Derivation of a fundamental diagram for urban traffic flow

12 The European Physical Journal B
defined as
∫x 1
dx ′ w(x ′ )f(x ′ )
x 0
∫x 1
. (A.1)
dx ′ w(x ′ )
x 0
In case of uniform arrivals of vehicles, we have a functional
relationship of the form f(x) =a + bx for the travel time,
and the weigth function is constant, i.e. w(x) =w. Here,
a = Ti 0,
x = ΔN i,andb =1/A i =1/(u i ̂Q i ). With x 0 =0
and x 1 = ΔNi
max , the formula for the average travel time
becomes
w[(ax 1 + bx 2 1 /2) − (ax 0 + bx 2 0 /2)]
= a + b x 1 + x 0
,
w(x 1 − x 0 )
2
(A.2)
wherewehaveused(x 2 1 − x 2 0 )=(x 1 − x 0 )(x 1 + x 0 ).
Inserting the above parameters, we obtain the previously
derived result
T i = T 0
i
+
ΔN
max
i
2u i ̂Qi
. (A.3)
When determining the average velocity Vi
av , the function
to average over is of the form f(x) =c/(a + bx), where
c = L i and the other parameters are as defined before. We
use the relationship
∫ x 1
dx ′ wc
a + bx ′ = wc (
)
ln |a + bx 1 |−ln |a + bx 0 |
b
x 0
= wc ∣ ∣ ∣∣∣
b ln a + bx 1 ∣∣∣
.
(A.4)
a + bx 0
Dividing this again by the normalization factor w(x 1 −x 0 )
and inserting the above parameters finally gives
V av
i
= L iu i ̂Qi
ΔN max
i
(
max
i
ln
1+ΔN ∣ u i ̂Qi Ti
0 ∣ ≈ L i
Ti
0 1 −
ΔN
max
i
2u i ̂Qi T 0
i
)
,
(A.5)
wherewehaveusedln(1+x) ≤ x − x 2 /2. This formula
corrects the naive formula
V av
i
≈ L i
T i
=
T 0
i
+
L i
ΔN max
1
2u i ̂Qi
≈ L i
T 0
i
(
1 −
)
max
ΔN1
, (A.6)
2u i ̂Qi Ti
0
wherewehaveused1/(1 + x) ≈ 1 − x. Therefore, the
above Taylor approximations of both formulas agree, but
higher-order approximations would differ. The formulas
in the main part of the paper result for Ni
av = Ni max /2,
which corresponds to the case δ i = 0 (i.e. f i − u i ).
Generalizing the above approach to the case δ i > 0,
we must split up the integrals into one over wc/(a + bx ′ )
extending from x 0 =0tox 1 = ΔNi
max and another one
over wc/a from x 1 = ΔNi max to x 2 =(1− u i )A i T cyc =
(1−u i )u i ̂Qi T cyc , where the specifications of a, b, andc are
unchanged. Taking into account Vi 0 = L i /Ti 0,thisgives
V av
i
= wL iu i ̂Qi ln |1+ΔN max
i /(u i ̂Qi T 0
i )| + Z
w(1 − u i )u i ̂Qi T cyc , (A.7)
where
Z = wV 0
i [(1 − u i )u i ̂Qi T cyc − ΔN max
i ]. (A.8)
Considering equation (15), we get
V av
i =
L i
(1 − u i )T cyc
ln
(
1+(1− f i ) T cyc
T 0
i
)
+ Vi
0 f i − u i
.
1 − u i
(A.9)
In second-order Taylor approximation, this results in
[ ( 1 −
Vi av ≈ Vi
0 fi
1 − (1 − f )
i)T cyc
1 − u i 2Ti
0 + f ]
i − u i
,
1 − u i
(A.10)
which can also be derived from equation (A.6), considering
equation (15) and the percentage of delayed vehicles,
which is given by equation (19).Thesameresultfollows
from
V av
i =
T 0
i
L i
+ T av
i
≈ V 0
i
together with equation (21).
References
(1 − T i
av )
Ti
0
(A.11)
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