- Email: [email protected]
A statistical theory of traffic flow on highways—III. Distributions of desired speeds
Transpn Res. Vol. 7, pp. 223-232.
Pergamon Press 1973. Printed in Great Britain
A STATISTICAL THEORY OF TRAFFIC FLOW ON HIGHWAYS-III. DISTRIBUTIONS OF DESIRED SPEEDSt FRANK C. ANDREWS Natural Sciences Bldg. I, University of California, (Received 21 September
Santa Cruz, Calif. 95060
1971; in revised form 16 February
1973)
1. INTRODUCTION CONTRIBUTIONS to the development
of a statistical theory of traffic flow on highways that is reasonably representative of actual traffic and at the same time mathematically tractable have been made by a large number of workers. (See reference list.) Theories of the type of Prigogine and co-workers (1960, 1962) and Munjal and Pahl (1969) and of this author (Andrews, 1970a, b) utilize a distribution of desired speeds for the set of cars on a highway at any given time. This distribution has been treated as independent of density, except perhaps for an ad hoc correction term reflecting drivers’ intentions changing in the direction of the average speed as the density increases (Prigogine et al. (1962), Munjal and Pahl (1969)). However, even in homogeneous flow if no driver changed his desired speed as he encountered varying traffic densities, the desired speed distribution should not be independent of density. This is because, as the density increases, the cars wanting to go fast get slowed down; thus, there is a greater fraction of them on the road than would be the case with dilute flow. In this paper, the distribution of desired speeds for the cars on a road at a given time is related to the distribution of desired speeds for the cars entering the road over a period of time. The distinction is simply that between a time-averaged and a space-averaged distribution. The implications of this for a theory of the density dependence of traffic behavior are discussed. 2.
DENSITY
DEPENDENCE
OF DISTRIBUTION
OF DESIRED
SPEEDS
Given homogeneous unidirectional traffic flow on a fixed section of road. Suppose each driver in the pool of cars (perhaps the nearby city) who uses the road has a fixed, densityindependent image of how fast he would like to drive. Let x’(u) du be the fraction of drivers in that pool who want to drive between v and v + dv. We presume that drivers leave the pool to enter the road at random, their rate of entry (balanced by rate of exit) establishing the traffic density on the road. Then, x”(v) dv is also the fraction of drivers entering the road (or crossing any other fixed point on it) in any suitable time interval that want to drive in dv. It is x’(v) which is independent of density if drivers wishes remain unchanged. 7 This research was performed for the System Development Corporation, Santa Monica, California, and prepared for the U.S. Department of Transportation, Federal Highway Administration, under Contract No. FH-11-7628. The substance of this report also appears in System Development Corporation Technical Memorandum TM-4638/006/00, July, 1971. 223 T.R.7/3-B
224
C.
FRANK
ANDREWS
We and all previous investigators have instead taken 4”(v) to be density-independent, where 4”(v) dv is the fraction of cars on the road that want to be going in dv. The difference between x0 and +” is that between time- and space-averages. Consider a simple example. Suppose half the drivers in a city want to go 30 m.p.h. on a section of road and half want to go 60 m.p.h. Suppose these speeds are independent of actual traffic density. At very low density, for each mile of road, half the cars will take 2 min and half will take 1 min. Thus, there will be twice as many cars on the road going 30 m.p.h. as going 60 m.p.h. The distribution x0 shows 0.50 of the cars at 30 m.p.h. and 0.50 at 60 m.p.h., but the distribution 4” shows 0.67 of the cars at 30 m.p.h. and 0.33 at 60 m.p.h. Now, at a moderate density suppose the cars wanting to go 30 m.p.h. were slowed to an average of 20 m.p.h. and those wanting to go 60 were slowed to an average of 30 m.p.h. The distribution x0 would of course be unchanged, but half the cars would now take 3 min to cover each mile and half would take 2 min. Thus 4” would show 0.60 of the cars wanting to go 30 m.p.h. and 0.40 wanting to go 60 m.p.h. Under jam conditions all cars are going the same slow speed, so 4” and x0 would show the same distribution, 0.50 wanting to go 30 m.p.h. and O-50 wanting to go 60 m.p.h. Thus, d”(v) always changes with density from its low-density value which is proportional to x”(v)v, to its jam-density value which is identical to x’(v). There is a dramatic increase in the fraction of cars desiring to go fast, and a decrease in the fraction wanting to go slow as the density increases to jam conditions.
3. RELATION
BETWEEN
x0 AND
4”
We now compute the relationship between x0 and +” in terms of queue densities defined (Andrews, 1970b) as follows: L-lg,(v,, vz,. .., v,) dvI dv,. . . dv, is the density of n-car queues on the road of length L moving with speed between v1 and v1 + dv,, in which the second car in the queue wants to go in dv 2, the third wants to go in dug, etc. Conservation of cars wanting to go v permits writing N+“(v) in terms of queues of ever greater lengths: m N4”(v) = gl(v) +
dv, gz(v, vl) +
s ”
dv, g,(v,,
v) +
s
dv2 g&,, s 01
dv,
"
0, ~2) +
dv,g&,
dv, s 0
m I
dv, g,(v, ~1, v,)
v
m
"
00 dv,
s 0
J’
0 "
+
m
”
7~2,~) +
...
(1)
s "1
The first term on the right-hand side of Equation 1 (multiplied by dv) is the number of single cars going their desired speed, between v and v + dv. The second term (times dv) is the number of cars going in their desired speed range, dv, at the head of 2-car queues. The third term is the number of cars wanting to go in dv but actually trailing in 2-car queues going any speed less than v. The fourth term is the number of cars going in dv at the head of 3-car queues, etc. Let tfo(v) dv be the average number of cars desiring to go between v and v + dv entering a long road in the time t. The average total number of cars (going all speeds) that cross any
225
A statistical theory of traffic flow on highways-III
fixed point on the road (including the entrance to the road) per unit time (given the symbol w) is m w=
dvf”(v).
(2)
s
0
Thus, x0 is given in terms off” simply by f “(4 m
x0(v) = s
= f “(4
(3)
w
dvf “(v)
0
Now, if L-’ g dv is the density of objects moving with speed between v and v + dv, such objects move a distance vt in the time t; therefore, the average number of them that pass a fixed point in the time t is vtL-lg dv. Conservation of cars wanting to go v that pass any fixed point on the road (including the entrance to the road) permits our writingf”(v) in terms of queues of ever greater length, in a manner analogous to Equation 1: f”(v) = wx”(v) = L-l gl(v)v+ L-’
J
dv, gz(v, vJv + L-’
m
+
L-l
m
dv,
s lJ L-l
s
s
dv,
0
s
dvz g,(v,, v, v&l
"1
m
0 +
dv, g,(v,, v)vl
m
”
dv, g,(v, ~1, vz)v + L-l
”s
J
0
”
dvl 0
s
dv, g,(v,, vz, v)vl + . . .
(4)
"1
where we have cancelled t dv from each term. The left-hand side of Equation 4 (times t dv) is the expected total number of cars wanting to go in dv that pass any fixed point in time t. The first term on the right-hand side is the number of these which are single cars going v. The second term is the number of these which lead 2-car queues going v. The third term is the number of these which trail in 2-car queues going all speeds slower than v; etc. If we now divide Equation 4 through by L-’ v, the resulting left-hand side can be added to the right-hand side of Equation 1, while the resulting right-hand side is subtracted from the right-hand side of Equation 1. The final result is
+fdv, 0
(1
-_%)g,(o,,v)+Sda, jdv~!l -$&>v,vz)
0
VI
(5)
226
FRANK C. ANDREWS
where we have replacedf”(u) by WX’(U) from Equation 3. The generalization of Equation 5 to include queues of arbitrary length is immediate. It shows the relationships between the distribution of desired speeds C+“(U)for the cars on the road (a density-dependent quantity) and the rate w at which cars enter the road and the distribution of desired speeds x’(u) for the cars using the road (a density-independent quantity). It is noteworthy that since x’(u) is independent of density, one might as well determine it under very dilute traffic conditions where each car is essentially going its desired speed. Under these circumstances, Equation 4 is approximately f”(v)
= L-1 g,(v)0
(6)
(P + 0)
and x”(v) is (7) (’ + ‘) 0
0
In Equation 7, Equation 6 was substituted; in Equation 8 it was recognized that so long as each car was moving singly, gl(u) is approximately equal to N+“(v); here, in the limit of vanishing density, I+“(V) is labeled &O(2)>. The final step simply defines ii” to be the average speed of the cars on the road (i.e. space-average) in the limit of low density. The result, Equation 8, must of course be independent of density. 4.
IMPLICATIONS
FOR
A THEORY
OF
TRAFFIC
FLOW
In Paper I, the beginnings were laid for a detailed statistical theory of traffic flow based on the density-independence of $“(a) [Andrews (1970a)l. Since no queuing arose in that paper, its results are unaffected by the conclusions of this paper. In Paper II [Andrews (1970b)], however, queuing arose, and formal expressions were found for the densities of various kinds of 2-car queues in terms of the density of l-car queues. Those results would in no sense be changed by the conclusions of this paper. However, in the examples (Sections 8 and 9) of Paper II, once the densities of queues were found, the space-average velocity 0 was found from Equation I-6 using Equation II-45 or Equation 11-55, which implies that 4”(v) is density-independent. Instead, since 4”(v) really does change with density, it would be better to study how the time average speed (v) (i.e. the average speed of the cars that pass a fixed point over a period of time) is affected by queuing. For this purpose it is convenient to define tf(v) dv to be the expected number of cars moving with velocity between a and u + dv across any fixed point in the time t. In terms of queue densities, we can immediately write m m 00 f(v)
= L-l gl(u)u + 2L-l
dv, g,(u, ol, UJD + . . . (9)
dv,
da, g,(v, vl)v + 3L-’ s ”
s ”
s 0
227
A statistical theory of traffic flow on highways-III
in terms of which (u) is m dv!f(v)u
( u) = w-l
(10)
s 0
since w-‘f(u) dv is the fraction of cars crossing convenient to call this fraction X(V) dv:
x(v)= 7 f(v)
We can writef(v),
j- daf(z$
~dvf”(~) 0
4, asf”(v)
m
s
plus a correction: to
0~
- L-’
do, g,(v,, v)vr + 2L-’ s
s
"
du, g,(a,, 0, v&l
- L-l
s
dv,g,(u,u,,v,)u s
"
co
du,
0
VI
00
do, I
"
a,
dv, 0
= ,’
0
" -L-l
(11)
”
da, g&,
s ”
f(u)
f(u)
=
0
using Equation
f(v) =.!-‘=(a) + L-l
a fixed point whose speed lies in dv. It is
du, g,(v,, 02, a)vl +
s
...
(12)
vi
This is completely analogous to Equation 11-45. The second term involving g, represents cars wanting to go v, but forced to slow down and queue behind cars going all speeds less than v. The first term involving g, represents an addition to the total number of cars going v caused by faster cars being forced to queue behind slower cars at v. Thus, we are visualizing a theory in which the entire input consists of the following: (1) The function x’(v), which describes the population of cars using the road; (2) Details of the car passing and queuing procedures, which also describe the population of cars using the road; and (3) w, which is no more than the reciprocal of the average time headway and which implicitly fixes the density of cars on the road. In Paper II, g, was expressed in terms of g, ; it now is necessary to express g, in terms of the input to the theory. For that purpose, Equation 4 may be used to whatever degree of accuracy is demanded. To lowest order in density, Equation 4 yields simply
Lf"(v) Lwx”(u)
g1(v) = 5. FIRST
= -.
V
EXAMPLE
To see how this works out, consider Equation II-46 for D, one uses Equations
(P +
V
OF PAPERS
00
dvvf”(v) + w--l L-l I
0 co ,. -1
L-1
II
of
dv, g,(v, ul) s
"
,"
JdvvJdv, g,(u,, vh
0
I and II. Instead
m
duo2 s
0
--w
I AND
(13)
the first example of Papers 10 and 12 to obtain
m ( v) = w-l
0)
0
(14)
228
C. ANDREWS
FRANK
Now, in the first example, I$=”was chosen to be (D, - D1)-’ for D1 6 v < Dz and zero elsewhere. If we use that instead as the limiting value of +a’ as p+O, thenf,“(v) =L-lN~ao (v)v in the limit p-+0, orfao(v) = L-lN(D, - D,)-l in the range D, < v < D,. We can find x’(v) from Equation 8: L-’ N(Dl - D1)-l v x0(v) =
2v
D,
= (Dz2 -
dvL-l N(Dz - D&-l
(15)
D12)
v
I Dl so the first example is redefined as follows: given that X?(V)= ~v/(D,~ - D12) in the range D1 < v < D, and is zero outside that range, and given a mechanism for passing as specified in Paper II, find (v) as a function of W, correct to the first term in which queuing plays a role.
The first term in Equation 14, (v)“, is immediate:
=D[.(D2:O”ll;)p) =8($I$).
(v)”
(16)
The remainder of Equation 14 is D, w -1 L-l
dvv2 s
dv, g,(v, vl) - w-l L-’ s
”
Dl
0
D,
D,
s
Dl
dvu s
dv, v1 g,(v,, v).
(17)
Dl
If the order of integration is interchanged in the second integral, this becomes D, D2 D, D, w -IL-’ s
Dl
dvv2
dvl g,(v, VJ - w-l L-’ s
I Dl
”
dv, vl
s “1
dvv g,(vl, v).
(18)
If the dummy integration variables in the second integral are interchanged, this becomes D, --w -1 L-l
.r D,
D,
dv
I ”
dv, v(vr - v) g,(v, vr).
(19)
If the dummy variables are changed from v and v1 to v and x = v1 - v, this becomes D, -1
--w
L-1
D,-V
dv s
Dl
dxvxg,(v, x).
(20)
s 0
In terms of v and x, Equation II-42 shows g,(v, x) = 3 Np’(D,
a result which was obtained
-
D,)-3
x
D, -
(d,,, + h)2 In ___
v
+ do,,2 In F
X
by replacing g, 3 by (N+“)3 = N3(D2 - D1)-3.
1 (21) However,
A statistical theory of traffic flow on highways-III
229
we would here replace g13 by L3w3x03v-3, using Equation 13, so for our purposes, right-hand side of Equation 21 should be multiplied by L3~3~“3v-3N-3(D2 - D,)3: g,(v, x) = 4Lw3x(Dz2 -
D2 (d,,, + A)’ In ___
D12)-3
v
+ doUt2In 2 E
X
If this is used in Equation
(22)
20, the result is
D1
(Dz2 -
.
D,2)-3 j’du~~x~xz[(d...+h)21n~+d,.‘ln~]
- 4wz(Dz2 -
= IF’
1
the
(23)
0
D12)-3 j.di.(Dl D1
-
v)" [(d,,,
+ h)2 -
dou: + 3d,,,Z ln f (Dz -
v)
1
(24) so, finally, < u> =
< v >” - ;
((4.t+4’
(D, - Dl) + 2d,,, + 3d,,‘? In A [E
(h2 - D12V3
(25) X(D2
-
(40,
DA4
which may be compared
+ Dz) + &
with Equations
6. SECOND
(D,’
-
D,2)-3 (160,
+ 9D2) d,,,
11-51.
EXAMPLE
OF PAPERS
I AND
II
In the second example of Papers I and II, only the three speeds cl, c2 andc, wereavailable to the cars, and in the limit of vanishing density one-third of the cars on the road want to go each speed. Thus, $p is the density of cars going each of the three speeds (to the lowest order in p). Thus 0J = 4 P(C1 + cz +
c3)
(26)
and we can replace p in Equations II-53 and II-54 by 3w/(c, + c2 + c3) in order to obtain a function of the appropriate variable, w. This means that the density of type-1 queues is
3 w3h
and the density
of type-II
+
~2 +
c3F3
N&,
3) +
4” (E)
(27)
queues is
4 w3(c1
+
c2 +
c3)-'
d,,,2(1,2)
Thus, instead of a density of w/(cl + c2 + c3) for cars going each of the three speeds, Equation 27 gives the density enhancement for cars going ci that want to go c2, and Equation 28 gives the density enhancement for cars going c1 that want to go c3.
230
FRANK C. ANDREWS
In the absence
of queuing, ( v)”
In the presence
=
of queuing,
the time average
speed is
em (Cl2 t_ c**+ cs2) =
(Cl2+ Q2 + cs2) (Cl +
c2
(29)
Gr’
this becomes A]2
(c*- 1 cl
c3 -
Cl
(30) which may be compared
with Equation
7.
H-55.
CONCLUSIONS
The approach of previous statistical theories of traffic flow on highways was to start with the following as input: d”(v) which describes the desired speeds of the drivers on the road and is taken to be independent of density p of cars on the road, and some kind of detailed picture of the passing and queuing behavior on the road. From this, one hoped to calculate the relative probabilities of various kinds of configurations of cars including queues of various types, and also to calculate the average speed of the cars on the road (i.e. the spaceaverage speed), all as functions of the density p. We now see that, since 4”(v) is expected not to be independent of p, the above approach is inappropriate. Instead, one should start with the following as input: x’(v) which describes the desired speeds of the drivers who use the road in a given period of time and which is taken to be independent of density, w which is the total number of cars which cross a fixed point on the road in unit time, and some kind of detailed picture of the passing and queuing behavior on the road. From this, one hopes to calculate the relative probabilities of various kinds of configurations of cars including queues of various types, and also to calculate the average speed of the cars using the road in a given time (i.e. the time-average speed), all as functions of w. The differences between the two approaches are significant. The distribution of desired speeds for the cars actually on a road will change with increasing density in such a way as to increase the fraction that want to go fast and decrease the fraction that want to go slow. Since the pressure to restore traffic to higher speeds depends on the high-speed region of #J”(V),there is always greater pressure under dense traffic conditions than one would expect from a naive theory for which 4”(v) was independent of p. This additional pressure will tend to keep traffic flowing faster, and thus will keep the q-k curve above its naive value, thus keep it nearly linear longer than would otherwise be predicted. Any serious statistical theory of traffic flow of the type we are considering will have to take account of this phenomenon. We have contrasted the results of the two approaches within the framework of a detailed theory previously presented, which treats traffic at densities so dilute that only 2-car queues need be considered.
A statistical theory of traffic flow on highways-III
231
REFERENCES ANDREWSF. C. (1970a). A statistical theory of traffic flow on highways-I. Steady-state flow in low-density limit. Trunspn Res. 4, 359-366. ANDREWSF. C. (1970b). A statistical theory of traffic flow on highways-II. Three-car interactions and the onset of queuing. Transpn Res. 4, 367-377. BARTLETTM. S. (1957). Some problems associated with random velocity. Pubis. Inst. Statist. Uniu. P&s 6, 261-270.
BREIMANLEO (1969). Point and trajectory processes in one-way traffic flow. Trunspn Res. 3, 251-264. CARLESONL. (1957). En matematisk model1 for landsvagstrafik. Nordisk Mat. Tidskr. 5,17&180. COWAN R. J. (1971). A road with no overtaking. Aastral. J. Statist. 13 (2), 94-116. GORDONW. J. and NEWELLG. F. (1964). Eauilibrium analysis of a stochastic model of traffic flow. Proc. Carnb. Phil. Sot. 60, 227-236.
.
_
HODGSONV. (1968). The time to drive through a no-passing zone. Trans. Sci. 2,252-264. HOLLANDH. J. (1967). A stochastic model for multilane traffic flow. Trans. Sci. 1, 184-205. MILLER A. J. (1959). Traffic flow treated as a stochastic process. Symposium on the theory of traffic flow, Detroit, publ. in Theory of Traffic Flow (1961), Elsevier Publ. Co., Amsterdam, 165-174. MILLERA. J. (1962). Road traffic flow considered as a stochastic process. Proc. Camb. Phil. Sot. 58,312-325. Corrigendum 59, 508 (1963). MORSEP. M. and YAFFEH. J. (1971). A aueuinn model for car oassinrr. Trans. Sci. 5.48-63. MUNJAL P. and PAHL J. (1969): An analysis of the Boltzmann-type statistical models. for multi-lane traffic flow. Transpn Res. 3, 151-163. NEWELLG. F. (1955). Mathematical models for freely-flowing highway traffic. Opns. Res. 3, 176-186. NEWELLG. F. (unpublished). Since the publication of Andrews (1970a, b) the author has learned that similar equations had previously been derived by Newell, but had not been published. NORMANN0. K. (1942). Results of highway-capacity studies. Public Roads 23, 57-81. PRIGOGINEI. and ANDREWSF. C. (1960). A Boltzmann-like approach for traffic flow. Opns. Res. 8,789-797. PRIG~GINEI., RFXBOISP., HERMANR. and ANDERSENR. L. (1962). On a generalized Boltzmann-like approach for traffic flow. Bull. Acad. r. Belg. Cl. Sri. 48, 805-814. WARDROPJ. G. (1952). Some theoretical aspects of road traffic research. Proc. Znstn. Civ. Engrs. 2, 325-378.
Abstract-The development of statistical theories of traffic flow on highways has often started with a model in which each driver retains a fixed desired speed. This has been taken to mean that the distribution of desired speeds for the cars on the road at a given time was independent of density. However, it is shown that this model really implies that the desired speed distribution for the cars crossing any fixed point on the road in a certain period of time will be densityindependent. The relation between the two distributions is computed, and the effect of the distinction on a theory of the density dependence of traffic behavior is discussed. The important input for a statistical theory of traffic flow seems to be a description of the desired speeds of the cars using the road in a period of time and a model of passing and queuing behavior. The number of cars crossing a fixed point in unit time appears to be more basic in such a theory than the density of cars.
R&urn&-L’elaboration de theories statistiques sur les flux de circulation sur les autoroutes a souvent pris comme point de depart un modele d’apres lequel chaque conducteur observe une vitesse souhaitable fixt. Cette hypothtse a et& adopt&e pour traduire le fait que la distribution des vitesses souhaitables pour les vthicules sur la route a un instent donnt, Btait independante de la densitd. Cependant, on dtmontre qu’en fait ce modele entraine que la distribution des vitesses souhaitables pour les vehicules traversant un point quelconque t?xe sur la route, durant un intervalle de temps donne, sera independante de la densite. La relation entre les deux distributions est calculee et l’on discute en outre de l’effet de la distinction sur une theorie de la dependance de la densite pour le comportement de la circulation. Les don&es importantes necessaires a l’btablissement d’une theorie statistique du flux de circulation se rev&lent 6tre la description des vitesses souhaitables des vehicules utilisant la route durant un intervalle de temps don& et un modele de comportement de franchissement et de file d’attente. Le nombre de vehicules passant par un point fixe don& durant l’unitt de temps apparait comme plus fondamental pour une telle theorie que la densite de vehicules.
232
FRANK C. ANDREWS Zusammenfassung-Die Entwicklung stochastischer Theorien des Verkehrsablaufs auf StraBen ist oft von einer Modellvorstellung ausgegangen, bei der jeder Fahrer eine feste Wunschgeschwindigkeit einhalt. Man hat dies so Interpretiert, dal3 die Verteilung der Wunschgeschwindigkeiten fur alle Fahrzeuge auf den betrachteten Streckenabschnitt zu einem gegebenen Zeitpunkt dichteunabhlngig ist. Die Modellvorstellung beinhaltet aber in Wirklichkeit, da13 die Verteilung der Wunschgeschwindigkeiten dichteunabhangig ftir die Zahrzeuge ist, die an einem beliebigen StraBenquerschnitt wahrend eines bestimmten Zeitintervalls beobachtet werden. Die Beziehungen zwischen den beiden Verteilungen werden berechnet. Weiterhin wird untersucht, welche Auswirkungen eine Unterscheidung auf eine Theorie der Dichteabhlngigkeit des Verkehrsverhaltenshaben kann. Wichtige F&gangsgrijDen fiir eine stochastische Theorie des Verkehrsflusses diirften die Wunschgeschwindigkeiten der Fahrzeuge w&rend eines Zeitintervalls sowie ein Model1 des Uberpolund Kolonnenfahr-verhaltens sein. Die Starke des Fahrzengstroms scheint im Rahmen einer solchen Theorie grundlegendere Bedeutung zu haben als die Verkehrsdichte.
Pergamon Press 1973. Printed in Great Britain
A STATISTICAL THEORY OF TRAFFIC FLOW ON HIGHWAYS-III. DISTRIBUTIONS OF DESIRED SPEEDSt FRANK C. ANDREWS Natural Sciences Bldg. I, University of California, (Received 21 September
Santa Cruz, Calif. 95060
1971; in revised form 16 February
1973)
1. INTRODUCTION CONTRIBUTIONS to the development
of a statistical theory of traffic flow on highways that is reasonably representative of actual traffic and at the same time mathematically tractable have been made by a large number of workers. (See reference list.) Theories of the type of Prigogine and co-workers (1960, 1962) and Munjal and Pahl (1969) and of this author (Andrews, 1970a, b) utilize a distribution of desired speeds for the set of cars on a highway at any given time. This distribution has been treated as independent of density, except perhaps for an ad hoc correction term reflecting drivers’ intentions changing in the direction of the average speed as the density increases (Prigogine et al. (1962), Munjal and Pahl (1969)). However, even in homogeneous flow if no driver changed his desired speed as he encountered varying traffic densities, the desired speed distribution should not be independent of density. This is because, as the density increases, the cars wanting to go fast get slowed down; thus, there is a greater fraction of them on the road than would be the case with dilute flow. In this paper, the distribution of desired speeds for the cars on a road at a given time is related to the distribution of desired speeds for the cars entering the road over a period of time. The distinction is simply that between a time-averaged and a space-averaged distribution. The implications of this for a theory of the density dependence of traffic behavior are discussed. 2.
DENSITY
DEPENDENCE
OF DISTRIBUTION
OF DESIRED
SPEEDS
Given homogeneous unidirectional traffic flow on a fixed section of road. Suppose each driver in the pool of cars (perhaps the nearby city) who uses the road has a fixed, densityindependent image of how fast he would like to drive. Let x’(u) du be the fraction of drivers in that pool who want to drive between v and v + dv. We presume that drivers leave the pool to enter the road at random, their rate of entry (balanced by rate of exit) establishing the traffic density on the road. Then, x”(v) dv is also the fraction of drivers entering the road (or crossing any other fixed point on it) in any suitable time interval that want to drive in dv. It is x’(v) which is independent of density if drivers wishes remain unchanged. 7 This research was performed for the System Development Corporation, Santa Monica, California, and prepared for the U.S. Department of Transportation, Federal Highway Administration, under Contract No. FH-11-7628. The substance of this report also appears in System Development Corporation Technical Memorandum TM-4638/006/00, July, 1971. 223 T.R.7/3-B
224
C.
FRANK
ANDREWS
We and all previous investigators have instead taken 4”(v) to be density-independent, where 4”(v) dv is the fraction of cars on the road that want to be going in dv. The difference between x0 and +” is that between time- and space-averages. Consider a simple example. Suppose half the drivers in a city want to go 30 m.p.h. on a section of road and half want to go 60 m.p.h. Suppose these speeds are independent of actual traffic density. At very low density, for each mile of road, half the cars will take 2 min and half will take 1 min. Thus, there will be twice as many cars on the road going 30 m.p.h. as going 60 m.p.h. The distribution x0 shows 0.50 of the cars at 30 m.p.h. and 0.50 at 60 m.p.h., but the distribution 4” shows 0.67 of the cars at 30 m.p.h. and 0.33 at 60 m.p.h. Now, at a moderate density suppose the cars wanting to go 30 m.p.h. were slowed to an average of 20 m.p.h. and those wanting to go 60 were slowed to an average of 30 m.p.h. The distribution x0 would of course be unchanged, but half the cars would now take 3 min to cover each mile and half would take 2 min. Thus 4” would show 0.60 of the cars wanting to go 30 m.p.h. and 0.40 wanting to go 60 m.p.h. Under jam conditions all cars are going the same slow speed, so 4” and x0 would show the same distribution, 0.50 wanting to go 30 m.p.h. and O-50 wanting to go 60 m.p.h. Thus, d”(v) always changes with density from its low-density value which is proportional to x”(v)v, to its jam-density value which is identical to x’(v). There is a dramatic increase in the fraction of cars desiring to go fast, and a decrease in the fraction wanting to go slow as the density increases to jam conditions.
3. RELATION
BETWEEN
x0 AND
4”
We now compute the relationship between x0 and +” in terms of queue densities defined (Andrews, 1970b) as follows: L-lg,(v,, vz,. .., v,) dvI dv,. . . dv, is the density of n-car queues on the road of length L moving with speed between v1 and v1 + dv,, in which the second car in the queue wants to go in dv 2, the third wants to go in dug, etc. Conservation of cars wanting to go v permits writing N+“(v) in terms of queues of ever greater lengths: m N4”(v) = gl(v) +
dv, gz(v, vl) +
s ”
dv, g,(v,,
v) +
s
dv2 g&,, s 01
dv,
"
0, ~2) +
dv,g&,
dv, s 0
m I
dv, g,(v, ~1, v,)
v
m
"
00 dv,
s 0
J’
0 "
+
m
”
7~2,~) +
...
(1)
s "1
The first term on the right-hand side of Equation 1 (multiplied by dv) is the number of single cars going their desired speed, between v and v + dv. The second term (times dv) is the number of cars going in their desired speed range, dv, at the head of 2-car queues. The third term is the number of cars wanting to go in dv but actually trailing in 2-car queues going any speed less than v. The fourth term is the number of cars going in dv at the head of 3-car queues, etc. Let tfo(v) dv be the average number of cars desiring to go between v and v + dv entering a long road in the time t. The average total number of cars (going all speeds) that cross any
225
A statistical theory of traffic flow on highways-III
fixed point on the road (including the entrance to the road) per unit time (given the symbol w) is m w=
dvf”(v).
(2)
s
0
Thus, x0 is given in terms off” simply by f “(4 m
x0(v) = s
= f “(4
(3)
w
dvf “(v)
0
Now, if L-’ g dv is the density of objects moving with speed between v and v + dv, such objects move a distance vt in the time t; therefore, the average number of them that pass a fixed point in the time t is vtL-lg dv. Conservation of cars wanting to go v that pass any fixed point on the road (including the entrance to the road) permits our writingf”(v) in terms of queues of ever greater length, in a manner analogous to Equation 1: f”(v) = wx”(v) = L-l gl(v)v+ L-’
J
dv, gz(v, vJv + L-’
m
+
L-l
m
dv,
s lJ L-l
s
s
dv,
0
s
dvz g,(v,, v, v&l
"1
m
0 +
dv, g,(v,, v)vl
m
”
dv, g,(v, ~1, vz)v + L-l
”s
J
0
”
dvl 0
s
dv, g,(v,, vz, v)vl + . . .
(4)
"1
where we have cancelled t dv from each term. The left-hand side of Equation 4 (times t dv) is the expected total number of cars wanting to go in dv that pass any fixed point in time t. The first term on the right-hand side is the number of these which are single cars going v. The second term is the number of these which lead 2-car queues going v. The third term is the number of these which trail in 2-car queues going all speeds slower than v; etc. If we now divide Equation 4 through by L-’ v, the resulting left-hand side can be added to the right-hand side of Equation 1, while the resulting right-hand side is subtracted from the right-hand side of Equation 1. The final result is
+fdv, 0
(1
-_%)g,(o,,v)+Sda, jdv~!l -$&>v,vz)
0
VI
(5)
226
FRANK C. ANDREWS
where we have replacedf”(u) by WX’(U) from Equation 3. The generalization of Equation 5 to include queues of arbitrary length is immediate. It shows the relationships between the distribution of desired speeds C+“(U)for the cars on the road (a density-dependent quantity) and the rate w at which cars enter the road and the distribution of desired speeds x’(u) for the cars using the road (a density-independent quantity). It is noteworthy that since x’(u) is independent of density, one might as well determine it under very dilute traffic conditions where each car is essentially going its desired speed. Under these circumstances, Equation 4 is approximately f”(v)
= L-1 g,(v)0
(6)
(P + 0)
and x”(v) is (7) (’ + ‘) 0
0
In Equation 7, Equation 6 was substituted; in Equation 8 it was recognized that so long as each car was moving singly, gl(u) is approximately equal to N+“(v); here, in the limit of vanishing density, I+“(V) is labeled &O(2)>. The final step simply defines ii” to be the average speed of the cars on the road (i.e. space-average) in the limit of low density. The result, Equation 8, must of course be independent of density. 4.
IMPLICATIONS
FOR
A THEORY
OF
TRAFFIC
FLOW
In Paper I, the beginnings were laid for a detailed statistical theory of traffic flow based on the density-independence of $“(a) [Andrews (1970a)l. Since no queuing arose in that paper, its results are unaffected by the conclusions of this paper. In Paper II [Andrews (1970b)], however, queuing arose, and formal expressions were found for the densities of various kinds of 2-car queues in terms of the density of l-car queues. Those results would in no sense be changed by the conclusions of this paper. However, in the examples (Sections 8 and 9) of Paper II, once the densities of queues were found, the space-average velocity 0 was found from Equation I-6 using Equation II-45 or Equation 11-55, which implies that 4”(v) is density-independent. Instead, since 4”(v) really does change with density, it would be better to study how the time average speed (v) (i.e. the average speed of the cars that pass a fixed point over a period of time) is affected by queuing. For this purpose it is convenient to define tf(v) dv to be the expected number of cars moving with velocity between a and u + dv across any fixed point in the time t. In terms of queue densities, we can immediately write m m 00 f(v)
= L-l gl(u)u + 2L-l
dv, g,(u, ol, UJD + . . . (9)
dv,
da, g,(v, vl)v + 3L-’ s ”
s ”
s 0
227
A statistical theory of traffic flow on highways-III
in terms of which (u) is m dv!f(v)u
( u) = w-l
(10)
s 0
since w-‘f(u) dv is the fraction of cars crossing convenient to call this fraction X(V) dv:
x(v)= 7 f(v)
We can writef(v),
j- daf(z$
~dvf”(~) 0
4, asf”(v)
m
s
plus a correction: to
0~
- L-’
do, g,(v,, v)vr + 2L-’ s
s
"
du, g,(a,, 0, v&l
- L-l
s
dv,g,(u,u,,v,)u s
"
co
du,
0
VI
00
do, I
"
a,
dv, 0
= ,’
0
" -L-l
(11)
”
da, g&,
s ”
f(u)
f(u)
=
0
using Equation
f(v) =.!-‘=(a) + L-l
a fixed point whose speed lies in dv. It is
du, g,(v,, 02, a)vl +
s
...
(12)
vi
This is completely analogous to Equation 11-45. The second term involving g, represents cars wanting to go v, but forced to slow down and queue behind cars going all speeds less than v. The first term involving g, represents an addition to the total number of cars going v caused by faster cars being forced to queue behind slower cars at v. Thus, we are visualizing a theory in which the entire input consists of the following: (1) The function x’(v), which describes the population of cars using the road; (2) Details of the car passing and queuing procedures, which also describe the population of cars using the road; and (3) w, which is no more than the reciprocal of the average time headway and which implicitly fixes the density of cars on the road. In Paper II, g, was expressed in terms of g, ; it now is necessary to express g, in terms of the input to the theory. For that purpose, Equation 4 may be used to whatever degree of accuracy is demanded. To lowest order in density, Equation 4 yields simply
Lf"(v) Lwx”(u)
g1(v) = 5. FIRST
= -.
V
EXAMPLE
To see how this works out, consider Equation II-46 for D, one uses Equations
(P +
V
OF PAPERS
00
dvvf”(v) + w--l L-l I
0 co ,. -1
L-1
II
of
dv, g,(v, ul) s
"
,"
JdvvJdv, g,(u,, vh
0
I and II. Instead
m
duo2 s
0
--w
I AND
(13)
the first example of Papers 10 and 12 to obtain
m ( v) = w-l
0)
0
(14)
228
C. ANDREWS
FRANK
Now, in the first example, I$=”was chosen to be (D, - D1)-’ for D1 6 v < Dz and zero elsewhere. If we use that instead as the limiting value of +a’ as p+O, thenf,“(v) =L-lN~ao (v)v in the limit p-+0, orfao(v) = L-lN(D, - D,)-l in the range D, < v < D,. We can find x’(v) from Equation 8: L-’ N(Dl - D1)-l v x0(v) =
2v
D,
= (Dz2 -
dvL-l N(Dz - D&-l
(15)
D12)
v
I Dl so the first example is redefined as follows: given that X?(V)= ~v/(D,~ - D12) in the range D1 < v < D, and is zero outside that range, and given a mechanism for passing as specified in Paper II, find (v) as a function of W, correct to the first term in which queuing plays a role.
The first term in Equation 14, (v)“, is immediate:
=D[.(D2:O”ll;)p) =8($I$).
(v)”
(16)
The remainder of Equation 14 is D, w -1 L-l
dvv2 s
dv, g,(v, vl) - w-l L-’ s
”
Dl
0
D,
D,
s
Dl
dvu s
dv, v1 g,(v,, v).
(17)
Dl
If the order of integration is interchanged in the second integral, this becomes D, D2 D, D, w -IL-’ s
Dl
dvv2
dvl g,(v, VJ - w-l L-’ s
I Dl
”
dv, vl
s “1
dvv g,(vl, v).
(18)
If the dummy integration variables in the second integral are interchanged, this becomes D, --w -1 L-l
.r D,
D,
dv
I ”
dv, v(vr - v) g,(v, vr).
(19)
If the dummy variables are changed from v and v1 to v and x = v1 - v, this becomes D, -1
--w
L-1
D,-V
dv s
Dl
dxvxg,(v, x).
(20)
s 0
In terms of v and x, Equation II-42 shows g,(v, x) = 3 Np’(D,
a result which was obtained
-
D,)-3
x
D, -
(d,,, + h)2 In ___
v
+ do,,2 In F
X
by replacing g, 3 by (N+“)3 = N3(D2 - D1)-3.
1 (21) However,
A statistical theory of traffic flow on highways-III
229
we would here replace g13 by L3w3x03v-3, using Equation 13, so for our purposes, right-hand side of Equation 21 should be multiplied by L3~3~“3v-3N-3(D2 - D,)3: g,(v, x) = 4Lw3x(Dz2 -
D2 (d,,, + A)’ In ___
D12)-3
v
+ doUt2In 2 E
X
If this is used in Equation
(22)
20, the result is
D1
(Dz2 -
.
D,2)-3 j’du~~x~xz[(d...+h)21n~+d,.‘ln~]
- 4wz(Dz2 -
= IF’
1
the
(23)
0
D12)-3 j.di.(Dl D1
-
v)" [(d,,,
+ h)2 -
dou: + 3d,,,Z ln f (Dz -
v)
1
(24) so, finally, < u> =
< v >” - ;
((4.t+4’
(D, - Dl) + 2d,,, + 3d,,‘? In A [E
(h2 - D12V3
(25) X(D2
-
(40,
DA4
which may be compared
+ Dz) + &
with Equations
6. SECOND
(D,’
-
D,2)-3 (160,
+ 9D2) d,,,
11-51.
EXAMPLE
OF PAPERS
I AND
II
In the second example of Papers I and II, only the three speeds cl, c2 andc, wereavailable to the cars, and in the limit of vanishing density one-third of the cars on the road want to go each speed. Thus, $p is the density of cars going each of the three speeds (to the lowest order in p). Thus 0J = 4 P(C1 + cz +
c3)
(26)
and we can replace p in Equations II-53 and II-54 by 3w/(c, + c2 + c3) in order to obtain a function of the appropriate variable, w. This means that the density of type-1 queues is
3 w3h
and the density
of type-II
+
~2 +
c3F3
N&,
3) +
4” (E)
(27)
queues is
4 w3(c1
+
c2 +
c3)-'
d,,,2(1,2)
Thus, instead of a density of w/(cl + c2 + c3) for cars going each of the three speeds, Equation 27 gives the density enhancement for cars going ci that want to go c2, and Equation 28 gives the density enhancement for cars going c1 that want to go c3.
230
FRANK C. ANDREWS
In the absence
of queuing, ( v)”
In the presence
=
of queuing,
the time average
speed is
em (Cl2 t_ c**+ cs2) =
(Cl2+ Q2 + cs2) (Cl +
c2
(29)
Gr’
this becomes A]2
(c*- 1 cl
c3 -
Cl
(30) which may be compared
with Equation
7.
H-55.
CONCLUSIONS
The approach of previous statistical theories of traffic flow on highways was to start with the following as input: d”(v) which describes the desired speeds of the drivers on the road and is taken to be independent of density p of cars on the road, and some kind of detailed picture of the passing and queuing behavior on the road. From this, one hoped to calculate the relative probabilities of various kinds of configurations of cars including queues of various types, and also to calculate the average speed of the cars on the road (i.e. the spaceaverage speed), all as functions of the density p. We now see that, since 4”(v) is expected not to be independent of p, the above approach is inappropriate. Instead, one should start with the following as input: x’(v) which describes the desired speeds of the drivers who use the road in a given period of time and which is taken to be independent of density, w which is the total number of cars which cross a fixed point on the road in unit time, and some kind of detailed picture of the passing and queuing behavior on the road. From this, one hopes to calculate the relative probabilities of various kinds of configurations of cars including queues of various types, and also to calculate the average speed of the cars using the road in a given time (i.e. the time-average speed), all as functions of w. The differences between the two approaches are significant. The distribution of desired speeds for the cars actually on a road will change with increasing density in such a way as to increase the fraction that want to go fast and decrease the fraction that want to go slow. Since the pressure to restore traffic to higher speeds depends on the high-speed region of #J”(V),there is always greater pressure under dense traffic conditions than one would expect from a naive theory for which 4”(v) was independent of p. This additional pressure will tend to keep traffic flowing faster, and thus will keep the q-k curve above its naive value, thus keep it nearly linear longer than would otherwise be predicted. Any serious statistical theory of traffic flow of the type we are considering will have to take account of this phenomenon. We have contrasted the results of the two approaches within the framework of a detailed theory previously presented, which treats traffic at densities so dilute that only 2-car queues need be considered.
A statistical theory of traffic flow on highways-III
231
REFERENCES ANDREWSF. C. (1970a). A statistical theory of traffic flow on highways-I. Steady-state flow in low-density limit. Trunspn Res. 4, 359-366. ANDREWSF. C. (1970b). A statistical theory of traffic flow on highways-II. Three-car interactions and the onset of queuing. Transpn Res. 4, 367-377. BARTLETTM. S. (1957). Some problems associated with random velocity. Pubis. Inst. Statist. Uniu. P&s 6, 261-270.
BREIMANLEO (1969). Point and trajectory processes in one-way traffic flow. Trunspn Res. 3, 251-264. CARLESONL. (1957). En matematisk model1 for landsvagstrafik. Nordisk Mat. Tidskr. 5,17&180. COWAN R. J. (1971). A road with no overtaking. Aastral. J. Statist. 13 (2), 94-116. GORDONW. J. and NEWELLG. F. (1964). Eauilibrium analysis of a stochastic model of traffic flow. Proc. Carnb. Phil. Sot. 60, 227-236.
.
_
HODGSONV. (1968). The time to drive through a no-passing zone. Trans. Sci. 2,252-264. HOLLANDH. J. (1967). A stochastic model for multilane traffic flow. Trans. Sci. 1, 184-205. MILLER A. J. (1959). Traffic flow treated as a stochastic process. Symposium on the theory of traffic flow, Detroit, publ. in Theory of Traffic Flow (1961), Elsevier Publ. Co., Amsterdam, 165-174. MILLERA. J. (1962). Road traffic flow considered as a stochastic process. Proc. Camb. Phil. Sot. 58,312-325. Corrigendum 59, 508 (1963). MORSEP. M. and YAFFEH. J. (1971). A aueuinn model for car oassinrr. Trans. Sci. 5.48-63. MUNJAL P. and PAHL J. (1969): An analysis of the Boltzmann-type statistical models. for multi-lane traffic flow. Transpn Res. 3, 151-163. NEWELLG. F. (1955). Mathematical models for freely-flowing highway traffic. Opns. Res. 3, 176-186. NEWELLG. F. (unpublished). Since the publication of Andrews (1970a, b) the author has learned that similar equations had previously been derived by Newell, but had not been published. NORMANN0. K. (1942). Results of highway-capacity studies. Public Roads 23, 57-81. PRIGOGINEI. and ANDREWSF. C. (1960). A Boltzmann-like approach for traffic flow. Opns. Res. 8,789-797. PRIG~GINEI., RFXBOISP., HERMANR. and ANDERSENR. L. (1962). On a generalized Boltzmann-like approach for traffic flow. Bull. Acad. r. Belg. Cl. Sri. 48, 805-814. WARDROPJ. G. (1952). Some theoretical aspects of road traffic research. Proc. Znstn. Civ. Engrs. 2, 325-378.
Abstract-The development of statistical theories of traffic flow on highways has often started with a model in which each driver retains a fixed desired speed. This has been taken to mean that the distribution of desired speeds for the cars on the road at a given time was independent of density. However, it is shown that this model really implies that the desired speed distribution for the cars crossing any fixed point on the road in a certain period of time will be densityindependent. The relation between the two distributions is computed, and the effect of the distinction on a theory of the density dependence of traffic behavior is discussed. The important input for a statistical theory of traffic flow seems to be a description of the desired speeds of the cars using the road in a period of time and a model of passing and queuing behavior. The number of cars crossing a fixed point in unit time appears to be more basic in such a theory than the density of cars.
R&urn&-L’elaboration de theories statistiques sur les flux de circulation sur les autoroutes a souvent pris comme point de depart un modele d’apres lequel chaque conducteur observe une vitesse souhaitable fixt. Cette hypothtse a et& adopt&e pour traduire le fait que la distribution des vitesses souhaitables pour les vthicules sur la route a un instent donnt, Btait independante de la densitd. Cependant, on dtmontre qu’en fait ce modele entraine que la distribution des vitesses souhaitables pour les vehicules traversant un point quelconque t?xe sur la route, durant un intervalle de temps donne, sera independante de la densite. La relation entre les deux distributions est calculee et l’on discute en outre de l’effet de la distinction sur une theorie de la dependance de la densite pour le comportement de la circulation. Les don&es importantes necessaires a l’btablissement d’une theorie statistique du flux de circulation se rev&lent 6tre la description des vitesses souhaitables des vehicules utilisant la route durant un intervalle de temps don& et un modele de comportement de franchissement et de file d’attente. Le nombre de vehicules passant par un point fixe don& durant l’unitt de temps apparait comme plus fondamental pour une telle theorie que la densite de vehicules.
232
FRANK C. ANDREWS Zusammenfassung-Die Entwicklung stochastischer Theorien des Verkehrsablaufs auf StraBen ist oft von einer Modellvorstellung ausgegangen, bei der jeder Fahrer eine feste Wunschgeschwindigkeit einhalt. Man hat dies so Interpretiert, dal3 die Verteilung der Wunschgeschwindigkeiten fur alle Fahrzeuge auf den betrachteten Streckenabschnitt zu einem gegebenen Zeitpunkt dichteunabhlngig ist. Die Modellvorstellung beinhaltet aber in Wirklichkeit, da13 die Verteilung der Wunschgeschwindigkeiten dichteunabhangig ftir die Zahrzeuge ist, die an einem beliebigen StraBenquerschnitt wahrend eines bestimmten Zeitintervalls beobachtet werden. Die Beziehungen zwischen den beiden Verteilungen werden berechnet. Weiterhin wird untersucht, welche Auswirkungen eine Unterscheidung auf eine Theorie der Dichteabhlngigkeit des Verkehrsverhaltenshaben kann. Wichtige F&gangsgrijDen fiir eine stochastische Theorie des Verkehrsflusses diirften die Wunschgeschwindigkeiten der Fahrzeuge w&rend eines Zeitintervalls sowie ein Model1 des Uberpolund Kolonnenfahr-verhaltens sein. Die Starke des Fahrzengstroms scheint im Rahmen einer solchen Theorie grundlegendere Bedeutung zu haben als die Verkehrsdichte.