Derivation of a fundamental diagram for urban traffic flow

6 The European Physical Journal B
into (26) [andwithf i = f 0 i , T cyc = T 0 cyc ,seeEq.(33)],
we get
L i ρ av
i (u i , {fj 0 av
})=ΔNi (u i , {fj 0 })
(1 − fi 0 = u i ̂Qi )2 Tcyc 0 ({f j 0})
, (39)
2(1 − u i )
which finally yields
u i (ρ av
i L i , {f 0 j }) =
(
̂Q 1+(1− fi 0 ) 2 i Tcyc 0 ({f j 0})
) −1
2ρ av .
i L i
(40)
Thiscanbeinsertedintoequation(26) togive
Ti
av (ρ av
i L i , {fj 0 }) =
u i (ρ av
i
ρ av
i L i
L i , {fj 0}) ̂Q i
3.1 Transition to congested traffic
= ρav i L i
+(1− f 0 T cyc({f 0
i
̂Q )2 j 0})
. (41)
i
2
The utilizations u i increase proportionally to the arrival
flows A i , i.e. they go up during the rush hour. Eventually,
∑
f j = ∑ (1 + δ j )u j → 1, (42)
j
j
which means that the intersection capacity is reached.
Sooner or later, there will be no excess capacities anymore,
which implies δ i → 0andf i → u i .Inthiscase,
we do not have any finite time periods ΔT i − T i , during
which there are no delayed vehicles and where the departure
flow O i (t) agrees with the arrival flow A i . Therefore,
O i (t) =γ i (t) ̂Q i according to equation (12), and certain relationships
simplify. For example, the utilization is given
as integral of the permeability over one cycle time T cyc ,
divided by the cycle time itself:
u i = 1
T cyc
t i0+T
∫ cyc
t i0
dt ′ γ i (t ′ ). (43)
Moreover, in the case of constant in- and outflows, i.e. linear
increase and decrease of the queue length, the average
number ΔNi
av of delayed vehicles is just given by half of
the maximum number of delayed vehicles 1
ΔN av
i
=
max
ΔNi
. (44)
2
1 Note that the formulas derived in this paper are exact
under the assumption of continuous flows. The fact that
vehicle flows consist of discrete vehicles implies deviations
from our formulas of upto 1 vehicle, which slightly affects
the average travel times as well. In Figure 2, for example,
the number of vehicles in the queue is Ni
max − 1 rather
than Ni
max . Therefore, the related maximum delay time is
(1 − u i)T cyc(Ni
max − 1)/Ni
max , which reduces the average delay
time Ti
av by (1 − u i)T cyc/(2Ni max ).
As a consequence, we have
T i ({u j })= L i
V 0
i
+ (1 − u i)T los
2 ( 1 − ∑ j u ). (45)
j
The average speed Vi
av of traffic stream i is often determined
by dividing the length L i of the road section reserved
for it by the average travel time T i = Ti
0 + Ti av ,
which gives
V av
i ({u j })=
(
L i
T i ({u j }) =
1
V 0
i
) −1
+ (1 − u i)T los
2L i (1 − ∑ j u ,
j)
(46)
and can be generalized with equation (28) tocaseswith
an efficiencies ɛ i ≠0:
V av
i ({u j },ɛ i )=
(
1
V 0
i
) −1
(1 − u i )T los
+(1− ɛ i )
2L i (1 − ∑ j u .
j)
(47)
Note, however, that the above formulas for the average
speed are implicitly based on a harmonic rather than
an arithmetic average. When correcting for this, equation
(46), for example, becomes
V av
i ({u j })=
∣
L i u i ̂Qi ∣∣∣∣
({u j }) ln 1+
ΔN max
i
ΔN
max
i ({u j })
u i ̂Qi T 0
i
∣ . (48)
As is shown in Appendix A, this has a similar Taylor approximation
as the harmonic average (47). The latter is
therefore reasonable to use, and it is simpler to calculate.
As expected from queuing theory, the average travel
time (45) diverges, when the sum of utilizations reaches
the intersection capacity, i.e. ∑ j u j → 1. In this practically
relevant case, the traffic light would not switch anymore,
which would frustrate drivers. For this reason, the
cycle time is limited to a finite value
T max
cyc ({u0 j })=
T los
1 − ∑ , (49)
j u0 j
where typically u 0 j ≤ u j. This implies that the sum of
utilizations must fulfill
∑
u j ≤ ∑ u 0 j =1− T los
T max . (50)
j
j
cyc
As soon as this condition is violated, we will have an increase
of the average number of delayed vehicles in time,
which characterizes the congested regime discussed in the
next section.
4 Fundamental relationships for congested
traffic conditions
In the congested regime, the number of delayed vehicles
does not reach zero anymore, and platoons cannot
be served without delay. Vehicles will usually have to