- Email: [email protected]
The instability of motorway traffic
Transpn. Res.-B. Vol. 28B, No. 2, pp. 175-186, 1994 Coovrinht 0 1994 Elsevier Science Ltd Printed-in &eat Britain. All rights reserved 0191-2615/94 $6.00 + .OO
Pergamon
THE INSTABILITY
University
OF MOTORWAY
PAOLO FERRARI School of Engineering of Pisa, Via Diotisalvi (Received
9 October
TRAFFIC
2, 56126 Pisa, Italy
1992; in revisedform 12 April 1993)
Abstract-This paper presents an analysis of the dynamic behaviour of the motorway traffic system, in order to single out its characteristics that are responsible for instability. It is shown that, besides the parameters considered in the traditional car-following theory, a great importance in determining instability of a vehicle platoon is the amplitude spectrum of the speed function of the leader vehicle. The results obtained provide a theoretical validation of the reliability theory of motorway traffic, which was founded on experimental basis, and by which the probability of instability of a traffic stream can be experimentally measured. 1. INTRODUCTION Instability is one of the most interesting phenomena of motorway traffic. A traffic stream is called unstable when, rapidly and without any obvious reason, marked and sudden falls of speed level take place, often resulting in a complete standstill. After the stop, the return to normal traffic conditions is quite difficult: the recovery of a speed value close to the initial one is often followed by a new fall, and so on repeatedly, giving rise to the classical stop-and-go phenomenon. Traffic instability was intensively studied during the past decades, both theoretically (e.g. Chandler, et al., 1958; Herman, et al., 1959; Newell, 1962) and experimentally (e.g. Edie and Foote, 1961; Treiterer and Myers, 1974), even if early theoretical studies of traffic “centered around the hope that traffic behaved in a stable way” (Newell, 1962, p. 38). The continuous theory (Lighthill and Whitham, 1955; Richards, 1956) considered traffic as a fluid characterized by three parameters, which are continuous functions of the space x and of the time t: the flow rate q, the density k and the space-mean speed v. This theory shows that in the t, x plane, the points characterized by the same values of parameters k, q and v are located along lines of constant slope c (the wave velocity), which are parallel to the tangent to the diagram q = q(k) in the point of abscissa k. It is possible that lines of different slope intersect in the t, x plane. This happens, for example, when a platoon of fast vehicles catches up to a slow one, or when at a certain moment a traffic hump takes place in a point of the carriageway. In these cases the intersection points are located along lines having slope w equal to that of the cord joining the points in the k, q diagram corresponding to the k values of the intersecting lines: w is the speed of the shock waves, i.e. the speed at which the discontinuities of k, q, v propagate along the carriageway. Taking into account the shock waves that absorb the discontinuities, traffic flow in this theory is always stable. A particular and important type of shock wave happens at a bottleneck, i.e. a section of reduced capacity. The theory predicts that flow past the bottleneck should stabilize at its capacity, and a shock wave propagates upstream. This conclusion is in contrast with experience, which shows that instability phenomena, characterized by a sequence of stop-start waves, are recurrent at bottlenecks (e.g. Edie and Foote, 1961). Traffic flow is stable even according to car-following theory, if it is supposed that each vehicle adjusts its speed immediately to that of the preceding one. This theory considers a sequence 1 . . . j . . . n of vehicles that proceed without overtaking along a single lane. Let v,(t) and xi(t) be the speed and position of the jth vehicle at time t, respectively. A relation exists between the speed v, of the jth vehicle and the distance dj = Xj_, - x, with respect to the preceding one: 175
176
P. FERRARI vj(t)
=
F[Xj_l(t)
(1)
Xi(t)]*
-
An example of eqn (1) is represented in Fig. 1. An equivalent form of (1) can be obtained by differentiating both sides: dv,( t) dt
where 2
=
F’Ixj-l(t) - xj(t)l[v,-l(t) - vj(t)l,
is the acceleration of the jth vehicle and F’ is the derivative of F with respect
to its argument. F’ is the slope of the tangent to the diagram v = F(d) (Fig. 1) at a given value d = Xj_, - Xj. This theory shows that, if the fluctuations of distance between vehicles are low so that F’ can be considered constant, speed fluctuations of the following vehicle vanish with time, and similarly speed fluctuations are damped along the platoon, so that flow is always stable (e.g. Chandler, et al., 1958). The main innovation of the car following theory, and one of the most important contributions to the theory of traffic flow, was the introduction of a time delay Tin the reaction of the following vehicle. In this case, by considering F’ as a constant CL,eqn (2) becomes: dVj( t) dt
= /A *
[Vj_l(t
-
T) -
vj(t
-
T)].
It was shown by Chandler et al. (1958) that the time lag T is the cause of traffic instability. Two kinds of instability were considered: local instability, due to the progressive increase with time of the amplitude of speed fluctuations of the following vehicle; and asymptotic instability, due to the progressive increase of the amplitude of speed fluctuations along a platoon. It was shown that flow is locally unstable when the product pT was greater than 7r/2, whereas it is asymptotically unstable when pT > l/2. Experimental measures carried out by many authors (e.g. Chandler, et al., 1958; Herman and Potts, 1962; Darroch and Rothery, 1972) indicate that pT in real life is always less than 7r/2, so that the only possible kind of instability is the asymptotic one. If we consider in Fig. 1 the point P where the slope of the tangent to the diagram v = F(d) is ~1 = 1/2T, the corresponding distance d* singles out a value k* = l/d* for the traffic density k, which is critical in the following sense. When k c k* the speed fluctuations of a vehicle are damped along the platoon, and flow is stable. If k > k* and there is a reduction in speed of a vehicle, this reduction will be amplified along the
Fig. 1. Relationship
v = F(d) between
speed v and distance
d.
Instability of motorway traffic
177
platoon, giving rise to a progressive decrease of distances between vehicles, thus to an increase of CL,which produces more and more decrease of speeds: if the platoon is long enough, a complete standstill will take place. The existence of a critical density, which depends on the relation v = F(d) and on the value of the time lag T,is the main result of the car-following theory, and one of the most important results of traffic theory. A consequence of this result is that, when flow rate tries to go over the carriageway capacity, and the critical density is exceeded, there is not a stable adjustment of flow to capacity-as the continuous theory predicts-but flow becomes unstable, and a sequence of stopstart waves takes place, as experience confirms. However some remarks have to be made about the method used by Chandler et al. (1958) in order to obtain the instability characteristics of a traffic stream. These authors investigated the stability question in terms of the Fourier components of the speed function v,(t) of the leader vehicle of the platoon, and considered as critical for the stability the least value of pT at which the amplitude of at least one of these components increases along the platoon. Components of low frequency, close to zero, give the greatest limitations on pT value. The increase of amplitude of these components from a vehicle to a successive one is very low, but the length of the platoon was considered infinite, so that the amplitude of vi(t) increases up to infinity along the platoon. However in real life platoons have finite length, so that these components, given the little increase of their amplitude, do not contribute to motorway instability. On the contrary, when pT is somewhat greater than l/2, the amplitude of components of v,(t) of higher frequency increases in more substantial measure, giving rise to marked fluctuations of speed level of traffic. For this reason instability does not depend only on pT value, but also on the amplitude spectrum of v#). Moreover v,(t), p and T,as well as the length of platoons, are random quantities. In particular v,(t) is a realization of a stochastic process, whose characteristics essentially depend on drivers’ behaviour; p is a random variable distributed around a mean that is an increasing function of traffic density, and T is a random variable whose mean essentially depends on environmental characteristics: it is high during dark hours and bad weather conditions, low when weather is good and light. For this reason instability is a stochastic phenomenon. The deterministic approach of the car-following theory is very important in order to single out the parameters on which instability depends and how these parameters cause instability. But deterministic values of traffic parameters at which flow becomes unstable do not exist: it is only possible to know the probability that in correspondence of given traffic parameters instability will take place during a certain time period, and this knowledge can be obtained only by studying traffic flow as a stochastic process. This paper is a contribution to research into how traffic parameters influence instability. Section 2 is dedicated to the study of dynamic behaviour of traffic flow. The influence of the amplitude spectrum of v,(t) on instability is examined in Section 3. Section 4 is dedicated to a discussion on the concept of critical density, and examines the link existing between the results obtained in this paper and those of the reliability theory of motorway traffic, by which the experimental measure of critical density and of the probability of instability can be obtained. 2. THE DYNAMICAL BEHAVIOUR OF TRAFFIC
Consider two vehicles, 1 and 2, which proceed with speed v,(t) and v,(t), one following the other, along a motorway lane. The dynamical behaviour of this system is defined by the difference-differential eqn (3):
dv, - = p * [v,(t - T) - vz(t dt
T)],
where parameters CLand T are characteristics of vehicle number 2; they vary randomly from one vehicle to another along a platoon, and even for the same vehicle with time,
178
P. FERRARI
around mean values that depend on traffic and environmental before. Taking the Laplace transform of (3a), we have:
conditions, as it was said
where V,(s) is the Laplace transform of vi(t). From (4) the following equation is obtained: 1/2(s) =
-*v,(s) pe s + pees*
= W(s) V,(s),
where -ST
W(s)
=
pe s + pews*
is the transfer function of the system. By using the typical definition of the dynamical system theory, different from that of the car-following theory, this system is said to be asymptotically stable if, given a transient input v,(t) + 0 when t + 03, even the output v2(f) --) 0 when t + 03. It is known (e.g. Kamen, 1980) that the necessary and sufficient condition for asymptotic stability is that the real part of the solutions of the characteristic equation: s + pees* = 0
(7)
is negative. The solutions of eqn (7) can be investigated by means of a method similar to that which was used by Hermann et al. (1959), and it can be shown that they lie on the ?r negative part of the complex plane-i.e. its real part is negative-if and only if pT < 2. In real life pT c
as was shown in the previous section; therefore motorway traffic is
asymptotically stable, and its instability depends on the increase of the amplitude of speed fluctuations from a vehicle to another along a platoon. The relation between v2(t) and v,(t) can be obtained by calculating the inverse of the Laplace transform (5):
b(t) =
&n-
s
‘+‘a pe-‘*V,
y-im
s
+
(s)
es,ds
p-S*
’
(8)
where y is a real number so that the integration path lies to the right of all the singularities of the Laplace transform.
As FT < s all singularities lie in the negative part of the
complex plane, so that it is possible to put y = 0. So eqn (8) becomes: V*(f) =
&
s m
_m G(w)Z(w)e’“‘dw,
(9)
where: G(w) = V,(h)
(10)
is the Fourier transform of the input v](t) and: -iId
Z(0)
=
1 iw + pe -jwT = 1 + iwe -‘“*/p
lJe
is the frequency response of the system.
(11)
179
Instability of motorway traffic
The complex variable 1 + iueiwT/p can be written: 1 + iueiuT/p = 1 - 0 sin UT + i 0 P CL
cos
UT,
(12)
so that the following expressions of module )Z 1 and phase 4 of Z(o) can be obtained:
IZ(@) I =
4(w)
)-I”
(l+L-2EsinwT cL p2
1 - w sin wT P
= *cos-’
l/2'
l+g-2EsinwT i CL CL2
(13)
1
where signs + and - hold when w < and > 0, respectively. By denoting I G(w) I and &(w) respectively module and phase of the input transform G(o), and + = -(+ + 6,), eqn (9) can be written:
s
:,
lG(w)l IZ(o)l 1cos (cot- a) + i sin (wt - +)]dw.
(14)
We deduce from (14) that the characteristics of v2(f) essentially depend on the characteristics of the amplitude spectra IG(w)1 and [Z(w)1 : if the frequency bands of high amplitude coincide to a large extent in these two spectra, there will be a high increase of the maximum of v2(t) with respect to vr(t). If these bands are different, a low increase will take place, or v2(f) will be damped. Therefore the increase of v2(f) does not depend only on the parameters p and T of the system, but also on the characteristics of the input v,(t). By introducing the variables Q = pT and x = wT into expressions (13), they are written in dimensionless form: [Z(x) I = [ 1 + (ST #J(x) = *cos-’
- 2 G sin xl-‘”
1 I1,
1 - g sin x IZ(x)
(1%
(16)
which define the amplitude and phase spectra of the response of the system defined by eqn (3a). The amplitude spectrum is reported in Fig. 2 for various values of Q. 3. THE INFLUENCE OF CHARACTERISTICS
OF THE SPEED FUNCTION OF THE
LEADER VEHICLE ON TRAFFIC STABILITY
Consider two vehicles that proceed with the same other, along a motorway lane. Suppose that at instant sudden decrease of speed, followed by a more gradual assumed that the speed variation v,(t) with respect to of time t, is: v,(t)
constant speed V, one following the t = 0 the first vehicle undergoes a recovery to the initial speed V. It is V of the first vehicle, as a function
= @tAv exp (1 - Pt),
(17)
where / Av 1 is the maximum absolute value of the speed shift with respect to V, l/p is the instant when this maximum is reached and ~Av is the mean value of the deceleration. The purpose of this section is to show the influence of P, and thus of the deceleration of the leader vehicle, on traffic stability. TR(B) 28:2-H
P. FERRARI
4-
3-
0
0.5
1
1.5
3
2.5
2
X= OT Fig. 2. Amplitude
spectra
of the response
of car-following
system for various
Qvalues.
The Fourier transform G(w) of the input v,(t) is (t r 0):
s 02
G(w)
=
/3Av o t exp [l - W
PAve + iu)ldt = tp + iwj2,
where e is the basis of natural logarithms. The module 1G(o) 1 and the phase 6, of G(w) are:
IGCw)l=
,,!~w21f$, =
-2
tan-’
(1%
(w/P).
By substituting (19) into (14), and taking into account that IZ(w) I = )Z( -co) (, 4(w) = -rp( - w), C+,(W) = -q5,( - w), we have the following exI G(w) I = lG(--41, pression of the speed variation v2(f) with respect to V of the second vehicle: v2(t)
=
cos (wt -
!Z!!!f! j,
Q)dw l/2’
?r (fl*
+
w2)
(20)
1 + (W/F)’ - 2 w sin CL
By introducing the dimensionless variables 77= v2/Av, 7 = t/T, b = PT, x = wT, eqn (20) becomes:
Q = pT,
s m
cos (x7 -
o
(b*
9)dx l/2’
+
x2)(
1 + (x/Q)* - 2 5 sen
Q
The input function (17), written in dimensionless form, is:
(21)
181
Instability of motorway traffic
v1
=
2 =h
(22)
exp ( 1 - br) .
In order to know the influence of the input characteristics on traffic stability, consider two input functions (22) having different b parameters: b = 0.1 and b = 0.4. These are, for example, the input functions of a system in which the time delay of the second vehicle is T = 1.5 s and the first vehicle undergoes a speed decrease of 20 kmph with, respectively, a low deceleration a Z 0.4 ms-’ and a high deceleration a z 1.4 msm2. These functions are reported with dashed lines in Figs. 3 and 4. The corresponding amplitude spectra:
IGWI = g--g
(23)
are reported, with dashed lines, in Figs. 5 and 6. It should be noted from these figures that when b = 0.1 the values 1G 1 > 1 range in the (0, 0.5) band of x, whereas when b = 0.4 they range in the band (0, 1.O). By comparing these spectra with those 1Z(x) 1 of the system response represented in Fig. 2, it can be noted that the latter show 1Z( values in the band (0, 0.5) close to 1 even for high Q values, whereas in the band (0, 1.0) there are IZ 1 values much greater than 1 when Q > 0.7. As a consequence of this fact we can argue that when b = 0.1 the maximum value of q given by (21) will be rather close to 1 even when Q is high, whereas when b = 0.4 the maximum of rl will be markedly higher than 1 for high Q values. This conjecture was confirmed by numerical integration of (21); the functions o(r) calculated for both the b values and for different Q are reported in Figs. 3 and 4. The output v2(t) of the system S, made up by the first and the second vehicle of a platoon is the input for the system S2 composed by the second and the third vehicle, and so on. The amplitude spectrum of the input for S, is given by I G2(x) ) = 1G(x)I1Z(x) I: its examination is important in order to know the progressive variations of q(r) function along a platoon. These spectra have been calculated for both the b values and for different Q, and they are reported in Figs. 5 and 6. It can be noted that, while for b = 0.1 the values I G2 I > 1 still range in the (0, 0.5) interval of x, when b = 0.4 there are high I G21 values in the band (0, 1.5) for Q > 0.7. From this result we can argue that when b = 0.1
b = 0.1
-A-
l st vehicle
Fig. 3. Relationships 7 = ~(7) relative to the 1st and the 2nd vehicle of a platoon, for Q values.
b = 0.I and
for various
P. FERRARI
182
b = 0.40
--0.6
Fig. 4. Relationships
7 = ~(7) relative
1 st vehicle 2 nd whack
to the 1st and the 2nd vehicle of a platoon, Q values.
for b = 0.4 and for various
the maximum 77is not increased in a way that has to be considered dangerous along a platoon, whereas when b = 0.4 marked increases of qrnaxwill happen when Q > 0.7. A confirmation of this conjecture is given by Fig. 7, where the ratios v4 = v,/v, relative to the 4th vehicle of the platoon are reported for both the values of b and for Q = 0.9, in the hypothesis that all p and Tarethe same along the platoon. The functions ~~(7) were calculated taking into account that the frequency response of a car-following system constituted by n vehicles with the same p and T values has the following module 1Z,, 1 and phase an: lZ,l = IGI IZI”-’
a’, = 6, + (n - 1)4.
4. THE CRITICAL
(24)
DENSITY
It was shown in the previous section that the progressive increase of speed fluctuations along a platoon, and thus the risk of instability, depends on traffic parameters p, T, 0.They randomly vary from a vehicle to another of the platoon, and even for the same vehicle with time. The average ii of p distribution is an increasing function of traffic density. The average 7 of T depends on environmental conditions: it is high during the dark hours and when weather is bad. The average p of /3 is an increasing function of the frequency of lane changes in dense traffic. For example, when a vehicle shifts from the inner-side lane to the off-side one of a motorway carriageway, its speed is usually less than that of the following vehicle on the off-side lane, which has to decelerate in order to avoid collision. The deceleration is low if the vehicle inserts itself into a large gap; but if traffic is dense and lane changes are frequent, little gaps are used, and in this case the deceleration, and thus 0 value, is high. It was shown in Section 1 that the classical theory of car-following defines as critical the density value over which flow is unstable: this value of critical density depends on T and on the relation between p and the distance between vehicles. This definition is not
183
Instability of motorway traffic
b = 0.1 2.5
---
1 st vehicle 2 nd vehicle
2.0
1.5
1.0
0.5
0’
0.5
1.0
1.5
2.0
2.5
3.0 x=01
Fig. 5. Amplitude spectra of the functions 7 = q(7) relative to the 1st and the 2nd vehicle of a platoon for b = 0.1 and for various Q values.
complete, because it is necessary to take into account also the /3 parameter. Moreover, it has to be noted that, given the stochastic nature of traffic parameters on which the instability depends, a deterministic value for the critical density does not exist. Consider a motorway carriageway stretch and let X be the event representing a progressive speed drop up to --a complete stop of circulation in the vehicle stream travelling along this stretch. Let P(X/k, T, 0) --equal the conditional probability that the event X takes place, given the average values k, T, 0 of density k and of parameters T and 6 of the stream, during a time period characterized by constant average traffic conditions. Because of the monotonic relation --between k and ji, it can be deduced from the results of the preceding section that &Y/k, T, 0) is an increasing function of k, as well of 7 and s’. Let Sr and 23,be two traffic streams travelling on the motorway stretch in different time periods: stream S, is characterized by high average values of T and 0, whereas they se low+ the stream S,. Consider the relations between the conditioned probability P(X/ k) and k for the two streams: according to what was said before, the probability &Y/k) of the stream S, is higher than that of S,, traffic density z being the same. Let us define critical density as the value k* of k corresponding to a threshold P* of the probability p(X/F) beyond which the risk of the stop of circulation is considered unacceptable. It is easy to verify that the critical density k-7 of S, is greater than that of k: of S,: i.e. critical density increases as the average 7 and /3 values of the stream decrease. The critical density is a fundamental concept of traffic theory, and it is also important from a practical point of view, because the choice of the best control strategy for a
184
P.
FERRARI
b= 0.4
-__
1 st vehicle 2 nd vehicle
2.5
2.0
1.5
1.0
0.5
d
0.5
1.0
1.5
2.0
2.5
3.0
= WT
Fig. 6. Amplitude spectra of the functions 7 = ~(7) relative to the 1st and the 2nd vehicle of a platoon, for 6 = 0.4 and for various Q values.
traffic stream depends on its value (Ferrari, 1991). But its estimation by using the carfollowing theory is very difficult, among other things because it is difficult to estimate the probability distributions of T and /3. For this reason it seems more convenient to obtain the critical density by studying the stochastic process of vehicle speed, in which the effects of probability distributions of p, T, @ are condensed. Taking into account the fact that instability first takes place on the off-side lane of a motorway carriageway, from a practical point of view and for control purposes it is sufficient to process the speed sequence surveyed in a cross section of this lane. Let Y,be the speed recorded at instant t, and 7, the speed level at the same instant, i.e. the conditioned mean of speed, given the preceding realization of the sequence. The reliability theory of motorway traffic (Ferrari, 1988) shows that the speed sequence is a succession of realizations of homogeneous integrated moving average processes of the first order, whose equation is: v, =
V,_I
+
a, -
(1 -
h)a,_1,
(25)
where a, is the shift of the speed v, from the corresponding level “J,, and X is a coefficient ranging between 0 and 1, which measures the influence that the shift a, at a certain instant t has on the level 7,+) at the subsequent instant. The shifts a, are random variables independently and normally distributed with zero mean and equal variance u*. Therefore, the process eqn (25) has two parameters: h and cr2. Parameter h is an increasing linear function of the logarithm of the average density F of each realization, and the relation
185
Instability of motorway traffic
t/T
Fig. 7. Relationships TJ= ~(7) relative to the 1st and the 4th vehicle of a platoon, for Q = 0.9 and for two values of b.
between h and Ink is independent of the circumstances and the place where data are recorded. On the contrary, parameter uz is a decreasing linear function of the logarithm of the average density of each realization, and the relationship between a2 and lnx varies, sometimes in a notable way, depending on the circumstances and on the place where the survey is carried out: in particular during the night the u2 values are much greater than during the day hours, density k being the same. There is a close correspondence between these parameters and those of the carfollowing theory presented in the preceding sections. Parameter X, like ~1,measures the influence that the speed fluctuations of a vehicle have on the speed of the successive one, as a function only of traffic density and independently of all other traffic parameters: therefore, both parameters represent a characteristic of the system, and do not depend on the input. The shifts a, of the speed from the process level in dense traffic are essentially due to the reactions of vehicles to speed fluctuations of the preceding one, and their variance uz depends on the whole of traffic characteristics. u2 decreases with density, because the lane changes frequency diminishes when density increases, and is a function, traffic density being the same, of the T and 0 distribution in traffic streams; its value, at a given average density k, is an increasing function of the average values of T and /3, because they increase when the environmental conditions become more difficult and the lane changes more frequent. The reliability theory shows that the critical density is given by the intersection of the straight line u2 = u2(lnk) with axis of k: it is greater for traffic streams having lower u2 values, density k being the same. That is in accordance with the result obtained above with reference to the influence of T and /3 values on critical density. Therefore, we can argue from the close correspondence between the two theories that the value of critical density experimentally estimated by using the reliability theory is a good measure of the critical density defined by the car-following theory presented in this paper. 5. CONCLUSIONS
The car-following theory presented in this paper identifies the causes of instability phenomena on motorways and indicates the traffic parameters on which instability depends. Its conclusions are in close accordance with those of reliability theory of motorway
186
P. FERRARI
traffic, which was established on an experimental basis, by processing a great deal of data recorded on various motorways in different traffic and environmental conditions. In this sense the theory presented in this paper can be considered a theoretical validation of the reliability theory. This conclusion seems very important, because the latter provides a method for the experimental measurement of the probability of instability and of the critical density, which are essential for traffic control and could not be estimated by the deterministic approach of the car-following theory. Acknowledgement-This PROMETHEUS.
research has been financed in part by the Consiglio Nazionale delle Ricerche, Progetto
REFERENCES Chandler R. E., Hermann R. and Montroll E. W. (1958) Traffic dynamics: Studies in car followina. Oons. Rex, 6, 165-184. Darroch J. N. and Rothery R. W. (1972) Car following and spectral analysis. Proceedings of the 5th International Symposium on the Theory of Traffic Flow and Transuortation. Elsevier. New York. DD.47-56. Edie L. C. and Foote R. S. (1961)~Experiments on single-lane-flow in tunnels. Proceedings af ihe 1st International Symposium on the Theory of Traffic Flow. Elsevier, Amsterdam, pp. 175-192. Ferrari P. (1988) The reliability of the motorway transport system. Transpn. Res., 22B, 291-310. Ferrari P. (1991) The control of motorway reliability. Transpn. Res., 25A, 419-427. Hermann R, Montroll E. W., Potts R. B., Rothery R. W. (1959) Traffic dynamics: Analysis of stability in car-following. Opns. Res., 7, 86-106. Hermann R. and Potts R. B. (1962) Single-lane traffic theory and experiments. Proceedings of the 1st International Symposium on the Theory of Traffic Flow. Elsevier, Amsterdam, pp. 120-146. Kamen E. W. (1980) On the relationship between zero criteria for two variable polinomials and asymptotic stability of delay differential equations. IEEE Trans., AC25,983-984. Lighthill M. J. and Whitham G. B. (1955) On kinematic waves 11, A theory of traffic flow on long crowded roads. Proc. Roy. Sot., 229A, 317-345. Newell G. F. (1%2) Theories of instability in dense highway traffic. JOpns. Res. Japan, 5.9-54. Richards P. I. (1956) Shock waves on the highway. Opns. Res.. 4, 42-51. Treiterer J. and Myers J. A. (1974) The hysteresis phenomenon in traffic flow. Proceedings of the 6th InternaGonalSymposium on Transport and Traffic Theory. Reed Pty Ltd, Artamon N.S.W., pp. 13-38.
Pergamon
THE INSTABILITY
University
OF MOTORWAY
PAOLO FERRARI School of Engineering of Pisa, Via Diotisalvi (Received
9 October
TRAFFIC
2, 56126 Pisa, Italy
1992; in revisedform 12 April 1993)
Abstract-This paper presents an analysis of the dynamic behaviour of the motorway traffic system, in order to single out its characteristics that are responsible for instability. It is shown that, besides the parameters considered in the traditional car-following theory, a great importance in determining instability of a vehicle platoon is the amplitude spectrum of the speed function of the leader vehicle. The results obtained provide a theoretical validation of the reliability theory of motorway traffic, which was founded on experimental basis, and by which the probability of instability of a traffic stream can be experimentally measured. 1. INTRODUCTION Instability is one of the most interesting phenomena of motorway traffic. A traffic stream is called unstable when, rapidly and without any obvious reason, marked and sudden falls of speed level take place, often resulting in a complete standstill. After the stop, the return to normal traffic conditions is quite difficult: the recovery of a speed value close to the initial one is often followed by a new fall, and so on repeatedly, giving rise to the classical stop-and-go phenomenon. Traffic instability was intensively studied during the past decades, both theoretically (e.g. Chandler, et al., 1958; Herman, et al., 1959; Newell, 1962) and experimentally (e.g. Edie and Foote, 1961; Treiterer and Myers, 1974), even if early theoretical studies of traffic “centered around the hope that traffic behaved in a stable way” (Newell, 1962, p. 38). The continuous theory (Lighthill and Whitham, 1955; Richards, 1956) considered traffic as a fluid characterized by three parameters, which are continuous functions of the space x and of the time t: the flow rate q, the density k and the space-mean speed v. This theory shows that in the t, x plane, the points characterized by the same values of parameters k, q and v are located along lines of constant slope c (the wave velocity), which are parallel to the tangent to the diagram q = q(k) in the point of abscissa k. It is possible that lines of different slope intersect in the t, x plane. This happens, for example, when a platoon of fast vehicles catches up to a slow one, or when at a certain moment a traffic hump takes place in a point of the carriageway. In these cases the intersection points are located along lines having slope w equal to that of the cord joining the points in the k, q diagram corresponding to the k values of the intersecting lines: w is the speed of the shock waves, i.e. the speed at which the discontinuities of k, q, v propagate along the carriageway. Taking into account the shock waves that absorb the discontinuities, traffic flow in this theory is always stable. A particular and important type of shock wave happens at a bottleneck, i.e. a section of reduced capacity. The theory predicts that flow past the bottleneck should stabilize at its capacity, and a shock wave propagates upstream. This conclusion is in contrast with experience, which shows that instability phenomena, characterized by a sequence of stop-start waves, are recurrent at bottlenecks (e.g. Edie and Foote, 1961). Traffic flow is stable even according to car-following theory, if it is supposed that each vehicle adjusts its speed immediately to that of the preceding one. This theory considers a sequence 1 . . . j . . . n of vehicles that proceed without overtaking along a single lane. Let v,(t) and xi(t) be the speed and position of the jth vehicle at time t, respectively. A relation exists between the speed v, of the jth vehicle and the distance dj = Xj_, - x, with respect to the preceding one: 175
176
P. FERRARI vj(t)
=
F[Xj_l(t)
(1)
Xi(t)]*
-
An example of eqn (1) is represented in Fig. 1. An equivalent form of (1) can be obtained by differentiating both sides: dv,( t) dt
where 2
=
F’Ixj-l(t) - xj(t)l[v,-l(t) - vj(t)l,
is the acceleration of the jth vehicle and F’ is the derivative of F with respect
to its argument. F’ is the slope of the tangent to the diagram v = F(d) (Fig. 1) at a given value d = Xj_, - Xj. This theory shows that, if the fluctuations of distance between vehicles are low so that F’ can be considered constant, speed fluctuations of the following vehicle vanish with time, and similarly speed fluctuations are damped along the platoon, so that flow is always stable (e.g. Chandler, et al., 1958). The main innovation of the car following theory, and one of the most important contributions to the theory of traffic flow, was the introduction of a time delay Tin the reaction of the following vehicle. In this case, by considering F’ as a constant CL,eqn (2) becomes: dVj( t) dt
= /A *
[Vj_l(t
-
T) -
vj(t
-
T)].
It was shown by Chandler et al. (1958) that the time lag T is the cause of traffic instability. Two kinds of instability were considered: local instability, due to the progressive increase with time of the amplitude of speed fluctuations of the following vehicle; and asymptotic instability, due to the progressive increase of the amplitude of speed fluctuations along a platoon. It was shown that flow is locally unstable when the product pT was greater than 7r/2, whereas it is asymptotically unstable when pT > l/2. Experimental measures carried out by many authors (e.g. Chandler, et al., 1958; Herman and Potts, 1962; Darroch and Rothery, 1972) indicate that pT in real life is always less than 7r/2, so that the only possible kind of instability is the asymptotic one. If we consider in Fig. 1 the point P where the slope of the tangent to the diagram v = F(d) is ~1 = 1/2T, the corresponding distance d* singles out a value k* = l/d* for the traffic density k, which is critical in the following sense. When k c k* the speed fluctuations of a vehicle are damped along the platoon, and flow is stable. If k > k* and there is a reduction in speed of a vehicle, this reduction will be amplified along the
Fig. 1. Relationship
v = F(d) between
speed v and distance
d.
Instability of motorway traffic
177
platoon, giving rise to a progressive decrease of distances between vehicles, thus to an increase of CL,which produces more and more decrease of speeds: if the platoon is long enough, a complete standstill will take place. The existence of a critical density, which depends on the relation v = F(d) and on the value of the time lag T,is the main result of the car-following theory, and one of the most important results of traffic theory. A consequence of this result is that, when flow rate tries to go over the carriageway capacity, and the critical density is exceeded, there is not a stable adjustment of flow to capacity-as the continuous theory predicts-but flow becomes unstable, and a sequence of stopstart waves takes place, as experience confirms. However some remarks have to be made about the method used by Chandler et al. (1958) in order to obtain the instability characteristics of a traffic stream. These authors investigated the stability question in terms of the Fourier components of the speed function v,(t) of the leader vehicle of the platoon, and considered as critical for the stability the least value of pT at which the amplitude of at least one of these components increases along the platoon. Components of low frequency, close to zero, give the greatest limitations on pT value. The increase of amplitude of these components from a vehicle to a successive one is very low, but the length of the platoon was considered infinite, so that the amplitude of vi(t) increases up to infinity along the platoon. However in real life platoons have finite length, so that these components, given the little increase of their amplitude, do not contribute to motorway instability. On the contrary, when pT is somewhat greater than l/2, the amplitude of components of v,(t) of higher frequency increases in more substantial measure, giving rise to marked fluctuations of speed level of traffic. For this reason instability does not depend only on pT value, but also on the amplitude spectrum of v#). Moreover v,(t), p and T,as well as the length of platoons, are random quantities. In particular v,(t) is a realization of a stochastic process, whose characteristics essentially depend on drivers’ behaviour; p is a random variable distributed around a mean that is an increasing function of traffic density, and T is a random variable whose mean essentially depends on environmental characteristics: it is high during dark hours and bad weather conditions, low when weather is good and light. For this reason instability is a stochastic phenomenon. The deterministic approach of the car-following theory is very important in order to single out the parameters on which instability depends and how these parameters cause instability. But deterministic values of traffic parameters at which flow becomes unstable do not exist: it is only possible to know the probability that in correspondence of given traffic parameters instability will take place during a certain time period, and this knowledge can be obtained only by studying traffic flow as a stochastic process. This paper is a contribution to research into how traffic parameters influence instability. Section 2 is dedicated to the study of dynamic behaviour of traffic flow. The influence of the amplitude spectrum of v,(t) on instability is examined in Section 3. Section 4 is dedicated to a discussion on the concept of critical density, and examines the link existing between the results obtained in this paper and those of the reliability theory of motorway traffic, by which the experimental measure of critical density and of the probability of instability can be obtained. 2. THE DYNAMICAL BEHAVIOUR OF TRAFFIC
Consider two vehicles, 1 and 2, which proceed with speed v,(t) and v,(t), one following the other, along a motorway lane. The dynamical behaviour of this system is defined by the difference-differential eqn (3):
dv, - = p * [v,(t - T) - vz(t dt
T)],
where parameters CLand T are characteristics of vehicle number 2; they vary randomly from one vehicle to another along a platoon, and even for the same vehicle with time,
178
P. FERRARI
around mean values that depend on traffic and environmental before. Taking the Laplace transform of (3a), we have:
conditions, as it was said
where V,(s) is the Laplace transform of vi(t). From (4) the following equation is obtained: 1/2(s) =
-*v,(s) pe s + pees*
= W(s) V,(s),
where -ST
W(s)
=
pe s + pews*
is the transfer function of the system. By using the typical definition of the dynamical system theory, different from that of the car-following theory, this system is said to be asymptotically stable if, given a transient input v,(t) + 0 when t + 03, even the output v2(f) --) 0 when t + 03. It is known (e.g. Kamen, 1980) that the necessary and sufficient condition for asymptotic stability is that the real part of the solutions of the characteristic equation: s + pees* = 0
(7)
is negative. The solutions of eqn (7) can be investigated by means of a method similar to that which was used by Hermann et al. (1959), and it can be shown that they lie on the ?r negative part of the complex plane-i.e. its real part is negative-if and only if pT < 2. In real life pT c
as was shown in the previous section; therefore motorway traffic is
asymptotically stable, and its instability depends on the increase of the amplitude of speed fluctuations from a vehicle to another along a platoon. The relation between v2(t) and v,(t) can be obtained by calculating the inverse of the Laplace transform (5):
b(t) =
&n-
s
‘+‘a pe-‘*V,
y-im
s
+
(s)
es,ds
p-S*
’
(8)
where y is a real number so that the integration path lies to the right of all the singularities of the Laplace transform.
As FT < s all singularities lie in the negative part of the
complex plane, so that it is possible to put y = 0. So eqn (8) becomes: V*(f) =
&
s m
_m G(w)Z(w)e’“‘dw,
(9)
where: G(w) = V,(h)
(10)
is the Fourier transform of the input v](t) and: -iId
Z(0)
=
1 iw + pe -jwT = 1 + iwe -‘“*/p
lJe
is the frequency response of the system.
(11)
179
Instability of motorway traffic
The complex variable 1 + iueiwT/p can be written: 1 + iueiuT/p = 1 - 0 sin UT + i 0 P CL
cos
UT,
(12)
so that the following expressions of module )Z 1 and phase 4 of Z(o) can be obtained:
IZ(@) I =
4(w)
)-I”
(l+L-2EsinwT cL p2
1 - w sin wT P
= *cos-’
l/2'
l+g-2EsinwT i CL CL2
(13)
1
where signs + and - hold when w < and > 0, respectively. By denoting I G(w) I and &(w) respectively module and phase of the input transform G(o), and + = -(+ + 6,), eqn (9) can be written:
s
:,
lG(w)l IZ(o)l 1cos (cot- a) + i sin (wt - +)]dw.
(14)
We deduce from (14) that the characteristics of v2(f) essentially depend on the characteristics of the amplitude spectra IG(w)1 and [Z(w)1 : if the frequency bands of high amplitude coincide to a large extent in these two spectra, there will be a high increase of the maximum of v2(t) with respect to vr(t). If these bands are different, a low increase will take place, or v2(f) will be damped. Therefore the increase of v2(f) does not depend only on the parameters p and T of the system, but also on the characteristics of the input v,(t). By introducing the variables Q = pT and x = wT into expressions (13), they are written in dimensionless form: [Z(x) I = [ 1 + (ST #J(x) = *cos-’
- 2 G sin xl-‘”
1 I1,
1 - g sin x IZ(x)
(1%
(16)
which define the amplitude and phase spectra of the response of the system defined by eqn (3a). The amplitude spectrum is reported in Fig. 2 for various values of Q. 3. THE INFLUENCE OF CHARACTERISTICS
OF THE SPEED FUNCTION OF THE
LEADER VEHICLE ON TRAFFIC STABILITY
Consider two vehicles that proceed with the same other, along a motorway lane. Suppose that at instant sudden decrease of speed, followed by a more gradual assumed that the speed variation v,(t) with respect to of time t, is: v,(t)
constant speed V, one following the t = 0 the first vehicle undergoes a recovery to the initial speed V. It is V of the first vehicle, as a function
= @tAv exp (1 - Pt),
(17)
where / Av 1 is the maximum absolute value of the speed shift with respect to V, l/p is the instant when this maximum is reached and ~Av is the mean value of the deceleration. The purpose of this section is to show the influence of P, and thus of the deceleration of the leader vehicle, on traffic stability. TR(B) 28:2-H
P. FERRARI
4-
3-
0
0.5
1
1.5
3
2.5
2
X= OT Fig. 2. Amplitude
spectra
of the response
of car-following
system for various
Qvalues.
The Fourier transform G(w) of the input v,(t) is (t r 0):
s 02
G(w)
=
/3Av o t exp [l - W
PAve + iu)ldt = tp + iwj2,
where e is the basis of natural logarithms. The module 1G(o) 1 and the phase 6, of G(w) are:
IGCw)l=
,,!~w21f$, =
-2
tan-’
(1%
(w/P).
By substituting (19) into (14), and taking into account that IZ(w) I = )Z( -co) (, 4(w) = -rp( - w), C+,(W) = -q5,( - w), we have the following exI G(w) I = lG(--41, pression of the speed variation v2(f) with respect to V of the second vehicle: v2(t)
=
cos (wt -
!Z!!!f! j,
Q)dw l/2’
?r (fl*
+
w2)
(20)
1 + (W/F)’ - 2 w sin CL
By introducing the dimensionless variables 77= v2/Av, 7 = t/T, b = PT, x = wT, eqn (20) becomes:
Q = pT,
s m
cos (x7 -
o
(b*
9)dx l/2’
+
x2)(
1 + (x/Q)* - 2 5 sen
Q
The input function (17), written in dimensionless form, is:
(21)
181
Instability of motorway traffic
v1
=
2 =h
(22)
exp ( 1 - br) .
In order to know the influence of the input characteristics on traffic stability, consider two input functions (22) having different b parameters: b = 0.1 and b = 0.4. These are, for example, the input functions of a system in which the time delay of the second vehicle is T = 1.5 s and the first vehicle undergoes a speed decrease of 20 kmph with, respectively, a low deceleration a Z 0.4 ms-’ and a high deceleration a z 1.4 msm2. These functions are reported with dashed lines in Figs. 3 and 4. The corresponding amplitude spectra:
IGWI = g--g
(23)
are reported, with dashed lines, in Figs. 5 and 6. It should be noted from these figures that when b = 0.1 the values 1G 1 > 1 range in the (0, 0.5) band of x, whereas when b = 0.4 they range in the band (0, 1.O). By comparing these spectra with those 1Z(x) 1 of the system response represented in Fig. 2, it can be noted that the latter show 1Z( values in the band (0, 0.5) close to 1 even for high Q values, whereas in the band (0, 1.0) there are IZ 1 values much greater than 1 when Q > 0.7. As a consequence of this fact we can argue that when b = 0.1 the maximum value of q given by (21) will be rather close to 1 even when Q is high, whereas when b = 0.4 the maximum of rl will be markedly higher than 1 for high Q values. This conjecture was confirmed by numerical integration of (21); the functions o(r) calculated for both the b values and for different Q are reported in Figs. 3 and 4. The output v2(t) of the system S, made up by the first and the second vehicle of a platoon is the input for the system S2 composed by the second and the third vehicle, and so on. The amplitude spectrum of the input for S, is given by I G2(x) ) = 1G(x)I1Z(x) I: its examination is important in order to know the progressive variations of q(r) function along a platoon. These spectra have been calculated for both the b values and for different Q, and they are reported in Figs. 5 and 6. It can be noted that, while for b = 0.1 the values I G2 I > 1 still range in the (0, 0.5) interval of x, when b = 0.4 there are high I G21 values in the band (0, 1.5) for Q > 0.7. From this result we can argue that when b = 0.1
b = 0.1
-A-
l st vehicle
Fig. 3. Relationships 7 = ~(7) relative to the 1st and the 2nd vehicle of a platoon, for Q values.
b = 0.I and
for various
P. FERRARI
182
b = 0.40
--0.6
Fig. 4. Relationships
7 = ~(7) relative
1 st vehicle 2 nd whack
to the 1st and the 2nd vehicle of a platoon, Q values.
for b = 0.4 and for various
the maximum 77is not increased in a way that has to be considered dangerous along a platoon, whereas when b = 0.4 marked increases of qrnaxwill happen when Q > 0.7. A confirmation of this conjecture is given by Fig. 7, where the ratios v4 = v,/v, relative to the 4th vehicle of the platoon are reported for both the values of b and for Q = 0.9, in the hypothesis that all p and Tarethe same along the platoon. The functions ~~(7) were calculated taking into account that the frequency response of a car-following system constituted by n vehicles with the same p and T values has the following module 1Z,, 1 and phase an: lZ,l = IGI IZI”-’
a’, = 6, + (n - 1)4.
4. THE CRITICAL
(24)
DENSITY
It was shown in the previous section that the progressive increase of speed fluctuations along a platoon, and thus the risk of instability, depends on traffic parameters p, T, 0.They randomly vary from a vehicle to another of the platoon, and even for the same vehicle with time. The average ii of p distribution is an increasing function of traffic density. The average 7 of T depends on environmental conditions: it is high during the dark hours and when weather is bad. The average p of /3 is an increasing function of the frequency of lane changes in dense traffic. For example, when a vehicle shifts from the inner-side lane to the off-side one of a motorway carriageway, its speed is usually less than that of the following vehicle on the off-side lane, which has to decelerate in order to avoid collision. The deceleration is low if the vehicle inserts itself into a large gap; but if traffic is dense and lane changes are frequent, little gaps are used, and in this case the deceleration, and thus 0 value, is high. It was shown in Section 1 that the classical theory of car-following defines as critical the density value over which flow is unstable: this value of critical density depends on T and on the relation between p and the distance between vehicles. This definition is not
183
Instability of motorway traffic
b = 0.1 2.5
---
1 st vehicle 2 nd vehicle
2.0
1.5
1.0
0.5
0’
0.5
1.0
1.5
2.0
2.5
3.0 x=01
Fig. 5. Amplitude spectra of the functions 7 = q(7) relative to the 1st and the 2nd vehicle of a platoon for b = 0.1 and for various Q values.
complete, because it is necessary to take into account also the /3 parameter. Moreover, it has to be noted that, given the stochastic nature of traffic parameters on which the instability depends, a deterministic value for the critical density does not exist. Consider a motorway carriageway stretch and let X be the event representing a progressive speed drop up to --a complete stop of circulation in the vehicle stream travelling along this stretch. Let P(X/k, T, 0) --equal the conditional probability that the event X takes place, given the average values k, T, 0 of density k and of parameters T and 6 of the stream, during a time period characterized by constant average traffic conditions. Because of the monotonic relation --between k and ji, it can be deduced from the results of the preceding section that &Y/k, T, 0) is an increasing function of k, as well of 7 and s’. Let Sr and 23,be two traffic streams travelling on the motorway stretch in different time periods: stream S, is characterized by high average values of T and 0, whereas they se low+ the stream S,. Consider the relations between the conditioned probability P(X/ k) and k for the two streams: according to what was said before, the probability &Y/k) of the stream S, is higher than that of S,, traffic density z being the same. Let us define critical density as the value k* of k corresponding to a threshold P* of the probability p(X/F) beyond which the risk of the stop of circulation is considered unacceptable. It is easy to verify that the critical density k-7 of S, is greater than that of k: of S,: i.e. critical density increases as the average 7 and /3 values of the stream decrease. The critical density is a fundamental concept of traffic theory, and it is also important from a practical point of view, because the choice of the best control strategy for a
184
P.
FERRARI
b= 0.4
-__
1 st vehicle 2 nd vehicle
2.5
2.0
1.5
1.0
0.5
d
0.5
1.0
1.5
2.0
2.5
3.0
= WT
Fig. 6. Amplitude spectra of the functions 7 = ~(7) relative to the 1st and the 2nd vehicle of a platoon, for 6 = 0.4 and for various Q values.
traffic stream depends on its value (Ferrari, 1991). But its estimation by using the carfollowing theory is very difficult, among other things because it is difficult to estimate the probability distributions of T and /3. For this reason it seems more convenient to obtain the critical density by studying the stochastic process of vehicle speed, in which the effects of probability distributions of p, T, @ are condensed. Taking into account the fact that instability first takes place on the off-side lane of a motorway carriageway, from a practical point of view and for control purposes it is sufficient to process the speed sequence surveyed in a cross section of this lane. Let Y,be the speed recorded at instant t, and 7, the speed level at the same instant, i.e. the conditioned mean of speed, given the preceding realization of the sequence. The reliability theory of motorway traffic (Ferrari, 1988) shows that the speed sequence is a succession of realizations of homogeneous integrated moving average processes of the first order, whose equation is: v, =
V,_I
+
a, -
(1 -
h)a,_1,
(25)
where a, is the shift of the speed v, from the corresponding level “J,, and X is a coefficient ranging between 0 and 1, which measures the influence that the shift a, at a certain instant t has on the level 7,+) at the subsequent instant. The shifts a, are random variables independently and normally distributed with zero mean and equal variance u*. Therefore, the process eqn (25) has two parameters: h and cr2. Parameter h is an increasing linear function of the logarithm of the average density F of each realization, and the relation
185
Instability of motorway traffic
t/T
Fig. 7. Relationships TJ= ~(7) relative to the 1st and the 4th vehicle of a platoon, for Q = 0.9 and for two values of b.
between h and Ink is independent of the circumstances and the place where data are recorded. On the contrary, parameter uz is a decreasing linear function of the logarithm of the average density of each realization, and the relationship between a2 and lnx varies, sometimes in a notable way, depending on the circumstances and on the place where the survey is carried out: in particular during the night the u2 values are much greater than during the day hours, density k being the same. There is a close correspondence between these parameters and those of the carfollowing theory presented in the preceding sections. Parameter X, like ~1,measures the influence that the speed fluctuations of a vehicle have on the speed of the successive one, as a function only of traffic density and independently of all other traffic parameters: therefore, both parameters represent a characteristic of the system, and do not depend on the input. The shifts a, of the speed from the process level in dense traffic are essentially due to the reactions of vehicles to speed fluctuations of the preceding one, and their variance uz depends on the whole of traffic characteristics. u2 decreases with density, because the lane changes frequency diminishes when density increases, and is a function, traffic density being the same, of the T and 0 distribution in traffic streams; its value, at a given average density k, is an increasing function of the average values of T and /3, because they increase when the environmental conditions become more difficult and the lane changes more frequent. The reliability theory shows that the critical density is given by the intersection of the straight line u2 = u2(lnk) with axis of k: it is greater for traffic streams having lower u2 values, density k being the same. That is in accordance with the result obtained above with reference to the influence of T and /3 values on critical density. Therefore, we can argue from the close correspondence between the two theories that the value of critical density experimentally estimated by using the reliability theory is a good measure of the critical density defined by the car-following theory presented in this paper. 5. CONCLUSIONS
The car-following theory presented in this paper identifies the causes of instability phenomena on motorways and indicates the traffic parameters on which instability depends. Its conclusions are in close accordance with those of reliability theory of motorway
186
P. FERRARI
traffic, which was established on an experimental basis, by processing a great deal of data recorded on various motorways in different traffic and environmental conditions. In this sense the theory presented in this paper can be considered a theoretical validation of the reliability theory. This conclusion seems very important, because the latter provides a method for the experimental measurement of the probability of instability and of the critical density, which are essential for traffic control and could not be estimated by the deterministic approach of the car-following theory. Acknowledgement-This PROMETHEUS.
research has been financed in part by the Consiglio Nazionale delle Ricerche, Progetto
REFERENCES Chandler R. E., Hermann R. and Montroll E. W. (1958) Traffic dynamics: Studies in car followina. Oons. Rex, 6, 165-184. Darroch J. N. and Rothery R. W. (1972) Car following and spectral analysis. Proceedings of the 5th International Symposium on the Theory of Traffic Flow and Transuortation. Elsevier. New York. DD.47-56. Edie L. C. and Foote R. S. (1961)~Experiments on single-lane-flow in tunnels. Proceedings af ihe 1st International Symposium on the Theory of Traffic Flow. Elsevier, Amsterdam, pp. 175-192. Ferrari P. (1988) The reliability of the motorway transport system. Transpn. Res., 22B, 291-310. Ferrari P. (1991) The control of motorway reliability. Transpn. Res., 25A, 419-427. Hermann R, Montroll E. W., Potts R. B., Rothery R. W. (1959) Traffic dynamics: Analysis of stability in car-following. Opns. Res., 7, 86-106. Hermann R. and Potts R. B. (1962) Single-lane traffic theory and experiments. Proceedings of the 1st International Symposium on the Theory of Traffic Flow. Elsevier, Amsterdam, pp. 120-146. Kamen E. W. (1980) On the relationship between zero criteria for two variable polinomials and asymptotic stability of delay differential equations. IEEE Trans., AC25,983-984. Lighthill M. J. and Whitham G. B. (1955) On kinematic waves 11, A theory of traffic flow on long crowded roads. Proc. Roy. Sot., 229A, 317-345. Newell G. F. (1%2) Theories of instability in dense highway traffic. JOpns. Res. Japan, 5.9-54. Richards P. I. (1956) Shock waves on the highway. Opns. Res.. 4, 42-51. Treiterer J. and Myers J. A. (1974) The hysteresis phenomenon in traffic flow. Proceedings of the 6th InternaGonalSymposium on Transport and Traffic Theory. Reed Pty Ltd, Artamon N.S.W., pp. 13-38.