Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts
aa r X i v : . [ phy s i c s . s o c - ph ] A p r Three-phase traffic theory and two-phasemodels with a fundamental diagram in thelight of empirical stylized facts
Martin Treiber a , ∗ , Arne Kesting a , and Dirk Helbing b , c a Technische Universit¨at Dresden, Institute for Transport & Economics,W¨urzburger Str. 35, D-01187 Dresden, Germany b ETH Z¨urich, UNO D11, Universit¨atsstr. 41, CH-8092 Z¨urich, Switzerland c Collegium Budapest – Institute for Advanced Study,Szenth´aroms´ag u. 2, H-1014 Budapest, Hungary
Abstract
Despite the availability of large empirical data sets and the long history of trafficmodeling, the theory of traffic congestion on freeways is still highly controversial.In this contribution, we compare Kerner’s three-phase traffic theory with the phasediagram approach for traffic models with a fundamental diagram. We discuss theinconsistent use of the term “traffic phase” and show that patterns demanded bythree-phase traffic theory can be reproduced with simple two-phase models, if themodel parameters are suitably specified and factors characteristic for real trafficflows are considered, such as effects of noise or heterogeneity or the actual freewaydesign (e.g. combinations of off- and on-ramps). Conversely, we demonstrate thatmodels created to reproduce three-phase traffic theory create similar spatiotemporaltraffic states and associated phase diagrams, no matter whether the parameters im-ply a fundamental diagram in equilibrium or non-unique flow- density relationships.In conclusion, there are different ways of reproducing the empirical stylized facts ofspatiotemporal congestion patterns summarized in this contribution, and it appearspossible to overcome the controversy by a more precise definition of the scientificterms and a more careful comparison of models and data, considering effects of themeasurement process and the right level of detail in the traffic model used. ∗ Corresponding author. Tel.: +49 351 463 36794; fax: +49 351 463 36809
Email address: [email protected] (Martin Treiber).
URL: (Martin Treiber).
Preprint submitted to Elsevier Science 2 September 2018
Introduction
The observed complexity of congested traffic flows has puzzled traffic modelersfor a long time (see Helbing (2001) for an overview). The most controversialopen problems concern the issue of faster-than-vehicle characteristic propaga-tion speeds (Daganzo, 1995; Aw and Rascle, 2000) and the question whethertraffic models with or without a fundamental diagram (i.e. with or withouta unique equilibrium flow-density or speed-distance relationship) would de-scribe empirical observations best. While the first issue has been intensivelydebated recently (see Helbing and Johansson (2009), and references therein),this paper addresses the second issue.The most prominent approach regarding models without a fundamental dia-gram is the three-phase traffic theory by (Kerner, 2004). The three phases ofthis theory are “free traffic”, “wide moving jams”, and “synchronized flow”.While a characteristic feature of “synchronized flow” is the wide scatter-ing of flow-density data (Kerner and Rehborn, 1996b), many microscopic andmacroscopic traffic models neglect noise effects and the heterogeneity of driver-vehicle units for the sake of simplicity, and they possess a unique flow-densityor speed-distance relationship under stationary and spatially homogeneousequilibrium conditions. Therefore, Appendix A discusses some issues concern-ing the wide scattering of congested traffic flows and how it can be treatedwithin the framework of such models.For models with a fundamental diagram, a phase diagram approach has beendeveloped (Helbing et al., 1999) to represent the conditions under which cer-tain traffic states can exist. A favourable property of this approach is the pos-sibility to semi-quantitatively derive the conditions for the occurence of thedifferent traffic states from the instability properties of the model under con-sideration and the outflow from congested traffic (Helbing et al., 2009). Thephase diagram approach for models with a fundamental diagram has recentlybeen backed up by empirical studies (Sch¨onhof and Helbing, 2009). Neverthe-less, the approach has been criticized (Kerner, 2002, 2008), which applies tothe alternative three-phase traffic theory as well (Sch¨onhof and Helbing, 2007,2009). While both theories claim to be able to explain the empirical data, par-ticularly the different traffic states and the transitions between them, the maindispute concerns the following points: • Both approaches use an inconsistent terminology regarding the definition oftraffic phases and the naming of the traffic states. • Both modeling approaches make simplifications, but are confronted withempirical details they were not intended to reproduce (e.g. effects of detailsof the freeway design, or the heterogeneity of driver-vehicle units). • three-phase traffic theory is criticized for being complex, inaccurate, and2nconsistent, and related models are criticized to contain too many param-eters to be meaningful (Helbing and Treiber, 2002; Sch¨onhof and Helbing,2007). • It is claimed that the phase diagram of models with a fundamental dia-gram would not represent the empirical observed traffic states and tran-sitions well (Kerner, 2004). In particular, the “general pattern” (GP) andthe “widening synchronized pattern” (WSP) would be missing. Moreover,wide moving jams should always be part of a “general pattern”, and ho-mogeneous traffic flows should not occur for extreme, but rather for smallbottleneck strengths.In the following chapters, we will try to overcome these problems. In Sec. 2 wewill summarize the stylized empirical facts that are observed on freeways inmany different countries and have to be explained by realistic traffic models.Afterwards, we will discuss and clarify the concept of traffic phases in Sec.3. In Sec. 4, we show that the traffic patterns of three-phase traffic theorycan be simulated by a variety of microscopic and macroscopic traffic modelswith a fundamental diagram, if the model parameters are suitably chosen.For these model parameters, the resulting traffic patterns look surprisinglysimilar to simulation results for models representing three-phase traffic theory,which have a much higher degree of complexity. Depending on the interestof the reader, he/she may jump directly to the section of interest. Finally, inSec. 5, we will summarize and discuss the alternative explanation mechanisms,pointing out possible ways of resolving the controversy.
In this section, we will pursue a data-oriented approach. Whenever possible,we describe the observed data without using technical terms used within theframework of three-phase traffic theory or models with a fundamental diagram.In order to show that the following observations are generally valid, we presentdata from several freeways in Germany, not only from the German freeway A5,which has been extensively studied before (Kerner, 1998; Kerner and Rehborn,1996a; Sch¨onhof and Helbing, 2007, 2009; Bertini et al., 2004; Lindgren et al.,2006). Our data from a variety of other countries confirm these observationsas well (Zielke et al., 2008).
In order to eliminate confusion arising from different interpretations of thedata and to facilitate a direct comparison between computer simulations and3bservations, one has to simulate the method of data acquisition and the sub-sequent processing or interpretation steps as well. We will restrict ourselveshere to the consideration to aggregated stationary detector data which cur-rently is the main data source of freeway traffic studies. When comparingempirical and simulation data, we will focus on the velocity V (and not thedensity), since it can be measured directly. In addition to the aggregation overone-minute time intervals, we will also aggregate over the freeway lanes. Thisis justified due to the typical synchronization of velocities among freeway lanesin all types of congested traffic (Helbing and Treiber, 2002).To simulate the measurement and interpretation process, we use “virtual de-tectors” recording the passage time and velocity of each vehicle. For eachaggregation time interval (typically 60 s), we determine the traffic flow Q asthe vehicle count divided by the aggregation time, and the velocity V as thearithmetic mean value of the individual vehicles passing in this time period.Notice that the arithmetic mean value leads to a systematic overestimation ofvelocities in congested situations and that there exist better averaging meth-ods such as the harmonic mean (Treiber et al., 2000a). Nevertheless, we willuse the above procedure because this is the way in which empirical data aretypically evaluated by detectors.Since freeway detectors are positioned only at a number of discrete loca-tions, interpolation techniques have to be applied to reconstruct the ob-served spatiotemporal dynamics at any point in a given spatiotemporal re-gion. If the detector locations are not further apart than about 1 km, it issufficient to apply a linear smoothing/interpolating filter, or even to plotthe time series of the single detectors in a suitable way (see, e.g. Fig. 1in Sch¨onhof and Helbing (2007)). This condition, however, severely restrictsthe selection of suitable freeway sections, which is one of the reasons whyempirical traffic studies in Germany have been concentrated on a 30 km longsection of the Autobahn A5 near Frankfurt. For most other freeway sectionsshowing recurrent congestion patterns, two neighboring detectors are 1-3 kmapart, which is of the same order of magnitude as typical wavelengths ofnon-homogeneous congestion patterns and therefore leads to ambiguities asdemonstrated by Treiber and Helbing (2002). Furthermore, the heterogene-ity of traffic flows and measurement noise lead to fluctuations obscuring theunderlying patterns.Both problems can be overcome by post-processing the aggregated detec-tor data (Cassidy and Windower, 1995; Coifman, 2002; Bertini et al., 2004;Mu˜noz and Daganzo, 2002; Belomestny et al., 2003; Treiber and Helbing,2002). Furthermore, Kerner et al. (2001) have proposed a method called“ASDA/FOTO” for short-term traffic prediction. Most of these methods, how-ever, cannot be applied for the present investigation since they do not pro-vide continuous velocity estimates for all points ( x, t ) of a certain spatiotem-4 ig. 1. Spatiotemporal dynamics of the average velocity on two different freeways.(a) German freeway A9 in direction South, located in the area North of Munich.Horizontal lines indicate two intersections (labelled “AK”), which cause bottlenecks,since they consume some of the freeway capacity. The traffic direction is shown byarrows. (b) German freeway A8 in direction East, located about 40 km East ofMunich. Here, the bottlenecks are caused by uphill and downhill gradients around“Irschenberg” and by an accident at x = 43 . poral region, or because they are explicitly based on models. (The methodASDA/FOTO, for example, is based on three-phase traffic theory.) We willtherefore use the adaptive smoothing method (Treiber and Helbing, 2002),which has recently been validated with empirical data of very high spatialresolution (Treiber et al., 2010). In order to be consistent, we will apply thismethod to both, the real data and the virtual detector data of our computersimulations. In this section, we will summarize the stylized facts of the spatiotemporalevolution of congested traffic patterns, i.e., typical empirical findings thatare persistently observed on various freeways all over the world. In order toprovide a comprehensive list as a testbed for traffic models and theories, wewill summarize below all relevant findings, including already published ones:(1)
Congestion patterns on real (non-circular) freeways are typically causedby bottlenecks in combination with a perturbation in the traffic flow.
An extensive study of the breakdown phenomena on the German free-ways A5-North and A5-South by Sch¨onhof and Helbing (2007), analyzingabout 400 congestion patterns, did not find examples where there was an5 ig. 2. Spatiotemporal velocity profiles for the German freeway A5 North nearFrankfurt/Main (both directions). Arrows indicate the direction of travel. apparent lack of a bottleneck. This is in agreement with former investi-gations of the Dutch freeway A9, the German freeway A8-East and West,and the German freeway A9-South (Treiber et al., 2000a). Nevertheless,it may appear to drivers entering a traffic jams on a homogeneous freewaysection that they are experiencing a ”phantom traffic jam”, i.e. a trafficjam without any apparent reason. In these cases, however, the trigger-ing bottleneck, which is actually the reason for the traffic jam, is locateddownstream, potentially in a large distance from the driver location (seeFig. 13 of Sch¨onhof and Helbing (2007) or in Fig. 1(a) of Helbing et al.(2009)).(2)
The bottleneck may be caused by various reasons such as isolated on-ramps or off-ramps, combinations thereof such as junctions or intersec-tions (Fig. 1(a) and 2), local narrowings or reductions of the number oflanes, accidents, or gradients. As an example, Fig. 1(b) shows a compos-ite congestion pattern on the German freeway A8-East caused by uphilland downhill gradients (“Irschenberg”) in the region 38 km ≤ x ≤
41 km,and an additional obstruction by an accident at x = 43 . The congestion pattern is either localized with a constant width of the or-der of 1 km, or it is spatially extended with a time-dependent extension.
Localized congestion patterns either remain stationary at the bottleneck,or they move upstream at a characteristic speed c cong . Typical values of c cong are between −
20 km / h and −
15 km / h, depending on the country andtraffic composition (Zielke et al., 2008), but not on the type of conges-tion. About 200 out of 400 breakdowns observed by Sch¨onhof and Helbing(2007) correspond to extended patterns.(4) The downstream front of congested traffic is either fixed at the bottleneck,or it moves upstream with the characteristic speed c cong (Helbing et al.,2009). Both, fixed and moving downstream fronts can occur within oneand the same congestion pattern. This can be seen in Fig. 1(a), where the6tationary downstream congestion front at x = 476 km (the location ofthe temporary bottleneck caused by an incident) starts moving upstreamat 17:30 h. Such a “detachment” of the downstream congestion front oc-curs, for example, when an accident site has been cleared, and it is oneof two ways in which the dissolution of traffic congestion starts (see nextitem for the second one).(5) The upstream front of spatially extended congestion patterns has no char-acteristic speed.
Depending on the traffic demand and the bottleneckcapacity, it can propagate upstream (if the demand exceeds the capac-ity) or downstream (if the demand is below capacity) (Helbing et al.,2009). This can be seen in all extended congestion patterns of Fig. 1(see also Sch¨onhof and Helbing (2009); Kerner (2004)). The downstreammovement of the congestion front towards the bottleneck is the secondand most frequent way in which congestion patterns may dissolve.(6)
Most extended traffic patterns show some “internal structure” propagat-ing upstream approximately at the same characteristic speed c cong . Conse-quently, all spatiotemporal structures in Figs. 1 and 2 (sometimes termed“oscillations”, “stop-and-go traffic”, or “small jams”), move in paral-lel (Smilowitz et al., 1999; Mauch and Cassidy, 2002; Zielke et al., 2008).(7)
The periods and wavelengths of internal structures in congested trafficstates tend to decrease as the severity of congestion increases.
This ap-plies in particular to measurements of the average velocity. (See, for ex-ample, Fig. 1(a), where the greater of two bottlenecks, located at theIntersection M¨unchen-Nord, produces oscillations of a higher frequency.Typical periods of the internal quasi-periodic oscillations vary betweenabout 4 min and 60 min, corresponding to wavelengths between 1 km and15 km (Helbing and Treiber, 2002).(8)
For bottlenecks of moderate strength, the amplitude of the internal struc-tures tends to increase while propagating upstream . This can be seenin all empirical traffic states shown in this contribution, and alsoin Sch¨onhof and Helbing (2009); Helbing et al. (2009). It can also be seenin the corresponding velocity time series, such as the ones in Fig. 12of Treiber et al. (2000a), in Zielke et al. (2008), or in all relevant timeseries shown in Chapters 9-13 of Kerner (2004). The oscillations mayalready be visible at the downstream boundary (Fig. 1(b)), or emergefurther upstream (Figs. 1(a), 2(a)). During their growth, neighboring per-turbations may merge (Fig. 1 in Sch¨onhof and Helbing (2009)), or prop-agate unaffected (Fig. 1). At the upstream end of the congested area,the oscillations may eventually become isolated “wide jams” (Fig. 2) orremain part of a compact congestion pattern (Fig. 1).(9)
Light or very strong bottlenecks may cause extended traffic patterns, whichappear homogeneous (uniform in space), see, for example, Figs. 1(d) and1(f) of Helbing et al. (2009). Note however that, for strong bottlenecks(typically caused by accidents), the empirical evidence has been contro-versially debated, in particular as the oscillation periods at high densities7each the same order of magnitude as the smoothing time window thathas typically been used in previous studies (cf. point 7 above). This makesoscillations hardly distinguishable from noise. See Appendix B for a fur-ther discussion of this issue.Note that the above stylized facts have not only be observed in Germany,but also in other countries, e.g. the USA, Great Britain, and the Netherlands(Zielke et al., 2008; Helbing et al., 2009; Wilson, 2008a; Treiber et al., 2010).Furthermore, we find that many congestion patterns are composed of severalof the elementary patterns listed above (Sch¨onhof and Helbing, 2007). Forexample, the congestion pattern observed in Fig. 2(b) can be decomposedinto moving and stationary localized patterns as well as extended patterns.The source of probably most controversies in traffic theory is an ob-served spatiotemporal structure called the “pinch effect” or “general pat-tern” (Kerner and Rehborn, 1996b), see Kerner (2004) for details and Fig. 1of Sch¨onhof and Helbing (2009) for a typical example of the spatiotemporalevolution. From the perspective of the above list, this pattern relates to styl-ized facts 6 and 8 , i.e., it has the following features: (i) relatively stationarycongested traffic ( pinch region ) near the downstream front, (ii) small pertur-bations that grow to oscillatory structures as they travel further upstream,(iii) some of these structures grow to form “wide jams”, thereby suppressingother small jams, which either merge or dissolve. The question is whetherthis congestion pattern is composed of several elementary congestion patternsor a separate, elementary pattern, which is sometimes called “general pat-tern” (Kerner, 2004). This will be addressed in Sec. 4.3. The concept of “phases” has originally been used in areas such as thermo-dynamics, physics, and chemistry. In these systems, “phases” mean differentaggregate states (such as solid, fluid, or gaseous; or different material composi-tions in metallurgy; or different collective states in solid state physics). Whencertain “control parameters” such as the pressure or temperature in the sys-tem are changed, the aggregate state may change as well, i.e. a qualitativelydifferent macroscopic organization of the system may result. If the transition Moreover, speed variations between ’stop and slow’ may result from problems inmaintaining low speeds (the gas and brake pedals are difficult to control in thisregime), and thus are different from the collective dynamics at higher speeds. Inany case, this is not a crucial point since there are models that can be calibrated togenerate homogeneous patterns for high bottleneck strengths (restabilization), ornot, see Eq. (1) in Sec. 4.1.2 below.
8s abrupt, one speaks of first-order (or “hysteretic”, history-dependent) phasetransitions. Otherwise, if the transition is continuous, one speaks of second-order phase transitions. In an abstract space, whose axes are defined by the control parameters, it isuseful to mark parameter combinations, for which a phase transition occurs,by lines or “critical points”. Such illustrations are called phase diagrams, asthey specify the conditions, under which certain phases occur.Most of the time, the terms “phase” and “phase diagram” are applied tolarge (quasi-infinite), spatially closed, and homogeneous systems in thermo-dynamic equilibrium, where the phase can be determined in any point of thesystem. When transferring these concepts to traffic flows, researchers have dis-tinguished between one-phase, two-phase, and three-phase models. The num-ber of phases is basically related to the (in) stability properties of the trafficflows (i.e. the number of states that the instability diagram distinguishes).The equilibrium state of one-phase models is a spatially homogeneous trafficstate (assuming a long circular road without any bottleneck). An examplewould be the Burgers equation (Whitham, 1974), i.e. a Lighthill–Whitham–Richard model (Lighthill and Whitham, 1955; Richards, 1956) with diffusionterm. Two-phase models would additionally produce oscillatory traffic statessuch as wide moving jams or stop-and-go waves, i.e. they require some in-stability mechanism (Wagner and Nagel, 2008).Three-phase models introduceanother traffic state, so-called “synchronized flow”, which is characterized bya self-generated scattering of the traffic variables. It is not clear, however,whether this state exists in reality in the absence of spatial inhomogeneities(freeway bottlenecks). Note, however, that the concept of phase transitions has also been transferredto non-equilibrium systems, i.e. driven, open systems with a permanent inflowor outflow of energy, inhomogeneities, etc. This use is common in systems the-ory. For example, one has introduced the concept of boundary-induced phasetransitions (Krug, 1991; Popkov et al., 2001; Appert and Santen, 2001). Fromthis perspective, the Burgers equation can show a boundary-induced phasetransition from a free-flow state with forwardly propagating congestion frontsto a congested state with upstream moving perturbations of the traffic flow.This implies that the Burgers equation (with one equilibrium phase) has twonon- equilibrium phases. Analogously, two-phase models (in the previouslydiscussed, thermodynamic sense) can have more than two non- equilibriumphases. However, to avoid confusion, one often uses the terms “(spatiotem- In order to measure whether a phase transition is continous or not, a suitable“order parameter” needs to be defined and measured. In fact, it even remains to be shown whether Kerner’s “three-phase” car-followingmodels (Kerner and Klenov, 2002, 2006) or other three-phase models really havethree phases in the thermodynamic sense pursued by Wagner and Nagel (2008). several congested traffic statesbesides free traffic flow (Treiber et al., 2000a). The phase diagram approachto traffic modeling proposed by Helbing et al. (1999) was originally presentedfor an open traffic system with an on-ramp. It shows the qualitatively differ-ent, spatiotemporal traffic patterns as a function of the freeway flow and thebottleneck strength.Note, however, that the resulting traffic state may depend on the history (e.g.the size of perturbations in the traffic flow), if traffic flows have the propertyof metastability.The concept of the phase diagram has been taken up by many authors and ap-plied to the spatiotemporal traffic patterns (non-equilibrium phases) producedin many models (Lee et al., 1998, 1999; Kerner, 2004; Siebel and Mauser,2006). Besides on-ramp scenarios, one may study scenarios with flow-conserving bottlenecks (such as lane closures or gradients) or with combina-tions of several bottlenecks. It appears, however, that the traffic patterns forfreeway designs with several bottlenecks can be understood, based on the com-bination of elementary traffic patterns appearing in a system with a single bot-tleneck and interaction effects between these patterns (Sch¨onhof and Helbing,2007; Helbing et al., 2009)The resulting traffic patterns as a function of the flow conditions and bottle-neck strengths (freeway design), and therefore the appearance of the phasediagram, depend on the traffic model and the parameters chosen. Therefore,the phase diagram approach can be used to classify the large number of trafficmodels into a few classes. Models with qualitatively similar phase diagramswould be considered equivalent, while models producing different kinds oftraffic states would belong to different classes. The grand challenge of traffictheory is therefore to find a model and/or model parameters, for which thecongestion patterns match the stylized facts (see Sec. 2.2) and for which thephase diagram agrees with the empirical one (Sch¨onhof and Helbing, 2007;Helbing et al., 2009). This issue will be addressed in Sec. 4For the understanding of traffic dynamics one may ask which of the twocompeting phase definitions (the thermodynamic or the non-equilibrium one)would be more relevant for observable phenomena. Considering the stylizedfacts (see Sec. 2), it is obvious that boundary conditions and inhomogeneitiesplay an important role for the resulting traffic patterns. This clearly favoursthe dynamic-phase concept over the definition of thermodynamic equilibriumphases: Traffic patterns are easily observable and also relevant for applications.(For calculating traveling times, one needs the spatiotemporal dynamics of the10raffic pattern, and not the thermodynamic traffic phase.) Moreover, thermo-dynamic phases are not observable in the strict sense, because real trafficsystems are not quasi-infinite, homogeneous, closed systems. Consequently,when assessing the quality of a given model, it is of little relevance whether ithas two or three physical phases, as long as it correctly predicts the observedspatiotemporal patterns, including the correct conditions for their occurrence.Nevertheless, the thermodynamic phase concept (the instability diagram) isrelevant for explaining the mechanisms leading to the different patterns. Infact, for models with a fundamental diagram, it is possible to derive the phasediagram of traffic states from the instability diagram, if bottleneck effects andthe outflow from congested traffic are additionally considered (Helbing et al.,1999).
In the following, we will show for specific traffic models that not only three-phase traffic theory, but also the conceptionally simpler two-phase models (asintroduced in Sec. 3) can display all stylized facts mentioned in Sec. 2, if themodel parameters are suitably chosen. This is also true for patterns that wereattributed exclusively to three-phase traffic theory such as the pinch effect orthe widening synchronized pattern (WSP).Considering the dynamic-phase definition of Sec. 3, the simplest system thatallows to reproduce realistic congestion patterns is an open system with abottleneck. When simulating an on-ramp bottleneck, the possible flow con-ditions can be characterized by the upstream freeway flow (“main inflow”)and the ramp flow, considering the number of lanes (Helbing et al., 1999).The downstream traffic flow under free and congested conditions can be deter-mined from these quantities. When simulating a flow-conserving (ramp-less)bottleneck, the ramp flow is replaced by the bottleneck strength quantifyingthe degree of local capacity reduction (Treiber et al., 2000b).Since many models show hysteresis effects, i.e. discontinuous, history-dependent transitions, the time-dependent traffic conditions before the onsetof congestion are relevant as well. In the simplest case, the response of thesystem is tested (i) for minimum perturbations, e.g. slowly increasing inflowsand ramp flows, and (ii) for a large perturbation. The second case is usuallystudied by generating a wide moving jam, which can be done by temporarilyblocking the outflow. Additionally, the model parameters characterizing thebottleneck situation have to be systematically varied and scanned through.This is, of course, a time-consuming task since producing a single point inthis multi-dimensional space requires a complete simulation run (or even toaverage over several simulation runs).11 .1 Two-phase models
Wagner and Nagel (2008) classify models with a fundamental diagram thatshow dynamic traffic instabilities in a certain density range, as two-phase mod-els. Alternatively, these models are referred to as “models within the funda-mental diagram approach”. Note, however, that certain models with a uniquefundamental diagram are one -phase models (such as the Burgers equation).Moreover, some models such as the KK model can show one-phase, two-phaseor three-phase behavior, depending on the choice of model parameters (seeSec. 4.2).A microscopic two-phase model necessarily has a dynamic acceleration equa-tion or contains time delays such as a reaction time. For macroscopic models,a necessary (but not sufficient) condition for two phases is that the modelcontains a dynamical equation for the macroscopic velocity.
We start with results for the gas-kinetic-based traffic model (Helbing, 1996;Treiber et al., 1999). Like other macroscopic traffic models, the GKT modeldescribes the dynamics of aggregate quantities, but besides the vehicle density ρ ( x, t ) and average velocity V ( x, t ), it also considers the velocity variance θ ( x, t ) = A ( ρ ( x, t )) V ( x, t ) as a function of velocity and density.The GKT model has five parameters v , T , τ , γ , and ρ max characterizingthe driver-vehicle units, see Table 1. In contrast to other popular second-order models (Payne, 1971; Kerner and Konh¨auser, 1993; Lee et al., 1999;Hoogendoorn and Bovy, 2000), the GKT model distinguishes between the de-sired time gap T when following other vehicles, and the much larger accelera-tion time τ to reach a certain desired velocity. Furthermore, the drivers of theGKT model “look ahead” by a certain multiple γ of the distance to the nextvehicle. The GKT model also contains a variance function A ( ρ ) reflecting sta-tistical properties of the traffic data. Its form can be empirically determined(see Table 1). For the GKT model equations, we refer to Treiber et al. (1999).We have simulated an open system with an on-ramp as a function of the mainflow and the ramp flow, using the two parameter sets listed in Table 1. In con-trast to the simulations in Helbing et al. (1999), we added variations of theon-ramp flow with an amplitude of 20 vehicles/h and a mean value of zero tocompensate for the overly smooth merging dynamics in macroscopic models,when mergings are just modeled by constant (or slowly varying) source termsin the continuity equation. For parameter set 1, we obtain the results of Fig. 3,i.e., the phase diagram found by Helbing et al. (1999) and by Lee et al. (1998).It contains five congested traffic patterns, namely pinned localized clusters able 1The two parameter sets for the GKT model (Treiber et al., 1999) used in this paper.The four last parameters specify the velocity variance prefactor A ( ρ ) = A +( A max − A ) / { tanh[( ρ − ρ c ) / ∆ ρ ] + 1 } Model parameter Value set 1 Value set 2Desired velocity v
120 km/h 120 km/hDesired time gap T τ
20 s 35 sAnticipation factor γ ρ max A for free traffic 0.008 0.01Variance prefactor A max for congested traffic 0.038 0.03Transition density free-congested ρ c . ρ max . ρ max Transition width ∆ ρ . ρ max . ρ max (PLC), moving localized clusters (MLC), triggered stop-and-go waves (TSG),oscillating congested traffic (OCT), and homogeneous congested traffic (HCT) .The OCT and TSG patterns look somewhat similar, and there is no discontin-uous transition between these patterns. This has been indicated by a dashedinstead of a solid line in the phase diagram. Furthermore, notice that the twolocalized patterns MLC and PLC are only obtained, when sufficiently strongtemporary perturbations occur in addition to the stationary on-ramp bottle-neck. Such perturbations may, for example, result from a temporary peak inthe ramp flow or in the main inflow (which may be caused by forming vehicleplatoons, when slower trucks overtake each other, see Sch¨onhof and Helbing(2007)). Furthermore, the perturbation may be an upstream moving trafficjam entering a bottleneck area (see Fig. 2(b) and Helbing et al. (1999)). Thiscase has been assumed here.When simulating the same system, but this time using parameter set 2 ofTable 1, we obtain the PLC, MLC, OCT and HCT states as in the first simu-lation, see Fig. 4 (the MLC pattern is not shown). However, instead of the TSGstate, we find two new patterns. For very light bottlenecks (small ramp flows),we observe a light form of homogeneous congested traffic that has the proper-ties of the widening synchronized pattern (WSP) proposed by Kerner (2004).Remarkably, this state is stable or metastable, otherwise moving jams shouldemerge from it in the presence of small-amplitude variations of the ramp flow.Although the WSP-properties of being extended and homogeneous in spaceare the same as for the HCT state, WSP occurs for light bottlenecks, whileHCT requires strong bottlenecks. Moreover, the two patterns are separatedin the phase diagram by oscillatory states that occur for moderate bottleneck13 ig. 3. Congested traffic patterns as a function of the dynamic phase space spannedby the main inflow and the ramp inflow for the GKT model with parameter set1 in Table 1. The dotted line indicates the maximum traffic demand for whichfree flow can be sustained. Below this line, congestion patterns can only be trig-gered by perturbations. For this purpose, a moving jam has been generated at thedownstream boundary in the two plots on the left-hand side. The abbreviationsdenote free traffic (FT), pinned localized cluster (PLC), moving localized cluster(MLC), homogeneous congested traffic (HCT), oscillatory congested traffic (OCT),and triggered stop-and-go (TSG) pattern.Fig. 4. Dynamic phase diagram of congested traffic patterns for the GKT modelas in Fig. 3, but this time using parameter set 2 from Table 1. While the TSGpattern is missing, the two additional patterns “pinch region” and WSP (wideningsynchronized pattern) are produced (see the main text for details). strengths. 14 Q ( v eh i c l e s / h ) ρ (vehicles/km) ρ ρ ρ ρ ρ cv Q ( v eh i c l e s / h ) ρ (vehicles/km) ρ ρ ρ ρ stablemetastableconv. unstablestable ( s t r i ng ) un s t ab l e set 1parameter (a) GKT model stable m e t a s t ab l e c on v . un s t ab l e (b) GKT model parameterset 2 stable Fig. 5. Fundamental diagrams and stability regimes of the GKT modelfor the two parameter sets in Table 1. The so-called critical densities ρ , ρ , ρ , and ρ correspond to the densities at which the transitionsstable ↔ metastable ↔ unstable ↔ metastable ↔ stable occur. For ρ > ρ cv , the instabil-ities are of a convective type. For the existence of a widening synchronized pattern(WSP), the critical density ρ must be on the “congested” side of the fundamentaldiagram. The second new traffic pattern is a congested state which consists of a sta-tionary downstream front at the on-ramp bottleneck, homogeneous, light con-gested traffic near the ramp, and velocity oscillations (“small jams” or OCT)further upstream. These are the signatures of the pinch effect. Similarly tothe transition TSG ↔ OCT in the dynamic phase diagram of Fig. 3, there isno sharp transition between light congested traffic and OCT.The corresponding stability diagrams shown in Fig. 5 for the two parametersets are consistent with these findings: In contrast to parameter set 1, param-eter set 2 leads to a small density range of metastable (rather than unstable)congested traffic near the maximum flow, which is necessary for the occurenceof the WSP. Furthermore, parameter set 2 leads to a wide density range ofconvectively unstable traffic, which favours the pinch effect as will be discussedin Sec. 4.3.2.Finally, we note that the transition from free traffic to extended congestedtraffic is of first order. The associated hysteresis (capacity drop) is reflectedin the phase diagram of Fig. 4 by the vertical distance between the dottedline and the line separating the PLC pattern from spatially extended trafficpatterns, and also by the large metastable density regime in the stability dia-gram (see Fig. 5). Note, that the optimal velocity model, in contrast, behavesnonhysteretic (Kerner and Klenov, 2006, Sec. 6.2), which is not true for themicroscopic models discussed in the next subsection.
In order to investigate the generality of the above results, we have simulatedthe same traffic system also with the intelligent driver model (IDM) as one15 ig. 6. Dynamic phase diagram of on-ramp-induced congested traffic patterns for theIntelligent Driver Model with the parameters given in Sec. 4.1.2. As in the previousdiagrams, the dashed line indicates the maximum traffic volume for which free flowcan be sustained. In order to trigger the congestion patterns at the bottleneck, amoving jam is introduced at the downstream boundary for the three simulationscorresponding to points below the dashed line (metastable regimes). representative of two-phase microscopic traffic models with continuous dy-namics (Treiber et al., 2000a).The IDM specifies the acceleration dv α /dt of vehicle α following a leader α − s α and the relative velocity∆ v α = v α − v α − ) as a continuous deterministic function with five modelparameters. The desired velocity v and the time gap T in equilibrium havethe same meaning as in the GKT model. The actual acceleration is limitedby the maximum acceleration a . The “intelligent” braking strategy generallylimits the decelerations, to the comfortable value b , but it allows for higherdecelerations if this is necessary to prevent critical situations or accidents. Fi-nally, the gap to the leading vehicle in standing traffic is represented by s .Notice that the sum of s and the (dynamically irrelevant) vehicle length l isequivalent to the inverse of the GKT parameter ρ max .It has been shown that the IDM is able to produce the five traffic patternsPLC, MLC, TSG, OCT, and HCT found in the GKT model with parameterset 1 (Treiber et al., 2000a). Here, we want to investigate whether the IDMcan also reproduce the “new” patterns shown in Fig. 4, i.e., the WSP andthe pinch effect. For this purpose, we slightly modify the simulation model ascompared to the assumptions made in previous publications: • Instead of a “flow-conserving bottleneck” we simulate an on-ramp. Sincethe focus is not on realistic lane-changing and merging models we simulate16 Q ( v eh i c l e s / h ) ρ (vehicles/km) ρ ρ cv = ρ ρ = ρ convectivelystring unstable stable m e t a s t ab l e IDM
Fig. 7. Fundamental diagram and stability regions of the IDM for the parameters v = 120 km / h, T = 1 s, s = 2 m, a = 1 . / s , and b = 1 . / s used in thispaper. In contrast to the original specifications by Treiber et al. (2000a), trafficflow at capacity is metastable rather than linearly unstable here, and the linearstring instability for higher densities is always of the convective type. See Fig. 5 forthe definition of the critical densities ρ i and ρ cv . here a main road consisting only of one lane and keep the merging rulesimple: As soon as an on-ramp vehicle reaches the merging zone of 600 mlength, it is centrally inserted into the largest gap within the on-ramp zonewith a velocity of 60% of the actual velocity of the leading vehicle on thedestination lane. • The IDM parameters have been changed such that traffic flow at max-imum capacity is meta stable rather than linearly un stable. This can bereached by increasing the maximum acceleration a . Specifically, we assume v = 120 km / h, T = 1 s, s = 2 m, a = 1 . / s , and b = 1 . / s . Fur-thermore, the vehicle length l is set to 6 m. Note that there is actuallyempirical evidence that flows are metastable at densities corresponding tocapacity (Helbing and Tilch, 2009; Helbing et al., 2009): A growing vehicleplatoon behind overtaking trucks is stable, as long as there are no significantperturbations in the traffic flow. However, weaving flows close to ramps canproduce perturbations that are large enough to cause a traffic breakdown,when the platoon reaches the neighborhood of the ramp.Figure 6 shows that, with one exception, the congestion patterns obtainedfor the IDM model with (meta-)stable maximum flow are qualitatively thesame as for the GKT model with parameter set 2 (see Fig. 4). As for theGKT model, all transitions from free to congested states are hysteretic, i.e.,the corresponding regions in the phase diagram extend below the dashed line,where free traffic can be sustained as well. In this case, free traffic downstreamof the on-ramp is at or below (static) capacity, and therefore metastable (cf.Fig. 7). Consequently, a sufficiently strong perturbation is necessary to triggerthe WSP, PLC, or OCT states. Specifically, for the WSP, PLC, and Pinch-OCT simulations of Fig. 6, the perturbations associated with the mergings at This value of l is reasonable for mixed traffic containing a considerable truckfraction. ρ and ρ do not exist. This finding, however, dependson the parameters. It can be analytically shown (Helbing et al., 2009) that aHCT state exists, if s < aT . (1)This means, when varying the minimum distance s and leaving all other IDMparameters constant (at the values given above), a phase diagram of the typeshown in Fig. 4 (containing oscillatory and homogeneous congested trafficpatterns) exists for s ≤ . s > . a and leaving all other IDM parameters at thevalues given above, the IDM phase diagram is of the type displayed in Fig. 6,if 0 .
93 m / s ≤ a < . / s , but of the type shown in the original phase dia-gram by Treiber et al. (2000a) (without a WSP state), if a < .
93 m / s , andof the type belonging to a single-phase model (with homogeneous traffic statesonly), if a > . / s .We obtain the surprising result that, in contrast to the IDM parameters cho-sen by Treiber et al. (2000a), homogeneous congested traffic of the WSP typecan be observed even for very small bottleneck strengths, while the pinch ef-fect is observed for intermediate bottleneck strengths and a sufficiently highinflow on the freeway into the bottleneck area. Furthermore, no restabiliza-tion takes place for strong bottlenecks, in agreement with what is demandedby Kerner (2008). Notice that the empirically observed oscillations are notperfectly periodic as in Fig. 6, but quasi-periodic with a continuum of ele-mentary frequencies concentrated around a typical frequency (correspondingto a period of about 3.5 min in the latter reference). In computer simulations,such a quasi-periodicity is obtained for heterogeneous driver-vehicle units withvarying time gaps.Clearly, the merging rule generates considerable noise at the on-ramp. It istherefore instructive to compare the on-ramp scenario with a scenario as-suming a flow-conserving bottleneck, but the same model and the same pa-rameters. Therefore, it is instructive to simulate a flow-conserving bottleneckrather than an on-ramp bottleneck. Formally, we have implemented the flow-conserving bottleneck by gradually increasing the time gap T from 1.0 s toa higher value T bottl within a 600 m long region as in Treiber et al. (2000a),keeping T = T bottl further downstream. The value of T bottl determines the ef-18 ig. 8. Congestion patterns caused (a) by an on-ramp bottleneck, and (b) by acomparable flow-conserving bottleneck (resulting in the same average traffic flowin the congested region). The simulations were performed with the IDM, using thesame parameters as specified in the main text before. fectively resulting bottleneck strength . We measure the bottleneck strengthas the difference of the outflow from wide moving jams sufficiently awayfrom the bottleneck and the average flow in the congested area upstreamof it (Treiber et al., 2000b).Performing exactly the same simulations as in Fig. 6, but replacing the on-ramp bottleneck by a flow-conserving bottleneck, we find essentially no differ-ence for most combinations of the main inflow and the bottleneck strength.However, a considerable fraction of the parameter space leading to a pincheffect in the on-ramp system results in a WSP state in the case of the flow-conserving bottleneck. Figure 8 shows the direct comparison for a main inflowof Q in = 2000 vehicles / h, and a ramp flow of Q rmp = 250 vehicles / h, corre-sponding to T bottl = 1 .
37 s in the flow-conserving system. It is obvious thatnonstationary perturbations are necessary to trigger the pinch effect, whichagrees with the findings for the GKT model.Complementary, we have also investigated other car-following models such asthe model of Gipps (1981), the optimal velocity model (OVM) of Bando et al.(1995), and the velocity difference model (VDM) investigated by Jiang et al.(2001). We have found that the Gipps model always produces phase diagramsof the type shown in Figs. 4 and 6 (see Fig. 11 below for a plot of the pincheffect). With the other two models, it is possible to simulate both types ofdiagrams, when the model parameters are suitably chosen.To summarize our simulation results, we have found that the pinch effectcan be produced with two-phase models with particular parameter choices.Furthermore, nonstationary perturbations clearly favour the emergence of thepinch effect. In practise, they can originate from lane-changing maneuversclose to on-ramps, thereby favouring the pinch effect at on-ramp bottlenecks,while it is less likely to occur at flow-conserving bottlenecks. Additionally,nonstationary perturbations can result from noise terms which are an integralpart of essentially all three-phase models proposed to date.19 ig. 9. Traffic patterns produced by the KK model in the open system with anon-ramp (merging length 600 m). (a) Inflow Q in = 2100 vehicles / h and ramp flow Q rmp = 150 vehicles / h; (b) Q in = 2050 vehicles / h, Q rmp = 550 vehicles / h (c) Q in = 2250 vehicles / h, Q rmp = 320 vehicles / h, and (d) Q in = 1350 vehicles / h, Q rmp = 750 vehicles / h.Fig. 10. (a) “Moving synchronized pattern” (for Q in = 2120 vehicles / hand Q rmp = 200 vehicles / h), and (b) “dissolving general pattern” (for Q in = 2150 vehicles / h and Q rmp = 250 vehicles / h), simulated with the KKmicro-model. The plot (c) shows the pinch effect for Q in = 1950 vehicles / h, Q rmp = 500 vehicles / h, and the synchronization distance parameter k = 1, for whichthe KK micro-model is reduced to a model with a unique fundamental diagram. .2 Three-phase models To facilitate a direct comparison of two- and three-phase models, we have simu-lated the same traffic system with two models implementing three-phase traffictheory, namely the cellular automaton of Kerner (2004) and the continuous-inspace model proposed by Kerner and Klenov (2002). In the following, we willfocus on the continuous model and refer to it as
KK micro-model . It is for-mulated in terms of a coupled iterated map, i.e., the locations and velocitiesof the vehicles are continuous, but the updates of the locations and velocitiesoccur in discrete time steps.To calculate one longitudinal velocity update, 19 update rules have to be ap-plied (see Kerner (2004), Eqs. (16.41), (16.44)-(16.48), and the 13 equationsof Table 16.5 therein). Besides the vehicle length, the KK micro-model has11 parameters and two functions containing five more constants: The desiredvelocity v free , the time τ which represents both, the update time step andthe minimum time gap, the maximum acceleration a , the deceleration b fordetermining the “safe” velocity, the synchronization range parameter k in-dicating the ratio between maximum and minimum synchronized flow understationary conditions at a certain density, the dimensionless sensitivity φ withrespect to velocity differences, a threshold acceleration δ that defines, whetherthe vehicle is in the state of “nearly constant speed”, and three probabilities p , p a , and p b defining acceleration noise and a slow-to-start rule. Addition-ally, the stochastic part of the model contains the two probability functions p ( v ) = 0 .
575 + 0 .
125 min(1 , . v ) (with v in units of m/s), and p ( v ) = 0 . v ≤
15 m / s, otherwise p ( v ) = 0 .
8. The KK micro-model includes furtherrules for lane changes and merges.We have implemented the longitudinal update rules according to the formula-tion in Kerner (2004), Section 16.3, and used the parameters from this refer-ence as well. Since we are interested in the longitudinal dynamics, we will usethe simpler merging rule applied already to the IDM in Sec. 4.1.2 of this paper.To test the implementation, we have simulated the open on-ramp system witha merging length of 600 m, as in the other simulations. This essentially pro-duced the phase diagram and traffic patterns depicted in Fig. 18.1 of Kerner(2004). (Due to the simplified merging rule assumed here, the agreement isgood, but not exact.)Figure 9 shows the patterns which are crucial to compare the KK micro-model with the two-phase models of the previous section. We observe thatthe WSP pattern (diagram (a)), the pinch effect (diagram (b)), and the OCT(diagram (d)) are essentially equivalent with those of the IDM (Fig. 6) orthe GKT model for parameter set 2 (see Fig. 4), but with the exceptionof the missing HCT states. Furthermore, the pattern shown in diagram (c)21esembles the triggered stop-and-go traffic (TSG) displayed in Fig. 3(b). Somedifferences, however, remain: • The oscillation frequencies of oscillatory patterns of the KK micro-modelare smaller than those of the IDM, and often closer to reality. How-ever, generalizing the IDM by considering reactions to next-nearest neigh-bors (Treiber et al., 2006b) increases the frequencies occurring in the IDMto realistic values. Note that the dynamics in the KK micro-model dependson next-nearest vehicles as well, so this may be an important aspect formicroscopic traffic models to be realistic. • The “moving synchronized patterns” in Fig. 10(a) (see also Fig. 18.1(d)in Kerner (2004)) differ from all other patterns in that their downstreamfronts (where vehicles leave the jams) and the internal structures within thecongested state propagate upstream at different velocities. Within the KKmicro-model, the propagation velocity of structures in congested traffic mayeven exceed 40 km/h (see, for example, Fig. 18.27 in (Kerner, 2004)), whilethere is no empirical evidence of this. Observations rather suggest that thedownstream front of congestion patterns is either stationary or propagatesat a characteristic speed (see stylized fact 4 in Sec. 2.2). • Another pattern which is sometimes produced by three-phase models isthe “dissolving general pattern” (DGP), where an emerging wide movingjam leads to the dissolution of synchronized traffic (Fig. 10(b), see alsoFig. 18.1(g) in Kerner (2004)). So far, we have not found any evidencefor such a pattern in our extensive empirical data sets. Congested trafficnormally dissolves in different ways (see stylized facts 4 and 5 ).Finally, we observe that the time gap T of the KK micro-model in stationarycar-following situations can adopt a range given by τ ≤ T ≤ kτ , where k isthe synchronization distance factor. By setting k = 1, the KK micro-modelbecomes a conventional two-phase model. When simulating the on-ramp sce-nario for the KK micro-model with k = 1, we essentially found the samepatterns (see Fig. 10(c) for an example). This suggests that there is actuallyno need of going beyond the simpler class of two-phase models with a uniquefundamental diagram. While the very first publications on the phase diagram of traffic states did notreport a pinch effect (or “general pattern”), the previous sections of this paperhave shown that this traffic pattern can be simulated by two-phase models, ifthe model parameters are suitably chosen. It also appears that nonstationaryperturbations at a bottleneck (which may, for example, result from frequentlane changes due to weaving flows) support the occurrence of a pinch effect.22his suggests to take a closer look at mechanisms, which produce this effect.We have identified three possible explanations, which are discussed in thefollowing. In reality, one may also have a combination of these mechanisms.
This mechanism is the one proposed by three-phase traffic theory. The startingpoint is a region with metastable congested (but flowing) traffic behind abottleneck, while sufficiently large perturbations trigger small oscillations inthe density or velocity that grow while propagating upstream. When theybecome fully developed jams, the outflow from the oscillations decreases, whichis modeled by some sort of slow-to-start rule : Once stopped or forced to driveat very low velocity, drivers accelerate more slowly, or keep a longer time gapthan they would do when driving at a higher velocity. In the KK micro-model,this effect is implemented by using velocity-dependent stochastic decelerationprobabilities p ( v ) and p ( v ). Other implementations of this effect are possibleas well, such as the memory effect (Treiber and Helbing, 2003), or a drivingstyle that depends on the local velocity variance (Treiber et al., 2006b). Eventhe parameters s and s of the IDM can be used to reflect this effect.In any case, as soon as the outflow from large jams becomes smaller than thatfrom small jams, most of the latter will eventually dissolve, resulting in onlya few “wide moving jams”. We call this the “depletion effect” . A typical feature of the pinch effect are small perturbations that grow to fullydeveloped moving jams. Therefore, it is expected that (linear or nonlinear)instabilities of the traffic flow play an essential role. However, another char-acteristic feature of the pinch effect is a stationary congested region near thebottleneck, called the pinch region (Kerner, 1998).The simultaneous observation of the stationary pinch region and growing per-turbations upstream of it can be naturally explained by observing that, in spa-tially extended open systems (such as traffic systems), there are two differenttypes of string instability (Huerre and Monkewitz, 1990; Kesting and Treiber,2008b). For the first type, an absolute instability , the perturbations will even-tually spread over the whole system. A pinch region, however, can only exist ifthe growing perturbations propagate away from the on-ramp (in the upstreamdirection), while they do not “infect” the bottleneck region itself. This corre-sponds to the second type of string instability called “convective instability” .Figure 11 illustrates convectively unstable traffic by a simulation of the bot-tleneck system with the model of Gipps (1981): Small perturbations caused23 ig. 11. Simulation of the on-ramp system with the Gipps model showing the pincheffect. The parameters v , a , b of this model have the same meaning as for the IDMand have been set to the same values (see main text). The update time ∆ t (playingalso the role of the time gap) has been set to ∆ t = 1 . by the merging maneuvers near the on-ramp at x = 10 km grow only in theupstream direction and eventually transform to wide jams a few kilometersupstream. The IDM simulations of Fig. 6 show this mechanism as well.The concept of convective instability, which has been introduced into the con-text of traffic modeling already some years ago (Helbing et al., 1999), is inagreement empirical evidence. It has been observed that, in extended con-gested traffic, small perturbations or oscillations may grow while propagat-ing upstream, whereas congested traffic is relatively stationary in the vicinityof the bottleneck (Kerner, 2004; Mauch and Cassidy, 2002; Smilowitz et al.,1999; Zielke et al., 2008). However, the congestion pattern emanating fromthe “pinch region” is not necessarily a fully developed “general pattern” inthe sense that it includes a pinch region, small jams, and a transition to widejams (Kerner, 2004). In fact, the pinch region is also observed as part of con-gestion patterns that include neither wide jams nor a significant number ofmerging events, see Fig. 1(a) for an example. This can be understood by as-suming that the mechanisms leading to the pinch region and to wide jamsare essentially independent from each other. One could therefore explain thepinch region by the convective instability, and the transition from small towide jams by the depletion effect (see Sec. 4.3.1). A third mechanism leading to similar results as the previous mechanismscomes into play at intersections and junctions, where off-ramps are locatedupstream of on-ramps (which corresponds to the usual freeway design). Fig-ure 12(a) illustrates this mechanism for the GKT model and the parameterset 1 in Table 1. The existence of a stationary and essentially homogeneouspinch region and a stop-and-go pattern further upstream can be explained, as-suming that the inflow Q rmp , on from the on-ramp (located downstream) mustbe sufficiently large such that a HCT or OCT state would be produced when24 ig. 12. Congested traffic at a combination of an off-ramp with an on-ramp, sim-ulated with the GKT model. The locally increased stability between the rampssupports the pinch and depletion effects, leading to a composite pattern consistingof a pinch region, narrow jams, and wide moving jams (see main text for details). simulating this on-ramp alone. Furthermore, the outflow Q rmp , off from the off-ramp must be such that an effective on-ramp of inflow Q rmp , eff = Q rmp , on − Q rmp , off (2)would produce a TSG state or an OCT state with a larger wavelength.Figure 12(b) shows a simulation of an off-ramp-on-ramp scenario with theGKT model and parameter set 2 in Table 1. Notice that, for the parame-ters chosen, a pinch effect is not possible at an isolated Table 2 gives an overview of mechanisms producing the observed spatiotem-poral phenomena listed in Sec. 2.2. So far, these have been either consideredincompatible with three-phase models or with two-phase models having a fun-25 able 2Overview of the main controversial traffic phenomena and their possible explana-tions. The term “three-phase model” has been used for models that are consistentwith Kerner’s theory, while two-phase models are conventional models that candisplay traffic instabilities such as second-order macroscopic models and most car-following models (see Sec. 4 for details).Phenomenon PossibleMechanism Examples and ModelsPinch region at abottleneck;small jams furtherupstream 1. Convective in-stability or meta-stability (i) Three-phase models(ii) Two-phase models with appropri-ate parameters2. Local changeof stabilityand capacity Off-ramp-on-ramp combinationsTransition fromsmall to wide jams 1. Depletionmechanism Slow-to-start rule and other forms ofintra-driver variability2. Mergingmechanism Different group velocities of the smallwavesHomogeneouscongested traffic atlow densities Maximum flow ismetastable or sta-ble Two- and three-phase models withsuitable parametersHomogeneouscongested traffic athigh densities Restabilization Severe bottleneck simulated with atwo-phase model with appropriate pa-rameters damental diagram. It is remarkable that the main controversial observation— the occurrence of the pinch effect or general pattern — is not only com-patible with three-phase models, but can also be produced with conventionaltwo-phase models. For both model classes, this can be demonstrated withmacroscopic, microscopic, and cellular automata models, if models and pa-rameters are suitably chosen.
It appears that some of the current controversy in the area of traffic model-ing arises from the different definitions of what constitutes a traffic phase. Inthe context of three-phase traffic theory, the definition of a phase is orientedat equilibrium physics, and in principle, it should be able to determine thephase based on local criteria and measurements at a single detector. Withinthree-phase traffic theory, however, this goal is not completely reached: Inorder to distinguish between “moving synchronized patterns” and wide mov-ing jams, which look alike, one needs the additional nonlocal criterium of26hether the congestion pattern propagates through the next bottleneck areaor not (Sch¨onhof and Helbing, 2007, 2009). In contrast, the alternative phasediagram approach is oriented at systems theory, where one tries to distinguishdifferent kinds of elementary congestion patterns, which may be consideredas non-equilibrium phases occurring in non-homogeneous systems (containingbottlenecks). These traffic patterns are distinguished into localized or spa-tially extended, moving or stationary (“pinned”), and spatially homogeneousor oscillatory patterns. These patterns can be derived from the stability prop-erties of conventional traffic models exhibiting a unique fundamental diagramand unstable and/or metastable flows under certain conditions. Models of thisclass, sometimes also called two-phase models, include macroscopic and car-following models as well as cellular automata.As key result of our paper we have found that features, which are claimed tobe consistent with three-phase traffic theory only, can also be explained andsimulated with conventional models, if the model parameters are suitably spec-ified. In particular, if the parameters are chosen such that traffic at maximumflow is (meta-)stable and the density range for unstable traffic lies completelyon the “congested” side of the fundamental diagram, we find the “wideningsynchronized pattern” (WSP), which has not been discovered in two-phasemodels before. Furthermore, the models can be tuned such that no homoge-neous congested traffic (HCT) exists for strong bottlenecks. Conversely, wehave shown that almost the same kinds of patterns, which are produced bytwo-phase models, are also found for models developed to reproduce three-phase traffic theory (such as the KK micro-model). Moreover, when the KKmicro-model is simulated with parameters for which it turns into a model witha unique fundamental diagram, it still displays very similar results. Therefore,the difference between so-called two-phase and three-phase models does notseem to be as big as the current scientific controversy suggests.For many empirical observations, we have found several plausible explanations(compatible and incompatible ones), which makes it difficult to determine theunderlying mechanism which is actually at work. In our opinion, convectiveinstability is a likely reason for the occurence of the pinch effect (or the gen-eral pattern), but at intersections with large ramp flows, the effect of off- andon-ramp combinations seems to dominate. To explain the transition to widemoving jams, we favour the depletion effect, as the group velocities of struc-tures within congested traffic patterns are essentially constant. For the widescattering of flow-density data, all three mechanisms of Table 2 do probablyplay a role. Clearly, further observations and experiments are necessary toconfirm or reject these interpretations, and to exclude some of the alterna-tive explanations. It seems to be an interesting challenge for the future todevise and perform suitable experiments in order to finally decide betweenthe alternative explanation mechanisms.27n our opinion, the different congestion patterns produced by three-phasetraffic theory and the alternative phase diagram approach for models witha fundamental diagram share more commonalities than differences. Moreover,according to our judgement, three-phase models do not explain more observa-tions than the simpler two-phase models (apart maybe from the fluctuationsof “synchronized flow”, which can, for example, be explained by the hetero-geneity of driver-vehicle units). The question is, therefore, which approach issuperior over the other. To decide this, the quality of models should be judgedin a quantitative way, applying the following established standard procedure(Greene, 2008; Diebold, 2003): • As a first step, mathematical quality functions must be defined. Note thatthe proper selection of these functions (and the relative weight that is givento them) depends on the purpose of the model. • The crucial step is the statistical comparison of the competing models basedon a new, but representative set of traffic measurements, using model pa-rameters determined in a previous calibration step. Note that, due to theproblem of over-fitting (i.e. the risk of fitting of noise in the data), a highgoodness of fit in the calibration step does not necessarily imply a goodfit of the new data set, i.e. a high predictive power (Brockfeld et al., 2003,2004). • The goodness of fit should be judged with established statistical methods,for example with the adjusted R-value or similar concepts considering thenumber of model parameters (Greene, 2008; Diebold, 2003). Given the samecorrelation with the data, a model containing a few parameters has a higherexplanatory power than a model with many parameters.Given a comparable predictive power of two models, one should select the sim-pler one according to Einstein’s principle that a model should be as simple aspossible, but not simpler . If one has to choose between two equally performingmodels with the same number of parameters, one should use the one which iseasier to interpret, i.e. a model with meaningful and independently measurableparameters (rather than just fit parameters). Furthermore, the model shouldnot be sensitive to variations of the model parameters within the bounds oftheir confidence intervals. Applying this benchmarking process to traffic mod-eling will hopefully lead to an eventual convergence of explanatory conceptsin traffic theory. For example, travel times may be the most relevant quantity for traffic forecasts,and macroscopic models or extrapolation models may be good enough to providereasonably accurate results at low costs. However, if the impact of driver assistancesystems on traffic flows is to be assessed, it is important to accurately reproducethe time-dependent speeds, distances, and accelerations as well, which calls for mi-croscopic traffic models. cknowledgements The authors would like to thank the
Hessisches Landesamt f¨ur Straßen- undVerkehrswesen and the
Autobahndirektion S¨udbayern for providing the freewaydata shown in Figs. 1 and 2. They are furthermore grateful to Eddie Wilson forsharing the data set shown in Fig. A.1, and to Anders Johansson for generatingthe plots from his data.
A Wide scattering of congested flow–density data
The discussion around three-phase traffic theory is directly related with thewide scattering of flow-density data within synchronized traffic flows. How-ever, it deserves to be mentioned that the discussion around traffic the-ories has largely neglected the fact that empirical measurements of widemoving jams show a considerable amount of scattering as well (see, e.g.Fig. 15 of Treiber et al. (2000a)), while theoretically, one expects to finda “jam line” (Kerner, 2004). This suggests that wide scattering is actu-ally not a specific feature of synchronized flow, but of congested trafficin general. While this questions the basis of three-phase traffic theory toa certain extent, particularly as it is claimed that wide scattering is adistinguishing feature of synchronized flows as compared to wide movingjams, the related car-following models (Kerner and Klenov, 2002), cellular au-tomata (Kerner et al., 2002; Jiang and Wu, 2005), and macroscopic models(Jiang et al., 2007) build in dynamical mechanisms generating such scatter-ing as one of their key features (Siebel and Mauser, 2006). In other models,particularly those with a fundamental diagram, this scattering is a simpleadd-on (and partly a side effect of the measurement process, see Sec. 2.1). Itcan be reproduced, for example, by considering heterogeneous driver-vehiclepopulations in macroscopic models (Wagner et al., 1996; Krauss et al., 1997;Banks, 1999; Treiber and Helbing, 1999; Hoogendoorn and Bovy, 2000) orcar-following models (Nishinari et al., 2003; Ossen et al., 2007; Igarashi et al.,2005; Kesting and Treiber, 2008a), by noise terms (Treiber et al., 2006b),or slowly changing driving styles (Treiber and Helbing, 2003; Treiber et al.,2006a).
B Discussion of homogeneous congested traffic
For strong bottlenecks (typically caused by accidents), empirical evidence re-garding the existence of homogeneous congested traffic has been somewhat29 ig. A.1. Homogeneous congested traffic on the high-coverage section of the Britishfreeway M42 ATM (averaged over 3 running lanes) (Wilson, 2008b). Note that nointerpolation or smoothing was applied to the data measured on November 27, 2008.The three-dimensional plots of the vehicle speed and the flow show measurements ofeach fifth detector only, otherwise the plots would have been overloaded. There is noclear evidence that perturbations in the vehicle speed or flow would grow upstream,i.e. against the flow direction that is indicated by arrows. ambiguous so far. On the one hand, when applying the adaptive smooth-ing method to get rid of noise in the data (Treiber and Helbing, 2002), thespatiotemporal speed profile looks almost homogeneous, even when the samesmoothing parameters are used as for the measurement of the other traffic pat-terns, e.g. oscillatory ones (Sch¨onhof and Helbing, 2007). On the other hand,it was claimed that data of the flow measured at freeway cross sections showan oscillatory behavior (Kerner, 2008). These oscillations typically have small30avelengths, which can have various origins: (1) They can result from the het-erogeneity of driver-vehicle units, particularly their time gaps, which is knownto cause a wide scattering of congested flow-density data (Nishinari et al.,2003). (2) They could as well result from problems in maintaining low speeds,as the gas and break pedals are difficult to control. (3) They may also be a con-sequence of perturbations, which can easily occur when traffic flows of severallanes have to merge in a single lane, as it is usually the case at strong bot-tlenecks. According to stylized fact 6 , all these perturbations are expected topropagate upstream at the speed c cong . In order to judge whether the patternis to be classified as oscillatory congested traffic or homogeneous congestedtraffic, one would have to determine the sign of the growth rate of perturba-tions, i.e. whether large perturbations grow bigger or smaller while travellingupstream.Recent traffic data of high spatial and temporal resolution suggest that ho-mogeneous congested traffic states do exist (see Fig. A.1), but are very rare.For the conclusions of this paper and the applicability of the phase diagramapproach, however, it does not matter whether homogeneous congested trafficactually exists or not. This is, because many models with a fundamental dia-gram can be calibrated in a way that either generates homogeneous patternsfor high bottleneck strengths or not (see Sec. 4).31 eferences Appert, C., Santen, L., 2001. Boundary induced phase transitions in drivenlattice gases with metastable states. Physical Review Letters 86 (12), 2498–1501.Aw, A., Rascle, M., 2000. Resurrection of ”second order” models of trafficflow. SIAM Journal on Applied Mathematics 60 (3), 916–938.Bando, M., Hasebe, K., Nakanishi, K., Nakayama, A., Shibata, A., Sugiyama,Y., 1995. Phenomenological study of dynamical model of traffic flow. Journalde Physique I France 5 (11), 1389-1399.Banks, J. H., 1999. Investigation of some characteristics of congested flow.Transportation Research Record 1678, 128–134.Belomestny, D., Jentsch, V., Schreckenberg, M., 2003. Completion and con-tinuation of nonlinear traffic time series: a probabilistic approach. Journalof Physics A: Mathematical and General 36 (45), 11369–11383.Bertini, R., Lindgren, R., Helbing, D., Sch¨onhof, M., 2004. Empirical anal-ysis of flow features on a German autobahn. In: Transportation ResearchBoard 83rd Annual Meeting, Washington DC. Washington, D.C., availableat Arxiv eprint cond-mat/0408138.Brockfeld, E., K¨uhne, R. D., Wagner, P., 2004. Calibration and validationof microscopic traffic flow models. Transportation Research Record 1876,62–70.Brockfeld, E., K¨uhne, R. D., Skabardonis, A., Wagner, P., 2004. Toward bench-marking of microscopic traffic flow models. Transportation Research Record1852, 124–129.Cassidy, M. J., Windower, J., 1995. Methodology for assessing dynamics offreeway traffic flow. Transportation Research Record 1484, 73–79.Coifman, B., 2002. Estimating travel times and vehicle trajectories on freewaysusing dual loop detectors. Transportation Research Part A 36 (4), 351–364.Daganzo, C. F., 1995. Requiem for second-order fluid approximations of trafficflow. Transportation Research Part B 29 (4), 277–286.Diebold, F., 2003. Elements of Forecasting. South-Western Publishing, Cinci-natti.Gipps, P. G., 1981. A behavioural car-following model for computer simula-tion. Transportation Research Part B 15 (2), 105–111.Greene, W. H., 2008. Econometric Analysis, particularly Chap. 7.4: Modelselection criteria. Prentice Hall, Upper Saddle River, NJ.Helbing, D., 1996. Gas-kinetic derivation of Navier-Stokes-like traffic equa-tions. Physical Review E 53 (3), 2366–2381.Helbing, D., 2001. Traffic and related self-driven many-particle systems. Re-views of Modern Physics 73 (4), 1067–1141.Helbing, D., Hennecke, A., Treiber, M., 1999. Phase diagram of traffic statesin the presence of inhomogeneities. Physical Review Letters 82 (21), 4360–4363.Helbing, D., Johansson, A. F., 2009. On the controversy around Daganzo’s32equiem for and Aw-Rascle’s resurrection of second-order traffic flow models.European Physical Journal B 69 (4), 549–562.Helbing, D., Tilch, B., 2009. A power law for the duration of high-flow statesand its interpretation from a heterogeneous traffic flow perspective. TheEuropean Physical Journal B 68 (4), 577–586.Helbing, D., Treiber, M., 2002. Critical discussion of ”synchronized flow”.Cooper@tive Tr@nsport@tion Dyn@mics 1, 2.1–2.24, (Internet Journal, ).Helbing, D., Treiber, M., Kesting, A., Sch¨onhof, M., 2009. Theoretical vs. em-pirical classification and prediction of congested traffic states. The EuropeanPhysical Journal B 69 (4), 583–598.Hoogendoorn, S., Bovy, P., 2000. Gas-kinetic modeling and simulation ofpedestrian flows. Transportation Research Record 1710, 28–36.Huerre, P., Monkewitz, P., 1990. Local and global instabilities in spatiallydeveloping flows. Annual Review of Fluid Mechanics 22 (1), 473–537.Igarashi, K., Takeda, K., Itakura, F., Abut, H., 2005. Is our driving behaviorunique? In: DSP for In-Vehicle and Mobile Systems. Springer, pp. 257–274.Jiang, R., Hu, M., Jia, B., Wang, R., Wu, Q., 2007. Spatiotemporal congestedtraffic patterns in macroscopic version of the Kerner–Klenov speed adapta-tion model. Physics Letters A 365 (1-2), 6–9.Jiang, R., Wu, Q., Zhu, Z., 2001. Full velocity difference model for a car-following theory. Physical Review E 64 (1), 017101.Jiang, R., Wu, Q.-S., 2005. Toward an improvement over Kerner-Klenov-Wolfthree-phase cellular automaton model. Physical Review E 72 (6), 067103.Kerner, B., 1998. Experimental features of self-organization in traffic flow.Physical Review Letters 81, 3797–3800.Kerner, B., 2002. Empirical macroscopic features of spatio-temporal trafficpatterns at highway bottlenecks. Physical Review E 65 (4), 046138.Kerner, B., 2008. A theory of traffic congestion at heavy bottlenecks. Journalof Physics A: Mathematical and General 41 (21), 215101.Kerner, B., Klenov, S., 2002. A microscopic model for phase transitions intraffic flow. Journal of Physics A: Mathematical and General 35 (3), L31–L43.Kerner, B., Klenov, S., 2006. Deterministic microscopic three-phase traffic flowmodels. Journal of Physics A: Mathematical and General 39 (8), 1775–1810.Kerner, B., Klenov, S., Wolf, D., 2002. Cellular automata approach to three-phase traffic theory. Journal of Physics A: Mathematical and General35 (47), 9971–10013.Kerner, B., Konh¨auser, P., 1993. Cluster effect in initially homogeneous trafficflow. Physical Review E 48 (4), R2335.Kerner, B., Rehborn, H., 1996a. Experimental features and characteristics oftraffic jams. Physical Review E 53 (2), 1297–1300.Kerner, B., Rehborn, H., 1996b. Experimental properties of complexity intraffic flow. Physical Review E 53 (5), R4275–R4278.Kerner, B., Rehborn, H., Aleksic, M., Haug, A., Lange, R., 2001. Online au-33omatic tracing and forecasting of traffic patterns. Traffic Engineering andControl 42 (10), 345–350.Kerner, B. S., 2004. The Physics of Traffic: Empirical Freeway Pattern Fea-tures, Engineering Applications, and Theory. Springer, Heidelberg.Kesting, A., Treiber, M., 2008a. Calibrating car-following models by using tra-jectory data: Methodological study. Transportation Research Record: Jour-nal of the Tranportation Research Board 2088, 148–156.Kesting, A., Treiber, M., 2008b. How reaction time, update time and adapta-tion time influence the stability of traffic flow. Computer-Aided Civil andInfrastructure Engineering 23 (2), 125–137.Krauss, S., Wagner, P., Gawron, C., 1997. Metastable states in a microscopicmodel of traffic flow. Physical Review E 55 (5), 5597–5602.Krug, J., 1991. Boundary-induced phase transitions in driven diffusive sys-tems. Physical Review Letters 67 (14), 1882–1885.Lee, H., Lee, H., Kim, D., 1998. Origin of synchronized traffic flow on highwaysand its dynamic phase transition. Physical Review Letters 81 (5), 1130.Lee, H. Y., Lee, H. W., Kim, D., 1999. Dynamic states of a continuum trafficequation with on-ramp. Physical Review E 59 (5), 5101–5111.Lighthill, M., Whitham, G., 1955. On kinematic waves: II. A theory of trafficon long crowded roads. Proc. Roy. Soc. of London A 229 (1178), 317–345.Lindgren, R. V., Bertini, R. L., Helbing, D., Sch¨onhof, M., 2006. Towarddemonstrating the predictability of bottleneck activation on German au-tobahns. Transportation Research Record 1965, 12–22.Mauch, M., Cassidy, M. J., 2002. Freeway traffic oscillations: observations andpredictions. In: Taylor, M. (Ed.), International Symposium of Traffic andTransportation Theory. Elsevier, Amsterdam.Mu˜noz, J., Daganzo, C., 2002. Fingerprinting traffic from static freewaysensors. Cooperative Transportation Dynamics 1, 1–11, (Internet Journal, ).Nishinari, K., Treiber, M., Helbing, D., 2003. Interpreting the wide scatteringof synchronized traffic data by time gap statistics. Physical Review E 68 (6),067101.Ossen, S., Hoogendoorn, S. P., Gorte, B. G., 2007. Inter-driver differences incar-following: A vehicle trajectory based study. Transportation ResearchRecord 1965, 121–129.Payne, H., 1971. Models of Freeway Traffic and Control. Simulation Councils,Inc.Popkov, V., Santen, L., Schadschneider, A., Sch¨utz, G. M., 2001. Empiricalevidence for a boundary-induced nonequilibrium phase transition. Journalof Physics A: Mathematical General 34 (6), L45–L52.Richards, P., 1956. Shock waves on the highway. Operations Research 4 (41),42–51.Sch¨onhof, M., Helbing, D., 2007. Empirical features of congested traffic statesand their implications for traffic modeling. Transportation Science 41 (2),1–32. 34ch¨onhof, M., Helbing, D., 2009. Critisism of three-phase traffic theory. Trans-poration Research Part B 43 (7), 784–797.Siebel, F., Mauser, W., 2006. Synchronized flow and wide moving jams frombalanced vehicular traffic. Physical Review E 73 (6), 66108.Smilowitz, K., Daganzo, C., Cassidy, M., Bertini, R., 1999. Some observationsof highway traffic in long queues. Transportation Research Record: Journalof the Transportation Research Board 1678, 225–233.Treiber, M., Helbing, D., 1999. Macroscopic simulation of widely scatteredsynchronized traffic states. Journal of Physics A: Mathematical and General32 (1), L17–L23.Treiber, M., Helbing, D., 2002. Reconstructing the spatio-temporal traffic dy-namics from stationary detector data. Cooperative Transportation Dynam-ics 1, 3.1–3.24, (Internet Journal, ).Treiber, M., Helbing, D., 2003. Memory effects in microscopic traffic modelsand wide scattering in flow-density data. Physical Review E 68 (4), 046119.Treiber, M., Hennecke, A., Helbing, D., 1999. Derivation, properties, and sim-ulation of a gas-kinetic-based, non-local traffic model. Physical Review E59 (1), 239–253.Treiber, M., Hennecke, A., Helbing, D., 2000a. Congested traffic states in em-pirical observations and microscopic simulations. Physical Review E 62 (2),1805–1824.Treiber, M., Hennecke, A., Helbing, D., 2000b. Microscopic simulation of con-gested traffic. In: Helbing, D., Herrmann, H., Schreckenberg, M., Wolf, D.(Eds.), Traffic and Granular Flow ’99. Springer, Berlin, pp. 365–376.Treiber, M., Kesting, A., Helbing, D., 2006a. Delays, inaccuracies and antici-pation in microscopic traffic models. Physica A 360 (1), 71–88.Treiber, M., Kesting, A., Helbing, D., 2006b. Understanding widely scatteredtraffic flows, the capacity drop, and platoons as effects of variance-driventime gaps. Physical Review E 74 (1), 016123.Treiber, M., Kesting, A., Wilson, R. E., 2010. Reconstructing the traffic stateby fusion of heterogenous data, Computer-Aided Civil and InfrastructureEngineering, accepted. Preprint physics/0900.4467physics/0900.4467