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An extended model for car-following
Transpn Res. Vol. 2, pp. 13-21. Pergamon Press 1968. Printed in Great Britain
AN EXTENDED MODEL FOR CAR-FOLLOWING STEN BEXELIUS Chalmers Tekniska Hijgskola, Giiteborg, Sweden (Received 9 November 1966; in revised form 6 April 1967)
SEVERALauthors have discussed different car-following models, see, for instance, Reuschel (1950), Pipes (1953), Chandler et al. (1958), Gazis et al. (1959, 1961), Herman ef al. (1959), Kometani and Sasaki (1959), Edie (1961) and Newell (1961,1962). In the past years, General Motors Research Laboratories have performed follow-the-leader experiments to fit actual data to the different models, as reported by Chandler et al. (1958), Gazis et al. (1959, 1961), Herman et al. (1959) and Herman and Rothery (1965). In one series they specially examined the influence of more than one preceding automobile (Herman and Rothery, 1965). The results were not significant, but it was concluded that in most cases a driver in a platoon observes only the car immediately ahead of him. This conclusion was drawn on a comparison between correlation coefficients. Buckley (1962) used material out of a timeheadway measurement in Los Angeles, carrying out an autocorrelation analysis on gaps in a traffic stream. The autocorrelation proved to be negative for large traffic volumes. This should imply that a driver observes at least the two nearest cars ahead. Most of the existing car-following models can be described in the following way:
(1) where x, is the position the sensitivity function, It is proposed here several of the preceding
of the nth vehicle, T is a constant time lag (or response time), X is and the dots express differentiation with respect to time t. to refine the model by the assumption that every driver reacts to vehicles, and the following form is suggested:
The special case, N = 1, gives equation (1). If X, = constant, the equation (2) is linear. We then may formulate stability criteria. A platoon of cars is stable when the amplitude of any small disturbance decreases when propagating down the line of cars. For the case N = 1, the stability condition is 2X, T < 1, as shown by Chandler et al. (1958). For the case N = 2, the calculation is carried out in the following way: If all sinusoidal disturbances are dampened, then any disturbance will be dampened, too. Therefore we study the following case: & = A,cos it = ReA,e*& It is well known that it is enough to consider the complex form ~?r= Aleid 13
14
STEN BEXELIUS
In the same way we may denote
the velocity f,
where {A,} are complex
constants
of the nth car
= A, eiWt
(depending
on w). For n > 2, A,, is determined
by
iwA,eiwT = h,(A,_l-A.)+;\,(A._.,-n.)
(3)
or
The condition for stability is A,,I,P>O for every H*# 0, when n-•fwJ. It cm bc noted that if A, = A, for M‘= 0, then all A,,, = A,, which is sensible. For a fixed NJ(4) is a special case of the following recurrence problem: Y?t = WV,-1+ k, where a and b are constants.
Its general
where B and C are determined characteristic equation
solution
2
(5)
is
by y,, and ~‘r, and where z1 and zz arc the roots of the 1 --az-b_-2 _ 0
In the case z1 = zz the general
solution
(7)
is
I1+t1C I’,! == ~__ ,, -1 For a derivation of (6) and (6’), see, for instance, Henrici (1964). increasing n the roots zr and z, must have absolute V~LICS > 1. The characteristic equation corresponding to (4) is xl-- ++_&&7’
--
h, z? X,+/\,+j,,.ei”+
If we want y,, -+O for
= O
(8)
or i1r3eiu’T -_1X+ X = ._.. _-L.-L :! L
;“+&
(9)
with the solution z _
h, + Jl(X, + 2X,)” + ill, eifr’7’4/\,] 2x, 2x,
or
For w = 0 we get =1=
For ul#O, z1 and z2 can be regarded
---
4+A2and
1.2 =
1
A2
as functions
of W. Because of the stability
condition,
An extended model for car-following
15
z, is the interesting one for small w. From (10) we get 22 M
i~(l
--
'l+q 2h2
+
iwT)4h,
1
w2
16hz2
'+g 2
(A 1 +2h)22
+g'(h
1 +2h)42
or
1
w2T
W"x, x,:h, h,+2h2+(h,+2h2)3
.Z2Z l+‘-p
(11)
and
or
The condition ] z21 > 1 now gives the necessary condition A, + 4x2 2T G (h,+ 2X,)2
(13)
In some cases another form is convenient 2T@,+2h2) < l+&
(14) 1
2
In the Appendix it is shown that (13) is a sufficient condition for all frequencies w#O. With X2= 0 we have the special case 2TX, f 1 (15) For non-linear models it is mathematically difficult to formulate stability criteria, but as long as the perturbations are small, it is possible to linearize the equations and use the above criteria. The “reciprocal-spacing” model suggested by Gazis et al. (1959) then gives: 2To16x,-x,+r where 01is a constant and x,=
O1 x7&-x,+1
This model pretends that below a certain value of the (mean) spacing the traffic flow becomes instable. Since in the steady-state flow every mean space corresponds to a certain velocity, we get a critical velocity below which the flow becomes instable. In studying steady-state flow one finds a velocity corresponding to maximum flow. It is called optimum velocity and is of special interest. The relation between the optimum and critical velocities is of importance since the maximum flow would not be attainable if it was unstable. Hence we find another form of the stability condition: vcrit < vopt
We will now calculate U,,it and oopt for the reciprocal-spacing model. Xn+Jt + T) = 2
x,(t)
_~,+l(t) Lw> - ~n+1(01
(16)
STEN BEXELIUS
16
By integration
we find x&t+
T) = 01In [x,(l) - ~,,+~(t)] + constant
and can write
where v is velocity, c is concentration or density concentration” corresponding to v = 0. The flow q (cars per time unit) is q =
co =
(cars per length
ncln-1
unit), and cj is the ‘Tarn
C.
c
and q,nax = c~(c~/e) where 01= ZJ,,~~(cf. Cazis et al., 1959). V,,it, is reached
when
2Ta = _1 = _1 e”f” c ci and so we get v,,~~ = (YIn 2Tolci = zJ,rt + z),,rt In 2Tq,,,, The condition
U,,it < v,,rJt may be changed
2mmx <
(17)
to 1
or
qlnax<-
1 2T
(18)
This simple relation is not special for the “reciprocal-spacing” general non-linear models described by Gazis et al. (1961), where xv&+IV
x1=a[x,(t)
model.
If one uses the
+ T)
- X.,+&)]l
the result remains the same for all sensible combinations of I and m. The relation ,naX < 1/2T is fundamental as long as drivers observe no other vehicle than the one 4 immediately ahead. We now turn to the case N = 2, and extend the reciprocal-spacing model in the following way: iJt+T) To attain
i,_,(t)
-i-,(t)
Xn-1(t)-x,(t)
%L-2(t) -&do
+PX,-‘2(t)-&L(f)
(19)
the steady state, we integrate. k,(t+
Assuming
= ci
T) = a In [x,-r(t)
-xJt>]
-t/3 In [x,_p(t) -x,(t)]
that we have a steady state with constant
+ constant.
spacing s and velocity v, we get
v = arlns+pln2s+constant or v = (cy+ p) In s f constant p does not influence
the form of equation
(20). Therefore
(20) the introduction
of more terms
An extended model for car-following
17
into the car-following equation does not influence the fitting of a model to observed steadystate data. However, it certainly intluences stability. The stability condition becomes
or
2T(a+/3) s
<1+-
B a+P
(21)
1 -i-@/a+/?) may be regarded as an index of stability: S. Greater values of S result in higher stability. If we introduce the concentration c and the jam concentration ci into the equations (20) and (21), we obtain v = (a+/3)ln3
(22)
c
and 2T(ol+/?)c
(23) equations in the case N = 1.
2Tqnmx
00pt
+
u0ptln
7
and the fundamental stability condition becomes 2Tq,,, < S (25) The question is to find sensible values of S. 01 = fi would result in S = 1.5. Higher values seem unlikely as long as N = 2, but if a driver could observe a greater number of cars driving ahead of him, S might reach higher values. The test runs undertaken by General Motors supplied T-values in the order of 1.5 set for normal drivers (Chandler et al., 1958). A series with professional drivers lead to smaller values of T (see Rothery et al., 1964). It is interesting to compare possible S- and T-values with observations of qmax. In the worst case, S = 1 and T = 1.5 set, we have a possible qmax of 1200 vehicles/hr. This value corresponds to values observed in road tunnels. In a very favourable case, S = 1.8 and T = 1.2 set might be possible. This allows a qmax of 2700 vehicleslhr. Such a high flow has never been reported and is probably impossible for other reasons than instability. However, S = 1.8 and T = 1.2 correspond to a case where the critical velocity is below the optimal. If we assume a qmax of 2000 vehicles/hr at Vopt = 35 m.p.h., we would have V
crit = 35-351n 2 ty i rz!
= 24.5 m.p.h.
One of the favourable characteristics of this flow is that its speed can oscillate around the optimal value and still the flow does not break down. This might be a correct description of modem freeway traffic. Beside the observations mentioned in the introduction, another indication was found for the assumption that drivers normally try to observe several cars ahead. The earlier models cannot explain observed maximum flows; one way to do this is to suppose a “multi’‘-following model.
STEN
18
BEXELIUS
REFERENCES BUCKLEYD. J. (1962). Road traffic headway distributions. Proc. Awt. Rd Res. Bd 1, 153-187. CHANDLERR. E., HERMANR. and MONTROLLE. W. (1958). Traffic dynamics: studies in car following. Ops Rex 6, 165-184. EDIE L. C. (1961). Car-folIowing and steady-state theory for noncongested traffic. Ops Res. 9, 66-76. GAZIS D., HERMAN R. and POTTS R. B. (1959). Car-following theory of steady-state traffic flow. Ops Res. 7, 499-505. GAZIS
D., HERMANR. and ROTHERYR. W. (1961). Nonlinear
follow-the-leader
models of traffic flow.
Ops Res. 9, 545-567. HENRICI P. (1964). EIements
of Numerical Analysis. John Wiley, New York. HERMAN R., MONTROLLE. W., POTTS R. B. and ROTHERYR. W. (1959). Traffic dynamics: analysis of stability in car following. Ops Res. 7, 86-108. HERMAN R. and ROTHERY R. W. (1965). Car following and steady-state flow. Proc. 2nd Znt. Symp. Theory of Road Traffic Flow. The Organisation for Economic Cooperation and Development, Paris. KOMETANIE. and SASAKIT. (1959). A safety index for traffic with linear spacing. Ops Res. 7, 704-720. NEWELL G. F. (1961). Nonlinear effects in the dynamics of car following. Ops Res. 9, 209-229. NEWELL G. F. (1962). Theories of instability in dense highway traffic. J. Ops Res. Sot. Japan 5, 9-54. PIPES L. A. (1953). An operational analysis of traffic dynamics. J. app/. Phys. 24, 274-281. REUSCHELA. (1950). Fahrzeugbewegungen in der Kolonne. Gst. ZngArch. 4, 193-215. ROTHERY R., SILVER R. and HERMAN R. (1964). Single-lane bus flow. Ops Res. 12, 913-933.
APPENDIX
Equation
(9) of the main text was x z”+ ‘z A,
=
in’ eiwT + h, + X, (Al)
A,
We want to find the conditions on the positive constants T, A, and &, under solutions z1 and z2 for all M?#0 obey the conditions 1z11 > 1, 1z2 1> 1. (Al) can be written
4 AI2= z2+ xz+4h2”
iw eiwT + A1+ A2 A2
Xl2
+4h,’
which the
(A21
The left-hand side of (A2) is an analytic function. We are interested in its mapping of 1z 1= 1. The function describes a translation followed by squaring. The unit circle can be transformed into one of the following shapes. o< -$2 Zm t
The diagrams in Fig. 1 are not to the same scale. The right-hand side of equation (A2) can be regarded as a complex function of the real variable II’. It is known as the Archimedes’ spiral. Its shape is shown in Fig. 2.
An extended model for car-following
19
Our original analytical condition can now be replaced by this graphical one. The spiral in Fig. 2 must not pass through the interior of the curve in Fig. 1. This is so, because every point inside the curve in Fig. 1 corresponds to at least one z with Iz] < 1, but no point outside the curve gives any such z. To study the problem we may use the curvature of the curves. For the spiral, one finds the curvature
+> = 2w1 +(J+J27-2/2>1 (1+ we T2)3’2 with c(0) = 2TA, and it is quite obvious that c(w) decreases when w increases.
FIG. 2.
For the curve in Fig. 1, we set z = e” and thus get a complex function of a real variable. For the curvature we get (XJ&J2 -I (6&/X2) COS t+ 8 t+ 413'2
k(t) = [(A1/A2)2+ (4h,/h2)cos
(43)
with (44) Since both curves start in the same point [(h,+2h2)/2h2]2, and leave in the same direction (parallel to the imaginary axis) the graphical condition requires the spiral to have at most the same curvature. Therefore 2TA
<
Xdh~ + 4x2)
2’ (&+2X2)2 or &+4x, 2TG (X,+2h2)2 which is the necessary condition found in the main text.
(A3
STEN BEXELWS
20
Let us now concentrate
on the limiting
case A, + 4h,
2T = (h,+2h,)3
For convenience set Al/h, = s. The two curves have the same curvature (s+4)/(~+2)~ There is a unique limiting circle with the same curvature Its radius is then (Sf2Y s+4
_ sz+4s+4 s+4
and so its diameter is > 2s. It is quite obvious that the spiral lies outside starting point). The curve in Fig. 1 is given by w(t) = ( ~(0) gives the starting
point.
2 ,
q+c”
with
at the starting point [(s+ 2)/212. going through the starting point.
>S
the circle (with
the exception
of the
O
1
We have
which shows that the point w(n) lies inside the limiting circle. If there is any point M’outside the limiting circle, there must also be an intersection for some 1 in the open interval (0 < t < n). The limiting circle is described by where
-_
0<+<2~
.
or, simplified,
and we get the equation
or
If we split up into real and imaginary scos~+-
parts we have
4cos4 s +s+4 s+4
ssin++-
4sin+
= scos t+cos2t
= ssint+sin2t
G46) (A7)
An extended model for car-following
21
(A6) may be written 4cos4
scos~+-
sf4
= scost-t-cos2t--
s s+4
648)
Squaring and adding (A7) and (A8) give
( ) 4
s+-
sf4
2
=s2+1+
s
2
2s
- s+4 (s cos t + cos 2f) + 2s(cos t cos 2t + sin t sin 2t) ( s+4 1
Rearranging gives 6s(s+4) -= (s+4)2
8scost-2scos22 s+4
or Cos2t-2cost+
1= 0
(cost-
1)2 = 0
and cost = 1 Hence, there is no other common point than the starting point, and so the curve from Fig. 1 has no point outside the limiting circle and thus no point outside the spiral. This shows that the condition (A5) is sufficient.
AN EXTENDED MODEL FOR CAR-FOLLOWING STEN BEXELIUS Chalmers Tekniska Hijgskola, Giiteborg, Sweden (Received 9 November 1966; in revised form 6 April 1967)
SEVERALauthors have discussed different car-following models, see, for instance, Reuschel (1950), Pipes (1953), Chandler et al. (1958), Gazis et al. (1959, 1961), Herman ef al. (1959), Kometani and Sasaki (1959), Edie (1961) and Newell (1961,1962). In the past years, General Motors Research Laboratories have performed follow-the-leader experiments to fit actual data to the different models, as reported by Chandler et al. (1958), Gazis et al. (1959, 1961), Herman et al. (1959) and Herman and Rothery (1965). In one series they specially examined the influence of more than one preceding automobile (Herman and Rothery, 1965). The results were not significant, but it was concluded that in most cases a driver in a platoon observes only the car immediately ahead of him. This conclusion was drawn on a comparison between correlation coefficients. Buckley (1962) used material out of a timeheadway measurement in Los Angeles, carrying out an autocorrelation analysis on gaps in a traffic stream. The autocorrelation proved to be negative for large traffic volumes. This should imply that a driver observes at least the two nearest cars ahead. Most of the existing car-following models can be described in the following way:
(1) where x, is the position the sensitivity function, It is proposed here several of the preceding
of the nth vehicle, T is a constant time lag (or response time), X is and the dots express differentiation with respect to time t. to refine the model by the assumption that every driver reacts to vehicles, and the following form is suggested:
The special case, N = 1, gives equation (1). If X, = constant, the equation (2) is linear. We then may formulate stability criteria. A platoon of cars is stable when the amplitude of any small disturbance decreases when propagating down the line of cars. For the case N = 1, the stability condition is 2X, T < 1, as shown by Chandler et al. (1958). For the case N = 2, the calculation is carried out in the following way: If all sinusoidal disturbances are dampened, then any disturbance will be dampened, too. Therefore we study the following case: & = A,cos it = ReA,e*& It is well known that it is enough to consider the complex form ~?r= Aleid 13
14
STEN BEXELIUS
In the same way we may denote
the velocity f,
where {A,} are complex
constants
of the nth car
= A, eiWt
(depending
on w). For n > 2, A,, is determined
by
iwA,eiwT = h,(A,_l-A.)+;\,(A._.,-n.)
(3)
or
The condition for stability is A,,I,P>O for every H*# 0, when n-•fwJ. It cm bc noted that if A, = A, for M‘= 0, then all A,,, = A,, which is sensible. For a fixed NJ(4) is a special case of the following recurrence problem: Y?t = WV,-1+ k, where a and b are constants.
Its general
where B and C are determined characteristic equation
solution
2
(5)
is
by y,, and ~‘r, and where z1 and zz arc the roots of the 1 --az-b_-2 _ 0
In the case z1 = zz the general
solution
(7)
is
I1+t1C I’,! == ~__ ,, -1 For a derivation of (6) and (6’), see, for instance, Henrici (1964). increasing n the roots zr and z, must have absolute V~LICS > 1. The characteristic equation corresponding to (4) is xl-- ++_&&7’
--
h, z? X,+/\,+j,,.ei”+
If we want y,, -+O for
= O
(8)
or i1r3eiu’T -_1X+ X = ._.. _-L.-L :! L
;“+&
(9)
with the solution z _
h, + Jl(X, + 2X,)” + ill, eifr’7’4/\,] 2x, 2x,
or
For w = 0 we get =1=
For ul#O, z1 and z2 can be regarded
---
4+A2and
1.2 =
1
A2
as functions
of W. Because of the stability
condition,
An extended model for car-following
15
z, is the interesting one for small w. From (10) we get 22 M
i~(l
--
'l+q 2h2
+
iwT)4h,
1
w2
16hz2
'+g 2
(A 1 +2h)22
+g'(h
1 +2h)42
or
1
w2T
W"x, x,:h, h,+2h2+(h,+2h2)3
.Z2Z l+‘-p
(11)
and
or
The condition ] z21 > 1 now gives the necessary condition A, + 4x2 2T G (h,+ 2X,)2
(13)
In some cases another form is convenient 2T@,+2h2) < l+&
(14) 1
2
In the Appendix it is shown that (13) is a sufficient condition for all frequencies w#O. With X2= 0 we have the special case 2TX, f 1 (15) For non-linear models it is mathematically difficult to formulate stability criteria, but as long as the perturbations are small, it is possible to linearize the equations and use the above criteria. The “reciprocal-spacing” model suggested by Gazis et al. (1959) then gives: 2To16x,-x,+r where 01is a constant and x,=
O1 x7&-x,+1
This model pretends that below a certain value of the (mean) spacing the traffic flow becomes instable. Since in the steady-state flow every mean space corresponds to a certain velocity, we get a critical velocity below which the flow becomes instable. In studying steady-state flow one finds a velocity corresponding to maximum flow. It is called optimum velocity and is of special interest. The relation between the optimum and critical velocities is of importance since the maximum flow would not be attainable if it was unstable. Hence we find another form of the stability condition: vcrit < vopt
We will now calculate U,,it and oopt for the reciprocal-spacing model. Xn+Jt + T) = 2
x,(t)
_~,+l(t) Lw> - ~n+1(01
(16)
STEN BEXELIUS
16
By integration
we find x&t+
T) = 01In [x,(l) - ~,,+~(t)] + constant
and can write
where v is velocity, c is concentration or density concentration” corresponding to v = 0. The flow q (cars per time unit) is q =
co =
(cars per length
ncln-1
unit), and cj is the ‘Tarn
C.
c
and q,nax = c~(c~/e) where 01= ZJ,,~~(cf. Cazis et al., 1959). V,,it, is reached
when
2Ta = _1 = _1 e”f” c ci and so we get v,,~~ = (YIn 2Tolci = zJ,rt + z),,rt In 2Tq,,,, The condition
U,,it < v,,rJt may be changed
2mmx <
(17)
to 1
or
qlnax<-
1 2T
(18)
This simple relation is not special for the “reciprocal-spacing” general non-linear models described by Gazis et al. (1961), where xv&+IV
x1=a[x,(t)
model.
If one uses the
+ T)
- X.,+&)]l
the result remains the same for all sensible combinations of I and m. The relation ,naX < 1/2T is fundamental as long as drivers observe no other vehicle than the one 4 immediately ahead. We now turn to the case N = 2, and extend the reciprocal-spacing model in the following way: iJt+T) To attain
i,_,(t)
-i-,(t)
Xn-1(t)-x,(t)
%L-2(t) -&do
+PX,-‘2(t)-&L(f)
(19)
the steady state, we integrate. k,(t+
Assuming
= ci
T) = a In [x,-r(t)
-xJt>]
-t/3 In [x,_p(t) -x,(t)]
that we have a steady state with constant
+ constant.
spacing s and velocity v, we get
v = arlns+pln2s+constant or v = (cy+ p) In s f constant p does not influence
the form of equation
(20). Therefore
(20) the introduction
of more terms
An extended model for car-following
17
into the car-following equation does not influence the fitting of a model to observed steadystate data. However, it certainly intluences stability. The stability condition becomes
or
2T(a+/3) s
<1+-
B a+P
(21)
1 -i-@/a+/?) may be regarded as an index of stability: S. Greater values of S result in higher stability. If we introduce the concentration c and the jam concentration ci into the equations (20) and (21), we obtain v = (a+/3)ln3
(22)
c
and 2T(ol+/?)c
(23) equations in the case N = 1.
2Tqnmx
00pt
+
u0ptln
7
and the fundamental stability condition becomes 2Tq,,, < S (25) The question is to find sensible values of S. 01 = fi would result in S = 1.5. Higher values seem unlikely as long as N = 2, but if a driver could observe a greater number of cars driving ahead of him, S might reach higher values. The test runs undertaken by General Motors supplied T-values in the order of 1.5 set for normal drivers (Chandler et al., 1958). A series with professional drivers lead to smaller values of T (see Rothery et al., 1964). It is interesting to compare possible S- and T-values with observations of qmax. In the worst case, S = 1 and T = 1.5 set, we have a possible qmax of 1200 vehicles/hr. This value corresponds to values observed in road tunnels. In a very favourable case, S = 1.8 and T = 1.2 set might be possible. This allows a qmax of 2700 vehicleslhr. Such a high flow has never been reported and is probably impossible for other reasons than instability. However, S = 1.8 and T = 1.2 correspond to a case where the critical velocity is below the optimal. If we assume a qmax of 2000 vehicles/hr at Vopt = 35 m.p.h., we would have V
crit = 35-351n 2 ty i rz!
= 24.5 m.p.h.
One of the favourable characteristics of this flow is that its speed can oscillate around the optimal value and still the flow does not break down. This might be a correct description of modem freeway traffic. Beside the observations mentioned in the introduction, another indication was found for the assumption that drivers normally try to observe several cars ahead. The earlier models cannot explain observed maximum flows; one way to do this is to suppose a “multi’‘-following model.
STEN
18
BEXELIUS
REFERENCES BUCKLEYD. J. (1962). Road traffic headway distributions. Proc. Awt. Rd Res. Bd 1, 153-187. CHANDLERR. E., HERMANR. and MONTROLLE. W. (1958). Traffic dynamics: studies in car following. Ops Rex 6, 165-184. EDIE L. C. (1961). Car-folIowing and steady-state theory for noncongested traffic. Ops Res. 9, 66-76. GAZIS D., HERMAN R. and POTTS R. B. (1959). Car-following theory of steady-state traffic flow. Ops Res. 7, 499-505. GAZIS
D., HERMANR. and ROTHERYR. W. (1961). Nonlinear
follow-the-leader
models of traffic flow.
Ops Res. 9, 545-567. HENRICI P. (1964). EIements
of Numerical Analysis. John Wiley, New York. HERMAN R., MONTROLLE. W., POTTS R. B. and ROTHERYR. W. (1959). Traffic dynamics: analysis of stability in car following. Ops Res. 7, 86-108. HERMAN R. and ROTHERY R. W. (1965). Car following and steady-state flow. Proc. 2nd Znt. Symp. Theory of Road Traffic Flow. The Organisation for Economic Cooperation and Development, Paris. KOMETANIE. and SASAKIT. (1959). A safety index for traffic with linear spacing. Ops Res. 7, 704-720. NEWELL G. F. (1961). Nonlinear effects in the dynamics of car following. Ops Res. 9, 209-229. NEWELL G. F. (1962). Theories of instability in dense highway traffic. J. Ops Res. Sot. Japan 5, 9-54. PIPES L. A. (1953). An operational analysis of traffic dynamics. J. app/. Phys. 24, 274-281. REUSCHELA. (1950). Fahrzeugbewegungen in der Kolonne. Gst. ZngArch. 4, 193-215. ROTHERY R., SILVER R. and HERMAN R. (1964). Single-lane bus flow. Ops Res. 12, 913-933.
APPENDIX
Equation
(9) of the main text was x z”+ ‘z A,
=
in’ eiwT + h, + X, (Al)
A,
We want to find the conditions on the positive constants T, A, and &, under solutions z1 and z2 for all M?#0 obey the conditions 1z11 > 1, 1z2 1> 1. (Al) can be written
4 AI2= z2+ xz+4h2”
iw eiwT + A1+ A2 A2
Xl2
+4h,’
which the
(A21
The left-hand side of (A2) is an analytic function. We are interested in its mapping of 1z 1= 1. The function describes a translation followed by squaring. The unit circle can be transformed into one of the following shapes. o< -$2 Zm t
The diagrams in Fig. 1 are not to the same scale. The right-hand side of equation (A2) can be regarded as a complex function of the real variable II’. It is known as the Archimedes’ spiral. Its shape is shown in Fig. 2.
An extended model for car-following
19
Our original analytical condition can now be replaced by this graphical one. The spiral in Fig. 2 must not pass through the interior of the curve in Fig. 1. This is so, because every point inside the curve in Fig. 1 corresponds to at least one z with Iz] < 1, but no point outside the curve gives any such z. To study the problem we may use the curvature of the curves. For the spiral, one finds the curvature
+> = 2w1 +(J+J27-2/2>1 (1+ we T2)3’2 with c(0) = 2TA, and it is quite obvious that c(w) decreases when w increases.
FIG. 2.
For the curve in Fig. 1, we set z = e” and thus get a complex function of a real variable. For the curvature we get (XJ&J2 -I (6&/X2) COS t+ 8 t+ 413'2
k(t) = [(A1/A2)2+ (4h,/h2)cos
(43)
with (44) Since both curves start in the same point [(h,+2h2)/2h2]2, and leave in the same direction (parallel to the imaginary axis) the graphical condition requires the spiral to have at most the same curvature. Therefore 2TA
<
Xdh~ + 4x2)
2’ (&+2X2)2 or &+4x, 2TG (X,+2h2)2 which is the necessary condition found in the main text.
(A3
STEN BEXELWS
20
Let us now concentrate
on the limiting
case A, + 4h,
2T = (h,+2h,)3
For convenience set Al/h, = s. The two curves have the same curvature (s+4)/(~+2)~ There is a unique limiting circle with the same curvature Its radius is then (Sf2Y s+4
_ sz+4s+4 s+4
and so its diameter is > 2s. It is quite obvious that the spiral lies outside starting point). The curve in Fig. 1 is given by w(t) = ( ~(0) gives the starting
point.
2 ,
q+c”
with
at the starting point [(s+ 2)/212. going through the starting point.
>S
the circle (with
the exception
of the
O
1
We have
which shows that the point w(n) lies inside the limiting circle. If there is any point M’outside the limiting circle, there must also be an intersection for some 1 in the open interval (0 < t < n). The limiting circle is described by where
-_
0<+<2~
.
or, simplified,
and we get the equation
or
If we split up into real and imaginary scos~+-
parts we have
4cos4 s +s+4 s+4
ssin++-
4sin+
= scos t+cos2t
= ssint+sin2t
G46) (A7)
An extended model for car-following
21
(A6) may be written 4cos4
scos~+-
sf4
= scost-t-cos2t--
s s+4
648)
Squaring and adding (A7) and (A8) give
( ) 4
s+-
sf4
2
=s2+1+
s
2
2s
- s+4 (s cos t + cos 2f) + 2s(cos t cos 2t + sin t sin 2t) ( s+4 1
Rearranging gives 6s(s+4) -= (s+4)2
8scost-2scos22 s+4
or Cos2t-2cost+
1= 0
(cost-
1)2 = 0
and cost = 1 Hence, there is no other common point than the starting point, and so the curve from Fig. 1 has no point outside the limiting circle and thus no point outside the spiral. This shows that the condition (A5) is sufficient.