Physica A 392 (2013) 5261–5282

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Criticism of generally accepted fundamentals andmethodologies of traffic and transportation theory:A brief reviewBoris S. Kerner ∗

Physik von Transport und Verkehr, Universität Duisburg–Essen, 47048 Duisburg, Germany

h i g h l i g h t s

• Fundamental empirical features of traffic breakdown.• Criticism on classical traffic and transportation theories.• Three-phase traffic theory.• Breakdown minimization principle.

a r t i c l e i n f o

Article history:Received 29 March 2013Received in revised form 21 May 2013Available online 27 June 2013

Keywords:Highway capacityLighthill–Whitham–Richards theoryGeneral Motors class of traffic flow modelsWardrop’ UE and SO principlesThree-phase traffic theoryBreakdown minimization principle

a b s t r a c t

It is explained why the set of the fundamental empirical features of traffic breakdown(a transition from free flow to congested traffic) should be the empirical basis for anytraffic and transportation theory that can be reliably used for control and optimizationin traffic networks. It is shown that the generally accepted fundamentals and method-ologies of the traffic and transportation theory are not consistent with the set of thefundamental empirical features of traffic breakdown at a highway bottleneck. To thesefundamentals and methodologies of the traffic and transportation theory belong (i)Lighthill–Whitham–Richards (LWR) theory, (ii) the General Motors (GM) model class (forexample, Herman, Gazis et al. GM model, Gipps’s model, Payne’s model, Newell’s optimalvelocity (OV) model, Wiedemann’s model, Bando et al. OV model, Treiber’s IDM, Krauß’smodel), (iii) the understanding of highway capacity as a particular (fixed or stochastic)value, and (iv) principles for traffic and transportation network optimization and control(for example, Wardrop’s user equilibrium (UE) and system optimum (SO) principles). Al-ternatively to these generally accepted fundamentals and methodologies of the traffic andtransportation theory, we discuss the three-phase traffic theory as the basis for traffic flowmodeling as well as briefly consider the network breakdown minimization (BM) principlefor the optimization of traffic and transportation networks with road bottlenecks.

© 2013 Published by Elsevier B.V.

Contents

1. Introduction.......................................................................................................................................................................................... 52622. Why should the empirical features of traffic breakdown be the empirical basis for any traffic and transportation theory?...... 52633. The set of the fundamental empirical features of traffic breakdown at highway bottlenecks ....................................................... 52634. Traffic breakdown at the bottleneck in the Lighthill–Whitham–Richards (LWR) model ............................................................... 52645. Traffic breakdown at the bottleneck in the models of General Motors (GM) model class.............................................................. 5266

∗ Tel.: +49 711 4793774.E-mail address: boris.kerner@uni-due.de.

0378-4371/$ – see front matter© 2013 Published by Elsevier B.V.http://dx.doi.org/10.1016/j.physa.2013.06.004

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6. Generally accepted understanding of the capacity of free flow at a bottleneck as a particular value ........................................... 52667. Traffic breakdown at the bottleneck in the three-phase theory ....................................................................................................... 5269

7.1. Probability of traffic breakdown at the bottleneck, Z-characteristic, and infinite number of highway capacities ........... 52697.2. The nature of traffic breakdown ............................................................................................................................................. 5271

8. Importance of capacity understanding for management and control in traffic networks.............................................................. 52729. Breakdown minimization (BM) principle........................................................................................................................................... 5272

9.1. Mathematical formulation of the BM principle ..................................................................................................................... 52729.2. When are traffic breakdowns at neighborhood bottlenecks independent events? ............................................................ 52739.3. About network optimization when breakdown has already occurred at a network bottleneck ....................................... 5273

10. The BM principle and classical Wardrop’s principles ........................................................................................................................ 527311. Achievements of the generally accepted classical traffic and transportation theories ................................................................... 527412. Why are the generally accepted classical traffic and transportation theories inconsistent with the features of real traffic? ..... 527413. Conclusions........................................................................................................................................................................................... 5275

Acknowledgments ............................................................................................................................................................................... 5275References............................................................................................................................................................................................. 5275

1. Introduction

Traffic researchers have developed a huge number of traffic theories for optimization and control of traffic andtransportation networks. In particular, to the generally accepted fundamentals and methodologies of the traffic andtransportation theory belong the following theories and associated methodologies.

(i) The Lighthill–Whitham–Richards (LWR) model [1,2].(ii) The General Motors (GM) model class. In traffic flow models within the framework of the GM model class traffic

breakdown is explained by a free flow instability that causes a growing wave of vehicle speed reduction propagatingupstream in traffic flow as introduced in the GM car-followingmodel byHerman, Gazis et al. [3–5] (see, e.g., Refs. [6–39]as well as references in reviews, books, and conference proceedings [40–86]).

(iii) The understanding of the highway capacity of free flow at a bottleneck as a particular (fixed or stochastic) value (see,e.g., Refs. [42,44,45,87–97]).

(iv) Principles for traffic and transportation network optimization and control in which travel cost should be minimized(examples are Wardrop’s user equilibrium (UE) and system optimum (SO) principles [98]).

These classical theories and methodologies are the basis for dynamic traffic assignment, traffic control and optimizationin traffic and transportation networks (see, e.g., Refs. [99–201] and references in reviews and books [48,202–209]),development and simulations of Intelligent Transportation Systems (e.g., Refs. [210–226]), traffic simulation tools (see, e.g.,Refs. [227–239]) as well as for other traffic engineering applications (e.g., Refs. [240–245]).

The fundamentals and associated methodologies of these theories have made a great impact on the understanding ofmany traffic phenomena. However, network optimization approaches based on these fundamentals and methodologieshave failed by their applications in the real world. Even several decades of a very intensive effort to improve and validatenetwork optimizationmodels have had no success. Indeed, there can be found no examples where on-line implementationsof the network optimizationmodels based on these fundamentals andmethodologies could reduce congestion in real trafficand transportation networks.

Under small enough network inflow rates, drivers move at their desired (or permitted) speeds. When inflow rates in atraffic network increase, as is well known, in an initial free flow traffic breakdown occurs resulting in traffic congestion.Rather than traffic congestion, the problem for the generally accepted fundamentals and methodologies of the traffic andtransportation theory is associated with the set of the fundamental empirical features of traffic breakdown at a highwaybottleneck found in real traffic data measured in different countries.

The objective of this brief reviewarticle is to explain the above-mentioned failure of applications of the generally acceptedfundamentals and methodologies for the control and optimization of traffic and transportation networks in the real world.We show that there is a fundamental problem for these fundamentals and methodologies that is as follows.

1. Generally accepted fundamentals and methodologies of the traffic and transportation theory are not consistent with theset of the fundamental empirical features of traffic breakdown at a highway bottleneck.

2. For this reason, the fundamentals andmethodologies of the traffic and transportation theory cannot be applied for reliablecontrol and optimization in traffic and transportation networks.

The term traffic breakdown at a highway bottleneck describes the phenomenon of the phase transition from an initial freeflow to congested traffic at the bottleneck [42,44,45,87–97,246–248]. During the breakdownvehicle speed sharply decreaseswhereas the flow rate can remain as large as in an initial free flow. For this reason, traffic breakdown is also called speeddrop or speed breakdown. In real traffic data, a typical duration of this phase transition at the bottleneck is equal to or lessthan 1 min [42,44,45,87–97,247,248]. During this short time interval, instead of an initial free flow at the bottleneck a stateof congested traffic occurs in a neighborhood of the bottleneck.

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After the breakdown has occurred, a spatiotemporal development of a congested traffic pattern begins. There can be adiverse variety of congested patterns resulting from the breakdown at the bottleneck (see, e.g., Refs. [42,44,45,87–97,247–372]). The most prominent phase transition in congested traffic is the emergence of moving jams.

Traffic breakdown can be considered independent of the subsequent spatiotemporal evolution of congested trafficresulting from the breakdown. The possibility of such a consideration follows fromahuge number of empirical studies of realmeasured traffic data (see, e.g., Refs. [87–97,249–276] and references in [42,247,248]). Indeed, the phase transition ‘‘trafficbreakdown’’ and phase transitions in congested traffic that determine the spatiotemporal evolution of a congested patternexhibit qualitatively different empirical spatiotemporal behavior as well as crucially different scales in time and space.(i) Different spatiotemporal behavior. Traffic breakdown at a highway bottleneck is a local phase transition from free flow

to congested traffic that occurs within amotionless region of traffic flow in a neighborhood of the bottleneck.1 In otherwords, the localized traffic flow region within which the speed drop occurs during traffic breakdown is fixed at thebottleneck. In contrast, the phase transition leading to the emergence of amoving jam in congested traffic occurswithina region of traffic flow thatmoves usually propagating upstream of the bottleneck along the road stretch about 1–5 kmuntil the moving jam has been developed.

(ii) Time scale. As above-mentioned, the characteristic time of traffic breakdown is about 1 min or less. In contrast, acongested pattern develops in space and time during a considerably longer time interval (10–120 min or longer). Inparticular, moving traffic jams develop in congested traffic often during about 4–20 min.

(iii) Space scale. Traffic breakdown occurs in a small neighborhood of the bottleneckwithin a traffic flow region that is about200–300 m long. In contrast, the pattern development occurs on a very long road stretch (3–30 km or longer) or evenseveral roads of the network and usually far away upstream of the bottleneck.

In this brief review article, the criticism of the generally accepted fundamentals and methodologies of the traffic andtransportation theory is solely based on an analysis whether the classical traffic and transportation theories are consistentwith the set of the fundamental empirical features of traffic breakdown at a highway bottleneck or not.

2. Why should the empirical features of traffic breakdown be the empirical basis for any traffic and transportationtheory?

Each traffic flow model and theory that can be reliably used for control and optimization in traffic networks shouldexplain the set of the empirical features of traffic breakdown, for the following well-known reasons.1. Capacity of free flow is restricted by traffic breakdown. Therefore, free flow capacity is determined by the empirical

features of traffic breakdown [42,44,45,87–97,247,248].2. The reliability of control and optimization of traffic and transportation networksdepends crucially onwhether traffic control

can prevent traffic breakdown or not. Thus, any control or optimization method should be consistent with the empiricalfeatures of traffic breakdown at a bottleneck [202–206,209].

3. The efficiency of dynamic traffic assignment in traffic and transportation networks depends crucially on whether theassignment can reduce traffic congestion in a network or not. Therefore, methods for dynamic traffic assignment intraffic and transportation networks should also be consistent with the empirical features of traffic breakdown at networkbottlenecks [202–206].

3. The set of the fundamental empirical features of traffic breakdown at highway bottlenecks

In empirical data, the characteristics of traffic breakdown (like breakdown probability) are found from a study oftraffic breakdown at a given bottleneck during many different days (and years) of traffic breakdown observations (see,e.g., empirical traffic data studies in Refs. [42,44,45,87–97,247–276]). The set of the fundamental empirical features of trafficbreakdown at a highway bottleneck disclosed from these studies is as follows [247,248].1. Traffic breakdown at a highway bottleneck is a local phase transition from free flow (F) to congested traffic whose

downstream front is usually fixed at the bottleneck location (see, e.g., Refs. [42,87–91,246,263] and references there).Such congested traffic we call synchronized flow (S). Within the downstream front of synchronized flow, vehiclesaccelerate from synchronized flow upstream of the bottleneck to free flow downstream of the bottleneck (Figs. 1–4).2

2. At the same bottleneck, traffic breakdown can be either spontaneous (Figs. 1 and 2) or induced (Figs. 3 and 4).3. The probability of traffic breakdown is an increasing flow-rate function [92–97].4. There is a well-known hysteresis phenomenon associated with traffic breakdown: when the breakdown has occurred at

some flow rates with resulting congested pattern formation upstream of the bottleneck, then a return transition to freeflow at the bottleneck is usually observed at smaller flow rates (Fig. 5) (see, e.g., Refs. [42,87–90,246,275] and referencesthere).

1 In the case of a moving bottleneck (that appears, for example, through a very slowmoving vehicle), amotionless region within which traffic breakdownoccurs in a neighborhood of the moving bottleneck is related to a co-ordinate system moving with the bottleneck velocity.2 Note that whereas there is a speed drop during traffic breakdown at the bottleneck, the flow rate in synchronized flow resulting from the breakdown

can remain almost as large as the flow rate in an initial free flow, which has been just before the breakdown has occurred (compare the flow rates in freeand synchronized flows at the bottlenecks shown in Figs. 1(c), 3(c), and 4(f)).

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Fig. 1. Empirical example of spontaneous traffic breakdowns at an on-ramp bottleneck (real measured traffic data of road detectors installed along three-lane freeway). (a) Averaged speed in space and time. (b), (c) Time-functions of average speed (b) and flow rate (averaged between freeway lanes) (c) at thebottleneck location. 1-min average data.Source: Taken from Ref. [247].

A spontaneous traffic breakdown occurs where there are free flows both upstream and downstream of the bottleneckbefore the breakdown has occurred (Figs. 1 and 2).

In an empirical example shown in Fig. 2, the flow rate in free flow exhibits many flow fluctuations while its mean valueincreases over time (Fig. 2(b)–(d)); consequently, the mean free flow speed decreases slightly (Fig. 2(f)–(h)). During sometime interval, a flow rate impulse appears in which the flow rate increases appreciably. This flow rate impulse within whichfree flow speed decreases propagates downstream (Fig. 2). When this flow rate impulse reaches the off-ramp bottleneck,traffic breakdown occurs at the bottleneck.While the downstream front of synchronized flow resulting from the breakdownis fixed at the bottleneck (labeled by the dashed line in Fig. 2(e)), the upstream front of the synchronized flow propagatesupstream (Fig. 2(e)–(h)).

In contrast with the spontaneous breakdown, an induced traffic breakdown is caused by a propagation of a congestedpattern that for example has earlier emerged at another downstream bottleneck (Figs. 3 and 4).

4. Traffic breakdown at the bottleneck in the Lighthill–Whitham–Richards (LWR) model

The basic idea of the classic LWR traffic flow theory [1,2,42,47,52,54] is as follows (Fig. 6): the maximum flow rate q0 onthe fundamental diagram (Fig. 6(b)) determines the free flow capacity at a bottleneck. If the sumof the flow rates in free flowupstream of the bottleneck (qon +qin in Fig. 6(a)) reaches themaximum flow rate q0, then a further increase in the upstreamflow rates must lead to traffic breakdown with resulting congestion formation upstream of the bottleneck (Fig. 6(c), (d)).

The LWRmodel consists of the law of conservation of the number of vehicles on the road inwhich a relationship betweenthe flow rate and density is given by the fundamental diagram. Because the LWR model has discontinuous solutions in theform of shock waves [1,2,42,47,52,54], its application for numerical simulations of large traffic and transportation networkshad been limited before Daganzo introduced a cell-transmission model that was consistent with the LWR theory [373,374].Therefore, Daganzo’s cell-transmission model is currently used for a huge number of traffic and transportation simulations,including simulations of traffic control and optimization in traffic and transportation networks (see, e.g., Refs. [375–389]).

With the use of numerical simulations of Daganzo’s cell-transmissionmodel [373], we have derived the features of trafficbreakdown in the LWR theory (Fig. 6) [248]. At a given flow rate qin in free flow upstream of the bottleneck that is smallerthan q0 (Fig. 6(a), (b)), there is a critical on-ramp inflow rate qon = q(d)

on = q0 − qin. When qon < q(d)on , i.e., condition

qon + qin < q0 (1)

is satisfied, then no congestion occurs at the bottleneck. When qon > q(d)on , i.e., condition

qon + qin > q0 (2)

is satisfied, then congested traffic occurs at the bottleneck with the subsequent shock wave propagation upstream of thebottleneck (Fig. 6(c), (d)). Themore the flow rate qon exceeds the value q

(d)on , the larger the absolute value of the shock velocity

|vshock| (Fig. 6(c)–(e)).The main disadvantage of the LWR theory in the explanation of traffic breakdown at a bottleneck is that the LWR

theory cannot show induced traffic breakdown (empirical feature 2 of Section 3, Figs. 3 and 4). This is demonstrated

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Fig. 2. Empirical example of spontaneous traffic breakdown at an off-ramp bottleneck. Real measured traffic data of the flow rate (a)–(d) and averagespeed (e)–(h) through road detectors installed along three-lane freeway. (a), (e) Total flow rate (a) and average speed (e) on the main road in space andtime. (b)–(d), (f)–(h) Time-functions of total flow rate across freeway (b)–(d) and average speed (f)–(h) measured at three locations of road detectors(x = 21.8 (b), (f), 20.9 (c), (g), and 19.9 km (d), (h); road location 21.8 km is about 1.1 km upstream of the beginning of the off-ramp merging region).1-min average data. The dashed arrows in the flow direction shown in (a), (e) mark the spatiotemporal propagation of a flow rate impulse and the relateddecrease in the free flow speed; this impulse of the flow rate increase and speed decrease in free flow is marked by dashed up-arrows at the locations ofroad detectors in (b)–(d), (f)–(h) as well as labeled by ‘‘flow rate increase’’ and ‘‘speed decrease’’ in (b), (f). The dashed arrow in the upstream directionshown in (e) marks the propagation of the upstream front of synchronized flow after the breakdown has occurred; the dashed down-arrow in (f)–(h)marks this upstream front of synchronized flow at the locations of road detectors. The dashed lines in (a), (e) mark the bottleneck location at which thedownstream front of synchronized flow resulting from the breakdown is localized. For a more detailed explanation of this spontaneous traffic breakdownat the off-ramp bottleneck see Section 3.3.1.2 of the book [248].

in Fig. 6(g), (h) in which at the same given flow rate qin we have chosen smaller qon satisfying condition (1). Therefore, nocongestion occurs firstly at the on-ramp bottleneck. Then, in simulations a wide moving jam has been induced downstreamof the on-rampbottleneck. As in empirical data (Fig. 3),while propagating upstreamof the on-ramp the jamcauses congestedtraffic at the on-ramp (Fig. 6(g), (h)). However, in contrast with empirical data in which congested traffic persists at thebottleneck independent of whether the jam propagates further (Fig. 3) or the jam dissolves (see Ref. [247]), in the LWRtheory congested traffic always dissolves after the jam dissolution. This is independent of the initial jam width WJ (in thelongitudinal direction) (Fig. 6(g), (h)) and on the value qon + qin as long as condition (1) is satisfied.3

3 Let us explain why even under condition (1) there is congestion at the bottleneck caused by the jam. First note that in the model the downstream jamfront moving with the velocity vg is associated with the line of the fundamental diagram for congested traffic (solid line with negative slope in Fig. 6(f)).In contrast, the upstream jam front moving with the velocity vup is associated with a line with negative slope in the flow—density plane that is below theline for congested traffic; when the jams are upstream of the bottleneck, the line related to the upstream jam front is a dashed line with negative slopeshown in Fig. 6(f). Because |vg | > |vup|, the jam width decreases over time (Fig. 6(g), (h)). Before the jam reaches the bottleneck, the flow rate in thejam outflow q(J)

out is equal to q0 . When the jam is upstream of the bottleneck, within the road region between the bottleneck and downstream jam frontcongested traffic occurs. The flow rate within this congestion is equal to the flow rate in the jam outflow, which decreases to q(J)

out = q0 −qon . This is becausein the LWR model the flow rate downstream of the bottleneck cannot exceed q0 (Fig. 6(b)). However, after the jam has dissolved, the flow rate upstreamof the bottleneck decreases to qin resulting in the dissolution of congestion due to condition (1).

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Fig. 3. Empirical example of traffic breakdown at the on-ramp bottleneck induced by wide moving jam propagation (real measured traffic data of roaddetectors installed along three-lane freeway). (a) Averaged speed in space and time. (b), (c) Time-functions of average speed (b) and flow rate (averagedbetween freeway lanes) (c) at the bottleneck location (x = 17.1 km). 1-min average data.Source: Taken from Ref. [247].

5. Traffic breakdown at the bottleneck in the models of General Motors (GM) model class

In microscopic traffic flow models firstly developed by Herman, Gazis et al. in 1958–1961, traffic breakdown has beenexplained by an instability of vehicular traffic flow associated with the over-deceleration effect (called also as driver’s over-reaction) [3,5]: if a vehicle begins to decelerate unexpectedly, then due to a finite driver reaction time the following vehiclestarts decelerationwith a delay.When the time delay is long enough, the driver of the following vehicle decelerates strongerthan needed to avoid a collision. As a result, the speed of the following vehicle becomes lower than the speed of the precedingvehicle. If this over-deceleration effect is realized for all following drivers, a growingwave of vehicle speed reduction appearsand increases in amplitude over time while propagating usually upstream in traffic flow.

This classical traffic flow instability has been incorporated in a huge number of traffic flow models like Gipps’s model[23,24], Payne’s model [21,22], Newell’s optimal velocity (OV)model [11], Wiedemann’s model [53], Whitham’s model [30],the Nagel–Schreckenberg (NaSch) cellular automaton (CA) model [31,32], Bando et al. OVmodel [33–35], Treiber’s IDM [36,60,70,71], Krauß’s model [37,38], the Aw–Rascle model [39] (these and many other different traffic flow models of the GMmodel class as well as results of their analyses can be found for example in Refs. [365,366,390–483] and reviews [42,49–51,53,55,56,58,60–62,65,67,70,71]).

Microscopic,macroscopic aswell as all other traffic flowmodels inwhich the traffic flow instability has been incorporatedbased on sometimes very different mathematical approaches can be considered as related to the GM model class. This isbecause all these traffic flow models as known now exhibit the same feature firstly found by Kerner and Konhäuser fromtheir study of a version of Payne’s macroscopic traffic flow model that belongs to the GM model class [392]: the modelinstability of the GMmodel class leads to a phase transition from free flow to a widemoving jam (F → J transition) (Fig. 7).4

However, rather than an F → J transition of the GMmodel class, in real data traffic breakdown is a phase transition fromfree flow to synchronized flow (F → S transition) (Figs. 1 and 2) [247,248]. Therefore, traffic flow models of the GM modelclass cannot show the fundamental empirical feature 1 of traffic breakdown (Section 3). A more detailed criticism of the GMmodel class for the description of traffic breakdown at highway bottlenecks is given in Section 10.3 of the book [248].

6. Generally accepted understanding of the capacity of free flow at a bottleneck as a particular value

As is well known, the capacity of free flow at a bottleneck is limited by traffic breakdown (see references for examplein Refs. [42,44,45,89–97]). The classical understanding of the capacity is as follows: at any time instant, there is a particularvalue of the capacity of free flow at a bottleneck.

4 Some of the models belonging to the GM model class like the OV model [33–35] and IDM [36,60,70,71] have a unique speed–density relationship,whereas the initial GM model of Herman, Montroll, Potts, and Rothery does not have [3]. Rather than a relation between steady speed and density in amodel, the common feature of any traffic flow model belonging to the GM model class is a free flow instability that causes an unlimited growing wave ofvehicle speed reduction propagating upstream in traffic flow. Note that to describe traffic beyond instabilities, Gazis, Herman and Rothery made a furtherdevelopment of the GM model in which steady-state solutions lie on a one-dimensional curve in the flow-density plane (fundamental diagram) [5] (seethe review by Nagel et al. [62] for more details).

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Fig. 4. Empirical example of traffic breakdown at the on-ramp bottleneck induced by the propagation of a moving synchronized flow pattern (MSP) (realmeasured traffic data of road detectors installed along three-lane freeway). (a), (b) Average speed (a) and total flow rate (b) on the main road in space andtime. (c)–(f) Time-functions of average speed (c), (e) and total flow rate (d, f) at the two road locations: 17.9 km (c), (d) and 17.1 km; location 17.1 km isabout 100 m downstream of the end of the merging region of the on-ramp bottleneck (e), (f). 1-min average data. Note that whereas the flow rate withinthe jam shown in Fig. 3(c) is very small, the flow rate within the MSP shown in (d) is almost as large as in the surrounded free flow. Another differencebetween the jam shown in Fig. 3 and the MSP shown here is as follows: the jam propagates through the bottleneck while maintaining the mean velocityof its downstream front (Fig. 3(a)); in contrast, the MSP is caught at the bottleneck (a) (catch effect).Source: Taken from Ref. [247].

Fig. 5. Example of a well-known empirical hysteresis phenomenon caused by traffic breakdown (F → S transition) and return transition from congestedtraffic to free flow (S → F transition) (real measured traffic data of road detectors).Source: Taken from Ref. [247].

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Fig. 6. Simulations of traffic breakdown at the on-ramp bottleneck on a single-lane road in the LWR theory [248]. (a) Qualitative schema of the on-rampbottleneck. (b) The fundamental diagram used in the simulations of Daganzo’s cell-transmission model. (c), (d) Congested traffic at the bottleneck undercondition (2) for two different values of the on-ramp inflow rate qon at a given flow rate qin upstream of the bottleneck: vehicle speed distributions in spaceand time that show shock wave emergence and propagation upstream of the on-ramp bottleneck. (e) Dependence of the velocity of the shock wave vshockon the flow rate qon . (f) The fundamental diagram taken from (b) together with a line (dashed line) related to the upstream front of wide moving jamsin (g), (h) when the jams are upstream of the bottleneck. (g), (h) Congested traffic at the bottleneck caused by jam propagation through the bottleneckunder condition (1): vehicle speed in space and time; the jam width (in the longitudinal direction) at t = 0 is equal to WJ = 2 (g) and 1.5 km (h).q0 = 1830 vehicles/h. qin = 1600 vehicles/h. qon = 270 (c), 600 (d) and 120 vehicles/h (g), (h).

Earlier it was assumed that there is a fixed value of capacity. Recently based on empirical data it has been found thattraffic breakdown exhibits a probabilistic nature [91–97] and it is assumed that capacity is a stochastic value: at the timeinstant, there is some particular value of capacity but we know this stochastic capacity with some probability only [94–97].

The generally accepted definition and understanding of the capacity of free flow at the bottleneck as a particular (eitherfixed or stochastic) value contradicts the set of the fundamental features of traffic breakdown that should determine thenature of capacity.

Indeed, let us assume that there is a particular (fixed or stochastic) highway capacity of free flow at a bottleneck. Underthis assumption, when the flow rate in free flow downstream of the bottleneck is smaller than the particular capacity, thenfree flow should persist at the bottleneck, i.e., traffic breakdown cannot be induced at the bottleneck. This assumption aboutthe nature of the particular capacity is inconsistent with empirical observations shown in Figs. 3 and 4: at the flow rateat which free flow is observed at the bottleneck, traffic breakdown is induced at the bottleneck when a wide moving jam(Fig. 3) or an MSP (Fig. 4) reaches the bottleneck location.

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Fig. 7. Simulations of traffic breakdown at the on-ramp bottleneck in the GMmodel class: speed in space and time during spontaneous wide moving jamemergence at the bottleneck. The flow rate on the main road upstream of the bottleneck qin is larger in (a) than that in (b), whereas the on-ramp inflowqon is smaller in (a) than that in (b) at the same other model parameters.Source: Taken from Ref. [248].

It is clear that the capacity of free flow at the bottleneck cannot depend on whether there is a jam, which has occurredoutside of the bottleneck and independent of the bottleneck existence, or not. This explains why the generally acceptedunderstanding of the nature of capacity as a particular value is invalid for real traffic.

7. Traffic breakdown at the bottleneck in the three-phase theory

Empirical data shows that in congested traffic, two qualitatively different traffic phases should be distinguished[247,248]: synchronized flow (S) and wide moving jam (J). Therefore, in the three-phase theory introduced by the authorthere are three phases: F, S, and J. The fundamental difference between the phases S and J in measured data of congestedtraffic that determines the phase definitions is as follows: while the downstream front of a wide moving jam propagatesthrough a highway bottleneck upstream with a characteristic mean velocity, the downstream front of synchronized flowdoes not exhibit this jam characteristic feature and this synchronized flow front is usually fixed at the bottleneck (Figs. 1–4).

The understanding of the three-phase theory can be clear from a consideration of the objective of this theory.• The main reason for the three-phase theory is the explanation of the set of the fundamental empirical features of traffic

breakdown [247,248],5 rather than of the features of traffic congestion.6

7.1. Probability of traffic breakdown at the bottleneck, Z-characteristic, and infinite number of highway capacities

The set 1–4 of the fundamental empirical features of traffic breakdown at a highway bottleneck (Section 3) has firstlybeen explained in the three-phase traffic theory by a phase transition from free flow to synchronized flow (F → S transition),which occurs in metastable free flow (Fig. 8) [247,248]. Thus in the three-phase theory the term traffic breakdown isequivalent to the term an F → S transition (traffic breakdown feature 1 of Section 3).

Due to themetastability of free flow at a highway bottleneck with respect to an F → S transition, in the three-phase the-ory both spontaneous (Fig. 8(a)) and induced (Fig. 8(b)) traffic breakdowns at the bottleneck are possible (traffic breakdownfeature 2 of Section 3). The probability of spontaneous traffic breakdown at the bottleneck found firstly in a microscopicthree-phase theory [488] (Fig. 8(c)) is a growing flow-rate function (traffic breakdown feature 3). This theoretical probabil-ity of spontaneous breakdown is well fitted by a function [488]

P (B)=

11 + exp[α(qP − qsum)]

, (3)

where the parameters α and qP depend on the on-ramp inflow rate qon and a time interval withinwhich traffic breakdown isstudied. Qualitatively the same growing flow-rate function for the breakdown probability has also been found in measuredtraffic data [94–96].

5 Themain criticismon the three-phase theorymade during last years, in particular in theworks byDaganzo et al. [378] and byHelbing, Treiber et al. [365,366,430] can be formulated as follows: the three-phase theory is not needed because the generally accepted traffic flow theories also succeed in simulatingthe most essential empirical features of traffic flow described by the three-phase theory. However, as explained above none of the generally acceptedtraffic flow models (see, e.g., Refs. [219–222,365,366,373–448,464–487] and reviews [42,49–51,53,55,56,58,60–62,65,67,70,71]) that include the modelsof Daganzo, Helbing, Treiber et al. (e.g., Refs. [60,70,71,219–222,365,366,373–380,393–416,426–432]) can simulate and explain the set of the fundamentalempirical features of traffic breakdown. The explanation of the fundamental empirical features of traffic breakdown (Section 3) is the main reason for thethree-phase theory [247,248]. Therefore, the statement ‘‘three-phase theory is not needed’’ is invalid.6 Because in the three-phase theory traffic breakdown at a highway bottleneck is an F → S transition, a peculiarity of the three-phase theory in the

explanation of traffic congestion is associated with the features of synchronized flow resulting from the breakdown. Although a diverse variety of complexspatiotemporal phenomena in synchronized flow have been found in the three-phase theory [247,248], one of the most important of them —moving jamemergence in synchronized flow (S → J transition) — is explained by the over-deceleration effect (driver’s over-reaction) due to a driver’s reaction time,i.e., by the same traffic flow instability discovered in the GMmodel [3–5]. This explains many similar results of the three-phase theory and the GMmodelclass in the description of the features of moving jam propagation in traffic congestion (see Section 11).

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Fig. 8. Explanations of the fundamental empirical features of traffic breakdown at a highway bottleneck with the three-phase theory [247,248]. (a),(b) Simulations of spontaneous (a) and induced (b) breakdown at the on-ramp bottleneck taken from Ref. [490]. (c) Simulations of the probability ofspontaneous traffic breakdown at the on-ramp bottleneck on a single-lane road taken from Ref. [488]. (d) Qualitative Z-speed–flow-rate characteristic fortraffic breakdown; F—free flow, S—synchronized flow (results of simulations of the Z-speed–flow-rate characteristic for traffic breakdown can be found inRefs. [247,248,490,489,491]; see, for example, Fig. 3.17(b) of Ref. [248]).

The metastability of free flow at the bottleneck is also responsible for a qualitative Z-characteristic for highway trafficin the speed–flow-rate plane found firstly in Ref. [489] (Fig. 8(d)). The Z-characteristic explains the hysteresis effect (trafficbreakdown feature 4 of Section 3). On this Z-characteristic, bottleneck states labeled by circles F and S (Fig. 8(d)) are relatedto the free flow and synchronized flow phases, respectively.

As stressed in Section 6, the understanding and definition of the capacity of free flow at a bottleneck as a particular (fixedor stochastic) value is not consistent with the empirical fact that traffic breakdown can be induced at the bottleneck (trafficbreakdown feature 2 of Section 3). In the three-phase traffic theory, the highway capacity of free flow at the bottleneck isdefined as follows [247,248].

• The infinite number of the flow rates in free flowdownstreamof the bottleneck atwhich traffic breakdown can be inducedat the bottleneck are the infinite number of highway capacities of free flow at the bottleneck.

This capacity definition is associated with the result of the three-phase theory that free flow at the bottleneck is in ametastable state with respect to traffic breakdown (F → S transition) at the flow rate downstream of the bottleneck, whichis within the flow rate range between theminimum andmaximum capacities (Fig. 8(d)) [247,248]. The term ‘‘infinite’’ in thecapacity definition stems from the fact that the flow rate is equal to the inverse of the average gross time headwaymeasuredat a road location, which can take on any real value between the minimum and maximum capacities.

Note that the minimum and maximum capacities, which limit the range of the infinite number of highway capacities offree flow at the bottleneck, do not depend on whether there is a congested pattern that propagates through the bottleneck,or not: if the flow rate in free flow downstream of the bottleneck is smaller than the minimum capacity, then no trafficbreakdown is possible at the bottleneck: traffic breakdown does not occur at the bottleneck, even if a localized congestedpattern (a wide moving jam in Fig. 3 or an MSP in Fig. 4) has reached the bottleneck.

If the flow rate in free flow downstream of the bottleneck is larger than the maximum capacity, then traffic breakdowndoes occur at the bottleneck. This traffic breakdown at the bottleneck does not depend on whether a congested patternpropagates through the bottleneck, or not.

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Fig. 9. Hypothesis about a discontinuous character of the probability of over-acceleration in the three-phase theory [247,248]. (a), (b) Qualitative Z-shapedfunctions of over-acceleration probability on the vehicle density (or occupancy) (a) and flow rate (b). F—free flow, S (hatched regions)—synchronized flow.

Thus in the three-phase theory (Fig. 8(c), (d)) [247,248], there is a broad range of flow rates q in free flow on a link of thetraffic network

Cmin ≤ q ≤ Cmax, (4)

where, respectively, Cmin and Cmax are the minimum and maximum capacities of free flow at a bottleneck on a networklink.7 Condition (4) determines the infinite number of capacities of free flow at a bottleneck. When

q < Cmin, (5)

no breakdown can occur in free flow. In this case, the generally accepted fundamentals of the traffic and transportationtheory are consistent with the features of free flow. When

q ≥ Cmax, (6)

the breakdown probability P (B)= 1 and, therefore, the breakdown does occur in this free flow with resulting traffic

congestion. If condition (6) is satisfied for each of the bottlenecks in the network, then all network links are congested. Inthis limit case, the generally accepted fundamentals of the traffic and transportation theory are also able to show some ofthe empirical features of traffic congestion.

7.2. The nature of traffic breakdown

In the three-phase traffic theory, the Z-characteristic for traffic breakdown, the increasing flow-rate function ofbreakdown probability and the associated range of the infinite number of highway capacities (Fig. 8(c), (d)) result froma competition between opposing tendencies occurring within a random local disturbance of an initial free flow. Within thisdisturbance the speed is lower and vehicle density is greater than the initial free flow. These opposing tendencies are asfollows [247].

(i) A tendency toward synchronized flow due to vehicle deceleration associated with a speed adaptation effect. The termspeed adaptation is related to car-following behavior inwhich the following vehicle should deceleratewhile approachinga slower moving preceding vehicle.

(ii) A tendency toward the initial free flow due to vehicle acceleration associated with an over-acceleration effect. The over-acceleration effect is a driver maneuver leading to a higher vehicle speed from initial car-following at a lower speed.The term over-acceleration emphasizes that this acceleration of the vehicle occurs from car-following even when thepreceding vehicle does not accelerate and it does not move with a higher speed than the speed of the following vehicle.

The competition between speed adaptation and over-acceleration explains the set of the fundamental empirical featuresof traffic breakdown (F → S transition) through a hypothesis of the three-phase theory that the probability of over-acceleration (denoted by POA) exhibits a discontinuous character [247,248].

• At the same density in free flow and synchronized flow, the over-acceleration probability is larger in free flow than insynchronized flow resulting in a Z-shaped density function of over-acceleration probability POA (Fig. 9(a)).

• At the same flow rate in free flow and synchronized flow, the over-acceleration probability is larger in free flow than insynchronized flow resulting in a Z-shaped flow-rate function of over-acceleration probability POA (Fig. 9(b)).

7 It should be mentioned that the values of the minimum capacity Cmin and the maximum capacity Cmax of free flow at the bottleneck can dependconsiderably on traffic parameters, like driver and vehicle characteristics (e.g., the percentage of long vehicles in traffic flow) as well as weather conditions.

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To understand the impact of the discontinuous character of over-acceleration probability on traffic breakdown, weassume firstly that an initial traffic phase at the bottleneck is synchronized flow. Due to the discontinuous character of over-acceleration probability, an instability can occur in synchronized flow that causes a growing local speed increase leading tothe free flow occurrence at the bottleneck. In contrast, let us now assume that an initial traffic phase at the bottleneck isfree flow. Then the over-acceleration maintains the free flow at the bottleneck as long as a local speed reduction occurs inthe free flow within which the speed adaptation overcomes the over-acceleration. In this case, traffic breakdown occurs atthe bottleneck.

The discontinuous character of over-acceleration probability leads to the increasing flow-rate function of breakdownprobability (Fig. 8(c)) as well as to a Z-characteristic for traffic breakdown at a highway bottleneck (Fig. 8(d)) [247,248].

Already in first microscopic three-phase traffic flow models [488,489] two mechanisms for over-acceleration have beenintroduced and studied.(i) Over-acceleration due to vehicle acceleration in car-following occurring without lane changing.(ii) Over-acceleration due to lane changing to a faster lane that is possible on a multi-lane road.

On single-lane roads, the over-accelerationmechanism occurringwithout lane changing does lead to the Z-characteristicfor traffic breakdown (F → S transition) as well as to the increasing flow-rate function of breakdown probability, as thishas firstly been found out in numerical simulations of Refs. [488,489].8 For multi-lane roads, the importance of the over-acceleration mechanism due to lane changing to a faster lane has been emphasized in Ref. [493].

Obviously the resulting discontinuous character of driver over-acceleration in traffic flow is determined by a combinationof all possible different mechanisms of over-acceleration. Over time a number of three-phase traffic models have beendeveloped (e.g., Refs. [490,491,493–535]). In some of these models the hypothesis of the three-phase theory about thediscontinuous character of the over-acceleration effect has also been incorporated.

8. Importance of capacity understanding for management and control in traffic networks

Both the case of small enough flow rate (5) and the case of very large flow rate (6) are not interesting for traffic controland optimization in vehicular networks for the following reasons.(i) At small enough flow rates (very small traffic demand),when condition (5) is satisfied for each of the links of the network,

no control and optimization is needed in such a traffic network in which no congestion can occur.(ii) At very large flow rates (very large traffic demand), when condition (6) is satisfied for each of the links of the network,

reliable traffic optimization is almost not possible in such a fully congested network.

Thus the infinite number of capacities of free flow at a bottleneck, which are within the flow rate range (4) betweenthe minimum and maximum capacities, are of the greatest interest for traffic control and optimization in vehicular trafficnetworks. However, none of the generally accepted fundamentals andmethodologies of the traffic and transportation theorycan show this feature of free flow resulting from the set 1–4 of the fundamental empirical features of traffic breakdown. Thisquestions any application of these fundamentals andmethodologies for a study, optimization, and control of real traffic andtransportation networks.9

9. Breakdownminimization (BM) principle

The network breakdownminimization (BM) principle [536] is consistent with the set 1–4 of the fundamental features oftraffic breakdown (Section 3). The BM principle for the optimization and control of traffic and transportation networks is asfollows [536].• The BM principle states that the optimum of a traffic network with N bottlenecks is reached when dynamic traffic

optimization and/or control are performed in the network in such a way that the probability for spontaneous occurrenceof traffic breakdown in at least one of the network bottlenecks during a given observation time reaches the minimumpossible value. The BM principle is equivalent to the maximization of the probability that traffic breakdown occurs atnone of the network bottlenecks.

9.1. Mathematical formulation of the BM principle

Assuming that at different bottlenecks traffic breakdown occurs independently of each other, the probability forspontaneous occurrence of traffic breakdown in at least one of the network bottlenecks during a given observation timecan be written as

Pnet = 1 −

Nk=1

1 − P (B,k) . (7)

8 See Figs. 2(a) and 3(b) of [489] and Fig. 18(b), (c) of Ref. [488]. Recently, Jin et al. [492] have found an F → S transition in their empirical study of realtraffic data measured on a single-lane road in China.9 An example of the application of these fundamentals for traffic control is a well-known on-ramp metering model ALINEA of Papageorgiou et al. [140–

143]: as explained in Section 10.6 of the book [248], ALINEA cannot prevent traffic breakdown at an on-ramp bottleneck.

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In accordance with the BM principle, the network optimum is reached at

minq1,q2,...,qM

{Pnet(q1, q2, . . . , qM)}. (8)

Here, M is the number of network links for which flow rates can be adjusted; qm is the link inflow rate for a link withindex m; m = 1, 2, . . . ,M , where M > 1; k is the bottleneck index, k = 1, 2, . . . ,N; N > 1; P (B,k) is the probability thatduring the observation time interval traffic breakdown occurs at the bottleneck with index k.

The above mathematical formulation of the BM principle is only valid when traffic breakdowns at neighborhoodbottlenecks can be considered as independent events. The latter is usually the case in real traffic networks as explainedbelow.

9.2. When are traffic breakdowns at neighborhood bottlenecks independent events?

Traffic breakdown at a highway bottleneck in a traffic network is a local phase F → S transition in a metastable freetraffic flow. With the largest probability, the F → S transition begins within a permanent local speed disturbance localizedin a neighborhood of the bottleneck [247]. Empirical studies allow us to assume that the length of the road in whichsuch a permanent speed disturbance is localized is about 200–300 m. Thus traffic breakdowns at neighborhood highwaybottlenecks can be considered independent events, i.e., the formulation of the BM principle through formulas (7), (8) is validonly when the permanent speed disturbances at the bottlenecks are not overlapping each other. This can approximately beconsidered to be satisfied when the distance between the neighborhood bottlenecks is noticeably larger than 300 m.

9.3. About network optimization when breakdown has already occurred at a network bottleneck

If traffic breakdownhas already occurred at a network bottleneck, then network optimization can consist of the followingstages.

(i) A spatial limitation of congestion growthwith the subsequent congestion dissolution at the bottleneck, if the dissolutionof congestion due to traffic management in a neighborhood of the bottleneck is possible.

(ii) The minimization of traffic breakdown probability with the BM principle in the remaining network, i.e., the networkpart that is not influenced by congestion.

10. The BM principle and classical Wardrop’s principles

The BM principle [536] is an alternative to well-known principles for vehicular network optimization and control basedon the minimization of travel costs (travel time, fuel consumption, etc.) or the maximization of traffic throughput [203,204,206]. In particular, the most prominent classical principles for the minimization of travel costs in a traffic network areWardrop’s UE and SO principles [98].

• Wardrop’s UE principle: traffic on a network distributes itself in such a way that the travel times on all routes used fromany origin to any destination are equal, while all unused routes have equal or greater travel times.

• Wardrop’s SO principle: the network-wide travel time should be a minimum.

Wardrop’s principles reflect either the wish of drivers to reach their destinations as soon as possible (UE) or the wish ofnetwork operators to reach the minimum network-wide travel time (SO).

However, when the flow rate on a link of a network is between the maximum and minimum capacities, there may be atleast two different states of a bottleneck on the link denoted by circles F and S shown in Fig. 8(d). The state F is related to thefree flow phase and the state S to the synchronized flow phase.10 Therefore, hypothetically assuming that each of the linkflow rates for each of the N network bottlenecks is between the associated minimum and maximum bottleneck capacities,we find that there may be

2N (9)

different bottleneck states in the network at the same distribution of the flow rates in the network. If we apply anoptimization algorithm associated with the minimization of travel cost in the network random transitions between thefree flow phase (F) and synchronized flow phase (S) may occur in traffic at different network bottlenecks during networkoptimization and/or control.11

10 There are the infinite number of different synchronized flow states inwhich the flow rate and speed can differ substantially from point to point (dashed2D-region in Fig. 8(d)). However, for the simplicity of a qualitative explanation made here and below, while referring to state S in Fig. 8(d), we emphasizeonly that traffic flow at the bottleneck is associated with the synchronized flow phase, rather than with the free flow phase. This is independent of thevalues of the flow rate and speed in the synchronized flow phase.11 Under congestion as has been shown by Wahle and Schreckenberg with colleagues [537] and Davis [501,502] usually no true Wardrop’s equilibriumcan be found.

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Rather than some travel costs, in the BM principle the objective function that should be minimized is the probability oftraffic breakdown in the network. Thus the objective function in the BMprinciple that should beminimized depends neitheron travel time nor other travel costs. The BM principle demands the minimization of the probability of the occurrence ofcongestion in the network. Under great traffic demand, the application of the BM principle should result in relatively smalltravel costs associated with free flow in a network.

11. Achievements of the generally accepted classical traffic and transportation theories

In the LWR model and associated kinetic macroscopic traffic flow models as well as in traffic flow models of the GMmodel class, diverse driver behavioral characteristics related to real traffic have been discovered and incorporated [40,42,44–56,58,60–66,69–71].

Most of the achievements of these classical approaches are also used in three-phase traffic flow models [247,248]. Inparticular, for simulations of moving jam emergence in synchronized flow (S → J transitions) three-phase traffic flowmodels [488,489,493] incorporate the over-deceleration effect (driver’s over-reaction) introduced by Herman, Gazis et al.[3,5]. A stochastic approach to traffic flowmodeling introduced by Nagel and Schreckenberg in the NaSch CA model [31,32]is the basic approach for the mathematical description of diverse driver’s delays in both stochastic Kerner–Klenov model[489,493] and KKW CA three-phase model [488].12 Additionally, the stochastic three-phase models [489,493] incorporatethe mathematical formulation for the description of a safe speed in traffic flow firstly introduced by Gipps [23] and furtherdeveloped by Krauß et al. [37] as well as lane changing rules developed by Nagel et al. [538].

Obviously, each of the traffic models can explain some real traffic phenomena and each of the models exhibits a limitedregion of the applicability for the explanation of real traffic and transportation phenomena.

1. The LWR theory [1,2,42,47,52,54] can show a transition to traffic congestion whose downstream front is fixed at abottleneck as observed in real data (compare Figs. 1 and 2 with Fig. 6(c), (d)).

2. Two-phase traffic flow models in the framework of the GM model class can explain at least the following empiricalfeatures of traffic congestion.• Characteristic parameters of wide moving jams and their propagation [392] (compare the jam shown in Fig. 3 with

jams shown in Fig. 7) (see also, e.g., Refs. [32,34,35,37,60,70,71,365,366,419,428–430,444–446]).• A broad spread of traffic data in the flow-density plane associated with congested traffic (e.g., Refs. [407,408,426,

431]).13

12. Why are the generally accepted classical traffic and transportation theories inconsistent with the features of realtraffic?

Because of the above-mentioned achievements of the generally accepted classical traffic and transportation theories[40,42,44–56,58,60–66,69–71] a question arises.

• Why does the author state that the generally accepted classical traffic and transportation theories are not consistent withthe set of the empirical features of traffic breakdown and, therefore, they are not applicable for a reliable description oftraffic breakdown, capacity, traffic control, and optimization of real traffic and transportation networks?

The failure of the LWR theory and two-phase traffic flow models of the GMmodel class is explained as follows.

1. The LWR theory fails because this theory cannot show induced traffic breakdown observed in real traffic (Figs. 3 and 4).2. Two-phase traffic flow models of the GM model class fail because traffic breakdown in the models of the GM class is an

F → J transition (Fig. 7). In contrast with thismodel result, real traffic breakdown is an F → S transition (Figs. 1, 2 and 5).

The possibility of induced traffic breakdown at a bottleneck (Figs. 3 and 4) leads to the conclusion that traffic at thebottleneck can be either in the free flow phase (F) or in the synchronized flow phase (S) at the same flow rate on a networklink (Fig. 8(d)). On the one hand, this fact is responsible for the existence of the range of the infinite number of highwaycapacities (Fig. 8(d)). On the other hand, the possibility of two different phases F and S at the same flow rate at the bottleneckmeans that there are two possible values for travel costs at the same link flow rate: one is related to the phase F and anotherone to the phase S (Fig. 8(d)).

This explains the above statement that the generally accepted principles for traffic and transportation networkoptimization and control (e.g., Wardrop’s UE and SO principles) are not consistent with the set of the fundamental empiricalfeatures of traffic breakdown at a highway bottleneck: nominimum travel costs and nomaximum traffic throughput can be

12 A detailed review of the Nagel–Schreckenberg approach to simulations of driver’s delays has recently been made in the book by Schadschneideret al. [69].13 In simulations, this broad spread of traffic data is usually associated with different driver behavioral characteristics (heterogeneous flow of differentdrivers and vehicles) used in a traffic flowmodel as well as with complex dynamics of moving jams and other dynamic spatiotemporal effects in congestedtraffic, rather than with steady states of a traffic flow model.

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found within the flow range between the minimum and maximum capacities within which one of the two different phasesF and S exists (Fig. 8(d)).

It must be noted that the existence of these two phases F and S does not result from the stochastic nature of traffic:even if there were no stochastic processes in vehicular traffic, the states F and S do exist at the same flow rate. For thisreason, the generally accepted stochastic approaches to traffic control and dynamic traffic assignment (see, e.g., Refs.[539–542] and references in reviews [202–206]),which donot assume the existence of the flow range between theminimumand maximum capacities of the three-phase theory, cannot resolve the above-discussed problem of the inconsistency ofclassical theories with the empirical features of real traffic breakdown.

However, the stochastic nature of traffic influences crucially on the probability of random transitions between the phasesF and S. At a given flow rate, this probability can change in several orders of magnitude when the stochastic characteristicsof traffic change.

13. Conclusions

(i) A traffic flow model that can be used for reliable control and optimization in traffic networks should explain the set1–4 of the fundamental empirical features of traffic breakdown in an initial free flow.

(ii) The set 1–4 of the fundamental empirical features of traffic breakdown can be explained by none of the generally ac-cepted classical traffic and transportation theories andmodels (e.g., the Lighthill–Whitham–Richardsmodel [1,2], Daganzo’scell-transmission model [373,374], the General Motors car-following model by Herman, Gazis et al. [3–5], Newell’s optimalvelocity (OV) model [11], Gipps’s model [23,24], Payne’s model [21,22], Wiedemann’s model [53], Whitham’s model [30],Bando et al. OV model [33–35], Treiber’s IDM [36,60,70,71], Krauß’s model [37,38], the Aw–Rascle model [39]).

(iii) The assumption of the generally accepted fundamentals of the traffic and transportation theories that at any timeinstant there is a particular capacity of free flow at a bottleneck is inconsistent with the empirical feature 2 of trafficbreakdown (see Section 3). This critical conclusion is valid even if this particular capacity is considered a stochastic value,which at a time instant is known with some probability only (see a more detailed critical discussion in Section 10.6 ofRef. [248]).

(iv) In contrast with the generally accepted fundamentals of the traffic and transportation theories, in the three-phasetheory there is a range of capacities of free flow (4), i.e., it is assumed that at any time instant there are the infinite numberof capacities of free flow at a bottleneck. This assumption allows us to explain the set 1–4 of the fundamental features oftraffic breakdown.

(v) Within the capacity range (4), principles and models for minimization of travel cost or maximization of trafficthroughput are inconsistent with the real features of traffic breakdown.

(vi) Within the capacity range (4), traffic breakdown at a bottleneck can occur with some probability regardless of trafficcontrol. However, the probability of traffic breakdown at a bottleneck can be controlled with the following objectives.

(a) The minimization of breakdown probability. This objective can be achieved through a decrease in speed fluctuations infree flow in a neighborhood of the bottleneck.

(b) The maximization of the minimum and maximum capacities at a bottleneck.

(vii) Future directions for traffic flowmodeling are associated with the further development of traffic flowmodels in theframework of the three-phase theory.

(viii) Future directions for traffic control and optimization theory can be associated with a combination of

• the minimization of breakdown probability in free flow at network bottlenecks based on the BM principle in those partsof a traffic network that are not influenced by congestion together with

• a spatial limitation or/and dissolution of congestion in congested parts of the network.

Acknowledgments

I thank our partners for their support in the project ‘‘Urban Space: User oriented assistance systems and networkmanagement’’, funded by the German Federal Ministry of Economics and Technology by resolution of the German FederalParliament. I would like to thank Hubert Rehborn, Sergey Klenov, Gerhard Hermanns, Florian Knorr, and Mario Aleksic forvery helpful suggestions.

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