2001:
occurs mostly at a bottleneck. Within the synchronized flow phase a further "self-compression" occurs and vehicle density increases while vehicle speed decreases. This self-compression is called "pinch effect". In "pinch" regions of synchronized flow, narrow moving jams emerge. If these narrow moving jams grow, wide moving jams will emerge labeled by S → J in Figure 9). Thus, wide moving jams emerge later than traffic breakdown (F → S transition) has occurred and at another road location upstream of the bottleneck. Therefore, when Kerner’s F → S → J phase transitions occurring in real traffic (Figure 9 (a)) are presented in the speed-density plane (Figure 9 (b)) (or speed-flow, or else flow-density planes), one should remember that states of synchronized flow and low speed state within a wide moving jam are measured at different road locations. Kerner notes that the frequency of the emergence of wide moving jams increases if the density in synchronized flow increases. The wide moving jams propagate further upstream, even if they propagate through regions of synchronized flow or bottlenecks. Obviously, any combination of return phase transitions (S → F, J → S, and J → F transitions shown in Figure 9) is also possible.
1093:
occurring even if the preceding vehicle does not drive faster than the vehicle and the preceding vehicle additionally does not accelerate. In Kerner’s theory, the probability of over-acceleration is a discontinuous function of the vehicle speed: At the same vehicle density, the probability of over-acceleration in free flow is greater than in synchronized flow. When within a local speed disturbance speed adaptation is stronger than over-acceleration, an F → S phase transition occurs. Otherwise, when over-acceleration is stronger than speed adaptation the initial disturbance decays over time. Within a region of synchronized flow, a strong over-acceleration is responsible for a return transition from synchronized flow to free flow (S → F transition).
1115:
free flow at the bottleneck leads to the emergence of congested traffic whose downstream front is fixed at the bottleneck (at least during some time interval), i.e., this congested traffic satisfies the definition for the synchronized flow phase. In other words, spontaneous traffic breakdown is always an F → S phase transition. (ii) Probability of this spontaneous traffic breakdown is an increasing function of the flow rates at the bottleneck. (iii) At the same bottleneck, traffic breakdown can be either spontaneous or induced (see empirical examples for these fundamental features of traffic breakdown in Secs. 2.2.3 and 3.1 of the book); for this reason, the F → S phase transition occurs in a
1150:
free flow with respect to the F → S phase transition is governed by the nucleation nature of an instability of synchronized flow. The explanation is a large enough local increase in speed in synchronized flow (called an S → F instability), which is a growing speed wave of a local increase in speed in synchronized flow at the bottleneck. The development of the S → F instability leads to a local phase transition from synchronized flow to free flow at the bottleneck (S → F transition). To explain this phenomenon Kerner developed a microscopic theory of the S → F instability. None of the classical traffic flow theories and models incorporate the S → F instability of the three-phase theory.
63:
556:) data related to congested traffic the "flow-interruption interval", i.e., a time headway between two vehicles following each other is observed, which is much longer than the mean time delay in vehicle acceleration from a wide moving jam (the latter is about 1.3–2.1 s), then the related flow-interruption interval corresponds to the wide moving jam phase. After all wide moving jams have been found through this criterion in congested traffic, all remaining congested states are related to the synchronized flow phase.
1992:
1711:
527:
straight through the freeway bottleneck. According to the definition , this pattern of congestion belongs to the "wide moving jam" phase. In contrast, the downstream front of the other pattern is fixed at a bottleneck. According to the definition , this pattern belongs to the "synchronized flow" phase (Figure 3 (a) and (b)). Other empirical examples of the validation of the traffic phase definitions and can be found in the books and, in the article as well as in an empirical study of
2232:, Kerner’s three-phase theory is a qualitative traffic flow theory that consists of several hypotheses. The hypotheses of Kerner’s three-phase theory should qualitatively explain spatiotemporal traffic phenomena in traffic networks found in real field traffic data, which was measured over years on a variety of highways in different countries. Some of the hypotheses of Kerner’s theory have been considered above. It can be expected that a diverse variety of different
590:
2256:
bottlenecks, and on moving bottlenecks, features of heterogeneous traffic flow consisting of different vehicles and drivers, jam warning methods, vehicle-to-vehicle (V2V) communication for cooperative driving, the performance of self-driving vehicles in mixture traffic flow, traffic breakdown at signals in city traffic, over-saturated city traffic, vehicle fuel consumption in traffic networks (see references in Sec. 1.7 of a review).
1236:
544:
macroscopic data has been measured in space and time, i.e., in an "off-line" study. This is because for the definitive distinction of the phases J and S through the definitions and a study of the propagation of traffic congestion through a bottleneck is necessary. This is often considered as a drawback of the traffic phase definitions and . However, there are local microscopic criteria for the distinction between the phases
1042:
transition from free flow to synchronized flow (called as F →S phase transition). This explanation is supported by available measurements, because in measured traffic data after a traffic breakdown at a bottleneck the downstream front of the congested traffic is fixed at the bottleneck. Therefore, the resulting congested traffic after a traffic breakdown satisfies the definition of the "synchronized flow" phase.
2168:
810:
55:
2071:
1680:(with free flow at this location, see Figure 8). This means that many wide-moving jams have similar features under similar conditions. These parameters are relatively predictable. The movement of the downstream jam front can be illustrated in the flow-density plane by a line, which is called "Line J" (Line J in Figure 8). The slope of Line J is the velocity of the downstream jam front
2059:
bottleneck and moves further upstream. In contrast to wide moving jams, the synchronized flow, even if it moves as an MSP, has no characteristic parameters. As an example, the velocity of the downstream front of the MSP might vary significantly and can be different for different MSPs. These features of SP and wide moving jams are consequences of the phase definitions and .
395:
1329:(see, for example, Sec. 17.2.2 of the book). The probability of a smaller disturbance in free flow is much higher than that of a larger disturbance. Therefore, the higher the flow rate in free flow at a bottleneck, the higher the probability of the spontaneous F → S phase transition. If the flow rate in free flow is lower than the minimum capacity
1066:
empirical example of an induced breakdown at a bottleneck leading to synchronized flow can be seen in Figure 3: synchronized flow emerges through the upstream propagation of a wide moving jam. The existence of empirical induced traffic breakdown (i.e., empirical induced F →S phase transition) means that an F → S phase transition occurs in a
518:
vehicle speeds across different lanes on a multilane road in this flow. In addition, there is a tendency towards synchronization of vehicle speeds in each of the road lanes (bunching of vehicles) in synchronized flow. This is due to a relatively low probability of passing. The term "synchronized" reflects this speed synchronization effect.
1445:, there is an infinite number of highway capacities of free flow at the bottleneck. The infinite number of flow rates, at which traffic breakdown can be induced at the bottleneck and the infinite number of highway capacities. These capacities are within the flow rate range between a minimum capacity and a maximum capacity (Figure 7).
2211:
bjects). ASDA/FOTO is a software tool able to process large traffic data volumes quickly and efficiently on freeway networks (see examples from three countries, Figure 11). ASDA/FOTO works in an online traffic management system based on measured traffic data. Recognition, tracking, and prediction of
1114:
Kerner’s explanation of traffic breakdown at a highway bottleneck by the F → S phase transition in a metastable free flow is associated with the following fundamental empirical features of traffic breakdown at the bottleneck found in real measured data: (i) Spontaneous traffic breakdown in an initial
1065:
In contrast, an induced F → S phase transition occurs through a region of congested traffic that initially emerged at a different road location downstream from the bottleneck location. Normally, this is in connection with the upstream propagation of a synchronized flow region or a wide moving jam. An
1061:
A spontaneous F →S phase transition means that the breakdown occurs when there has previously been free flow at the bottleneck as well as both up- and downstream of the bottleneck. This implies that a spontaneous F → S phase transition occurs through the growth of an internal disturbance in free flow
593:
Figure 4: Hypothesis of Kerner’s three-phase traffic theory about 2D region of steady states of synchronized flow in the flow—density plane: (a) Qualitative representation of free flow states (F) and 2D region of homogeneous synchronized flow (dashed region S) on a multi-lane road in the flow-density
577:
Kerner’s hypothesis is that homogeneous synchronized flow can occur anywhere in a two-dimensional region (2D) of the flow-density plane (2D-region S in Figure 4(a)). The set of possible free flow states (F) overlaps in vehicle density with the set of possible states of homogeneous synchronized flow.
2255:
The Kerner-Klenov stochastic three-phase traffic flow model in the framework of Kerner’s theory has further been developed for different applications. In particular, to simulate on-ramp metering, speed limit control, dynamic traffic assignment in traffic and transportation networks, traffic at heavy
1136:
The empirical nucleation nature of traffic breakdown at highway bottlenecks cannot be explained by classical traffic theories and models. The search for an explanation of the empirical nucleation nature of traffic breakdown (F → S phase transition) at a highway bottleneck has been the motivation for
1101:
a slower vehicle (over-acceleration) and deceleration to the speed of a slower-moving vehicle ahead (speed adaptation). Overtaking supports the maintenance of free flow. "Speed adaptation" on the other hand leads to synchronized flow. Speed adaptation will occur if overtaking is not possible. Kerner
1074:
free flow means that when small perturbations occur in free flow, the state of free flow is still stable, i.e., free flow persists at the bottleneck. However, when larger perturbations occur in free flow in a neighborhood of the bottleneck, the free flow is unstable and synchronized flow will emerge
493:
occurring in many systems of natural science (like gas plasma, electron-hole plasma in semiconductors, biological systems, and chemical reactions): Both the wide moving jam and a wide autosoliton exhibit some characteristic features, which do not depend on initial conditions at which these localized
477:
reflects the fact that if a moving jam has a width (in the longitudinal road direction) considerably greater than the widths of the jam fronts, and if the vehicle speed inside the jam is zero, the jam always exhibits the characteristic feature of maintaining the velocity of the downstream jam front
1589:
A moving jam will be called "wide" if its length (in direction of the flow) clearly exceeds the lengths of the jam fronts. The average vehicle speed within wide moving jams is much lower than the average speed in free flow. At the downstream front, the vehicles accelerate to the free flow speed. At
2292:
The above criticism has been responded to in a recent study of data measured in the US and the United
Kingdom, which confirms conclusions made based on measurements on the Bundesautobahn 5 in Germany. Moreover, there is a recent validation of the theory based on floating car data. In this article
2158:
In many freeway infrastructures, bottlenecks are very close to each other. A congestion pattern whose synchronized flow covers two or more bottlenecks is called an
Expanded Pattern (EP). An EP could contain synchronized flow only (called ESP: Expanded Synchronized Flow Pattern)), but normally wide
1625:
Kerner’s empirical results show that some characteristic features of wide moving jams are independent of the traffic volume and bottleneck features (e.g. where and when the jam formed). However, these characteristic features are dependent on weather conditions, road conditions, vehicle technology,
1448:
The range of highway capacities at a bottleneck in Kerner’s three-phase traffic theory contradicts fundamentally the classical understanding of stochastic highway capacity as well as traffic theories and methods for traffic management and traffic control which at any time assume the existence of a
1149:
Kerner developed the three-phase theory as an explanation of the empirical nature of traffic breakdown at highway bottlenecks: a random (probabilistic) F → S phase transition that occurs in the metastable state of free flow. Herewith Kerner explained the main prediction, that this metastability of
2247:
can show and explain traffic breakdown by an F → S phase transition in the metastable free flow at the bottleneck was the Kerner-Klenov model introduced in 2002. The Kerner–Klenov model is a microscopic stochastic model in the framework of Kerner’s three-phase traffic theory. In the Kerner-Klenov
2058:
The difference between the SP and the wide moving jam becomes visible in that when a WSP or MSP reaches an upstream bottleneck the so-called "catch-effect" can occur. The SP will be caught at the bottleneck and as a result a new congested pattern emerges. A wide-moving jam will not be caught at a
2017:
To further illustrate S → J phase transitions: in Kerner’s three-phase traffic theory Line J divides the homogeneous states of synchronized flow in two (Figure 8). States of homogeneous synchronized flow above Line J are meta-stable. States of homogeneous synchronized flow below Line J are stable
1995:
Figure 9: Empirical example of cascade of F → S → J phase transitions in Kerner’s three-phase traffic theory: (a) The phase transitions occurring in space and time. (b) The representation of the same phase transitions as those in (a) in the speed-density plane (arrows S → F, J → S, and J → F show
1960:
In contrast to wide moving jams, both the flow rate and vehicle speed may vary significantly in the synchronized flow phase. The downstream front of synchronized flow is often spatially fixed (see definition ), normally at a bottleneck at a certain road location. The flow rate in this phase could
1123:
free flow is as follows. Small enough disturbances in metastable free flow decay. However, when a large enough disturbance occurs at the bottleneck, an F → S phase transition does occur. Such a disturbance that initiates the F → S phase transition in metastable free flow at the bottleneck can be
2000:
In 1998, Kerner found out that in real field traffic data the emergence of a wide moving jam in free flow is observed as a cascade of F → S → J phase transitions (Figure 9): first, a region of synchronized flow emerges in a region of free flow. As explained above, such an F → S phase transition
1140:
In particular, in two-phase traffic flow models in which traffic breakdown is associated with free flow instability, this model instability leads to the F → J phase transition, i.e. in these traffic flow models traffic breakdown is governed by spontaneous emergence of a wide moving jam(s) in an
1041:
In measured data, congested traffic most often occurs in the vicinity of highway bottlenecks, e.g., on-ramps, off-ramps, or roadwork. A transition from free flow to congested traffic is known as traffic breakdown. In Kerner’s three-phase traffic theory traffic breakdown is explained by a phase
543:
In Sec. 6.1 of the book has been shown that the traffic phase definitions and are the origin of most hypotheses of three-phase theory and related three-phase microscopic traffic flow models. The traffic phase definitions and are non-local macroscopic ones and they are applicable only after
517:
The term "synchronized flow" is meant to reflect the following features of this traffic phase: (i) It is a continuous traffic flow with no significant stoppage, as often occurs inside a wide moving jam. The term "flow" reflects this feature. (ii) There is a tendency towards synchronization of
2259:
Over time several scientific groups have developed new mathematical models in the framework of Kerner’s three-phase theory. In particular, new mathematical models in the framework of Kerner’s three-phase theory have been introduced in the works by Jiang, Wu, Gao, et al., Davis, Lee, Barlovich,
526:
Measured data of averaged vehicle speeds (Figure 3 (a)) illustrate the phase definitions and . There are two spatial-temporal patterns of congested traffic with low vehicle speeds in Figure 3 (a). One pattern propagates upstream with an almost constant velocity of the downstream front, moving
1092:
Kerner explains the nature of the F → S phase transitions as a competition between "speed adaptation" and "over-acceleration". Speed adaptation is defined as the vehicle's deceleration to the speed of a slower moving preceding vehicle. Over-acceleration is defined as the vehicle acceleration
3226:
Kerner, Boris S; Rehborn, Hubert; Schäfer, Ralf-Peter; Klenov, Sergey L; Palmer, Jochen; Lorkowski, Stefan; Witte, Nikolaus (2013). "Traffic dynamics in empirical probe vehicle data studied with three-phase theory: Spatiotemporal reconstruction of traffic phases and generation of jam warning
573:
state of synchronized flow of identical vehicles and drivers in which all vehicles move with the same time-independent speed and have the same space gaps (a space gap is the distance between one vehicle and the one behind it), i.e., this synchronized flow is homogeneous in time and space.
1302:, then even small disturbances in free flow at a bottleneck will lead to a spontaneous F → S phase transition. On the other hand, only very large disturbances in free flow at the bottleneck will lead to a spontaneous F → S phase transition, if the flow rate is close to a minimum capacity
1096:
There can be several mechanisms of vehicle over-acceleration. It can be assumed that on a multi-lane road the most probable mechanism of over-acceleration is lane changing to a faster lane. In this case, the F → S phase transitions are explained by an interplay of acceleration while
2299:
This criticism has been responded to in a review as follows. The most important feature of Kerner’s theory is the explanation of the empirical nucleation nature of traffic breakdown at a road bottleneck by the F → S transition. The empirical nucleation nature of traffic breakdown
2018:
states in which no S → J phase transition can occur. Metastable homogeneous synchronized flow means that for small disturbances, the traffic state remains stable. However, when larger disturbances occur, synchronized flow becomes unstable, and an S → J phase transition occurs.
34:
between 1996 and 2002. It focuses mainly on the explanation of the physics of traffic breakdown and resulting congested traffic on highways. Kerner describes three phases of traffic, while the classical theories based on the fundamental diagram of traffic flow have two phases:
1173:
Spontaneous traffic breakdown, i.e., a spontaneous F → S phase transition, may occur in a wide range of flow rates in free flow. Kerner states, based on empirical data, that because of the possibility of spontaneous or induced traffic breakdowns at the same freeway bottleneck
578:
The free flow states on a multi-lane road and states of homogeneous synchronized flow are separated by a gap in the flow rate and, therefore, by a gap in the speed at a given density: at each given density the synchronized flow speed is lower than the free flow speed.
340:
2045:
Frequently the upstream front of a SP propagates upstream. If only the upstream front propagates upstream, the related SP is called
Widening Synchronised Flow Pattern (WSP). The downstream front remains at the bottleneck location and the width of the SP increases.
2272:
in
Germany. It may be that this road has this pattern, but other roads in other countries have other characteristics. Future research must show the validity of the theory on other roads in other countries around the world. Second, it is not clear how the data was
2215:
Further applications of the theory are seen in the development of traffic simulation models, a ramp metering system (ANCONA), collective traffic control, traffic assistance, autonomous driving, and traffic state detection, as described in the books by Kerner.
217:
1087:
Figure 6: Explanation of traffic breakdown by a Z-like non-linear interrupted function of the probability of overtaking in Kerner’s three-phase traffic theory. The dotted curve illustrates the critical probability of overtaking as a function of traffic
513:
In "synchronized flow," the downstream front, where the vehicles accelerate to free flow, does not show this characteristic feature of the wide moving jam. Specifically, the downstream front of the synchronized flow is often fixed at a bottleneck.
1572:
of a range of highway capacities in Kerner’s theory changes crucially methodologies for traffic control, dynamic traffic assignment, and traffic management. In particular, to satisfy the nucleation nature of traffic breakdown, Kerner introduced
468:
reflects the jam propagation as a whole localized structure on a road. To distinguish wide moving jams from other moving jams, which do not characteristically maintain the mean velocity of the downstream jam front, Kerner used the term
585:
as to the space gap to the preceding vehicle, within the range associated with the 2D region of homogeneous synchronized flow (Figure 4(b)): the driver accepts different space gaps at different times and does not use one unique gap.
1894:; otherwise, the jam dissolves over time. Depending on traffic parameters like weather, percentage of long vehicles, et cetera, and characteristics of the bottleneck where the F → S phase transition can occur, the minimum capacity
3545:
Hartenstein, Hannes (2010). "Vehicular
Traffic Flow Theory: Three, Not Two Phases [review of "Introduction to Modern Traffic Flow Theory and Control: The Long Road to Three-Phase Traffic Theory; Kerner, B.S.; 2009) ]".
2358:
2176:
1432:
Metastability of free flow means that for small disturbances free flow remains stable (free flow persists), but with larger disturbances the flow becomes unstable and an F → S phase transition to synchronized flow occurs.
380:
335:
in free flow (dotted line in Figure 2) divides the empirical data on the flow-density plane into two regions: on the left side data points of free flow and on the right side data points corresponding to congested traffic.
333:
1839:
while the jam propagates in free flow: Indeed, if the jam propagates through free-flow (i.e., both upstream and downstream of the jam free flows occur), then a wide moving jam can persist, only when the jam inflow
1590:
the upstream jam front, the vehicles come from free flow or synchronized flow and must reduce their speed. According to the definition the wide moving jam always has the same mean velocity of the downstream front
2049:
It is possible that both upstream and downstream front propagates upstream. The downstream front is no longer located at the bottleneck. This pattern has been called Moving
Synchronised Flow Pattern (MSP).
2041:
A congestion pattern of synchronized flow (Synchronized Flow
Pattern (SP)) with a fixed downstream and a not continuously propagating upstream front is called Localised Synchronized Flow Pattern (LSP).
2296:
Other criticisms have been made, such as that the notion of phases has not been well defined and that so-called two-phase models also succeed in simulating the essential features described by Kerner.
2067:
An often occurring congestion pattern is one that contains both congested phases, and . Such a pattern with and is called
General Pattern (GP). An empirical example of GP is shown in Figure 9 (a).
1968:, Kerner’s three-phase traffic theory assumes that the hypothetical homogeneous states of synchronized flow cover a two-dimensional region in the flow-density plane (dashed regions in Figure 8).
1128:
nature. Kerner considers the empirical nucleation nature of traffic breakdown (F → S phase transition) at a road bottleneck as the empirical fundamental of traffic and transportation science.
1058:
Kerner notes using empirical data that synchronized flow can form in free flow spontaneously (spontaneous F →S phase transition) or can be externally induced (induced F → S phase transition).
1141:
initial free flow (see Kerner’s criticism on such two-phase models as well as on other classical traffic flow models and theories in
Chapter 10 of the book as well as in critical reviews,).
482:
has nothing to do with the width across the jam, but actually refers to its length being considerably more than the transition zones at its head and tail. Historically, Kerner used the term
2248:
model, vehicles move in accordance with stochastic rules of vehicle motion that can be individually chosen for each of the vehicles. Some months later, Kerner, Klenov, and Wolf developed a
645:
The hypothesis of Kerner’s three-phase traffic theory about the 2D region of steady states of synchronized flow is contrary to the hypothesis of earlier traffic flow theories involving the
1428:
2260:
Schreckenberg, and Kim (see other references to mathematical models in the framework of Kerner’s three-phase traffic theory and results of their investigations in Sec. 1.7 of a review).
1007:
909:
3364:
Gao, Kun; Jiang, Rui; Hu, Shou-Xin; Wang, Bing-Hong; Wu, Qing-Song (2007). "Cellular-automaton model with velocity adaptation in the framework of Kerner's three-phase traffic theory".
3075:
Gao, Kun; Jiang, Rui; Hu, Shou-Xin; Wang, Bing-Hong; Wu, Qing-Song (2007). "Cellular-automaton model with velocity adaptation in the framework of Kerner's three-phase traffic theory".
274:
2787:
Kerner, Boris S (2015). "Microscopic theory of traffic-flow instability governing traffic breakdown at highway bottlenecks: Growing wave of increase in speed in synchronized flow".
2557:
Rehborn, Hubert; Klenov, Sergey L; Palmer, Jochen (2011). "An empirical study of common traffic congestion features based on traffic data measured in the USA, the UK, and
Germany".
870:
911:
the vehicle adapts its speed to the speed of the preceding vehicle without caring what the precise space gap is. The dashed region of synchronized flow is taken from Figure 4(b).
800:
919:, the driver tends to adapt his speed to the speed of the preceding vehicle without caring what the precise gap is, so long as this gap is not smaller than the safe space gap
944:
756:
639:
211:
1946:
1892:
1833:
1775:
1678:
1865:
1983:
Wide moving jams do not emerge spontaneously in free flow, but they can emerge in regions of synchronized flow. This phase transition is called an S → J phase transition.
1919:
1806:
1748:
1563:
1536:
1509:
1482:
1354:
1327:
1300:
1273:
1230:
1203:
552:
without a study of the propagation of congested traffic through a bottleneck. The microscopic criteria are as follows (see Sec. 2.6 in the book): If in single-vehicle (
184:
837:
721:
2153:
2126:
2099:
1705:
1651:
1615:
447:
3439:
H. Rehborn, S. Klenov, "Traffic Prediction of Congested Patterns", In: R. Meyers (Ed.): Encyclopedia of Complexity and Systems Science, Springer New York, 2009.
1017:
In the framework of the three-phase theory the hypothesis about 2D regions of states of synchronized flow has also been applied for the development of a model of
2592:
R.-P. Schäfer et al, "A study of TomTom’s probe vehicle data with three-phase traffic theory". Traffic Engineering and Control, Vol 52, No 5, Pages 225–231, 2011
2363:
1574:
1377:
968:
695:
675:
612:
157:
137:
3032:
Jiang, Rui; Wu, Qing-Song (2004). "Spatial–temporal patterns at an isolated on-ramp in a new cellular automata model based on three-phase traffic theory".
1165:
free flow. Probably the most important consequence of that is the existence of a range of highway capacities between some maximum and minimum capacities.
1441:
Thus the basic theoretical result of three-phase theory about the understanding of the stochastic capacity of free flow at a bottleneck is as follows:
3402:
1084:
279:
233:, as used in classical traffic theory, cannot adequately describe the complex dynamics of vehicular traffic. He instead divides congestion into
2659:
Kerner, Boris S (2013). "Criticism of generally accepted fundamentals and methodologies of traffic and transportation theory: A brief review".
1124:
called a nucleus for traffic breakdown. In other words, real traffic breakdown (F → S phase transition) at a highway bottleneck exhibits the
2909:
Kerner, Boris S (2014). "Three-phase theory of city traffic: Moving synchronized flow patterns in under-saturated city traffic at signals".
1565:
can depend considerably on traffic parameters (the percentage of long vehicles in traffic flow, weather, bottleneck characteristics, etc.).
3444:
H. Rehborn, J. Palmer, "Using ASDA and FOTO to generate RDS/TMC traffic messages", Traffic Engineering and Control, July 2008, pp. 261–266.
3589:
646:
1617:, even if the jam propagates through other traffic phases or bottlenecks. The flow rate is sharply reduced within a wide moving jam.
2323:
2289:
is used, but as said, only loop detector measurements are used. How the data in between was gathered or interpolated, is not clear.
62:
1808:
characterizes an F → S phase transition at a bottleneck, i.e., a traffic breakdown. In contrast, the outflow of a wide moving jam
1153:
Initially developed for highway traffic, Kerner expanded the three phase theory for the description of city traffic in 2011–2014.
3599:
581:
In accordance with this hypothesis of Kerner’s three-phase theory, at a given speed in synchronized flow, the driver can make an
813:
Figure 5: Qualitative explanation of car-following in Kerner’s three-phase traffic theory: A vehicle accelerates at a space gap
460:
is meant to reflect the characteristic feature of the jam to propagate through any other state of traffic flow and through any
1359:
The infinite number of highway capacities at a bottleneck can be illustrated by the meta-stability of free flow at flow rates
2328:
2293:
one can also find methods for spatial-temporal interpolations of data measured at road detectors (see article’s appendixes).
2268:
The theory has been criticized for two primary reasons. First, the theory is almost completely based on measurements on the
1106:(Figure 6): at a given vehicle density, the probability of overtaking in free flow is much higher than in synchronized flow.
2159:
moving jams form in the synchronized flow. In those cases, the EP is called EGP (Expanded General Pattern) (see Figure 10).
594:
plane. (b) A part of the 2D-region of homogeneous synchronized flow in the space gap-speed plane (dashed region S). In (b),
3594:
2734:
Kerner, Boris S (2016). "Failure of classical traffic flow theories: Stochastic highway capacity and automatic driving".
3335:
Lieu, Henry (2005). "The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory".
2979:
Kerner, Boris S; Klenov, Sergey L; Wolf, Dietrich E (2002). "Cellular automata approach to three-phase traffic theory".
1178:
there is a range of highway capacities at a bottleneck. This range of freeway capacities is between a minimum capacity
3443:
2591:
3486:"Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts"
3267:"Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts"
229:
Data show a weaker relationship between flow and density in congested conditions. Therefore, Kerner argues that the
2368:
2285:, which span the whole length of the road under investigation. These trajectories can only be measured directly if
1384:
2313:
973:
970:
in car following in the framework of Kerner’s three-phase theory can be any space gap within the space gap range
875:
2543:
Boris S. Kerner, “Breakdown in Traffic Networks: Fundamentals of Transportation Science”, Springer, Berlin, 2017
1022:
2524:
247:
3584:
641:, are respectively a synchronization space gap and safe space gap between two vehicles following each other.
3485:
3266:
1964:
Because the synchronized flow phase does not have the characteristic features of the wide moving jam phase
3515:
3417:
3296:
1991:
1653:(in the upstream direction) is a characteristic parameter, as is the flow rate just downstream of the jam
842:
761:
398:
Figure 3: Measured data of speed in time and space (a) and its representation on the time-space plane (b)
2244:
2944:
Kerner, Boris S; Klenov, Sergey L (2002). "A microscopic model for phase transitions in traffic flow".
1710:
3507:
3462:
3451:"Introduction to Modern Traffic Flow Theory and Control: The Long Road to Three-Phase Traffic Theory"
3438:
3373:
3344:
3288:
3236:
3184:
3127:
3084:
3041:
2998:
2918:
2867:
2806:
2753:
2668:
2622:
2566:
2498:
2404:
1161:
In three-phase traffic theory, traffic breakdown is explained by the F → S transition occurring in a
922:
734:
617:
489:
189:
3520:
3422:
3301:
2304:
be explained with earlier traffic flow theories including two-phase traffic flow models studied in.
1924:
1870:
1811:
1753:
1656:
2603:
Kerner, Boris S (2018). "Physics of automated driving in framework of three-phase traffic theory".
2527:
Introduction to Modern Traffic Flow Theory and Control: The Long Road to Three-Phase Traffic Theory
2318:
1843:
230:
93:
The word "wide" is used even though it is the length of the traffic jam that is being referred to.
3563:
3533:
3497:
3314:
3278:
3208:
3174:
3057:
3014:
2988:
2961:
2891:
2857:
2830:
2796:
2769:
2743:
2711:
2612:
2474:
2353:
2333:
2249:
2240:
2233:
2225:
419:
3165:(2004). "Mechanical Restriction versus Human Overreaction Triggering Congested Traffic States".
449:
is maintained. This is the characteristic feature of the wide moving jam that defines the phase
1897:
1784:
1726:
1714:
Figure 8: Three traffic phases on the flow-density plane in Kerner’s three-phase traffic theory
1541:
1514:
1487:
1460:
1332:
1305:
1278:
1251:
1208:
1181:
343:
Figure 2: Flow rate versus vehicle density in free flow and congested traffic (fictitious data)
162:
3389:
3200:
3143:
3100:
2883:
2822:
2638:
2286:
2033:
Very complex congested patterns can be observed, caused by F → S and S → J phase transitions.
589:
528:
1626:
percentage of long vehicles, etc.. The velocity of the downstream front of a wide moving jam
3555:
3525:
3470:
3427:
3381:
3352:
3306:
3244:
3192:
3135:
3092:
3049:
3006:
2953:
2926:
2875:
2814:
2761:
2703:
2676:
2630:
2574:
2466:
2439:
2412:
2269:
816:
700:
2220:
Mathematical models of traffic flow in the framework of Kerner’s three-phase traffic theory
2167:
2131:
2104:
2077:
1683:
1629:
1593:
1083:
425:
2373:
1235:
2694:
Kerner, Boris S (2015). "Failure of classical traffic flow theories: A critical review".
1835:
determines a condition for the existence of the wide moving jam, i.e., the traffic phase
3511:
3466:
3377:
3348:
3292:
3240:
3188:
3131:
3088:
3045:
3002:
2922:
2871:
2810:
2757:
2672:
2626:
2570:
2408:
2070:
1961:
remain similar to the one in free flow, even if the vehicle speeds are sharply reduced.
2278:
1362:
1239:
Figure 7: Maximum and minimum highway capacities in Kerner’s three-phase traffic theory
1018:
953:
680:
660:
649:, which supposes a one-dimensional relationship between vehicle density and flow rate.
597:
339:
142:
122:
3053:
3010:
3578:
3061:
3018:
2965:
2957:
2773:
2478:
2343:
2175:
One of the applications of Kerner’s three-phase traffic theory is the methods called
1162:
1120:
1116:
1071:
1067:
3567:
3537:
3318:
3212:
2895:
2834:
2715:
2443:
394:
2395:
Kerner, B. S (1998). "Experimental Features of Self-Organization in Traffic Flow".
2348:
2338:
2282:
2274:
2229:
1356:, there will be no traffic breakdown (no F →S phase transition) at the bottleneck.
1132:
The reason for Kerner’s theory and his criticism of classical traffic flow theories
31:
27:
3196:
2542:
2236:
of traffic flow can be developed in the framework of Kerner’s three-phase theory.
809:
220:
Figure 1: Measured flow rate versus vehicle density in free flow (fictitious data)
159:(in vehicles per unit distance). This relationship stops at the maximum free flow
54:
3248:
3162:
2930:
2765:
2680:
2578:
2416:
461:
216:
116:
3385:
3139:
3096:
2879:
2818:
2634:
1577:(BM principle) for the optimization and control of vehicular traffic networks.
1457:
there is a range of highway capacities, which are between the minimum capacity
244:
In congested traffic, the vehicle speed is lower than the lowest vehicle speed
3529:
3431:
3310:
2707:
2212:
and are performed using the features of Kerner’s three-phase traffic theory.
1125:
1119:
free flow at a highway bottleneck. As explained above, the sense of the term
1098:
276:
encountered in free flow, i.e., the line with the slope of the minimal speed
3559:
2459:
Transportation Research Record: Journal of the Transportation Research Board
328:{\displaystyle v_{\text{free}}^{\min }={\frac {q_{\max }}{k_{\text{crit}}}}}
3393:
3204:
3147:
3104:
2887:
2826:
2642:
2171:
Figure 11: Traffic patterns in the ASDA/FOTO application in three countries
522:
Explanation of the traffic phase definitions based on measured traffic data
3179:
2993:
2457:
Kerner, Boris (1999). "Congested Traffic Flow: Observations and Theory".
2252:(CA) traffic flow model in the framework of Kerner’s three-phase theory.
657:
In Kerner’s three-phase theory, a vehicle accelerates when the space gap
2163:
Applications of three-phase traffic theory in transportation engineering
464:
while maintaining the velocity of the downstream jam front. The phrase
3475:
3450:
3356:
1453:
highway capacity. In contrast, in Kerner’s three-phase traffic theory
43:. Kerner’s theory divides congested traffic into two distinct phases,
2243:
of traffic flow in the framework of Kerner’s three-phase theory that
1921:
might be smaller (as in Figure 8), or greater than the jam’s outflow
677:
to the preceding vehicle is greater than a synchronization space gap
560:
Kerner’s hypothesis about two-dimensional (2D) states of traffic flow
486:
from a qualitative analogy of a wide moving jam in traffic flow with
23:
2470:
2801:
2748:
2617:
2359:
Traffic congestion: Reconstruction with Kerner’s three-phase theory
3502:
3283:
2862:
2166:
2069:
1990:
1709:
1234:
1082:
808:
588:
393:
338:
215:
61:
53:
2848:
Kerner, Boris S (2011). "Physics of traffic gridlock in a city".
1013:
Autonomous driving in the framework of three-phase traffic theory
418:
A so-called "wide moving jam" moves upstream through any highway
3118:
Davis, L. C (2004). "Multilane simulations of traffic phases".
1079:
Physical explanation of traffic breakdown in three-phase theory
539:
Traffic phase definition based on empirical single-vehicle data
422:. While doing so, the mean velocity of the downstream front
3161:
Lee, Hyun Keun; Barlovic, Robert; Schreckenberg, Michael;
1987:"Jam without obvious reason" – F → S → J phase transitions
2054:
Catch effect of synchronized flow at a highway bottleneck
2037:
Classification of synchronized flow traffic patterns (SP)
1719:
Minimum highway capacity and outflow from wide moving jam
379:
in congested traffic are observed outcomes in universal
1110:
Discussion of Kerner’s explanation of traffic breakdown
3484:
Treiber, Martin; Kesting, Arne; Helbing, Dirk (2010).
3265:
Treiber, Martin; Kesting, Arne; Helbing, Dirk (2010).
391:
are defined through the definitions and as follows:
3229:
Physica A: Statistical Mechanics and Its Applications
2911:
Physica A: Statistical Mechanics and Its Applications
2736:
Physica A: Statistical Mechanics and Its Applications
2661:
Physica A: Statistical Mechanics and Its Applications
2559:
Physica A: Statistical Mechanics and Its Applications
2134:
2107:
2080:
1927:
1900:
1873:
1846:
1814:
1787:
1756:
1729:
1686:
1659:
1632:
1596:
1544:
1517:
1490:
1463:
1387:
1365:
1335:
1308:
1281:
1275:: If the flow rate is close to the maximum capacity
1254:
1211:
1184:
1070:
state of free flow at a highway bottleneck. The term
976:
956:
925:
878:
845:
819:
764:
737:
703:
683:
663:
620:
600:
428:
360:
and of the phases J and S in congested traffic": -->
282:
250:
192:
165:
145:
125:
115:
In free traffic flow, empirical data show a positive
2390:
2388:
2195:
nalyse (Automatic tracking of wide moving jams) and
727:in Figure 5); the vehicle decelerates when the gap
2430:Kerner, Boris S (1999). "The physics of traffic".
2147:
2120:
2093:
1940:
1913:
1886:
1859:
1827:
1800:
1769:
1742:
1699:
1672:
1645:
1609:
1557:
1530:
1503:
1476:
1422:
1371:
1348:
1321:
1294:
1267:
1224:
1197:
1145:The main prediction of Kerner’s three-phase theory
1001:
962:
938:
903:
864:
831:
794:
750:
715:
689:
669:
633:
606:
441:
327:
268:
205:
178:
151:
131:
1244:Highway capacities and metastability of free flow
2729:
2727:
2725:
2538:
2536:
2520:
2518:
2516:
2514:
2512:
2510:
2494:
2492:
2490:
2488:
1906:
1793:
1735:
1550:
1523:
1496:
1469:
1412:
1393:
1341:
1314:
1287:
1260:
1217:
1190:
1137:the development of Kerner’s three-phase theory.
1102:states that the probability of overtaking is an
308:
293:
261:
171:
139:(in vehicles per unit time) and vehicle density
51:, bringing the total number of phases to three:
2654:
2652:
3490:Transportation Research Part B: Methodological
3410:Transportation Research Part B: Methodological
3271:Transportation Research Part B: Methodological
3034:Journal of Physics A: Mathematical and General
2981:Journal of Physics A: Mathematical and General
2946:Journal of Physics A: Mathematical and General
381:spatial-temporal features of real traffic data
2074:Figure 10: Measured EGP at three bottlenecks
1621:Characteristic parameters of wide moving jams
1423:{\displaystyle C_{\min }\leq q<C_{\max }.}
8:
1723:Kerner emphasizes that the minimum capacity
478:(see Sec. 7.6.5 of the book). Thus the term
1867:is equal to or larger than the jam outflow
1104:interrupted function of the vehicle density
1002:{\displaystyle g_{\text{safe}}\leq g\leq G}
904:{\displaystyle g_{\text{safe}}\leq g\leq G}
653:Car following in three-phase traffic theory
3260:
3258:
3519:
3501:
3474:
3421:
3403:"Criticism of three-phase traffic theory"
3300:
3282:
3178:
2992:
2861:
2800:
2747:
2616:
2552:
2550:
2364:Kerner’s breakdown minimization principle
2139:
2133:
2112:
2106:
2085:
2079:
1932:
1926:
1905:
1899:
1878:
1872:
1851:
1845:
1819:
1813:
1792:
1786:
1761:
1755:
1734:
1728:
1691:
1685:
1664:
1658:
1637:
1631:
1601:
1595:
1549:
1543:
1522:
1516:
1495:
1489:
1468:
1462:
1411:
1392:
1386:
1364:
1340:
1334:
1313:
1307:
1286:
1280:
1259:
1253:
1216:
1210:
1189:
1183:
981:
975:
955:
930:
924:
883:
877:
856:
844:
818:
776:
775:
763:
742:
736:
702:
682:
662:
625:
619:
599:
433:
427:
317:
307:
301:
292:
287:
281:
260:
255:
249:
197:
191:
170:
164:
144:
124:
3401:Schönhof, Martin; Helbing, Dirk (2009).
2281:), but draws his conclusions on vehicle
2277:. Kerner uses fixed-point measurements (
2384:
1025:, self-driving or autonomous vehicle).
269:{\displaystyle v_{\text{free}}^{\min }}
2696:Elektrotechnik und Informationstechnik
2063:General congested traffic pattern (GP)
1169:Maximum and minimum highway capacities
186:with a corresponding critical density
1750:and the outflow of a wide moving jam
7:
1781:of free flow: the minimum capacity
1248:There is a maximum highway capacity
1062:in a neighbourhood of a bottleneck.
865:{\displaystyle g<g_{\text{safe}}}
795:{\displaystyle g<g_{\rm {safe}}}
647:fundamental diagram of traffic flow
569:Homogeneous synchronized flow is a
3548:IEEE Vehicular Technology Magazine
1437:Discussion of capacity definitions
1046:Empirical spontaneous and induced
786:
783:
780:
777:
565:Steady states of synchronized flow
531:(floating car data is also called
14:
2529:, Springer, Berlin, New York 2009
2503:, Springer, Berlin, New York 2004
2324:Intelligent transportation system
950:in Figure 5). Thus the space gap
731:is smaller than a safe space gap
1779:qualitatively different features
1575:breakdown minimization principle
348:Definitions and of the phases
939:{\displaystyle g_{\text{safe}}}
751:{\displaystyle g_{\text{safe}}}
634:{\displaystyle g_{\text{safe}}}
371:In Kerner's theory, the phases
206:{\displaystyle k_{\text{crit}}}
2329:Microscopic traffic flow model
1941:{\displaystyle q_{\text{out}}}
1887:{\displaystyle q_{\text{out}}}
1828:{\displaystyle q_{\text{out}}}
1770:{\displaystyle q_{\text{out}}}
1673:{\displaystyle q_{\text{out}}}
839:and decelerates at space gaps
498:The "synchronized flow" phase
1:
3197:10.1103/PhysRevLett.92.238702
1860:{\displaystyle q_{\text{in}}}
1996:possible phase transitions).
403:The "wide moving jam" phase
3249:10.1016/j.physa.2012.07.070
3054:10.1088/0305-4470/37/34/001
3011:10.1088/0305-4470/35/47/303
2931:10.1016/j.physa.2013.11.009
2766:10.1016/j.physa.2016.01.034
2681:10.1016/j.physa.2013.06.004
2579:10.1016/j.physa.2011.07.004
2417:10.1103/PhysRevLett.81.3797
1157:Range of highway capacities
3616:
3590:Transportation engineering
3386:10.1103/PhysRevE.76.026105
3140:10.1103/PhysRevE.69.016108
3097:10.1103/PhysRevE.76.026105
2958:10.1088/0305-4470/35/3/102
2880:10.1103/PhysRevE.84.045102
2819:10.1103/PhysRevE.92.062827
2635:10.1103/PhysRevE.97.042303
2369:Transportation forecasting
872:, whereas under condition
20:Three-phase traffic theory
3530:10.1016/j.trb.2010.03.004
3432:10.1016/j.trb.2009.02.004
3311:10.1016/j.trb.2010.03.004
2708:10.1007/s00502-015-0340-3
2444:10.1088/2058-7058/12/8/30
2314:Active traffic management
1952:Synchronized flow phase (
1914:{\displaystyle C_{\min }}
1801:{\displaystyle C_{\min }}
1743:{\displaystyle C_{\min }}
1558:{\displaystyle C_{\max }}
1531:{\displaystyle C_{\min }}
1504:{\displaystyle C_{\max }}
1477:{\displaystyle C_{\min }}
1349:{\displaystyle C_{\min }}
1322:{\displaystyle C_{\min }}
1295:{\displaystyle C_{\max }}
1268:{\displaystyle C_{\max }}
1232:of free flow (Figure 7).
1225:{\displaystyle C_{\max }}
1198:{\displaystyle C_{\min }}
179:{\displaystyle q_{\max }}
3449:Davis, L. Craig (2010).
2245:mathematical simulations
915:If the gap is less than
494:patterns have occurred.
102:state in space and time.
3600:Road traffic management
3560:10.1109/MVT.2010.937837
3167:Physical Review Letters
2397:Physical Review Letters
2264:Criticism of the theory
1205:and a maximum capacity
2501:The Physics of Traffic
2172:
2155:
2149:
2122:
2095:
1997:
1942:
1915:
1888:
1861:
1829:
1802:
1771:
1744:
1715:
1701:
1674:
1647:
1611:
1559:
1532:
1505:
1478:
1424:
1373:
1350:
1323:
1296:
1269:
1240:
1226:
1199:
1089:
1029:Traffic breakdown – a
1003:
964:
940:
912:
905:
866:
833:
832:{\displaystyle g>G}
796:
752:
717:
716:{\displaystyle g>G}
691:
671:
642:
635:
608:
443:
399:
344:
329:
270:
221:
207:
180:
153:
133:
119:between the flow rate
67:
59:
16:Theory of traffic flow
2170:
2150:
2148:{\displaystyle B_{3}}
2123:
2121:{\displaystyle B_{2}}
2096:
2094:{\displaystyle B_{1}}
2073:
1994:
1943:
1916:
1889:
1862:
1830:
1803:
1772:
1745:
1713:
1702:
1700:{\displaystyle v_{g}}
1675:
1648:
1646:{\displaystyle v_{g}}
1612:
1610:{\displaystyle v_{g}}
1560:
1533:
1506:
1484:and maximum capacity
1479:
1425:
1374:
1351:
1324:
1297:
1270:
1238:
1227:
1200:
1086:
1021:vehicle (called also
1004:
965:
941:
906:
867:
834:
812:
797:
753:
718:
692:
672:
636:
609:
592:
444:
442:{\displaystyle v_{g}}
397:
342:
330:
271:
219:
208:
181:
154:
134:
65:
57:
3595:Mathematical physics
2132:
2105:
2078:
2022:Traffic patterns of
1925:
1898:
1871:
1844:
1812:
1785:
1754:
1727:
1684:
1657:
1630:
1594:
1542:
1515:
1488:
1461:
1385:
1363:
1333:
1306:
1279:
1252:
1209:
1182:
974:
954:
923:
876:
843:
817:
762:
735:
701:
681:
661:
618:
598:
426:
356:in congested traffic
280:
248:
190:
163:
143:
123:
3512:2010arXiv1004.5545T
3467:2010PhT....63c..53K
3378:2007PhRvE..76b6105G
3349:2005PhT....58k..54K
3293:2010arXiv1004.5545T
3241:2013PhyA..392..221K
3189:2004PhRvL..92w8702L
3132:2004PhRvE..69a6108D
3089:2007PhRvE..76b6105G
3046:2004JPhA...37.8197J
3003:2002JPhA...35.9971K
2923:2014PhyA..397...76K
2872:2011PhRvE..84d5102K
2811:2015PhRvE..92f2827K
2758:2016PhyA..450..700K
2673:2013PhyA..392.5261K
2627:2018PhRvE..97d2303K
2571:2011PhyA..390.4466R
2409:1998PhRvL..81.3797K
2319:Fundamental diagram
2234:mathematical models
1570:at any time instant
1443:At any time instant
1176:at any time instant
1075:at the bottleneck.
297:
265:
231:fundamental diagram
78:Synchronized flow (
3126:(1 Pt 2): 016108.
2354:Traffic congestion
2334:Traffic bottleneck
2250:cellular automaton
2241:mathematical model
2226:mathematical model
2173:
2156:
2145:
2118:
2091:
1998:
1938:
1911:
1884:
1857:
1825:
1798:
1767:
1740:
1716:
1697:
1670:
1643:
1607:
1581:Wide moving jams (
1555:
1528:
1501:
1474:
1420:
1369:
1346:
1319:
1292:
1265:
1241:
1222:
1195:
1090:
1019:autonomous driving
999:
960:
936:
913:
901:
862:
829:
792:
748:
713:
687:
667:
643:
631:
604:
533:probe vehicle data
439:
400:
345:
325:
283:
266:
251:
222:
213:. (See Figure 1.)
203:
176:
149:
129:
68:
60:
3476:10.1063/1.3366241
3366:Physical Review E
3357:10.1063/1.2155762
3120:Physical Review E
3077:Physical Review E
2850:Physical Review E
2789:Physical Review E
2667:(21): 5261–5282.
2605:Physical Review E
2403:(17): 3797–3800.
2287:floating car data
1935:
1881:
1854:
1822:
1764:
1667:
1372:{\displaystyle q}
1023:automated driving
984:
963:{\displaystyle g}
933:
886:
859:
745:
690:{\displaystyle G}
670:{\displaystyle g}
628:
607:{\displaystyle G}
529:floating car data
323:
320:
290:
258:
235:synchronized flow
225:Congested traffic
200:
152:{\displaystyle k}
132:{\displaystyle q}
85:Wide moving jam (
58:Synchronized flow
45:synchronized flow
41:congested traffic
3607:
3571:
3541:
3523:
3505:
3480:
3478:
3435:
3425:
3407:
3397:
3360:
3323:
3322:
3304:
3286:
3262:
3253:
3252:
3223:
3217:
3216:
3182:
3180:cond-mat/0404315
3158:
3152:
3151:
3115:
3109:
3108:
3072:
3066:
3065:
3029:
3023:
3022:
2996:
2994:cond-mat/0206370
2976:
2970:
2969:
2941:
2935:
2934:
2906:
2900:
2899:
2865:
2845:
2839:
2838:
2804:
2784:
2778:
2777:
2751:
2731:
2720:
2719:
2691:
2685:
2684:
2656:
2647:
2646:
2620:
2600:
2594:
2589:
2583:
2582:
2554:
2545:
2540:
2531:
2522:
2505:
2496:
2483:
2482:
2454:
2448:
2447:
2427:
2421:
2420:
2392:
2270:Bundesautobahn 5
2154:
2152:
2151:
2146:
2144:
2143:
2127:
2125:
2124:
2119:
2117:
2116:
2100:
2098:
2097:
2092:
2090:
2089:
1979:phase transition
1947:
1945:
1944:
1939:
1937:
1936:
1933:
1920:
1918:
1917:
1912:
1910:
1909:
1893:
1891:
1890:
1885:
1883:
1882:
1879:
1866:
1864:
1863:
1858:
1856:
1855:
1852:
1834:
1832:
1831:
1826:
1824:
1823:
1820:
1807:
1805:
1804:
1799:
1797:
1796:
1776:
1774:
1773:
1768:
1766:
1765:
1762:
1749:
1747:
1746:
1741:
1739:
1738:
1706:
1704:
1703:
1698:
1696:
1695:
1679:
1677:
1676:
1671:
1669:
1668:
1665:
1652:
1650:
1649:
1644:
1642:
1641:
1616:
1614:
1613:
1608:
1606:
1605:
1564:
1562:
1561:
1556:
1554:
1553:
1537:
1535:
1534:
1529:
1527:
1526:
1510:
1508:
1507:
1502:
1500:
1499:
1483:
1481:
1480:
1475:
1473:
1472:
1429:
1427:
1426:
1421:
1416:
1415:
1397:
1396:
1378:
1376:
1375:
1370:
1355:
1353:
1352:
1347:
1345:
1344:
1328:
1326:
1325:
1320:
1318:
1317:
1301:
1299:
1298:
1293:
1291:
1290:
1274:
1272:
1271:
1266:
1264:
1263:
1231:
1229:
1228:
1223:
1221:
1220:
1204:
1202:
1201:
1196:
1194:
1193:
1037:phase transition
1008:
1006:
1005:
1000:
986:
985:
982:
969:
967:
966:
961:
948:speed adaptation
945:
943:
942:
937:
935:
934:
931:
910:
908:
907:
902:
888:
887:
884:
871:
869:
868:
863:
861:
860:
857:
838:
836:
835:
830:
801:
799:
798:
793:
791:
790:
789:
757:
755:
754:
749:
747:
746:
743:
722:
720:
719:
714:
696:
694:
693:
688:
676:
674:
673:
668:
640:
638:
637:
632:
630:
629:
626:
613:
611:
610:
605:
583:arbitrary choice
510:
509:
505:
448:
446:
445:
440:
438:
437:
415:
414:
410:
368:
367:
363:
334:
332:
331:
326:
324:
322:
321:
318:
312:
311:
302:
296:
291:
288:
275:
273:
272:
267:
264:
259:
256:
239:wide moving jams
212:
210:
209:
204:
202:
201:
198:
185:
183:
182:
177:
175:
174:
158:
156:
155:
150:
138:
136:
135:
130:
100:is defined as a
3615:
3614:
3610:
3609:
3608:
3606:
3605:
3604:
3575:
3574:
3544:
3521:10.1.1.186.2970
3483:
3448:
3423:10.1.1.475.3565
3405:
3400:
3363:
3334:
3331:
3326:
3302:10.1.1.186.2970
3264:
3263:
3256:
3225:
3224:
3220:
3160:
3159:
3155:
3117:
3116:
3112:
3074:
3073:
3069:
3031:
3030:
3026:
2978:
2977:
2973:
2943:
2942:
2938:
2908:
2907:
2903:
2847:
2846:
2842:
2786:
2785:
2781:
2733:
2732:
2723:
2693:
2692:
2688:
2658:
2657:
2650:
2602:
2601:
2597:
2590:
2586:
2565:(23–24): 4466.
2556:
2555:
2548:
2541:
2534:
2523:
2508:
2497:
2486:
2471:10.3141/1678-20
2456:
2455:
2451:
2429:
2428:
2424:
2394:
2393:
2386:
2382:
2374:Two-fluid model
2310:
2266:
2222:
2165:
2135:
2130:
2129:
2108:
2103:
2102:
2081:
2076:
2075:
2065:
2056:
2039:
2031:
2015:
2005:The physics of
1989:
1981:
1958:
1928:
1923:
1922:
1901:
1896:
1895:
1874:
1869:
1868:
1847:
1842:
1841:
1815:
1810:
1809:
1788:
1783:
1782:
1757:
1752:
1751:
1730:
1725:
1724:
1721:
1687:
1682:
1681:
1660:
1655:
1654:
1633:
1628:
1627:
1623:
1597:
1592:
1591:
1587:
1545:
1540:
1539:
1518:
1513:
1512:
1491:
1486:
1485:
1464:
1459:
1458:
1439:
1407:
1388:
1383:
1382:
1361:
1360:
1336:
1331:
1330:
1309:
1304:
1303:
1282:
1277:
1276:
1255:
1250:
1249:
1246:
1212:
1207:
1206:
1185:
1180:
1179:
1171:
1159:
1147:
1134:
1112:
1081:
1056:
1039:
1015:
977:
972:
971:
952:
951:
926:
921:
920:
879:
874:
873:
852:
841:
840:
815:
814:
771:
760:
759:
738:
733:
732:
699:
698:
679:
678:
659:
658:
655:
621:
616:
615:
596:
595:
567:
562:
541:
524:
511:
507:
503:
501:
500:
458:wide moving jam
429:
424:
423:
416:
412:
408:
406:
405:
369:
365:
361:
359:
358:
313:
303:
278:
277:
246:
245:
227:
193:
188:
187:
166:
161:
160:
141:
140:
121:
120:
113:
49:wide moving jam
17:
12:
11:
5:
3613:
3611:
3603:
3602:
3597:
3592:
3587:
3585:Road transport
3577:
3576:
3573:
3572:
3542:
3481:
3446:
3441:
3436:
3398:
3361:
3330:
3327:
3325:
3324:
3254:
3235:(1): 221–251.
3218:
3173:(23): 238702.
3153:
3110:
3067:
3024:
2971:
2936:
2901:
2840:
2779:
2721:
2702:(7): 417–433.
2686:
2648:
2595:
2584:
2546:
2532:
2506:
2484:
2449:
2422:
2383:
2381:
2378:
2377:
2376:
2371:
2366:
2361:
2356:
2351:
2346:
2341:
2336:
2331:
2326:
2321:
2316:
2309:
2306:
2279:loop detectors
2265:
2262:
2224:Rather than a
2221:
2218:
2164:
2161:
2142:
2138:
2115:
2111:
2088:
2084:
2064:
2061:
2055:
2052:
2038:
2035:
2030:
2020:
2014:
2003:
1988:
1985:
1980:
1970:
1957:
1950:
1931:
1908:
1904:
1877:
1850:
1818:
1795:
1791:
1760:
1737:
1733:
1720:
1717:
1694:
1690:
1663:
1640:
1636:
1622:
1619:
1604:
1600:
1586:
1579:
1568:The existence
1552:
1548:
1525:
1521:
1498:
1494:
1471:
1467:
1438:
1435:
1419:
1414:
1410:
1406:
1403:
1400:
1395:
1391:
1368:
1343:
1339:
1316:
1312:
1289:
1285:
1262:
1258:
1245:
1242:
1219:
1215:
1192:
1188:
1170:
1167:
1158:
1155:
1146:
1143:
1133:
1130:
1111:
1108:
1080:
1077:
1055:
1044:
1038:
1027:
1014:
1011:
998:
995:
992:
989:
980:
959:
929:
900:
897:
894:
891:
882:
855:
851:
848:
828:
825:
822:
806:in Figure 5).
788:
785:
782:
779:
774:
770:
767:
741:
712:
709:
706:
686:
666:
654:
651:
624:
603:
566:
563:
561:
558:
540:
537:
523:
520:
499:
496:
436:
432:
404:
401:
357:
346:
316:
310:
306:
300:
295:
286:
263:
254:
226:
223:
196:
173:
169:
148:
128:
112:
105:
91:
90:
83:
76:
15:
13:
10:
9:
6:
4:
3:
2:
3612:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3582:
3580:
3569:
3565:
3561:
3557:
3553:
3549:
3543:
3539:
3535:
3531:
3527:
3522:
3517:
3513:
3509:
3504:
3499:
3495:
3491:
3487:
3482:
3477:
3472:
3468:
3464:
3460:
3456:
3455:Physics Today
3452:
3447:
3445:
3442:
3440:
3437:
3433:
3429:
3424:
3419:
3415:
3411:
3404:
3399:
3395:
3391:
3387:
3383:
3379:
3375:
3372:(2): 026105.
3371:
3367:
3362:
3358:
3354:
3350:
3346:
3343:(11): 54–56.
3342:
3338:
3337:Physics Today
3333:
3332:
3328:
3320:
3316:
3312:
3308:
3303:
3298:
3294:
3290:
3285:
3280:
3276:
3272:
3268:
3261:
3259:
3255:
3250:
3246:
3242:
3238:
3234:
3230:
3222:
3219:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3181:
3176:
3172:
3168:
3164:
3157:
3154:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3114:
3111:
3106:
3102:
3098:
3094:
3090:
3086:
3083:(2): 026105.
3082:
3078:
3071:
3068:
3063:
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3028:
3025:
3020:
3016:
3012:
3008:
3004:
3000:
2995:
2990:
2986:
2982:
2975:
2972:
2967:
2963:
2959:
2955:
2951:
2947:
2940:
2937:
2932:
2928:
2924:
2920:
2916:
2912:
2905:
2902:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2864:
2859:
2856:(4): 045102.
2855:
2851:
2844:
2841:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2803:
2798:
2795:(6): 062827.
2794:
2790:
2783:
2780:
2775:
2771:
2767:
2763:
2759:
2755:
2750:
2745:
2741:
2737:
2730:
2728:
2726:
2722:
2717:
2713:
2709:
2705:
2701:
2697:
2690:
2687:
2682:
2678:
2674:
2670:
2666:
2662:
2655:
2653:
2649:
2644:
2640:
2636:
2632:
2628:
2624:
2619:
2614:
2611:(4): 042303.
2610:
2606:
2599:
2596:
2593:
2588:
2585:
2580:
2576:
2572:
2568:
2564:
2560:
2553:
2551:
2547:
2544:
2539:
2537:
2533:
2530:
2528:
2525:B.S. Kerner,
2521:
2519:
2517:
2515:
2513:
2511:
2507:
2504:
2502:
2499:B.S. Kerner,
2495:
2493:
2491:
2489:
2485:
2480:
2476:
2472:
2468:
2464:
2460:
2453:
2450:
2445:
2441:
2437:
2433:
2432:Physics World
2426:
2423:
2418:
2414:
2410:
2406:
2402:
2398:
2391:
2389:
2385:
2379:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2344:Traffic model
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2311:
2307:
2305:
2303:
2297:
2294:
2290:
2288:
2284:
2280:
2276:
2271:
2263:
2261:
2257:
2253:
2251:
2246:
2242:
2237:
2235:
2231:
2227:
2219:
2217:
2213:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2169:
2162:
2160:
2140:
2136:
2113:
2109:
2086:
2082:
2072:
2068:
2062:
2060:
2053:
2051:
2047:
2043:
2036:
2034:
2029:
2025:
2021:
2019:
2012:
2008:
2004:
2002:
1993:
1986:
1984:
1978:
1974:
1971:
1969:
1967:
1962:
1955:
1951:
1949:
1929:
1902:
1875:
1848:
1838:
1816:
1789:
1780:
1777:describe two
1758:
1731:
1718:
1712:
1708:
1692:
1688:
1661:
1638:
1634:
1620:
1618:
1602:
1598:
1584:
1580:
1578:
1576:
1571:
1566:
1546:
1519:
1511:. The values
1492:
1465:
1456:
1452:
1446:
1444:
1436:
1434:
1430:
1417:
1408:
1404:
1401:
1398:
1389:
1380:
1366:
1357:
1337:
1310:
1283:
1256:
1243:
1237:
1233:
1213:
1186:
1177:
1168:
1166:
1164:
1156:
1154:
1151:
1144:
1142:
1138:
1131:
1129:
1127:
1122:
1118:
1109:
1107:
1105:
1100:
1094:
1085:
1078:
1076:
1073:
1069:
1063:
1059:
1053:
1049:
1045:
1043:
1036:
1032:
1028:
1026:
1024:
1020:
1012:
1010:
996:
993:
990:
987:
978:
957:
949:
946:(labelled by
927:
918:
898:
895:
892:
889:
880:
853:
849:
846:
826:
823:
820:
811:
807:
805:
802:(labelled by
772:
768:
765:
739:
730:
726:
723:(labelled by
710:
707:
704:
684:
664:
652:
650:
648:
622:
601:
591:
587:
584:
579:
575:
572:
564:
559:
557:
555:
551:
547:
538:
536:
534:
530:
521:
519:
515:
506:
497:
495:
492:
491:
485:
481:
476:
472:
467:
463:
459:
454:
452:
434:
430:
421:
411:
402:
396:
392:
390:
386:
383:. The phases
382:
378:
374:
364:
355:
351:
347:
341:
337:
314:
304:
298:
284:
252:
242:
240:
236:
232:
224:
218:
214:
194:
167:
146:
126:
118:
110:
106:
104:
103:
99:
94:
88:
84:
81:
77:
74:
70:
69:
64:
56:
52:
50:
46:
42:
38:
33:
30:developed by
29:
25:
21:
3551:
3547:
3496:(8–9): 983.
3493:
3489:
3458:
3454:
3413:
3409:
3369:
3365:
3340:
3336:
3277:(8–9): 983.
3274:
3270:
3232:
3228:
3221:
3170:
3166:
3163:Kim, Doochul
3156:
3123:
3119:
3113:
3080:
3076:
3070:
3040:(34): 8197.
3037:
3033:
3027:
2987:(47): 9971.
2984:
2980:
2974:
2949:
2945:
2939:
2914:
2910:
2904:
2853:
2849:
2843:
2792:
2788:
2782:
2739:
2735:
2699:
2695:
2689:
2664:
2660:
2608:
2604:
2598:
2587:
2562:
2558:
2526:
2500:
2462:
2458:
2452:
2438:(8): 25–30.
2435:
2431:
2425:
2400:
2396:
2349:Traffic wave
2339:Traffic flow
2301:
2298:
2295:
2291:
2283:trajectories
2275:interpolated
2267:
2258:
2254:
2238:
2230:traffic flow
2223:
2214:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2183:utomatische
2180:
2174:
2157:
2066:
2057:
2048:
2044:
2040:
2032:
2027:
2023:
2016:
2010:
2006:
1999:
1982:
1976:
1972:
1965:
1963:
1959:
1953:
1836:
1778:
1722:
1624:
1588:
1582:
1569:
1567:
1454:
1450:
1447:
1442:
1440:
1431:
1381:
1358:
1247:
1175:
1172:
1160:
1152:
1148:
1139:
1135:
1113:
1103:
1095:
1091:
1064:
1060:
1057:
1051:
1047:
1040:
1034:
1030:
1016:
947:
916:
914:
804:deceleration
803:
728:
725:acceleration
724:
656:
644:
582:
580:
576:
571:hypothetical
570:
568:
553:
549:
545:
542:
532:
525:
516:
512:
490:autosolitons
487:
483:
479:
474:
470:
465:
457:
455:
450:
417:
388:
384:
376:
372:
370:
353:
349:
243:
238:
234:
228:
114:
108:
101:
97:
95:
92:
86:
79:
72:
48:
44:
40:
36:
32:Boris Kerner
28:traffic flow
19:
18:
3227:messages".
2742:: 700–747.
2465:: 160–167.
2199:orecasting
1455:at any time
1054:transitions
758:, i.e., at
697:, i.e., at
554:microscopic
473:. The term
420:bottlenecks
117:correlation
107:Free flow (
71:Free flow (
3579:Categories
3416:(7): 784.
3329:References
2952:(3): L31.
2917:: 76–110.
2802:1511.04912
2749:1601.02585
2618:1710.10852
2239:The first
2013:transition
1451:particular
1163:metastable
1126:nucleation
1121:metastable
1117:metastable
1099:overtaking
1072:metastable
1068:metastable
466:moving jam
462:bottleneck
3554:(3): 91.
3516:CiteSeerX
3503:1004.5545
3461:(3): 53.
3418:CiteSeerX
3297:CiteSeerX
3284:1004.5545
3062:122146399
3019:119372458
2966:118445685
2863:1108.4310
2774:119138694
2479:108899410
2177:ASDA/FOTO
1399:≤
994:≤
988:≤
896:≤
890:≤
456:The term
37:free flow
3568:21113397
3538:18335270
3394:17930102
3319:18335270
3213:13974469
3205:15245199
3148:14995668
3105:17930102
2896:22249347
2888:22181213
2835:21537585
2827:26764764
2716:30041910
2643:29758629
2308:See also
1088:density.
3508:Bibcode
3463:Bibcode
3374:Bibcode
3345:Bibcode
3289:Bibcode
3237:Bibcode
3185:Bibcode
3128:Bibcode
3085:Bibcode
3042:Bibcode
2999:Bibcode
2919:Bibcode
2868:Bibcode
2807:Bibcode
2754:Bibcode
2669:Bibcode
2623:Bibcode
2567:Bibcode
2405:Bibcode
2207:raffic
3566:
3536:
3518:
3420:
3392:
3317:
3299:
3211:
3203:
3146:
3103:
3060:
3017:
2964:
2894:
2886:
2833:
2825:
2772:
2714:
2641:
2477:
2302:cannot
2191:ynamik
2128:, and
502:": -->
407:": -->
24:theory
3564:S2CID
3534:S2CID
3498:arXiv
3406:(PDF)
3315:S2CID
3279:arXiv
3209:S2CID
3175:arXiv
3058:S2CID
3015:S2CID
2989:arXiv
2962:S2CID
2892:S2CID
2858:arXiv
2831:S2CID
2797:arXiv
2770:S2CID
2744:arXiv
2712:S2CID
2613:arXiv
2475:S2CID
2380:Notes
1379:with
488:wide
98:phase
22:is a
3390:PMID
3201:PMID
3144:PMID
3101:PMID
2884:PMID
2823:PMID
2639:PMID
2463:1678
2026:and
1538:and
1405:<
983:safe
932:safe
885:safe
858:safe
850:<
824:>
769:<
744:safe
708:>
627:safe
614:and
548:and
504:edit
484:wide
480:wide
475:wide
471:wide
409:edit
387:and
375:and
362:edit
352:and
319:crit
289:free
257:free
237:and
199:crit
47:and
39:and
3556:doi
3526:doi
3471:doi
3428:doi
3382:doi
3353:doi
3307:doi
3245:doi
3233:392
3193:doi
3136:doi
3093:doi
3050:doi
3007:doi
2954:doi
2927:doi
2915:397
2876:doi
2815:doi
2762:doi
2740:450
2704:doi
2700:132
2677:doi
2665:392
2631:doi
2575:doi
2563:390
2467:doi
2440:doi
2413:doi
2228:of
2187:tau
1934:out
1907:min
1880:out
1821:out
1794:min
1763:out
1736:min
1666:out
1551:max
1524:min
1497:max
1470:min
1413:max
1394:min
1342:min
1315:min
1288:max
1261:max
1218:max
1191:min
535:).
309:max
294:min
262:min
172:max
66:Jam
26:of
3581::
3562:.
3550:.
3532:.
3524:.
3514:.
3506:.
3494:44
3492:.
3488:.
3469:.
3459:63
3457:.
3453:.
3426:.
3414:43
3412:.
3408:.
3388:.
3380:.
3370:76
3368:.
3351:.
3341:58
3339:.
3313:.
3305:.
3295:.
3287:.
3275:44
3273:.
3269:.
3257:^
3243:.
3231:.
3207:.
3199:.
3191:.
3183:.
3171:92
3169:.
3142:.
3134:.
3124:69
3122:.
3099:.
3091:.
3081:76
3079:.
3056:.
3048:.
3038:37
3036:.
3013:.
3005:.
2997:.
2985:35
2983:.
2960:.
2950:35
2948:.
2925:.
2913:.
2890:.
2882:.
2874:.
2866:.
2854:84
2852:.
2829:.
2821:.
2813:.
2805:.
2793:92
2791:.
2768:.
2760:.
2752:.
2738:.
2724:^
2710:.
2698:.
2675:.
2663:.
2651:^
2637:.
2629:.
2621:.
2609:97
2607:.
2573:.
2561:.
2549:^
2535:^
2509:^
2487:^
2473:.
2461:.
2436:12
2434:.
2411:.
2401:81
2399:.
2387:^
2203:f
2101:,
2009:→
1975:→
1948:.
1853:in
1707:.
1050:→
1033:→
1009:.
453:.
241:.
96:A
3570:.
3558::
3552:5
3540:.
3528::
3510::
3500::
3479:.
3473::
3465::
3434:.
3430::
3396:.
3384::
3376::
3359:.
3355::
3347::
3321:.
3309::
3291::
3281::
3251:.
3247::
3239::
3215:.
3195::
3187::
3177::
3150:.
3138::
3130::
3107:.
3095::
3087::
3064:.
3052::
3044::
3021:.
3009::
3001::
2991::
2968:.
2956::
2933:.
2929::
2921::
2898:.
2878::
2870::
2860::
2837:.
2817::
2809::
2799::
2776:.
2764::
2756::
2746::
2718:.
2706::
2683:.
2679::
2671::
2645:.
2633::
2625::
2615::
2581:.
2577::
2569::
2481:.
2469::
2446:.
2442::
2419:.
2415::
2407::
2209:O
2205:T
2201:O
2197:F
2193:A
2189:D
2185:S
2181:A
2179:(
2141:3
2137:B
2114:2
2110:B
2087:1
2083:B
2028:J
2024:S
2011:J
2007:S
1977:J
1973:S
1966:J
1956:)
1954:S
1930:q
1903:C
1876:q
1849:q
1837:J
1817:q
1790:C
1759:q
1732:C
1693:g
1689:v
1662:q
1639:g
1635:v
1603:g
1599:v
1585:)
1583:J
1547:C
1520:C
1493:C
1466:C
1418:.
1409:C
1402:q
1390:C
1367:q
1338:C
1311:C
1284:C
1257:C
1214:C
1187:C
1052:S
1048:F
1035:S
1031:F
997:G
991:g
979:g
958:g
928:g
917:G
899:G
893:g
881:g
854:g
847:g
827:G
821:g
787:e
784:f
781:a
778:s
773:g
766:g
740:g
729:g
711:G
705:g
685:G
665:g
623:g
602:G
550:S
546:J
508:]
451:J
435:g
431:v
413:]
389:S
385:J
377:S
373:J
366:]
354:S
350:J
315:k
305:q
299:=
285:v
253:v
195:k
168:q
147:k
127:q
111:)
109:F
89:)
87:J
82:)
80:S
75:)
73:F
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.