".. ,1:, !,
~,
iI ~
138
~
,
I. Prigogine. R. Herman and R. Anderson,
I,
observed. Further. bythethefunctional collective regime relation form of T determines
I,
uniquely the flow in the
1 qcollective
Discussion of trafficstream measurements
= T'
and definitions :It
Likewise, T is also determined from the experimental flow curve and once specified, according to our present ideas, it would have to combine with the f. to yield the individual portion we of the flow curve. Further, knowing T as a function of concentration findobserved that
I
Ctransition T(ctransition)
L.C.EDlE The Port of New York Authority, New York, U.S.A.
= I, ABSTRACT
and thus we see thatare the all concentration, the transition point interrelated. speed and the value of the integral I at
dI
I ,tII
, II~~
An examination of this model with respect to experimental results indicates a gratifying qualitative agreement with respect to the main features of the flow-concentration curve. The detailed aspects of this problem and some of the difficulties are to be presented elsewhere.
il-
,I "'-
'''-
REFERENCES (1)
PRIGOGlNE, I., A Boltzmann-like approach to the statistical theory of traffic flow, Theory of Traffic Flow, Amsterdam, 1961 (Elsevier Publish. ing Company) pp.158-164.
(2)
PRIGOGINE, I. and F. C. ANDREWS,A Boltzmann-like traffic flow, Opns.Res. 1960,8,789-97.
(3)
ANDERSON, R. L., R. HERMAN and I. PRIGOGINE,On the statistical distribution function theory of traffic flow, Opns.Res. 1962, 10, 180-96.
(4)
PRIGOGINE, I., R. HERMAN and R. L. ANDERSON,On individual and collective flow, Proc.of the Belg.Roy.Acad., 1963.
(5)
PRIGOGINE, I., P. RESIBOIS, R. HERMAN and R. L. ANDERSON, On a generalized approach for traffic flow, Proc.of the BeIg. Roy. Acad.,1Boltzmann-like 963. __
(6)
BALESCU, R., I. PRIGOGlNE, R. HERMAN and R. ANDERSON, to be published.
~ c:; ii,...
'''==
e~
il: .""
approach for
The flow, concentration and speed of a traffic stream are meaningful only as averages. The kinds of averages to be employed vary. In some instances it is correct to use arithmetic averages, in others harmonic averages, and in still others only space or time averages are suitable. In this paper, the correct type of average to use for each situation is derived from definitions of flow, concentration and speed of the traffic stream which are tied to two different methods of measurement. New definitions, independent of method of measurement and applicable to all kinds of measurements, are then proposed. They consider each vehicle trajectory through a given area of space and time to be a vector (xi, ti) where xi is the distance traveled and ti is the time taken by the ith vehicle. Average flow and concentration values are derived by averaging such vectors jointly in space and time. This approach unifies the former definitions and eliminates any ambiguity or question about methods of averaging. While developed for one-dimensional flow of roadway vehicles, it could be extended to other types of vehicles and to additional dimensions of space.
'
INTRODUCTION
"
I, ,"
I: ii H "
"
,",l
II'
}, \
In studying road traffic as a stream, Le., a fluid continuum, one is primarily interested in three characteristics of streams, viz., their flow (quantity per unit time), concentration (quantity per unit space), and speed (space per unit time) and how they vary in space and time. In pursuit of this interest, several methods of observation have been devised by various experimenters. Since road traffic is actually made up of discrete vehicles, rather than being in fact a continuous fluid, these methods generally are concerned with measurements made on individual vehicles to determine the speed or transit time of the vehicle plus either its time headway or its spacing behind the vehicle ahead. The need to convert these discrete measurements into the desired continuum characteristics has led to precise definitions of the latter in terms of the former for two common types of measurements, one type made at a point in space and the other at a point in time. It is the objective of this paper to explore certain implications of these definitions and td unify them into a single definition.
:1
140
L
t
L. C.Edie
Traffic
and definitions
141
-5
-10
-IS
PRESENT DEFINITIONS
stream measurements
Lighthill and Whitham1 have supplied precise definitions of the flow q and the concentration k at a given point x and at a given time t by giving instructions for measuring them over a short increment of roadway dx, whose midpoint is located at x, for a long time period T, whose mid-point is located at t, as follows:
iii, II
.t ::II:
2:
'-' '..
BC; ~ .
: !I:: , ..,--
j=::
\f:
~ii! f--' 1r:: 1
..
'll: , , ,
it! I '1
I; Ii
" !'
al 1m
II,
,
k='Edti Tdx
(2)
u=9._ndx k - 'E dti'
x
10
(3)
In connection with (3), Lighthill and Whitham pointed out that it results in the space-mean speed, which had previously been demonstrated by Wardrop2 to be consistent with an average condition of q = u.k. In addition to showing how to compute the space-mean speed from observations made at a point, Wardrop also showed how to compute the stream speed from a different set of measurements, using two aerial photographs of a long section on roadway X taken a short time apart: In notation similar to the above, the stream speed was given by Wardrop as ~dxi u=n~'
15
~
T
to-
Fig. 1 Measurement of a traffic stream at point (x, t) by two methods.
()4
where n is the number of vehicles on X at time t; and dxi (i = 1,2,3,... n) is
the sum of the distances traveled by the n vehicles on X during the interval dt between photographs. The definitions of concentration and flow for aerial observations are obviously the following: n (5) k=X ~ dxi q = u. k = ""Xdf
H 1\1
(1)
where n is the number of vehicles crossing ~ during time T, and dti (i = 1,2, 3, ... n) is the sum of the times taken by the n vehicles to cross. The speed of the stream was defined as
Gat .., Id:
q=n/T
(6)
The foregoing six equations give two sets of definitions, both of which are operational, Le., associated with how the measurements are made. One set is concerned with stream behavior on a small increment of roadway over a long period of time; the other with behavior during a small increment of time averaged over a long section of roadway. Both methods are illustrated in Fig. 1. It is easier to conceptualize equations 1-6 if they are not expressed in terms of differentials dx and dt but instead are expressed in terms of the speed u{ and the headway hi or the spacing si of each vehicle (i = 1,2,3, . . .n). The modified definitions are given below, where all summations are from i = 1 to
i
=n.
'
Short roadway-long q
= ~= ~ ~.1 ' ~ dti
1 ~ U'
~ 1 u:
k=Tdx =r ndx u
=~
dti
=ri}'
n
= ~!.
'
(10)
(7)
Short time-long roadway ~dxi ~ui ~ui q=Xdt =X-=~Si'
(8)
n '1-, k=X=~Si
(11)
(9)
u = n dt = ---n-.
time
'E dxi
~ ui
(12)
ui In the revised form, it is clear that the stream speed is the harmonic mean of vehicle speeds measured at a point and the arithmetic mean of vehicle speeds measured at an instant. It is also apparent that q at a point can be taken as the reciprocal of the arithmetic mean time headway and that k at an instant can be taken as the reciprocal of the arithmetic mean spacing. Traffic engineers will recognize equation (10) as that given in the Highway Capacity Manual: namely, average speed divided by average spacing.3 But how to average k over time at a point or q over space at an instant is not self-evident.
r
i.
,)1,
142
I.
If one has computed averages of qi, ki and ui for groups of ni vehicles passing a point and wishes to combine two or more of them, what kinds of
L. C. Edie
averages
Traffic stream measurements
should be taken for each?
If ti
= ni!qi
is the duration
of the ith flow
level, the formulas for combining two values or any number of values are as follows, as derived from the original definitions.
il;,
Short roadway observations : 11
n nl + n2 q=-=-=
!
,,1,I
.
II,
k
T
tl + t2 nl + ~ - ul u2
--
1I'
u
, !!!!
-
tl + t2
n
nl + n2
- ~!. ui
nl
ul
+
n2
u2
2:; ti
Measurements tl kl + t2k2 =-,2:;tiki tl + t2 2:; ti tl ql + t2q2
-
,counted,but with a slight shifting of the trajectories of 6 and -7 we might !obtainvalues of 14/T or 12/T. There,is a random source of error which :dependson just where T is taken or on small fluctuations in vehicle traject~ries. A similar source or"error exists for the 'observation in the :Xdt area. TABLE I
Average Flow q, Concentration k and Speed u for n Values of qi, ki, ui' (All summations are from i = 1 to i = n) (13)
=-,2:;tiqi
ti + t2
-
T
+
tl kl
--.Z; tiqi
-
t2k2
Z;~ki
(14) l at a point q (15)
I
k
General Z;tiqi 2:;ti tiki ti
n equal slices Z; qi n
n equal groups n
n single vehicles. n
2:; l/qi
2:;hi
2:;ki n
2:;ki/qi
2:;l/ui
2:; l/qi
Z; hi
~
,; I ,II!
Short time observations
lie
"
"
combined for different road sections
I u
If xi = ni/ki is the distance over which the concentration ki was measured, the formulas for combining are
"...I!
~... ' il::J :' ~,
q---ui
- si -
Ii
(==: ~~ : !he' ".
k
n
-
X ui
u=-=
n
~i .III'!
"~
~ I,
I, II Ii
II I 11"
II
". 11I1
I: II
It-
nlul
+ n2u2
xl + X2 nl + n2
-
-
Xl
+ x2
nlul
-
+ n2u2
nl + n2
xl ql + x2q2
Xl + X2
xlkl + x2k2 Xl
+ x2 xlql
--,2:;xiqi
- 2:;Xi
-~,2:;Xiki -
(16)
2:; tNi
qi
2:; tiki
2:; ki
q (17)
L xiqi L xi
2:; Xi
+ x2~
xlkl + x2k2
~ xiqi
=-.
~ xiki
(18)
n
k
2:;xiki 2:;xi
-
-
n
Z; ki!qi
2:; I/Ui
qi n
k qi/ki k l/ki
Z;ui
ki
n
-n
Measurements at an instant
Z; xiqi
I
143
combined for different times
tlql + t2~
2:;!. ui
and definitions
n qi
Z;si
2:;l/ki
2:;si
-2:;qi/ki
2:;ui
u Thus continuum values for the same roadway taken at adjacent time n n 2:;ki Xiki periods may be combined using a time-weighted average of flow and concentration. Average speed (space-mean) is the harmonic mean or the timeweighted flow divided by the time-weighted concentration. For adjacent sec.* Expressed in terms of ui' and either hi = headwaytime, or 8i = spacing. tions of roadway measured during the same time period continuum values may be combined using a space-weighted average of flow and concentration. AverA more serious question arises if one uses too long a strip of roadway dx age speed (space-mean) is the arithmetic mean or the space-weighted flow as shown by the dashed lines. The number of vehicles which fully cross dx divided by the space-weighted concentration. during T is then only a fraction of those arriving and departing during this These formulas are retabulated in Table I, which includes rules for avertime, and the original definition cannot be used. One definition is not useful for aging in three special cases; namely, for equal slices of time or space, for measurements over a long roadway and the other is not useful for measureequal size groups of vehicles and for the virtual values of single vehicles. ments over a long time. The definitions given originally by Wardrop and by Lighthill and Whitham have been' put into a variety of forms of possible use to traffic engineers and experimenters concerned with different circumstances. The various forms are REVISED DE FINITIONS consistent with the original definitions, but difficulties may arise in applying . the rules. For early investigators, observations made on a short section. of roadway Referring again to Fig. 1, one can see that, for the observation on dx for over time seemed the easiest way to take data and such measurements were time T, vehicles from -7 to +5 inclusive traverse dx. The measured flow is adequate to the development of present-day continuum theories. It is becoming therefore q =l3/T. Vehicle #6 fails to make it across and therefore is not
. ,
144
L. C. Edie
Traffic stream measurements
and definitions
145
In connection with computing averages using the proposed definitions, one increasingly desirable, however, to study behavior over longer sections of road. way for various periods of time. It has also become desirable to observe pIa. ,observes,with reference to the above and Table I, that in all cases flow and toons of vehicles traveling in space and time and to convert these observations 'concentration are combined by using space-time means and speed is always into continuum variables. How to do these precisely is not self-evident. There, ithi!meanflow divided by the mean concentration. Thus to combine n values of appears to be a need for definitions applicable to vehicle trajectories through ;~i'ki,ui for domains of areas ~, the following identical rules apply: any space-time domain. (23) I. aiqi I. aiki L aiqi If the numerator and denominator of equation (1) are multiplied by dx, we q=_'!; ai k=-,L ai u=-' !; aiki obtain ndx (19) I: In connection with generality, one can show, as follows, how the' original q=Tdx' (twosets of definitions can be derived from the proposed definitions. The numerator is the aggregate distance traveled by all n vehicles on dx, .!!. Le., the aggregate vehicular transport, and the denominator is the area of a Z rectangular space-time domain of space length dx and time length T in which :; n they traveled. Equation (2) k = !; dti/Tdx, already gives the concentration as the total time spent on dx by the n vehicles divided by the area of the spacetime domain. Equation (6) q = L dxi/Xdt, gives the flow as the total distance traveled by the n vehicles on X divided by the area of the rectangular space-time domain Xdt. If equation (5) k = n/X is multiplied by dt in both numerator and denominator, it gives the concentration as total time spent divided by the area of the space-time domain. It is, therefore, feasible to combine the original two sets of definitions into .. one set which is independent of methods of measurement. First, we define a space-time domain as any enclosed portion of a space-time plane such as shown in Fig. 1, the area of which is measured in unit rectangles., If we measure space in miles and time in hours, the area of a space-time domain is measured in mile-hours. The flow q of a traffic stream (measured in vehicles/hour), may then be defined as the aggregate distance traveled by all vehicles (measured in vehicle. miles) passing through a space-time domain divided by the area of the domain.
.
4111
I_II
"
..." r 11 I":: j2:: AtI~j "'11 f4-" ~.-II " jIIO!"
,1 '1.==:: i~:~'1
i'~"
,-
~I" ';I ~ ,.
II'
q
I
~
....
(20)
where xi is the distance traveled in the area A by the ith vehicle. The concentration k of a traffic stream (measured in vehicles/mile) is the aggregate time spent by all vehicles (measured in vehicle-hours) in passing through a space-time domain divided by the area of the domain. k
= !;Ati
.. (t T
t-
(21)
Fig. 2 Trajectories
where ti is the time spent in the domain by the ith vehicle. The speed of a traffic stream in a given space-time domain is the aggregate distance traveled divided by the aggregate time spent by all vehicles traversing it, Le.,
--t
of a group of n vehicles over a distance X.
Fig. 2 shows a space-time domain bounded by a section of roadway of (22) length X and by the trajectories of vehicles 0 and n. We assume that one vehicle is just outside this domain and one vehicle is barely inside, or that both are half in 'and half out and count as one vehicle. If the curvature of the trajectoThese definitions are strange in relation to our general concepts of flow ries can be neglected the area is and concentration, but they are consistent with previous definitions, independ(24) ent of particular methods of measurement, explicit and unvarying in methods '" L+L A=TX_X2 of averaging, and have a reduced sensitivity to random errors which result 2 Uo un from measurements assumed to have been made at a point or for an instant. U
(
= L;i
-~ LX' - !; t.1
(
!
)
146
Traffic stream measurements
L. C.Edie
Vehicle
where T is the total time of interest and Uo and un are the average speeds In traversing X for the two boundary vehicles. 2: xi nX q
"I
= x- =
e
1)
~uo X2
.~ dltl _II
-. "
I =~ 4I!'!" C" C~
-II -II
,. """,.
-II ]~II
... " ... , r.:: !~~~ "Ii
... "
'\,,, ,;;
i!
u
r
' ..
I,
2: xi T""f.""
1
~;. Similarly,
n
= -y2:-
,
(27)
)
(
=
2: Xi T""f.""
1
=
X--
n T l
(
-+2uo
2:ul
-+un T C 1 ) ) = X-- 2uo
I
(29)
un 1
1 un
)
. "
f
.c 2000 1
.
(30)I
(31)
When T approaches 0, the above equations become identical with the original definitions for an instant of time. These equations can be used where T is significantly greater than 0 to compute average stream values existing in a platoon of n vehicles for the given time period. The concentration in such a platoon depends only on the speeds of the first and last vehicles.
intervah.
lSOO
2: Ui
= 11
polnU en at 5-..cond
,
1000-
~
~
!..
\
" \ 'rI I
SOO
~ Plow Aero..
lnter..ction
:.....-
,: .. .
APPLICATIONS TO AERIAL OBSERVATION OF PLATOONS
~! ...
Ii
,"-
(t)
(28)
XT-"'I'r-+~uo T2 C
T_
-I Fig. 3 Trajectories of a group of n vehicles over time T. L
2S00
1 1 A=XT-- T2 - +2uoun 2:.! 2:xi ui
q =-r 2:ti k=-r=
Vehicle n
(26)
ui or the harmonic mean speed. Equations (25)-(27) can be used to compute average stream values existing in a platoon of n vehicles traversing a roadway of length X. The flow carried by such a platoon Is independent of the speeds of Intermediate vehicles and depends only on the first and last vehicles, whereas k and u depend on all vehicles. Fig. 3 shows a space-time domain bounded by a period of time T and by the trajectories of vehicles 0 and n. The area and stream characteristics described in Fig. 3 are then:
U If
=
fJ;
)
(
"
.u.
un
which reduces to equation (1), because as X ~ 0; q 2:X 2: ti ui k = A = TX--X2 -1 +-1 2 Uo un. which reduces to equation (8) as X approaches O.
'I
0
E
(25)
TX---n - +-
and definitions
Fig. 4 shows th. behavior within a platoon of 13 vehicles passing through an intersection in one lane for a period of 30 seconds. Each point represents
I
o. o
so
-
" ..... 100 concentration
150 (It)
.
2bO Vehicl../Hl1.
Fig. 4 Platoon flow at an intersection when not completely stopped by the traffic signal.
'SO
147
148
L. C.Edie
Traffic stream measurements
an average over 5'seconds.
and definitions
149
2500
In the first 5 seconds, the first vehicle was just
starting through the intersection on a green light, the next three vehicles were stopped behind the first and the rest of the vehic1.es were approaching those stopped. During the last five seconds all but the last vehicle had crossed or entered the intersection. One sees in this curve a flow-concentration relationship much different from the usual ones for averages at a point over long periods of time. The dashed lines connect points showing the flow-concentration values existing in the intersection. They are interesting by showing a con' stant concentration with flow increasing from 45 vehicles/hour in the first 5 seconds to more than 2000 vehicles/hour in the sixth 5-second period, with an average of 1450.
~ Painu an ae ,~:)ne-"cond lntervall.
2000
1500
2500
.
~ phtaon "T
4811 GI= '1
,
f
Stopping
point. are IncetVah.
~ at
1000
~ ..
S-a.cond
2000
. llatoon
>
Expandlna
500
....
I'~.
~, ,
=:
1500
2:
c:
-. I'-II
$0
. r=:
!!:
.....
;;:
, ... 141 I !J'
I~f: I :.= ~ i lib II' ~ i... II "
" " ~" "",.. JS:'
~.
.~~
I 'fI " '! ~
f
,
...1...
1000
,,/
1\0 (k)
200 -
-
2S0
Vehidu/Mlh
Fig. 6 Platoons compressing and expanding in a traffic wave at the George Washington Bridge.
I I I I
,
500
11)0-Concentration
, ;J.
I
caused most of the vehicles to come to a full stop for periods of one or a few seconds. These curves describe a situation which is partially like that in Fig. 4 and part like Fig.5. It is interesting in its dissymmetry, however, between the J Flow Acr'o" Inhnectlon compression behavior and expansion behavior. In this case the mean deceleration was about 20 per cent higher than at the intersection, a result which might 50 100 1'50 reasonably occur because a driver does not anticipate a full stop in the center Concentution (k) of a bridge as well as he anticipates it at an intersection with a traffic signal. The data used in Figs. 4- 6were obtained by high altitude aerial photography Fig. 5 Platoons stopping and starting at an intersection. with photos taken at 5-second intervals. For Figs. 4 and 5 no smoothing of the data was made. Fig. 6 was developed, however, by drawing smooth curves through the coordinate (x, t) points for each vehicle. The points plotted in Fig. 6 Fig. 5 shows behavior within a platoon of 14 vehicles coming to a stop at are estimates of platoon behavior at one-second intervals. Without averaging an intersection and 9 vehicles starting up from a stopped queue. Both are in space and time, one would not obtain such smooth transitions from second to shown on the same chart to reveal the close ~ymmetry between the two beha- second. At a point, one-second intervals would either pass one vehicle or no viors; thus suggesting similar absolute rates of deceleration and acceleration vehicle so that the second-by-second values of flow would be either 3600 at this location. These dynamic q-k curves are also quite different from vehicles per hour or zero. steady-state curves obtained at a point, but are similar to steady-state curves derived by the Road Research Laboratory from experiments on a closed 100p.1 They agree even more closely with an assumption that speed is a linear funcAPPLICATION TO DATA AT A POINT tion of spacing. Fig. 6 shows behavior within platoon of 6 and 5 vehicles caught in a decele . It is feasible to deal with single vehicles for some purposes. The proposed ration-accelerati0i wave on a lane of the George Washington Bridge. The wav definitions suggest implications not yet explored with respect to the virtual I
I I~
!
150
.
fer.
,,411' ,1
~i
-;1 IC ....
,-
.=: I:=
f
~
I ~
I, Ii II
~
Traffic stream measurements
L. C. Edie
traffic flows of single vehicles. Methods for dealing with single vehicles in some statistical manner enable estimates to be made of average virtual values! which are of considerable interest. Such virtual values are important in studying points of a roadway which cannot be properly studied with real flow values obtained by averaging over time intervals or groups of vehicles. In situations like those existing in tunnels, real flow values may not reveal the roadway capacity at the observed point because they are influenced too much by condItions upstream or downstream from the point of observation. Traffic behavior in a tunnel upstream and downstream from a bottleneck was reported by Edle and Foote some years ago.5 Another reason for the use of virtual values is that averages taken over even a relatively few vehicles in sequence tend to fluctuate considerably and to hide detail effectE; of individual vehicle speeds on rates of flow.. The usual method of computing virtual values is to classify vehicles by speed and to compute the arithmetic mean spacing of the vehicles foundin eachli speed class. The formulas in Table I under single vehicles may be used, and the concentration for vehicles at the same speed becomes simply the reciprocal of the average spacing, independently of whether data is taken at a point in space or a point in time. When virtual values are computed for single vehicles, it is not obvious that the formulas used imply the assignment of a particular space~time domain to each vehicle. For example, the formulas in Table I for a point in space imply the assignment of a space-time domain on the approach or upstream side of tb point of observation. This results from the use of the time headway in front of the vehicle to compute its virtual flow. If, instead, one used the headway behind the vehicle, there is an implicit assumption of a space-time domain on the exit or downstream side of the point. In the first case one is estimating the rate of inflow to the point and in the second the rate of outflow. For an estimate of conditions at the instant the vehicle is observed one would need to average the two headways. Some of the relationships between the headways used in computing virtual flow values and the space-time domains implicit in such use are given in Figs. 7 and 8. Fig.7a illustrates a method of associating two time headways with the vehicle they encompass. Fig.7b associates the speeds Of two adjacent vehicles with the headway time between them. An estimate of the throughflow Fig. 7a involves an arithmetic mean of the two headways or an harmonic mean of the two virtual flows, and in Fig. 7b it involves the use of an arithmetic me of the two transit times or an harmonic mean of the two speeds. Fig. 8 illustrates the type of virtual flows measured by associating variou time intervals with a particular vehicle when both entering and exiting time intervals are observed. It might be noted that conditions within the measured zone a-b are described by the departure headway time hi" rather than the approach headway time hi, although the latter seems to be more generally employed. Figs. 7 and 8 do not include all possible correct ways of computing virtual values for single or paired vehicles. In particular they do not illustrate the way to treat two vehicles as a platoon, but how to do this should be obvious from the previous discussion. . In general, one would expect little difference between one speed-headway combination and another. However, there may be cases where differences are not small or where small differences would be significant. Fig. 9 illustrates one case where the difference between estimated inflow and estimated outflow
I--
\+1--i
1 Outflow of Ut'
iT - - - --~' II I (1-1) I'
(
(1)
--.;;-
ql
78.- Average H.adway A..ociated
with.
:
/
Particular
. +1
Outflow .peed of flow q
(1-1)
'
U
.
't. (1+1). h1+1 --t
: (1)
/ !--
Vehicle (1).
u1_1
_
u-r---r-2 - +-
__ _1_ -' 1 hi -.t Inflow.peedof flowq -~. 7b.- Avarage Speed A..octated
hl+1
hl + hl+1 ~
1 Inflow of ut'
--i
r---'
ql -
h-
(ll1)
_u _L_J h1
and definitions
with.
ul_1
u1
U- u1 Particular
H.adway hi-
Fig. 7 Virtual inflow and outflow for a single measure of headway time at a point. x
t-
to-hI+l -t ';--7--'
h'1
, /
I
b
o
h1
Re8d...y A..ocleted h1 h1+l hI h~+l
nth
Ul
T~e
of Virtual Flow lnto
It out of Flow lnto It
out
Flow Me..ured
Polnt
Polnt of
(e)
.. (b)
It
Fig. 8 Virtual inflow and outflow of vehicle (i) for two measures of headway time.
151
152
L. C.Edie
Traffic stream measurements
2500
GENERAL
and definitions
153
DISCUSSION
8 I
:
2000
1500
~Cr1tl'" ~
i4l1 d~ ,-,.
1000
!..
/
=:~7' 23 H P H
~..;ri~: .
500
~
W'.
InflowV.11l" Outflow
V.l""8
[~~~ '.'~I.
~fr: Ice: I~:" .1=2
,i "'1"
tl=:: ...~ 1:"
,e
I ~
It
I~
Ib
o r o
2>
50 Concentration
75 (It)
100
125
Vehlcl../Hile
Fig. 9 Inflow-outflow analysis of bottleneck at the Holland Tunnel during police waving experiments.
for a given speed may be significant. This figure shows data taken at the bottleneck location in the Holland Tunnel when police were waving vehicles to move faster. Under these circumstances, the sudden drop in flow, previously observed to Occur in tunnels at a critical speed and spacing,6 has been amplified by the police action. An analysis of both inflow and outflow helps to identi fy more closely the conditions under which this sudden drop in flow occurs. It also may help to understand the predominant behavior of vehicles traveling at other speeds. If a traffic stream near capacity flow at a bottleneck is alternately in a state of expansion and compression, successive headways would tend to increase or decrease with time depending On whether the vehicle's speed is above or below optimum. At high speeds compression should produce decreasing headways and expansion increasing ones. For slow speeds the Opposite would be expected. Thus a difference between virtual inflow and outflow suggests an imbalance between expansion and compression, Le., acceleration vs. deceleration. There is one exception and it Occurs at the optimum speed at which either expansion or compression would tend to produce increas ing headways. For high speed one expects a slower foll()wing vehicle and therefore a compression; for slow speeds, the Opposite. Thus, generally, outflow should be greater than inflow at all speeds, other than the critical speed. Fig. 9 is compatible with this hypothesis except for one point, but it should be noted than around critical speed comprise more data than at . critical.that points other
In view of the foregoing it appears that definitions of flow, concentration andspeed which are usable for any kind of observations are feasible and have practical applications. Without them, the computed results of observations maybe biased, erroneous or hide some significant feature of the traffic stream. In reviewing some of the published literature, and some of the com.putationsmade in the past at the Port Authority, one can observe errors of ,twentyper cent or more in averages taken over a number of vehicles or :periods of time for observations at a point. Most of such errors involve the .incorrect use of arithmetic averages when harmonic or time or space averages wouldbe correct. Such errors now arise mostly in averaging values of q and 'ksince Wardrop made it clear2 that speeds observed at a point in space should :beaveraged harmonically. As with speed, the use of arithmetic averages for q andk values also results in overestimates. If the value of flow is derived from a product of u. k, and both u and k are overestimated, a large error can result. In connection with the reduction of random and non-random errors, if the proposed definitions are applied to rectangular space-time domains, the measurement accuracy increases with the number of vehicles involved on each axis. With point or instant observations slight shifts of the time period or space length can affect the count by one or two vehicles causing differences in measurements Of.!., n or! n for n vehicles. However, if one employs a rectangular domain covering several vehicles in both directions a slight shift in any dimension would have a small effect on the measurement. FUTURE WORK
This paper has covered only a small part of the question of traffic stream measurement and definition. The definitions proposed herein for q, k and u suggest a new way of looking at these stream values. The flow q has become theunit rate of vehicle transport in vehicle-miles per mile-hour. In a steady-state condition on a roadway of length X, the rate of transport would be qX, in vehicle-miles per hour, a quantity which might well be defined as the roadway production rate. Over a time T, the total production achieved by such a roadway would be P
= qXT
measured
in vehicle-miles.
Similarly, traffic concentration becomes the unit rate at which delay is accruing to vehicles on the roadway, measured in vehicle-hours per' mile-hour. In a steady-state condition on roadway X, the rate is kX vehicle-hours per hour, and total delay in time T is D = kXT vehicle-hours. These terms might be useful because they are additive. If the production and delay for one roadway are Pl> Dl and for a second are P2, D2' the production and delay for both roadways are P1 + P2, Dl + D2. If one doubles roadway length or width and the production more than doubles, the overall roadway productivity has been increased. Another consideration for future study would be the normalization of flow and concentration for different vehicle lengths so that q would be expressed in miles of vehicle per hour and k in miles of vehicle per mile. Or flow might be better normalized for weight so that q would be mass flow in vehicle-tons per hour and k would be mass concentration in vehicle-tons per mile. Both types of normalizing would tend to reduce fluctuations resulting from different vehicle sizes and weights, but require more sophisticated instrumentation.
154
L. C. Edie
A further thought would be to look at the momentum and kinetic energy behavior in traffic streams. If m is the average mass of a vehicle, the momen'~ turn of a steady-state traffic stream would be mqX in vehicle-ton-miles per Overtaking in free traffic hour. This is the same as the mass transport rate or production for the road. way. The kinetic energy would be l~mquX for a steady state. It is interesting; W. LEUTZBACH AND P. EGERT to speculate whether the maximization of the kinetic energy of the stream may in some circumstances be preferable to a maximization of its momentum or Lehrstuhl und Institut fur Verkehrswesen an der Technischen Hochschule rate of flow. Fridericiana, Karlsruhe, Germany.
,"
.
Dusseldorf, Germany
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ACKNOWLEDGEMENTS
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The author wishes to thank Justin Dickins of the Operations Standards Division and Bob Foote of the Tunnels and Bridges Department for supplying the field data used to develop Figs. 4-6 and 9, and to thank Wesley Hurley and Walter Helly for helpful discussion.
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REFERENCES
ABSTRACT The distributions of speeds are described. The theoretical occur in a traffic stream where tween vehicles is derived. The pared with observations.
of vehicles in space and time rate at which overtakings there is no interference betheoretical results are com-
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(1)
(2)
WARDROP, J. G. Some theoretical aspects of road traffic reasearch. Proc.Instn Civ.Engrs, Part n, London 1952,325-78.
(3)
BUREAU OF PUBLIC ROADS. Highway Capacity Manual Washington, D.C., 1950. (Dept. of Commerce, U.S. Govt. Printing Office). WARDROP, J. G. Experimental speed/flow relations in a single lane. Proc.2nd Int. Symposium on Theory of Road Traffic Flow, Paris, 1965, (O.E.C.D.) pp.l04-19. EDIE, L.C.and R.S.FOOTE. Traffic flow in tunnels. Proc.Highw.Res. Bd, 1958,37, 334-44. EDIE, L. C. Car following and steady-state theory for non-congested traffic. Opns Res., 1961,9 (1),66-76.
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(4)
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LIGHTHILL, M. J. and G. B. WHITHAM. On kinematic waves, n. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. A, 1955,229, 217-345.
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There is "free" road traffic when drivers of vehicles are able to act independently, i.e. when the actions of one driver interfere little or not at all with those of another. Hence "free" describes traffic which is entirely different from the pattern treated in the "Car-following Theory", where it is assumed that motorists behave interdependently. , It is obvious that these definitions describe only border cases. In 1954 Egertl defined the latter type of traffic as "interlinked" traffic. Actual traffic does not correspond to either of the border types. It is neither "free" nor "interlinked" in the true sense of the definitions but partly free and partly interdependent, that is to say, "partly interlinked". Even though science has abolished the principle of "natura non saltat", the above statement remains valid when it cernes to characteristics of a universe of independent events. The theory of free traffic deals with such characteristics. Thus its results may well apply in areas where the actual assumptions for the theory are no more strictly valid. Therefore it is to be expected that these results will retain at least an approximate value in the area of partly interlinked traffic. To give a more detailed definition of "free road traffic" the following conditions should be present:
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(i) Time headways between vehicles should be independent. (ii) Distance headways should be independent. (Iii) Vehicle-speeds should be independent. Both time and distance headways should be d~l$tributed exponentially (number of vehicles should have a poisson distribution in both space and time). M and D are constants. t is a measure of time, and x a distance measured along the road. Then the probability of observing a vehicle at a fixed point within the time interval (t, t + dt) is (1) dp = M . dt + o(dt)
