic conservation equa
tion.
D
y
namic mod
e
ls are
onl
y suitable for
the crowded, eq
uilibrium and st
able traffic
flow
.
First-orde
r
continuum
model
The model a
ssumes the
relationship between velocit
y
-
density
in e
quilibrium state.
It can simulate
the form of
traffic shock and the dissipation
of
congestion. But it can’t simulate the
tr
affic flow
in non-
equilibr
i
um state.
There
contains
no rela
xation
time.
Second-orde
r
continuum
models
These models assume a
d
y
namic relation
ship bet
w
een t
r
af
fic
velocity
-densit
y
.
It can simulate
phenome
na
like the
stop-and-go and t
he
propagation o
f
n
onlinear
w
a
ves.
There contain
relaxation time.
2.3. Mesos
c
opic
Models
Meso
scopi
c
model
s in
clu
de ga
s-kin
e
tic mo
d
e
ls a
n
d
hybrid m
o
dels. Th
ese
model
s
usu
a
lly comb
ine different
model
s amo
ng macr
o
s
co
pic and mi
croscopi
c mod
e
ls. Mesosco
p
ic
model
s de
scribe si
ngle ve
hicle’
s dyna
mic respon
se
to the variat
ion of the tra
ffic flow de
nsity
(flow
rate
or
velocity). Th
e
s
e
model
s t
r
eat traffic flo
w
a
s
‘
p
latoo
n
s
’ fo
rmed
by
a set of n
e
a
r
b
y
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7744
vehicle
s
to d
e
scrib
e
the b
ehavior
of the inflow
a
nd
outflow of ea
ch plato
on. T
he de
scriptio
n of
the moveme
nt of the veh
i
cle
s
in these model
s
is simila
r with them in macroscopi
c mod
e
ls,
whi
c
h me
an
s vehicl
es i
n
the sam
e
platoon h
a
v
e a same
spee
d. The
classificatio
n
of
meso
scopi
c
model
s is sho
w
n in Figu
re
3.
Figure 3. Cla
ssifi
cation Fig
u
re of Me
so
scopi
c Mod
e
ls
2.4. Gas
Kin
e
tic
Models
Prigogine m
odel:
P
r
igogi
ne-He
rman
model propo
sed
the
first meso
scopi
c model of
traffic
flow
[15]
in 1971. Such
model ded
uces that the L
W
R
mo
del is
a limitation ca
se a
c
cordi
ng
to
its
k
i
n
e
t
ic th
eo
r
y
. T
h
e mode
l us
es
a p
a
rtia
l d
i
ffe
rentia
l equ
ation to
expre
s
s the
spatio-tem
poral
evolution of t
he velo
city and de
nsity of
vehicle
s
. Th
en they imp
o
r
t an a
pproximation relatio
n
to
clo
s
e the Bol
z
man
n
eq
uati
on in o
r
de
r to
obtain t
he m
odel eq
uation
.
In a histori
c
al view, the g
a
s
kineti
c
model
contri
bute
s
o
n
the basi
s
of
theoret
ical d
e
rivation of the macro
s
copi
c equ
ation.
The conserv
a
tion equ
atio
n contai
ns th
e relaxation t
e
rm
rel
f
t
and the intera
ction
term
int
f
t
:
int
re
l
f
dx
dv
f
f
ff
tx
d
t
v
d
t
t
t
(15)
If the quantity of vehicle
s
remain
s the same,
whi
c
h
mean
s there
has n
o
othe
r
entran
c
e
s
and exits. Th
e con
s
e
r
vatio
n
equatio
n is:
()
()
f
dx
dv
ff
tx
d
t
v
d
t
(
1
6
)
That mean
s the location x and the velo
city v
determin
e
vehicle’
s st
ate in this mo
del.
Fonta
n
a mo
del:
Fo
ntana
[16] extende
d
the Pri
gogin
e
mod
e
l in
19
75. He a
s
su
mes th
at
all vehi
cle
s
h
a
ve thei
r in
di
vidual exp
e
ct
ed velo
city.
The l
o
cation
x
, veloc
i
ty
v
an
d expe
cted
veloc
i
ty
0
v
determine the
state of the vehicle.
The governing eq
uation
is as follo
w:
0
0
()
()
(
)
(
)
tr
dv
f
fv
dv
f
ff
tx
v
d
t
v
d
t
t
(17)
Helbing model:
The
ga
s kinetic b
a
sed
model
s ha
d
no
furthe
r im
provem
ent d
ue to the
mathemati
c
al
difficulty of it’s ga
s kin
e
tic fram
e until
Helbi
ng prop
ose
d
his m
o
del [17] in 19
95.
Helbi
ng bi
ng
s in the
interaction
betwe
en the a
c
cele
ration of ve
hi
cle
N
and
1
N
to form the
kineti
c
-b
ased
continu
u
m m
odel, a so-cal
led Hel
b
ing
model. He de
fines P as th
e
traffic pre
s
su
re,
()
e
v
as the dyna
mic equili
briu
m velocity in his mod
e
l:
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Re
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11
()
e
vv
P
vv
v
tx
x
(
1
8
)
The well
-kno
wn traffic software MASTER adopt
s Hel
b
ing mod
e
l for the advant
age
s of
its fast comp
utation and strong r
obu
stn
e
ss, etc. This model i
s
able to simula
te stop-a
n
d
-
g
o
wave an
d no
nlinea
r dyna
mic phe
nom
e
non such as
con
g
e
s
tion of
synch
r
oni
zati
on, etc.
Hoo
gen
doo
rn
and B
o
vy propo
sed
a g
e
nerali
z
e
d
g
a
s kin
e
tic traffic flo
w
mo
del
[18] in
1999, which became an u
n
itive fram
ework of me
so
scopi
c mod
e
ls.
2.5. H
y
brid
Models
Hybrid
mo
d
e
ls are
u
s
u
a
lly the combi
nation m
odel
s mixed
with macroscopi
c,
meso
scopi
c
and mi
cro
s
co
pic mo
del
s. For exam
ple,
Burgho
ut prese
n
ts a hyb
r
id me
so
scop
ic-
microsco
pic model
th
at applie
s
mi
croscopi
c simu
lation to
are
a
s
of spe
c
ific inte
re
st while
simulatio
n
a large
surro
u
n
d
ing network in less detai
l with a mesoscopi
c mod
e
l [19]. McCrea
present
s a
hybrid approach
co
m
b
ined the complementary feat
ures and
capab
ilities of
both
contin
uum m
a
thematical model
s and
kno
w
le
dge
-b
as
e
d
model
s to describe
effectively traffic
flow in ro
ad
netwo
rks [20]
. Depalm
a
mixes micr
o
s
co
pic meth
od with macrosco
pic meth
od [2
1].
In his model,
movement
s o
f
vehicles a
r
e
modeled in
macro
s
copi
c way with the policy of vehicle
s
modele
d
in
microsco
pic
way. Depalm
a
mo
del
and
Sch
w
e
r
dtfeg
e
r m
odel
[22
]
are
ad
opte
d
by
softwa
r
e ME
TROPO
L
IS and DYNEM
O
.
2.6. Microsco
pic
Models
We
usually
call micro
s
cop
i
c mo
del
s th
e entity-ba
se
d mod
e
ls.
Th
ese
mod
e
ls focu
s
on
the individu
al
vehicle
s
’ m
o
deling to
de
scrib
e
thei
r m
o
vements an
d interactio
ns. These mo
d
e
ls
attempt to describe th
e overall
cha
r
a
c
te
ristics of
the
system by int
egratin
g the
cha
r
a
c
teri
stic of
each in
dividu
al vehi
cle. T
h
at is,
ea
ch ve
hicle
gath
e
rs
informatio
n of
su
rroun
ding
one
s a
nd th
e
n
gene
rate
s its
own
drivin
g
strategy to fo
rms th
e a
c
tual
traffic flo
w
.
Vehicle’
s i
ndi
vidual b
ehavi
o
rs
su
ch like
ca
r-followin
g
, lane-chan
ging a
nd overta
king
can actu
al re
flected in the
s
e mod
e
l. Being
different from
macro
s
copi
c model
s, mi
crosco
pi
c mo
dels do
not
con
s
id
er
abo
ut the
spe
c
ifi
c
situation of th
e feature
s
of traffic flow, su
ch
a
s
traffic flow rate, de
nsity and velocity.
Macro
s
copi
c model
s conta
i
n ca
r-follo
win
g
m
odel
s, su
b-mi
cro
s
copi
c mod
e
ls a
n
d
particl
e
hoppi
ng mod
e
ls (al
s
o
kno
w
n as
cellul
a
r automaton
model
s). Microscopi
c mod
e
ls’ cla
s
sifica
tion
is sh
own in Figure 4.
Figure 4. Cla
ssifi
cation Fig
u
re of Micro
s
copi
c Mod
e
ls
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Vol. 12, No. 11, Novem
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14: 77
38 – 775
7
7746
2.7. Car-follo
w
i
ng
Models
Car-follo
wing
mode
s
aim t
o
stu
d
y the
p
r
opa
gating
of
the traffic flo
w
o
n
a
si
ngl
e lan
e
.
Pipes p
r
op
osed the earli
est car-foll
o
wi
n
g
model an
d its theory in 1
953 [23]. It uses mathe
m
ati
c
al
model to a
nal
yze its dyn
a
m
ic theo
ry in
orde
r to
simul
a
te vehicle’
s f
o
llowin
g
be
ha
vior on a
sin
g
l
e
lane. Such m
odel
s re
sea
r
ch on the state
s
of traffic
flow mainly in the synchro
n
ized flow of traffic,
whi
c
h are d
e
fined in three-p
h
a
s
e tra
ffic theory.
The syn
c
hroni
zed flo
w
has characte
ri
stics
inclu
d
ing con
d
itionality, retarda
n
ce and
transitivit
y, which ma
ke
s the state of traffic prop
agat
ing
backward int
e
rmittently an
d co
ntinuo
usl
y
like
pul
se d
o
.
The
s
e
mo
dels are “stimulus-respon
se”
model
s that rese
arch on v
ehicl
e’s follo
wing b
ehavio
rs
in
synchro
n
ize
d
flow by
analyzi
ng driv
er’s
respon
se
s to different stim
ulation
s
. The form is
:
ca
r-f
o
llowing re
spo
n
se =
sensitiv
ity × stimulus.
Car-follo
wing
models
co
ntain linea
r car-followi
n
g
model
s, nonlin
ear
ca
r-follo
win
g
model
s and
car-foll
o
wi
ng
model
s ba
se
d on fuzzy inferen
c
e
syste
m
.
2.7.1.
Linear car
-fo
llo
w
i
ng Mod
e
ls
Pipes mo
del,
the earlie
st prop
osed car-followi
ng mo
del, is a re
prese
n
tative linear
car-
followin
g
mod
e
l introdu
ce
d in our pa
pe
r.
Pipes model
：
Th
e mod
e
l
define
s
()
st
as th
e dista
n
ce b
e
t
ween ve
hicl
e
N
and
1
N
that two vehi
cle
s
wo
n’t crash
whe
n
ve
hicle n
br
e
a
ks; T as the
reactio
n
time, durin
g whi
c
h
the
velocity of ve
hicle
N+1
do
s n
o
t chan
ge
. The
sc
hem
atic di
agram
of Pipe
s mo
del i
s
sho
w
n
i
n
Figure 5:
Figure 5. Sch
e
matic Di
ag
ram of Pipes
Model
The mo
del
a
s
sume
s the
brea
kin
g
di
st
ance of vehi
cle n
3
d
equal
s to the b
r
ea
king
distan
ce of vehicl
e n+1
2
d
. The gap b
e
twe
en two vehi
cl
es is:
11
()
()
()
(
)
nn
n
st
x
t
x
t
T
x
t
T
L
The type t differential,
we
can g
e
t
1
()
n
x
tT
and
1
()
()
nn
x
tx
t
, sepa
rately the
accele
ration
(re
actio
n
) of
vehicl
e
1
n
at time
tT
and
the velo
city d
i
fference b
e
t
w
ee
n
vehicle
n
and
1
n
at time
t
:
11
1
()
(
)
(
)
nn
n
x
tT
x
t
x
t
T
(
1
9
)
We n
o
tice th
at the rea
c
tio
n
of vehicle
1
n
is propo
rtion
a
l to the velocity differen
c
e
betwe
en
n
and
1
n
at time
t
. Thus we n
a
me th
e model line
a
r
ca
r-follo
win
g
model.
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Re
vie
w
of the Urba
n Traffi
c Modeling
(Zh
u
Song)
7747
The eq
uation
23
dd
in the mode
l is assu
med,
whi
c
h do
es
not exist in p
h
ysical truth.
Hen
c
e, resea
r
ch
ers ad
opt
the rea
c
tion
coefficient
to repla
c
e the
se
nsitivity
1
T
and then form
the gene
ral type of linear
car-foll
o
wi
ng
model
s:
11
(
)
()
()
nn
n
x
tT
x
t
x
t
(
2
0
)
That is: car-following
reacti
on = sensitivit
y(or re
actio
n
coeffici
ent) ×stimulu
s.
There are va
riou
s rea
c
tion
coeffici
ents
assume
d in d
i
fferent mod
e
l
s
. Some mo
dels
assume
as a con
s
tant (e. g.,
a
). Some model
s assu
me
in distrib
u
tion function
su
ch
as:
12
12
,
,
ad
d
c
bd
d
c
(
2
1
)
a
,
b
,
c
is
c
o
ns
tant.
2.7.2.
Non
-
linear Car-follo
w
i
ng Models
Nonli
nea
r car-followi
ng mo
dels
co
ntain
model
s imp
r
o
v
ed ba
sed o
n
linea
r car-followin
g
model
s su
ch
as Ga
zi
s mod
e
l, OV model, etc.
Gazis mod
e
l:
The a
s
sum
p
tion that the acceleration
(re
actio
n
) of vehicle
1
n
in Pipes
model relate
s only with two vehicle
s
’ re
lative velocity which is
not agre
ed
with Gazi
s. Thu
s
,
he
prop
osed a
nonlin
ear
car-followi
ng mo
del in whi
c
h
the rea
c
tion
coeffici
ent
wa
s inv
e
r
s
ely
prop
ortio
nal to the gap b
e
twee
n two
vehicle
s
in 1
959 [24]. Ga
zis d
e
fine
s that
a
as the
prop
ortio
nality coefficient
whi
c
h is p
r
o
portion
al
to the critic
al veloc
i
ty of traffic
flow
m
V
and
inversely pro
portion
al to the gap of two vehicle
s
f
V
:
1
2
mf
aV
V
.G
a
z
is
mo
de
l is
s
h
ow
n
as
follow:
11
1
(
)
()
()
()
(
)
nn
n
nn
a
x
tT
x
t
x
t
xt
x
t
(
2
2
)
Gazi
s p
r
o
p
o
s
es the
ge
neral form
of ca
r-fo
llo
win
g
m
odel
s in
his
sub
s
e
que
nt rese
arch
[25]. He defin
es
1
1
()
()
()
m
n
l
nn
xt
T
x
txt
as the se
nsitivity,
m
,
l
are
con
s
tants.
1
11
1
()
(
)
()
()
()
(
)
m
n
nn
n
l
nn
xt
T
x
tT
a
x
t
x
t
xt
x
t
(24)
It is the no
nli
near mod
e
l formul
a when
0
m
and
1
l
; Yet it is the
gen
eral form of
linear m
odel
whe
n
0
m
and
0
l
.
Gipps mode
l (Safe
t
y-distance mod
e
l):
Gipp
s m
o
d
e
l assum
e
s
a
safety dista
n
c
e that
vehicle
s
al
wa
ys ke
ep in th
eir follo
wing
behavio
r to
a
v
oid crashing
. Thus th
e m
odel i
s
kno
w
n as
the ca
r-foll
o
wing mod
e
l ba
sed
on
safety distan
ce
[26]
.The ori
g
inal fo
rm of the mod
e
l is exp
r
e
s
se
d
in differential
equatio
ns of
basi
c
Newto
n
i
an moti
on bu
t not in the form of stimulus-re
sp
on
se:
22
11
1
0
()
(
)
()
(
)
(
)
nn
n
l
n
n
x
tx
t
x
t
x
t
T
x
t
Tb
(25)
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