計測自動制御学会東北支部第 266 回研究集会（2011.7.20）
資料番号 266-12

Scientific Study of Road Traffic Flow (3)5. Development of Theory for Time-Domain Analysis-Fundamentals of Road Traffic Characterizations-

Tasuku Takagi
Professor Emeritus, Tohoku University
Keywords：road traffic, time-domain, Weber’ law, Fechner’s law, t-s diagram,高木相
連絡先：〒981-0952

仙台市青葉区中山 5-2-20(Nakayama 5-2-20, Sendai, Japan
E-mail: tasuku@sirius.ocn.ne.jp

Continued from “Scientific Study of Road Traffic
Flow (1)” presented at the previous meeting of
262nd SICE Tohoku at Tohoku University on Dec.
23, 2010, Second version (2) subtitled“-Measurement
andExampleof
Data
Analysis-“ was presented at Tohoku Bunka Gakuen
University on March 11 when devastating
earthquake and tsunami hit around northern
Tohoku area, the oral presentation ceased due tothe disaster. This paper begins with Section 5
because the present series have finished Chapters 1,
2, 3 in (1) and Chapter 4 in (2). Thus in thisunderstand presentation (3) mentions Section 5. T
The content (subtitle) of the previous presentation
is as follows:(1) Road Traffic Flow Fundamentals
from Weber-Fechner Law(2) Measurement and Example of Data Analysis

X(m)

400
350
300
250
200
150
100
50
0

30
250

350

450

550 t(s) 650

Fig.5-1 Variation ofwith respect to elapsed
time(The original data is shown in Fig.5-4)
understand the traffic flow dynamics. The meaningof the time-domain analysis is that it makes possible
to see the dynamical phenomena on the axis of time(denoted) of which unit we use in our study isalways second. The dynamical phenomena should be
expressed on the graph of which abscissa is scaledwith the elapsed time in seconds. Fig.5-1 is the
typical example of this type of graph (graph).
We are able to understand that this type ofgraph has full of contents, which is able tocharacterize the car movement behaviors against
the elapsed time
. Clearly we can see from
Fig.5-1 that the distance-headways ( ) between
successively moving cars have a tendency like thatshown by the arrows in the figure. That is to say,

5. Development of Theory for Time-Domain
Analysis -Fundamentals of Road Traffic
Characterizations5.1 Example of Time-Domain DataFirst of all, we will see the graph Fig.5-1 which is apart of Fig.5-4 in this Section.
The reason why the author shows Fig.5-1 first of allis that the readers may be easily able to1every driver in average has a habit to chase
Since the condition of() is very
foregoing car with faster speed than that of strict condition and as we saw in 5.1 in this Section,
foregoing car, and reaches the shortest this condition cannot be sustainable for long time indistance-headway () about 30m as the practical flow like the case of not so dense flow.
shown in Fig.5-1 when the car speed becomes the The speed can be thought in general as,same with the speed of foregoing car (). The then (3.7) is written likestate of the minimum distance-headway in this casecould not be sustained long time and the=phenomena like Fig.5-1 cyclically occur, which may
=
.
(5.1)
depend on car density. In case where car density
increased, the state of shortest distance headway
We have noted in (3.4) of Section 3 thatis
may continue and the cyclical movement of(3.4) (from Section 3)
should become slow.
Readers may remember when they read 4.5.3 inThus (5.1) can be written as
Section 4 where the distance-headway becomes(5.2)( 27m, whenis assumed to be 10m), but in caseof Fig.5-1().
Theoretically this condition appears at the
5.3 Relationship between
condition of, which is the condition of
5.3.1
as a time function
maximum flow (the minimum time-headway).
We have to clarify the formula (5.2) which can
Following subsections will discuss the details of the connect the statistical data to the time domain
above mentioned.
behavior of the road traffic flow. As we have seen andmentioned before, the averaged driver has a habit of
5.2 Fechner’s Law under condition ofchasing the foregoing car with higher speed than that
5.2.1 Prologof foregoing car. Should we say a little mathematically,
To begin with, we shall admit that the analyses we can write as a function of time, i.e.
, andever made in the world on the road traffic flow were can be written likealmost all statistical ones and also we should admit,(5.3)
this because no appropriate ways exist in analyses whereis an arbitral initial distance-headway at
without statistics. We cannot forget that almost all.
can be written as
physical laws were born from the results of limitless,(5.4)
number of experiences or investigations, and the where is the time-headway.
Weber-Fechner Law was also the same. However, so
Referring Fig.3-1 ((2) in this series) along with
far as we concern the road traffic flow, we inevitably related sentences,can be written as
have to discuss it with respect to elapsing time i.e.
, (5.5)time-domain. Our interest in our live activity is
where
and
are foregoing and following
always bound by time and we cannot be indifference
car speed, respectively. From driver’s chasing habit,
to time. Nevertheless, we have not yet a good
the relationship between
isachievement in dealing with the traffic flow in() in average. The
terms of time domain. Present study tries for theapproaching speed of the distance-headway to its
first time to make a theoretical approach toshortest of e depends on the averaged slopes offormulate the traffic flow behavior with respect tothe arrows in Fig.5-1. The averagediselapsing time. The key is the condition ofnegative when distance-headway
is large, but
5.2.2 Fechner’s exponential formula underfinally becomes equal to zeroat theThe Fechner’s law in road traffic flow was shown
shortest limit of. Fig.5-2 shows this situation
again here as follows;
conceptually. As shown in Fig.5-1, the actual(3.7) (from Section 3)
distance-headway in this case is, that2

means averaged
This is the highest
efficiency state.
After the highest efficiency state, there are twocases that are shown by A and B in Fig.5-2, where A
is the case going to the flow of less density and B tocongestion.
X

shown in Fig.5-4).
The state ofat the speed
gives
the highest transportation efficiency with highest car
density. Like that shown in Fig.5-2, this condition canbe separated due to the condition of flow in near
future, so that it may go to congestion or high speed
lower density flow. Such flow dynamics will be later
discussed by using time-space diagram (t-s diagram).

5.3.3 Approach to laminar flow in actual flow
In the actual flow, what process does involve inapproaching the laminar flow will be shown here.
X
The most comprehensive result of examination isshown in Fig.5-3. As we have seen (3.4) in Section 3,A
the variation of
can be substituted b.
The situation thatis going to toward zero in theBnormalflow(nocongestion)means
is also goingeX 0
X0
toward zero, where the perfect laminar flow ist
0 ti 1 tiestablished. We have the smallest distance-headway
with the speed
at this situation, and theFig.5-2 Conceptual tendency ofapproaching time-headway becomesasin 3.5 andto the shortest distance-headway . After that:
3-2.
A: goes to high speed with less density flow. B:
We have examined in Fig.5-1 that the shortest
goes to congestion
distance-headway
is about 30m in average. In
this condition, the speeds of all cars become the samewith , and the flow becomes laminar flow.
5.3.2 Time-domain effect of
We consider here about how speed
and600
time-headway
affect on the movement of
in a time-domain.
500
ΔX
Here, we shall put the timing by. In
400
Fig.5-2,
and
----)
300
where denotes the car’s number and.
200
The following matters come into being:
X(m)

X0

100
0

0

at

and

10

15

20

T(s)

.
Since

5

(

Fig. 5-3
plots shows
decreases as
decreases (Tohoku Expressway)), then,In Fig.5-3, both smallest values ofare. The details will be shown in Fig.5-4
which is the same data with that of Fig.5-3 but thedata of are limited less than 5 seconds ((5.6)
As totally(-------) is negative,
shrinks with respect to time until
becomeszero which means all
approach tothat issimultaneously all
become equal value of(but the actual data are different as3

entireflow
independentlyoftime-headway when the flow is smooth.

X= 27.749T + 1.2282

200

the

180

5.4 Bunching flow and density
5.4.1 Periodical phenomena of distance-headway
Fig.5-1 was the part (t=300~550s) of Fig.5-5, where
the same phenomena can be seen that occurs
periodically. We can naturally assume thephenomena are the driver’s natural habit. The
phenomena shown here must stem from the resultof driver’s unconscious behavior to follow theforegoing car as close as possible. The closest
distance-headway isin the theory, where thetime-headway T becomes minimum which isdenoted by. As we see details in Fig.5-5,
appear periodically. In Fig.5-5, the arrows indicate
the variations of . Like that shown in Fig.5-1,
almost all occasions goes to be small from thepeak values. We should note that the slopes of
upward arrows are steeper than those of thedownward arrows, which means, as we naturally
consent, thatis seldom to increase one after
another, on the contrary to this,decreases
generally one after another. The bunching
phenomena in road traffic flow can be interpreted
from the above behaviors.

160

Δ
X

140
X(m)

120
100
80
60
Xam=
40
30m

20

Xmin=
15m 0

0

2

Tmin = 0.6s

4

T(s)

6

Tam=1.3s

Fig.5-4

plots (

From Fig.5-4, we can see rather precise values ofthe small parts of, where the bothminimum(= 0.6s,=15m) andboth
are shown (The value of
was decided frommeasured data of Fig.5-1 (theoretically,should be equal to). To meet with,corresponded to the average ofless than 3
seconds, and it became 1.3s.
The value of 30m foris not so different for

any other roads in spite of urban, suburban roads.
However

800

depends of roads, the typical value

700
600
X(m)

should be around 1.5 s for expressway and 2s forurban and suburban roads.
As references, the minimum values of(,shown, but they do not have theessential meaning because they are occasional.
● Definition of Laminar Flow
Theoretically the definition of laminar flow
should be the case when
. However,perfect state ofdoes not appear as
shown in Fig.5-4, so that we should define itby the limitation of the time-headway. The
author defined the flow less than 3 seconds oftime-headway should be the laminar flow for
any style of roads, although no definite reason
exists. Roughly speaking, if we look at Fig.5-3,
the laminar flow nearly exists among an

500
400
300
200
100
0

0

200

400

600

800

t(s)

1000

Fig.5-5
data showing waving phenomena(Tohoku Expressway. Hitokita)
However, more visible way to comprehend thebunching phenomena can be seen from the cardensity.
5.4.2 Flow density and bunching phenomena
4

The above periodical variation ofmust becomprehensively shown by the car density variation.
The density denoted bywill be defined as,

analyzing traffic flow, and this will be used in later
Section for qualitative (some time quantitative)
analyses for the fundamental behaviors of roadtraffic flow. (V. Section 8)
For the time being, the t-s diagram will be
introduced here for understanding the conceptual
traffic flow phenomena. Fig.5-7 is indicating how
the traffic flow goes to the minimum
time-headways.

(5.7),

which comes from the Fechner’s law in (3.7) in
Section 3. The definition of (5.7) is rigorous because
is the shortest distance-headway when cars are
stopping () to make a queue, andis thedistance-headway when cars are running ().
The typical case iswhich can be seen atintersection when signal is red. At this condition
becomes one () (see more detail discussions in
5.8.1 in this Section). At present discussions on theflow in Fig.5-5, the speed was not so different for allcars, so that we need not take into account of theeffect of speed on the density (see Appendix 5.1). In
order to get clear characteristics, we took themoving average method of n=5 (5 data are averaged
sequentially, in this case is denoted by) wastaken for extracting the features of Fig.5-5. The
density of cars was observed at one point on theroad with respect to time. Fig.5-6 is the result ofdensity variations. As shown in Fig.5-6, periodic or
cyclic variations of the density are clearly shown,which means in the actual flow, the bunchingoccurs cyclically.

ｘ
9

eX 0
7
6
5
4

X45

3
X12

2

T12

T45

1
0

1

①

②

1

2

③
3

④
4

5

⑤ ⑥
6

7

8

9

10

t

Fig.5-7 Example of t-s diagram
explaining how each car moves with a
group

0.8
0.6
kn5

Tmin

8

0.4
0.2

Fig.5-7 shows how both the distance-headway
and the time-headway
are reaching to theshortest one i.e.
respectively on thetime-space (t-s) plane. This type of graph (t-sdiagram) is a very important tool for traffic analyses.
In Fig.5-7, six cars are indicated by black points, ①～④ are one group and ⑤, ⑥,--are of anothergroup. In the graph, the ordinate x denotes
distance and its scale (0~9) is arbitrary and also theabscissa t denotes time t and its scale (0~10) is also
arbitrary. The dotted lines are indicating the
trajectories of individual car movement. The slope of
trajectory indicates the speed , which can bederived by

0
0

200

400

600

800

1000

t(s)

Fig.5-6(car-density variation of flow in
Fig.5-5) The otherandare shown in Appendix 5-1)
5.5 Traffic flow in time-space diagram (t-s diagram)
The group flow dynamics for cars can be clearlyshown by using so called t-s diagram. By using it,
we can intuitively comprehend the time-domain
movements of cars on a two-dimensional paper.
Here we introduce the time-space diagram(abbreviated by t-s diagram), which is inevitable for5The situation of cars ⑤,⑥--- will be the same with①~④. If we draw this situation of Fig.5-7 at thetiming of t=7 on road, it will be like Fig.5-8.

(5.8).
The time-headway
is the distance between
trajectories, for instanceindicates thetime-headway between car ① and ② . The
distance-headway
is shown on the ordinate
along withon abscissa.
As we see in Fig.5-7, the speed of car ① is set
constant (slope =of ① is constant), and ②
follows ①, ③ follows ②, and ④ follows ③. The
cars ⑤ , ⑥ ,---, are another group, and thetime-headway
between ④ and ⑤ is wider
than those of the others (group ①~④), and we see
the same thing of the distance headway
as
discussed in Fig.5-8. The two groups symbolize thephenomena in the case of Fig.5-5 or Fig.5-6 (bothshow the cyclic changes of distance-headway anddensity respectively as we will see next 5.6).

⑥

④

⑤

③

②

X45

①

eX

Fig.5-8 Final situation of Fig.5-7 is shown as onroad situation of t-s diagram in t=7
Naturally, if the car density increases,
in
Fig.5-8 becomes short (its limit is). We should
notice again that the above mentioned phenomena
are independent of the car speed, because thecondition of
can be satisfied at any
run-speed for any roads.
5.7 From maximum flow to congestion
5.7.1 Cue to congestion
The car flow shown by the row shown in Fig.5-8(cars ①~④) is the situation of the condition of
The highest density at the normal flow
without congestion can be seen as the row of cars
moving with the same speed shown like ①~④ in
Fig.5-7 at the timing of t=7. If we could have thissituation with many cars, the flow of better
efficiency will be realized, The maximum flow rateshouldbeable
toobtainwhen
alldistance-headways in a row of moving cars become
the shortest in average, i.e.
(about 30m). Somedrivers may have had the experience closed to thissituation. But in many cases, this situation oftenchanges
and
goes
to
tight
congestion

5.6 Process going to high density flow
We will be able to visibly see that how the traffic
flow goes to bunching state of which ultimate is thatof the shortest time-headway and simultaneously of
the shortest distance-headway. Both the shortest
time-headway and the shortest distance-headway
have been denoted byand e respectively. If
we look at Fig.5-7, we easily understand the processof howandare produced by andFig.5-7 was drawn for that the car ① is moving
with constant speed (its trajectory is a straight line)
and is followed by ② of which speed is faster thanthat of ① (trajectory of ② is steeper than that of①). The relationships of speed of ②, ③ and ④ are
the same as the relationship between ① and ②,
respectively. So the speed of ④ is the fastest. Eachcar speed of ② to ④ is being kept nearly constantuntil t=6, but from t=6 to 7, each speed becomes thesame with that of ①, then all cars move with thesame speed after t=7, then the differences of speed
become 0 ((), (), see also Fig.5-4). At
this situation, the ultimate state of car flow ofis realized, which is the case of highest car
density in the normal flow. But when the density is
not so high, flow becomes a bunching row as
mentioned in 5.4, which can be easily understand if
we look at another group of cars of ⑤,⑥--- in
Fig.5-7, which are far apart from the group of ①~④.

Ｘ
’

Ｘ０
ｑ

A

Ｘ０
Ｘ０

B
t

Fig.5-9 Arrow A is the case of going sparse flow,
and B is the case of going to complete
congestion.
6

After the flow got the shortest distance-headway
e as we saw in Fig.5-2, to where the arrow goes?
There are two ways, one is the case where nocongestion occurs, and another case is that traffic
goes to congestion. The former case, (indicated by A
in Fig.5-9), occurs when the traffic density
decreased. The latter case, indicated by B (inFig.5-9), occurs when the flow density increased orsome disturbance happened, for example a car in
the row made its speed down even if it is small, andthe flow goes to dead. From the theoretical point ofview, the destiny of the flow, death or alive, is clear,but from practical situation, the cue is very criticaland not known here.
So far as the flow rate is concerned, it is indicatedby(dotted line) in Fig.5-9, which is defined in(5.9) in the next 5.7.2).

5.8 30seconds averaged parameters in time-domain
5.8.1 Car density k ()
We have seen the definition of car density
briefly in (5.7). Here, we describe it in details.
Traditionally car density has been defined as thenumber of cars per 1km for each lane. But this isdifficult to measure and almost impossible inpractical, because it should be done from highattitude observation and count the number of cars
which varies with time to time. Thus, we cannot
pursue it in the time-domain.
In this study, the density k is defined as,then becomes as we have seen in (5.7) in 5.4.1;(5.7) (from 5.4.1)
The above (5.7) is of course the formula ofstatistical base, and becomes 1 () when,because the argumentin the exponentialfunction becomes equal to zero (isdimensionless number and, at the highest density
occurs when, and.
● (5.7) is the relationship between and .
This relationship has been called
correlation which had been discussed in theearly days in the research of traffic flow as
the one of the main research subjects. Manymodel formulae have been proposed byresearchers in the world, but all the modelcannot be fitted with the actual data ofrelationship. We should notice allmodels proposed so far are empirical ones
and not scientific. The researches on thissubject had been made in 1930’s to 60’s.
Today, this item looks out of interest.

5.7.2 Distance-headway in congestion
The maximum flow condition was the case where
the distance-headway was the shortest (),
and simultaneously the time-headway T became
the shortest () as we saw above.
The ultimate congestion is the case where cars ina row cannot move i.e.
.
Again we refer the Fechner’s formula of(3.7) (from Section 3)
If
in (3.7), becomes equal to(),
because.
The arrow in Fig.5-2 goes to two ways that can beshown in Fig.5-9. Two arrows A and B; A goes toapart away from shortest distance-headway,
and B goes to congestion.
is thedistance-headway at speed. The flow rate
(dotted curve) is conceptual flow variation.
isdefined as.
●(5.10)We denote the density
which is averaged
density in 30 seconds. As an example, we haveshown the moving averaged data in Fig.5-6. Herewe will show the 30seconds averaged density
denoted byinstead of the moving average (was used in Fig.5-6).
Fig.5-10 shows the statistical, original
relationship of the same data with Fig.4.4 in Section
4.

(5.9).

The flow stops in the case of ultimate
congestion
where
the
averagedistance-headway becomes smallest valueofof which value is assumed to beabout 10m in case of passenger car.
Time-distance at this condition (becomes infinity because all cars stop (see(Ap 3-2) in Section 3).
7

and
The flow rate with 30seconds averagedisapplied for Tohoku Expressway, Azuma (Fig.4-4),
where
is

X = 1e-0.069V

1.6
1.4
1.2

(5.12).

k1
0.8

Fig.5-12 shows
and

0.6
0.4

Fig.5-12(a) t-q30

0.2
0
0

10

20

30

40

50

of

TohokuFig.5-10
relationship
Expressway (Azuma)

q30(s-1)

V (m/s)As shown in Fig.5-10, data are widely dispersed
but as shown by the centered line which computer
approximated, it looks very good approximationthat can be expressed by (5.11). The numericalformula became(5.11),
where =0.069 is the same with that ofrelationship of this data shown in Fig.4-6 whereof which round off number is 0.069.
Fig.5-11 shows the time-domain variation ofwith respect to time t.

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0

200

400

600

800

1000

t(s)

V30(m/s)

Fig.5-12(b) t-V30
35
30
25
20
15
10
5
0
0

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

200

400

600

800

1000

800

1000

t(s)
Fig.5-12(c) t-T30
7
6
5
0

200

400

600

800

1000

T30(s)kgraph along with
.

1200

4
3
2

t(s)

1
0

Fig. 5-11 t-(Tohoku Expressway, Azuma)

0

200

400

600

t(s)

As shown in Fig.5-11, the variation of the density
can be clearly seen by 30seconds averaged data
where high region of( 0.6) may be recognized
as the congestion state.
5.8.2 Time variation of flow rate () along with8

X30(m)Fig.5-12(d)the time variations of those parameters. That is;andcan indicate Congestion State,
andcan indicate Maximum Flow State.

t-X30

160
140
120
100
80
60
40
20
0
0

200

400

600

800

5.9 Concluding remarks
We discussed in this Section about the
time-domain behavior of the road traffic flow.
Basing upon the Weber-Fechner law, thestatistically maximum flow condition of
was clearly explained in the time-domain. The
process with respect to time for both going to flow
maximum state and congestion state could becomprehensively explained. Those mentioned
concepts may be useful in future optimizations ofroad traffic flow.
The flowand speed should be thefinal essential parameters for evaluating the roadquality which will be how fast, how many cars orvehicles and how much loads can be carried to thedestinations. Those practical matters will be
discussed in the next Section 6. We have had, fromthe discussions in the present Section 5, the basic
tools for discussing the matters mentioned above.

1000

t(s)

The gothic marks shown in Fig.5-12 mean as
follows:▲: congestion states●: maximum flow states.
a) From(maximum flowrate) is clear, but congestion state isunclear. The value ofis about 0.4-1(s ),=33m and=2.52s.
(=)=13.2m/s (47.6km/h),
while the result from the statisticalanalysis was 14.5m/s (52km/h). Bothvalues are almost the same.
b) Fromcongestion state is clear,but flow maximum state is unclear..
c) Fromminimum(isclear of which inverse is the flow
maximum (), butcongestion state is unclear.
d) From t- , congestion state is clear, but
maximum flow state is unclear. (SeeFig.4-4(c))
Table 5-1 is the summary of the above a), b), c) andd).

Acknowledgments
The author expresses his great thanks toProfessor Shosuke at the Tohoku Bunka Gakuen
University for his earnest review of the manuscript,
and corrected mistypes.
Appendix 5.1
and(Tohoku Expressway,
Hitokita)
Fig. (a), (b) and (c) show the time-domaincharacteristics of the moving averaged data (n=5)
for
(
and
with respect totime t. The flow was very smooth and no congestion
existed.

Table 5-1 Parameters and their information
Congestion

Flow

State

Unclear

Clear

Clear

UnclearUnclear

Clear=(a) t-Xn5
=

400
300

Xn5(m)

Maximum

We will see from the above Table that the
information given by the parametersandisessential to know the flow situation, if we observe

200
100
0

9

(b) t-Vn5
40

Vn5(m/s)

35
30
25
20
15
10
5
0

Tn5(s)(c) t-Tn5
10
9
8
7
6
5
4
3
2
1
0
0

200

400

600

800

t(s)

1000

Fig. App. 5-2,
andatsmooth flow (Tohoku Expressway (Hitokita))
Here we see that the distance-headway
isgoverned by almost the time-headway when theflow is smooth (speed does not change drastically so
the variation of and with respect to time arealmost the same).

10