aa r X i v : . [ phy s i c s . s o c - ph ] S e p The Physics of Traffic and Regional Development
Dirk Helbing ∗ Institute for Economics and Traffic, Dresden University of Technology,Andreas-Schubert-Str. 23, 01062 Dresden, Germany
Kai Nagel † Transport Systems Analysis and Transport Telematics,Technical University of Berlin, Salzufer 17–19 SG 12, 10587 Berlin, Germany (Dated: September 4, 2018)This contribution summarizes and explains various principles from physics which are used for thesimulation of traffic flows in large street networks, the modeling of destination, transport mode, androute choice, or the simulation of urban growth and regional development. The methods stem frommany-particle physics, from kinetic gas theory, or fluiddynamics. They involve energy and entropyconsiderations, transfer the law of gravity, apply cellular automata and require methods from evo-lutionary game theory. In this way, one can determine interaction forces among driver-vehicle units,reproduce breakdowns of traffic including features of synchronized congested flow, or understandchanging usage patterns of alternative roads. One can also describe daily activity patterns basedon decision models, simulate migration streams, and model urban growth as a particular kind ofaggregation process.
I. INTRODUCTION
From the point of view of a physicist, cars on a highwayor pedestrians in a mall are many particle systems. Yet,the particles of those systems are more complicated thantypical particles in physical systems. A question thereforeis in how far the methods from physics can be applied tosystems with these more complicated particles.It turns out that in fact quite a lot can be explainedwith the methods from physics, which combine simplemicroscopic modeling assumptions with advanced math-ematics and/or advanced computational science. For ex-ample, those systems display a surprising variety ofstates, such as jammed, laminar, or oscillating. Thosestates can be explained using several concepts, frommicroscopic to fluid-dynamical. Physics methods havecontributed significantly to better understanding thosestates, and to alleviate them where possible.Traffic usually unfolds on networks , rather than onflat two-dimensional space. And indeed, traffic has somesimilarity to the steady-state flow of electrons in a fusenetwork with non-linear resistors. Besides the fact thatthe traffic dynamics can be more complicated than thissteady-state behavior, as pointed out in the last para-graph, the main difference between human travelers andelectrons in a network is that human travelers typicallyhave an individual destination. This introduces an im-portant additional non-linearity, and also brings gametheory into the picture, in particular the concept of aNash Equilibrium (NE). Game theory states that a sys-tem is at a NE when no player can gain by unilaterally ∗ Electronic address: helbing@trafficforum.org;URL: † Electronic address: [email protected]; URL: switching strategies. What is more of an interest to physi-cists is that certain types of learning systems can havea NE as fixed point attractor – they can, however, alsohave other types of attractors, such as periodic or chaoticones. And indeed, route choice experiments with humansubjects confirm that in many cases the players approacha NE, but also that this NE does not have to be stableand displays behavior reminiscent of intermittency andvolatility clustering.This and other evidence demonstrates that the routechoice behavior of real world people can be approximatedby a fastest path algorithm, which is relatively cheapin terms of computational complexity. From a practicalpoint of view, the next problem then becomes to gener-ate the destinations of the travelers. Two different ap-proaches are discussed in the paper: (i) gravity models,which make the probability to select a destination de-pendent on the distance; and (ii) activity-based demandgeneration, which attempts to generate complete dailyplans for all travelers in a system. And once more thesituation is similar to physics: While it may be hopelessto realistically describe the behavior of individual peo-ple in the system, there is some hope that the macro-scopic (“emergent”) behavior of many travelers togethercan be derived from simple microscopic rules, akin to thederivation of the ideal gas equation. Also, behavioral in-variances such as the energy consumption of the human– not the vehicle – can sometimes be found and used.Intuitively, the next important question is then how thecity itself develops – where humans find residences, whereretail and other industries find locations, and where thecity grows when the space inside its current boundariesis no longer sufficient. Once more, models from physics,this time growth models based on the cellular automataparadigm, can offer important insight. The use of suchmodels is related to the fact that some quantities of real-world cities, such as the area, the perimeter, or the sizedistribtion, display fractal properties. An alternative ap-proach is to use models from Synergetics, with its asso-ciated master equation, as the starting point to describemigration dynamics.Similar to the molecular dynamics approach in physics,it is now possible to build large scale “agent-based” simu-lation systems of real world transport systems which arebased on the “first principles” introduced above. Sincea typical metropolitan area has about 10 inhabitants,this is well within the range of computational feasibility,although some computational efficiency is lost because ofthe more complicated particles. The main challenge cur-rently is to integrate the different levels, once more a chal-lenge that is similar to multi-scale modeling in physics.Results achieved so far indicate that simulation mod-els based on those principles are, albeit just at the begin-ning of their development, at least as good as establishedmethods. In consequence, one can start to move thosemodels into the realm of real-world infrastructure plan-ning. In addition, one would expect that additional scien-tific insights, better implementation techniques allowingfaster turnaround, and the increasingly better availabil-ity of electronic data will make them considerably betterin the future. II. FLUID-DYNAMIC AND GAS-KINETICTRAFFIC MODELS
Every driver has probably encountered the widespreadphenomenon of so-called “phantom traffic jams”, forwhich there is no visible reason such as an accident ora bottleneck [1]. So, why are vehicles sometimes stoppedalthough everyone likes to drive fast? [2]The first well-known approach to describe densitywaves in traffic flows goes back to Lighthill and Whitham[3] in 1955. They start with the typical continuity equa-tion, ∂ρ ( x, t ) ∂t + ∂Q ( x, t ) ∂x = 0 , (1)where ρ ( x, t ) denotes density and Q ( x, t ) denotes flow orthroughput. The equation reflects that vehicles are notgenerated or lost in the absence of ramps, intersections,or accidents. As usual, one also has the relation Q ( x, t ) = ρ ( x, t ) V ( x, t ), where V ( x, t ) is the velocity at place x andtime t .Lighthill and Whitham postulated that the traffic flow Q ( x, t ) could be specified as a function of the den-sity ρ ( x, t ) only. This assumes an instantaneous adapta-tion of Q ( x, t ) to some equilibrium flow-density relation Q e ( ρ ( x, t )). The corresponding curve Q e ( ρ ) = ρV e ( ρ ) isoften called the “fundamental diagram” and obtained asa fit to empirical data.Because of ∂Q e ( ρ ) /∂x = ( dQ e /dρ ) ∂ρ/∂x , one obtains ∂ρ∂t + C ( ρ ) ∂ρ∂x = 0 , (2) which is a non-linear wave equation where the propaga-tion velocity C ( ρ ) = dQ e ( ρ ) dρ = V e ( ρ ) + ρ dV e ( ρ ) dρ ≤ V e ( ρ ) (3)of so-called kinematic waves depends on the vehicle den-sity only. Kinematic waves have the property that theykeep their amplitude, while their shape changes until shock waves (i.e. discontinuous changes in the density)have developed. The densities ρ + and ρ − immediatelyupstream and downstream of a shock front determine itspropagation speed S ( ρ + , ρ − ) = Q e ( ρ + ) − Q e ( ρ − ) ρ + − ρ − = ∆ Q e ∆ ρ . (4)Note that Eq. (4) is just the discrete version of Eq. (3).Experimental observations of traffic patterns show ad-ditional features which cannot be reproduced by theabove traffic model. While traffic flow appears to bestable with respect to perturbations at small and largedensities, there is a linearly unstable range [ ρ c2 , ρ c3 ]at medium densities, where already small disturbancesof uniform traffic flow give rise to traffic jams. Be-tween the stable and linearly unstable density ranges,one finds meta- or multi-stable ranges [ ρ c1 , ρ c2 ) and( ρ c3 , ρ c4 ], since there exists a density-dependent, criti-cal amplitude ∆ ρ c , so that the resulting traffic patternis path- or history-dependent [4–7] (see Fig. 1). Whilesubcritical perturbations fade away, supercritical pertur-bations cause a breakdown of traffic flow (“nucleation ef-fect”). Consequently, traffic displays critical points, non-equilibrium phase transitions, noise-induced transitions,and fluctuation-induced ordering phenomena . One mayview this as non-equilibrium analogue of the phase transi-tions between vapour (free flow), water (“synchronized”,queued traffic flow above freeway capacity), and ice (jamswith standing vehicles and zero flow). Note that noise-like effects enter the above deterministic models only viathe boundary conditions, but some models take care ofthem by additional fluctuation terms or by distinguish-ing different driver-vehicle types, i.e. partial flows withnon-linear interactions.According to Krauß [8], traffic models show the ob-served hysteretic phase transition related with meta-stable traffic and high flows, if the typical maximal ac-celeration is not too large and the deceleration strengthis moderate. In such kinds of models, ρ c1 is the densitywhere the flow-density relation J ( ρ ) = 1 T (cid:18) − ρρ jam (cid:19) (5)of traffic with fully developed traffic jams (the so-called“jam line”) intersects with the free branch of the flow-density diagram Q e ( ρ ). T denotes the net time gap (timeclearance) in congested traffic and ρ jam the density insideof traffic jams. The intuitive interpretation of Eq. (5) ρ ρ ρ∆ρ c (b)(a)(c) ρ c1 ρ c2 ρ c3 ρ c4 ρ max ρ c1 ρ c2 ρ c3 ρ c4 ρ max ρ c1 ρ c2 ρ c3 ρ c4 ρ max ρρρ∆ρ (x) ∆ρ (x)x x0 0Complex sequences of traffic jams or of "dipole-layers"Sta-ble State Sta-ble State"Metastable" State "Metastable" StateUnstable State Localized StructuresNonlocalized Structures "Dipole Layers"Traffic Jams xx x
FIG. 1: Schematic illustration of (a) the chosen initial pertur-bation ∆ ρ ( x ) and the density-dependent perturbation am-plitudes ∆ ρ c required for jam formation in the meta-stabledensity regime, (b) the related instability diagram of homo-geneous (and slightly inhomogeneous) traffic flow (predict-ing stable traffic at small and high densities, linearly unsta-ble traffic at medium densities, and meta-stable traffic in be-tween) and (c) the finally resulting traffic patterns ρ ( x ) de-pending on the respective density regime. (Simplified diagramafter [5, 6], see also [4].) is that traffic in the congested regime is composed ofmoving areas and of areas where vehicles are stopped.The fraction of the moving areas is given by 1 − ρ/ρ jam .Moreover, the outflow Q out from traffic jams is a self-organized constant [4, 7, 9], which lies between Q e ( ρ c1 )and Q e ( ρ c2 ) [10]. The jam line corresponds to the flow-density relation for traffic patterns with a self-organized,stationary profile [4]. These propagate with the velocity C = dJ/dρ = − / ( T ρ jam ) ≈ −
15 km/h, which is anothertraffic constant. The explanation of this is the following:Once a traffic jam is fully developed, vehicles leave thedownstream jam front at a constant rate, while new onesjoin it at the upstream front. This makes the jam moveupstream with constant velocity.In order to reproduce such emergent traffic jamsand meta-stable density regimes, the Lighthill-Whithammodel from Eqs. (1) to (4) needs to be generalized. Forthis, it is helpful that fluid-dynamic models can be de-rived from gas-kinetic models, which relate to models of interactive driver behavior [11, 12]. The gas-kineticapproach has been introduced by Prigogine et al. [13–15] and is inspired by kinetic gas theory . It describesthe spatio-temporal change of the phase space density(= vehicle density × velocity distribution). The relatedequations are either of Boltzmann-like type (for point-like vehicles or low densities) or of
Enskog-like type, ifvehicular space requirements at moderate and high den-sities are taken into account (see [11] and referencestherein). They allow the systematic derivation of macro-scopic equations for the vehicle density ρ ( x, t ), the av-erage velocity V ( x, t ), the velocity variance Θ( x, t ), etc.This hierarchy of equations is usually closed after the ve-locity or variance equation, although the separation oftime scales assumed by the underlying approximationsis weak. Nevertheless, the observed traffic dynamics israther well reproduced by the resulting coupled partialdifferential equations. The density equation is just thecontinuity equation ∂ρ∂t + ∂ ( ρV ) ∂x = ν + − ν − , (6)where ν + and ν − denote on- and off-ramp flows, respec-tively. The velocity equation can be cast into the form ∂V∂t + V ∂V∂x = − ρ ∂P∂x + 1 τ ( V e − V ) . (7)In theoretically consistent macroscopic traffic modelssuch as the gas-kinetic-based traffic model, the “trafficpressure” P and the velocity V e are non-local functionsof the density ρ , the average velocity V , and the varianceΘ [16, 17]. Several well-known models are special casesof Eq. (7): • The Lighthill-Whitham model of Eqs. (1) to (4)is obtained in the (unrealistic) limit τ → τ . In that case V ( x, t ) = V e ( ρ ( x, t )), and therefore Q ( x, t ) = ρ ( x, t ) V e ( ρ ( x, t )) =: Q e ( ρ ( x, t )), which depends ondensity only as was postulated for the LW model. • Payne’s macroscopic traffic model [18, 19] is ob-tained for P ( ρ ) = [ V max − V e ( ρ )] / (2 τ ) and V e = V e ( ρ ), where V max denotes the maximum speed (theaverage velocity at very low densities). • Kerner’s and Konh¨auser’s model [4, 20] is a vari-ant of K¨uhne’s model [21] and corresponds to thespecifications P = ρ Θ − η ∂V /∂x and V e = V e ( ρ ),where Θ and η are positive constants. The corre-sponding equation is a Navier-Stokes equation witha viscosity term η ∂ V /∂x and an additional re-laxation term [ V e ( ρ ) − V ] /τ describing the delayedadaptation to the velocity-density relation V e ( ρ ).Both for the Payne model and for the Kerner-Konh¨ausermodel, the linearly unstable regime [ ρ c2 , ρ c3 ] is deter-mined by the densities ρ fulfilling the condition ρ (cid:12)(cid:12)(cid:12)(cid:12) dV e ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12) > s dP ( ρ ) dρ . (8)In consequence, both the Payne model and the Kerner-Konh¨auser model have linearly unstable ranges if dV e ( ρ ) /dρ is large, while the Lighthill-Whitham modelis marginally stable.Beyond linear stability analysis, in some cases it is alsopossible to find analytical expressions for the large ampli-tude instability in the meta-stable ranges [ ρ c1 , ρ c2 ) and( ρ c3 , ρ c4 ] (Fig. 1). For these more complicated aspects,the interested reader is referred to [22].In simple terms, the reason for emergent traffic jamsis as follows: Due to the finite adaptation time (= re-action + acceleration time), a small disturbance in thetraffic flow Q e can cause an overreaction (overbraking)of a driver, if the safe vehicle speed V e ( ρ ) (which sat-isfies dV e ( ρ ) /dρ ≤
0) drops too rapidly with increasingvehicle density ρ . At high enough densities ρ , this willgive rise to a chain reaction of the followers, as othervehicles will have approached before the original speedcan be regained. This feedback can eventually cause theunexpected standstill of vehicles known as traffic jam. III. CONGESTED TRAFFIC STATES INTHEORY AND EMPIRICAL DATA
With this knowledge one can understand the variouscongested traffic states observed on freeway sections withbottlenecks [10, 23–26]. Given that the flow Q and density ρ are measured per lane, a bottleneck due to ramp flowsis determined by ν + = Q rmp / ( IL ), where L is the usedlength of the on-ramp and I the number of freeway lanes.The corresponding bottleneck strength is∆ Q = Q rmp I . (9)
A. Theoretical Phase Diagram of Traffic States
Let Q up denote the traffic flow per lane upstream ofthe bottleneck and Q tot = Q up + ∆ Q = Q up + Q rmp I (10)the total capacity required downstream of the ramp.Then, we expect to always observe free traffic (FT) belowthe first instability threshold at density ρ c1 , i.e. if Q tot = Q up + ∆ Q < Q e ( ρ c1 ) . (11)Traffic flow will always be congested, if the maximumflow Q max = max ρ Q e ( ρ ) (the capacity) is exceeded, i.e. Q tot = Q up + ∆ Q > Q max . (12)At least if the density related with the maximum flow Q max lies between ρ c1 and ρ c2 , traffic states between thetwo diagonal lines Q up + ∆ Q = Q e ( ρ c1 ) and Q up + ∆ Q = Q max in the Q up -vs-∆ Q phase space can be either con-gested or free, depending on the initial and boundary conditions [26] (see Fig. 2). While homogeneous free flowmay persist over long time periods, large perturbationstend to produce congested states. Extended congestedtraffic can emerge above the line Q up = Q out − ∆ Q (13)in the phase diagram, i.e. if Q tot is greater than the dy-namic capacity Q out . This line does not have to be paral-lel to the previously mentioned phase boundaries, as Q out may depend on the bottleneck strength ∆ Q [1, 27]. For Q e ( ρ c1 ) ≤ Q tot = Q up + ∆ Q < Q out , congested trafficstates are always localized, i.e. they never grow over longsections of the freeway, as the outflow can cope with theoverall traffic volume.The traffic flow Q cong resulting in the congested areaplus the inflow or bottleneck strength ∆ Q are normallygiven by the outflow Q out , i.e. Q cong = Q out − ∆ Q (14)(if vehicles cannot enter the freeway downstream ofthe congestion front). One can distinguish the followingcases: Homogeneous congested traffic (HCT) such as typ-ical traffic jams during holiday seasons can occur, if thedensity ρ cong associated with the congested flow Q cong = Q e ( ρ cong ) (15)lies in the stable or meta-stable range Q cong < Q e ( ρ c3 ) , i.e. ∆ Q > Q out − Q e ( ρ c3 ) . (16)Oscillating forms of congested traffic can emerge, if∆ Q ≤ Q out − Q e ( ρ c4 ) and Q up > Q out − ∆ Q . (17)That is, lower bottleneck strengths tend to produce lessserious congestion, namely oscillating rather than homo-geneous congested flow. We either find oscillating con-gested traffic (OCT), triggered stop-and-go traffic (TSG),or moving localized clusters (MLC) [26]. In contrast toOCT, stop-and-go traffic is characterized by a sequenceof moving jams, between which traffic flows freely. Thisstate can either emerge from a spatial sequence of ho-mogeneous and oscillating congested traffic [28], whichis also called the “pinch effect” [29], or it can be causedby the inhomogeneity at the ramp. In the latter case,each traffic jam triggers another one by inducing a smallperturbation in the inhomogeneous freeway section (seeFig. 2), which propagates downstream as long as it issmall, but turns back when it has grown large enough (“boomerang effect”) . This growth requires the down-stream traffic flow to be linearly unstable. If it is meta-stable (when the traffic volume Q tot is sufficiently small),small perturbations will fade away. Therefore, if Q e ( ρ c1 ) ≤ Q tot = Q up + ∆ Q ≤ Q out ≤ Q e ( ρ c2 ) , (18)one expects to find localized traffic states, either a sin-gle moving localized cluster (MLC), or a pinned localized PLC480 490Location(km) 7 8Time (h)1000Velocity (km/h)
TSG15 16Time (h)480490Location (km)1000Velocity (km/h)
OCT470 480Location(km) 10 11 12Time (h)1000Velocity (km/h)
HCT470 480Location(km) 14 15 16Time (h)1000Velocity (km/h) ( v eh i c l e s / h / l ane ) Q up ∆ Q(vehicles/h/lane)2200200018001600140012001000 2000 400 600 800 1000 1200
OCTPLC,= Q ∆ +alwaysFT Q out −Q900 Q out Q c4 c3 MLC,FT FT FT OCTHCTFT ~~ Q t o t c = Q ~~ Q t o t = Q ou t ~~ Q t o t = Q m a x ~~Q up Q c1 ~~ ~~OCT ges ted OCTTSG, OCT always con tot up
Q Q ,HCT HCTTSG,OCTMLC,FT ,, ,,
MLC15 16Time (h)470480490Location(km)1000Velocity (km/h)
FIG. 2: Empirical representatives of different kinds of congested traffic observed on the German freeway A5 close to Frankfurt(after [2, 26]) and theoretical phase diagram of traffic states (after [26]) in the presence of one bottleneck as a function of theupstream flow Q up and the bottleneck strength ∆ Q (center right). cluster (PLC) at the location of the ramp. The latterrequires the traffic flow in the upstream section to bestable, i.e. Q up < Q e ( ρ c1 ) , (19)so that no traffic jam can survive there. In contrast, mov-ing localized clusters and triggered stop-and-go waves re-quire [26] Q up ≥ Q e ( ρ c1 ) . (20)The simulation results displayed in Figure 2 summarize,for freeways with a single bottleneck, which states aretypically found in different areas of the phase diagram.Apart from the detailed shape and exact location of thephase boundaries, this phase diagram is expected to beuniversal for all microscopic and macroscopic, stochasticand deterministic traffic models with the same instability diagram (with stable, meta-stable, and unstable densityranges) [1]. Results for more complex freeway geometriesare available as well. B. Wide Scattering of Traffic Data inHeterogeneous Traffic
In the congested traffic regime, flow-density data arecharacterized by a surprisingly wide scattering [30] (seeFig. 3a). This has led people to question the applicabilityof the jam line (5) [31–33]. However, it can be shownthat, taking into account the variation of the average nettime gap (time clearance) T , the variations in the datacan be reproduced with a correlation coefficient of 0.92(compared to 0.35 when only the density is varied).A closer analysis reveals a large variation of time gapsbetween vehicles. The time gaps of trucks are particularly E m p . F l o w Q ( t ) ( v eh / h l ane ) Density ρ (t) (vehicles/km lane)(a) 6008001000120014001600180040 50 60 70 80 90 100 J ( ρ ( t ) , T ( t ) , ρ j a m ( t )) ( v eh / h l ane ) Density ρ (t) (vehicles/km lane)(b) FIG. 3: The two-dimensional scattering of empirical flow-density data in synchronized traffic flow of high density ρ ≥ ρ , but alsothe empirically measured variation of the average time gap T [and the maximum density ρ max ] is taken into account(see (b)). The pure density-dependence J ( ρ ) (thick blackline) is linear and cannot explain a two-dimensional scatter-ing. However, variations of the average time gap T changeits slope − / ( ρ max T ) (see arrows), and about 95% of thedata are located between the thin lines J ( ρ, T ± T, /l ) =(1 − ρl ) / ( T ± T ), where l = 3 . T = 2 .
25 s the average time gap, and ∆ T = 0 .
29 sthe standard deviation of T . The predicted form of this areais club-shaped, as demanded by Kerner [31–33]. (After [35].) large. Simulations of heterogeneous traffic with differentkinds of vehicles (i.e. different parameter values) suggestthat at least part of the scattering of flow-density datamay be explained by the mixture of cars and trucks [34]and of different driving behaviors (see Fig. 4). Althoughheterogeneous traffic can be treated by macroscopic mod-els (see Fig. 4), a simulation of individual vehicles is muchmore easy, flexible, and efficient, in particular if one likesto simulate network traffic of drivers with different ori-gins and destinations. Therefore, we will focus on some“microscopic” traffic models in the following. IV. DRIVEN MANY-PARTICLE SYSTEMSAND MICROSCOPIC TRAFFIC SIMULATION
The observations in freeway traffic can not only be de-scribed by fluid-dynamic or “macroscopic” traffic models.They are also well reproduced by “microscopic” models[27]. Note that a theoretical connection between both ap-proaches exists. It is called the micro-macro link [12].
A. Car-Following Models
Microscopic models are often follow-the-leader models specifying the acceleration dv i /dt of each single vehicle i as a function F i of their speed v i , their distance headway d i = x i − − x i with respect to the leading vehicle i − v i = v i − v i − : dv i dt = F i ( d i , v i , ∆ v i ) + ξ i ( t ) . (21) ξ i ( t ) are random variations in the acceleration behav-ior, while F i may be interpreted as average situation-dependent acceleration force. A typical example is thenon-integer car-following model dv i ( t + ∆ t ) dt = − ∆ v i ( t ) τ [ v i ( t + ∆ t )] ℓ [ d i ( t )] m (22)with the reaction time ∆ t ≈ .
3s and the parameters τ ≈ ∆ t/ . ℓ ≈ .
8, and m ≈ . τ /T > /
2. (The time delay ∆ t onboth sides facilitates to determine analytical solutions.)A simpler model is the optimal velocity model dv i ( t ) dt = 1 τ h V (cid:0) d i ( t ) (cid:1) − v i ( t ) i , (23)where V ( d i ) is the “optimal” velocity-distance relationand τ the adaptation time [37, 38]. This model has anunstable range for dV ( d i ) /dd i > / (2 τ ). The respectivenon-linearly coupled differential equations (or stochasticdifferential equations, if fluctuations are taken into ac-count) are numerically solved as in molecular dynamics.Let us now define the maximum, desired, or free veloc-ity by V max = max V ( d i ). The expression f ( s i ) = V ( s i + l i ) − V max τ (24)reflects the interaction force f as a function of the netdistance (clearance) s i = d i − l i = ( x i − − x i − l i ) betweentwo successive vehicles, where l i denotes the length ofvehicle i . With this, Eq. (23) can be reformulated in termsof the “social force model” dv i dt = V max − v i τ + f ( s i ) + ξ i ( t ) . (25) B. Interaction Forces Among Vehicles
The difficulty is now to specify the interaction forcesamong vehicles in an appropriate way. Their exact speci-fication is essential for realistic traffic simulations, whichare required for the design of efficient and reliable trafficoptimization measures such as intelligent on-ramp con-trols, driver assistance systems, lane-changing assistants,etc. If the fluctuation term was zero (i.e. ξ i ( t ) = 0), theinteraction force could simply be determined from thevelocity-density relation. Since the average of the head-ways d i = s i + l i determines the inverse density 1 /ρ , wewould simply find the relation f ( s i ) = [ V e (1 / ( s i + l i )) − V max ] /τ .However, things are considerably more difficult, whenrandom fluctuations are not negligible. One may thinkof applying scattering theory to determine the interac-tion potential U ( s ) = − df ( s ) /ds from statistical distri-butions, but learning about human interactions requiresa somewhat different approach. Progress has recently F r a c t i on o f T r u cks (a) Time t (h) F r equen cy Time Headway (s)(b) ρ >30 veh./km, right laneTrucksCars F l o w ( v eh ./ h / l ane ) Density (veh./km/lane)(c)
FIG. 4: (a) The empirical truck fraction varies considerably in the course of time. (b) The time headways of long vehicles(“trucks”) are on average much higher than those of short vehicles (“cars”). (c)-(e) Assuming a fundamental diagram forcars (solid line), a separate one for trucks (dashed line), weighting them according to the measured truck fraction, and usingempirical boundary conditions, allows one to reproduce the observations in a (semi-)quantitatively way [34]: (c) Free traffic (atlow densities) is characterized by a (quasi-)one-dimensional curve. Data of congested traffic upstream of a bottleneck are widelyscattered in a two-dimensional area. (d)
Immediately downstream of the bottleneck, one observes homogeneous-in-speed statesreflecting recovering traffic. (e) Further downstream, the data points approach the curve describing free traffic. Dark symbolscorrespond to empirical one-minute data, light ones to corresponding simulation results. (After [25].) been made by means of random matrix theory, a pow-erful method from quantum and statistical physics. It al-lows one to determine the interaction potentials via netdistance distributions of vehicles.Random matrix theory [39] has been developed formany-particle systems exposed to a ”thermal bath” ofa given reciprocal temperature β , i.e. to random forces ofa certain variance and statistics. The resulting velocityand net distance distributions allow one to draw conclu-sions about the interaction potential U ( s ) = − df ( s ) /ds ,as this determines their shapes. Although a generaliza-tion to arbitrary potentials U ( s ) is possible, here we willfocus our attention on power-law potentials U ( s ) ∝ s − α (26)for simplicity. The exponent α > s is the net distance (clearance) between two successivecars. Such a potential describes the repulsive tendency ofdrivers to keep a safe distance to the respective car ahead.The related net distance distributions can be calculatedto be P ( ρs ) = Ae − β ( ρs ) − α e − Bρs , (27)where ρ is the vehicle density and A , B repre-sent normalization constants determined by requiring R ∞ P ( ρs ) d ( ρs ) = 1 and h ρs i ≡ R ∞ ρsP ( ρs ) d ( ρs ) = 1 . Originally, the applied random matrix method is anequilibrium concept assuming the conservation of en-ergy H = T + V , i.e. a transformation of potential en-ergy V = P Ni =1 U ( | x i − − x i − l i | ) into kinetic energy T = P Ni =1 m i v i / x i , l i , m i , and v i being the loca-tion, length, mass, and velocity of the i th vehicle. It canhowever be shown [40] that this is a reasonable approx-imation for an ensemble of N vehicles i in a stationarystate at a given density ρ and inverse “temperature” (ve-locity variance) β , while the stop-and-go regime between20 and 40 vehicles per kilometer and lane must be ex-cluded from an investigation of this kind.Milan Krbalek and one of us have separately analyzedeight density intervals of Dutch single vehicle data in thefree low-density regime and eight density intervals in thecongested traffic regime. In accordance with the theoret-ical predictions, the velocity distributions fit Gaussiandistributions very well [41]. Moreover, the best fit of thenet distance distributions is obtained for the integer pa-rameter α = 4 in the free traffic regime, while we find anexcellent agreement with α = 1 in the congested regime(see Figs. 5, 6). This is not only a strong support of stud-ies questioning a uniform behavior of drivers in all trafficregimes [42]. It also offers an interpretation of the myste-rious fractional distance-scaling exponent α + 1 ≈ . ρ ∈ (0;2.5) F r e qu e n cy D i s t r i bu t i on P ρ ∈ (10;12.5) ρ ∈ (40;45) ρ ∈ (60;65) Scaled Net Distance ρ s FIG. 5: Representative distributions of net distances (clear-ances) among successive cars for various densities ρ , deter-mined from Dutch single-vehicle data (after [44]). The solidfit curves correspond to the theoretical distributions (27) forthe empirical values of ρ and β . In the free traffic regime, theonly fit parameter is α = 4 (above, blue), while α = 1 in thecongested regime (below, red). Note that the net distance distribution (27) agrees withthe Pearson III type of distribution [45] which has beensuggested to fit empirical time headway distributions al-ready before our theoretical explanation had been found.Therefore, the determination of interaction potentials infreeway traffic contributes to the challenging problem ofunderstanding time headway or distance distributions ofcars [45, 46]. In addition, the identification of behavioralregimes points to adaptive driver behavior.
V. SIMULATION OF LARGE TRAFFICNETWORKS WITH CELLULAR AUTOMATAA. Cellular Automata Rules for Links
An alternative approach to car-following models arerule-based cellular automata (CA). For the CA tech-nique, road links are divided into cells, say of length7.5 m, which can contain at most one car each, and thereis one array of cells for each lane, see Fig. 7(a). Movementis performed by jumping from one cell to another, wherethe new cell is determined by a set of “driving rules.”A good update time step is 1 sec (justified by reactiontime arguments). Taking this together means, for exam-ple, that a jump of 5 cells in a time step models a speedof 5 × . S qu a r e d D ev i a t i on χ
40 50 60 70 800.000.020.040.060.080.100.120.140.160.180.20
Density ρ (vehicles/km/lane) α =1 α =2 α =3 α =4 FIG. 6: Sum of squared deviations between the empirical andtheoretical net distance distributions for various fit parame-ters α (after [44]). The best fit is α = 4 in the free trafficregime and α = 1 in the congested regime. ization, and movement. The deterministic speed calcula-tion rule first computes a new speed for each car basedon its current speed and closeness to the car in front ofit. An example for such a rule is v safe ,t = min[ v t − + 1 , v max , g t ] , where v t − + 1 models acceleration, v max is a speed limit,and g t is the gap, i.e. the number of empty spaces tothe vehicle ahead. See Fig. 7(a) for an illustration ofgap. Next, the randomization rule introduces a proba-bility p noise for each car so that instead of following theabove speed deterministically, it may instead drive onevelocity level more slowly: v ′ t = ( max[0 , v safe ,t −
1] with probability p noise v safe ,t else. . In the movement rule, each particle is moved forward: x t +1 = x t + v ′ t . To illustrate the above rules, Fig. 7(a) shows traffic mov-ing to the right. Using a p noise value of 0.2, the leftmostvehicle accelerates to velocity 2 with probability 0.8 andstays at velocity 1 with probability 0.2. The middle ve-hicle slows down to velocity 1 with probability 0.8 andto velocity 0 with probability 0.2. The rightmost vehicleaccelerates to velocity 3 with probability 0.8 and staysat velocity 2 with probability 0.2. Velocities are in “cellsper time step.” All vehicles are then moved according totheir velocities using the movement rule described above.Fig. 8 shows, in a so-called space-time diagram for traf-fic, the emergence of a traffic jam. Lines show configu-rations of a segment of road in second-by-second timesteps, with time increasing in the downward direction;cars drive from left to right. Integer numbers denote thevelocities of the cars. For example, a vehicle with speed“3” will move three sites (dots) forward. Via this mech-anism, one can follow the movement of vehicles from leftto right, as indicated by an example trajectory. B. Lane Changing
Typical rules for lane changing (see Fig. 7(b)) include areason to change lanes and a safety criterion for changinglanes. First, there needs to be a reason why a vehiclewants to change lanes, for example that the other lane isfaster, or that it needs to get into a certain lane in order tomake an intended turn at the end of the link. A possiblerule to model the first is “check if the (forward) gap inthe other lane is larger than the gap in the current lane.”Second, the vehicle needs to check that there is reallyenough space in the destination lane. A possible rule isthat the forward gap in the other lane needs to be largerthan v t , and the backward gap in the other lane needs tobe larger than the velocity of the oncoming vehicle.Figure 7(b) illustrates the lane changing rules. Onlylane changes to the left lane are considered. In situationI, the leftmost vehicle on the bottom lane will change tothe left because the forward gap on its own lane, 1, issmaller than its velocity, 3; the forward gap in the otherlane, 10, is larger than the gap on its own lane, 1; theforward gap in the target lane is large enough; and thebackward gap is large enough. In situation II, the secondvehicle from the right on the right lane will not performa lane change because the gap backwards on the targetlane is not sufficient.Due to the complexity of the dynamics, it is inconve-nient to do both lane changing and car following in oneparallel update. It is however possible, and convenientfor parallel computing, to do the update in two com-pletely parallel sub-timesteps: First, all lane changes areexecuted in parallel based on information from time t , re-sulting in intermediate information, at time t + . Then,all car following rules are executed in parallel, based oninformation from time t + , which results in informationfor time t +1. C. Intersections
The typical modeling substrate for transportation isthe network, which is composed of links and nodes thatmodel roads and intersections, respectively. Since the fullsimulation of an intersection is rather complicated, forlarge scale applications one attempts to reduce the com-plexity by resolving all conflicts at the entering of theintersection. Under such a simplification, it is still possi-ble to model things like left turns against oncoming traf-fic, but the decision is made before the vehicle enters theintersection, not inside the intersection. In consequence, gap vehicle with velocity 1 cell per time-step (a) forward gapbackward gap gap forward gapgap Situation ISituation II
52 3 backward gap (b)
15 3 53 2 1 2 3 gap = 3 * velocity(oncoming vehicle) (c)
FIG. 7: Illustration of cellular automata driving rules (from[50]). (a) Definition of gap and one-lane update. (b) Lane-changing. (c) Left turn against oncoming traffic. complications inside the intersection, such as the inter-esting dynamics of a large roundabout in Paris, are notmodeled by this approach. Nevertheless, the simplifiedapproach is quite useful for large scale applications.Given this, there are only two remaining cases for traf-fic: Protected and unprotected turns. A protected turnmeans that a signal schedule takes care of possible con-flicts, and thus the simulation does not have to worryabout them. In this case, it is enough if the simulationmakes vehicles stop when a signal is red, and lets themgo when a signal is green.In unprotected turns, conflicts are resolved by the le-gal rules, not by signals. For example, a vehicle makinga left turn needs to give priority to all oncoming vehicles(Fig. 7(c)). This entails that one needs, for each turningdirection, to encode which other oncoming lanes have thepriority. Once this is done, a vehicle that intends to doa certain movement only needs to check if there is a ve-hicle approaching on any of these interfering lanes. If so,the vehicle under consideration has to stop, otherwise itcan proceed. As said before, once the vehicle has entered0 ......4.....1..2.....3...2....3..........5..........5......5.......2..1...2......1..2.....4............4.................3....1.2....2.....1...2.......4............4................1..1..3....2....1....2.........5............4...4.........1..2....2...3...2.....3............4...............5......2...2....3....1..3......4.............5................3...3...3.....1..1....4.......4..............5.4............3...2...3...2..1.......5.......4................4...........2..2....1..1.1...........4......5................4.........1..2...2..1.2..............5.......4.....4.........4......2...3...1.1..2.................4............5.........4....2....1.0.2...2...................4.............5........2..3...01...2...2....................5...............4.....2...00.1....3...3.......................4..............2...0.01..2......3...3................5.......4............1.0.0.1...3.......3...3..................5......5.........00.1..2.....3.......3...4....................5......3....00..1...3......3.......4....4.....................4....0.01...1.....4......4........4....5.....................01.0.1...2........4......5........4......................1.00..1....3..........4.......4...........................000...2......3...........5.......4......5................000.....3.......3.............5.......4......5...........001........3.......4...............5......5.......4......00.1..........3........5............5..........5......1..01..2............3..........5............5..........2..1.0.2...3.............3............5.....4......4.......1.00...2....3.............3...................4......3....001.....2.....3.............3....................5.....1.00.1......3......4.............4........4.............1.000..1........3.......5.............5..
FIG. 8: Emergence of a jam “out of nowhere” (“phantomtraffic jam”), simulated with a stochastic cellular automaton. the intersection, other vehicles are no longer regarded.Figure 7(c) shows an example of a left turn against on-coming traffic. The turn is accepted because on all threeoncoming lanes, the gap is larger or equal to three timesthe first oncoming vehicle’s velocity.
D. Modeling Queues
Sometimes, even the relatively simple CA rules arecomputationally too slow. Another factor of ten in the ex-ecution speed can often be gained by simplifying the dy-namics even more. In the so-called queue model [47, 48],links have no internal structure any more, they can beconsidered as vehicle reservoirs. These reservoirs have astorage constraint, N max , which varies from link to link,and is based on the physical attributes of the link, such aslength and number of lanes. Once the storage constraintis exhausted, no more vehicles can enter the link untilsome other vehicles have left the link.Vehicles can only leave the link if their destination linkhas space, if they have spent the time it takes to traversethe link, and if the flow capacity constraint is fulfilled.The flow capacity constraint is a maximum rate withwhich vehicles can leave the link.Except for the storage constraint and its consequences,this is just a regular M/M/1 queuing model. The storageconstraint does, however, necessitate one adaptation ofthe model: Since destination links can be full, there isnow competition for space on the destination link. A goodoption is to give that space randomly to incoming links,with a bias toward incoming links with higher capacity[49].With the queue model and using parallel computing,it is possible to simulate a full day of all car traffic of allof Switzerland (approx. 7 mio inhabitants) in less than5 minutes [49]. Although data movement is still a chal- lenge, this makes it possible to run completely micro-scopic (agent-based) studies of large metropolitan areas[117]. VI. ENTROPY LAWS AND DESTINATIONCHOICE
The simulation of traffic in street networks requiresdata on the number V kl of trips between origins k anddestinations l as a function of the time of the day, theweekday etc. These origin-destination data are relativelyscarce and expensive. However, the number Q k = P l V kl of trips starting in a city k or quarter of a city andthe number Z l = P k V kl of trips ending in l is rela-tively well-known, as this requires only 2 n instead of n data, where n denotes the number of dinstinguished ori-gins/destinations. Apart from this, Q k and Z l can beautomatically measured by detectors, while the determi-nation of the entries V kl of the traffic flow matrix requiresto obtain origin-destination pairs, i.e. to ask drivers.Therefore, the relative frequencies p kl = V kl /V fromthe origins k to the destinations l , where V = P k,l V kl ,are often determined via the “most likely” distributionunder the constraints X l p kl = Q k /V = p k and X k p kl = Z l /V = p l . (28)This is obtained by minimizing the “information gain” I = − X k,l p kl ln p kl b kl (29)compared to some “natural distribution” b kl . This is anal-ogous to maximizing entropy H = − P kl p kl ln p kl incases where b kl is an equidistribution. The only differenceis that (29) takes into account a “resistance function” b kl ,which reflects the “natural distribution” as a functionof the distance and other behaviorally relevant variablesin the absence of constraints like (28). The appropriatespecification of b kl will be discussed in Sec. VI A.Instead of minimizing (29) under the constraints (28),it is easier to minimize L = − X k,l p kl ln p kl b kl + X k λ k p k − X l p kl ! + X l µ l p l − X k p kl ! , (30)where the additional terms vanish when the constraints(28) are fulfilled. Differentiation with respect to p k ′ l ′ gives − ln p k ′ l ′ b k ′ l ′ − − λ k ′ − µ l ′ = 0 (31)1or p kl = b kl e − λ k − / | {z } = g k e − µ l − / | {z } = h l . (32)The 2 n Lagrange parameters λ k and µ l are now deter-mined in a way that the 2 n constraints (28) are satis-fied. This requires the use of numerical iteration proce-dures. For an overview of numerical solution algorithmssee Refs. [51–53]. These references also suggest ways totreat the choice of destinations and transport modes si-multaneously. A. Energy Laws and Resistance Function
According to the last paragraphs, the distribution oftraffic significantly depends on the “natural distribution” b kl . Robert K¨olbl and one of us have investigated theelectronically available empirical data of the UK NationalTravel Survey during the years 1972–1998 [54]. When wedistinguished between different daily modes of transport j , we found that the average modal travel time t j perday and person remained almost constant over the 27years of observation, despite variations by a factor 3.8between different modes. More specifically, the averagetravel times t j were 40 min during a day with walking( j = w), without any usage of other means of transport,42 min for cyclists ( j = c), 67 min for stage bus users( j = b), 75 min for car drivers ( j = d), 59 min for carpassengers ( j = p), and 153 min for train passengers( j = t).When we scaled the t -axis of the modal travel-timedistributions P j ( t ) dt by the average travel times t j , wediscovered that, within the statistical variation, the re-sulting distributions collapsed onto one single curve P ( t j ) dt j ≈ N exp (cid:18) − αt j − t j γ (cid:19) dt j (33)with t j = t/t j , two fit parameters α and γ , and the nor-malisation constant N = N ( α, γ ) (see Fig. 9a). This im-plies a universal law of human travel behaviour and theresistance function or natural distribution b kl = N exp (cid:18) − αw kl − w kl γ (cid:19) , (34)where w kl = t kl /t j is the travel time t kl from k to l divided by the average travel time t j for transport mode j . In the semi-logarithmic representation ln P ( t j ) =ln N − α/t j − t j /γ of (33), the term α/t k is relevant onlyfor short time scales up to t j ≈ .
5, while the linear rela-tionship ln P ( t j ) = ln N − t j /γ dominates clearly over awide range of scaled travel times t j (see Fig. 9b). There-fore, we suggest the following interpretation: The dom-inating term P j ( t j ) dt j ∝ exp( − t j /γ ) dt j corresponds tothe canonical energy distribution P ′ ( E j ) dE j = N ′ exp( − E j / ¯ E ) dE j , (35) F r equen cy D i s t r i bu t i on P ( t j ) Scaled Travel Time t / t j _ walkers(a) cyclistsbus userscar driverscar passengersfit curve F r equen cy D i s t r i bu t i on P ( t j ) Scaled Travel Time t / t j _ (b) walkerscyclistsbus userscar driverscar passengersfit curve FIG. 9: Scaled time-averaged travel-time distributions for dif-ferent modes of transport (a) in linear and (b) in semiloga-rithmic representation. Within the statistical variation (androunding errors for small frequencies, which are magnified inthe semilogarithmic plot), the mode-specific data could all befitted by one universal curve (33) with α = 0 . γ = 0 .
7, andnormalization constant N ( α, γ ) = 2 .
5. The few railway datawere not significant because of their large scattering. (From[59].) where E j = p j t with p j = ¯ E/ ( γt j ) stands for theenergy consumed by the physical activity of travellingwith transport mode i for a time period t . The term P j ( t j ) ∝ exp( − α/t j ) possibly represents the additionalamount of energy required for the mobilization of shorttrips, i.e. the Simonson effect [55]. With an average travelenergy budget of ¯ E = 615 kJ, this interpretation is con-sistent with values of the energy consumption p j per unittime for certain activities obtained by ergonomic mea-surements of the related O -consumption [56–58].Based on entropy maximization, the canonical distri-bution can be shown to be the most likely distribution,given that the average energy consumption ¯ E per day by2an ensemble of travellers is fixed for the area of investi-gation. This agrees well with the investigated data andgeneralizes the concept of a constant travel-time budget[60–67]. In addition, the above theory can be combinedwith trip distribution models [68–73], e.g. the multino-mial logit model P ∗ j = e β ′ V j P j ′ e β ′ V j ′ , (36)where β ′ is a parameter (see Sec. VII B) and the mode-dependent energy consumption enters the utility function V j as a (negative) cost term, which was shown to be asignificant variable of human travel behaviour.In summary, when daily travel time distributions bydifferent modes of transport such as walking, cycling, busor car travel are appropriately scaled, they turn out tohave a universal functional relationship. A closer inves-tigation reveals a law of constant average energy con-sumption by the physical activity of daily travelling. Theadvantage of the behavioral law (33) is its expected long-term validity over more than 25 years even under chang-ing conditions. It will, therefore, prove to be importantfor urban, transport, and production planning: Com-pared to previous models, it facilitates improved quanti-tative conclusions about trip distributions, modal splitsand induced traffic after the more reliable determinationof fewer parameters, which are constant over typical plan-ning horizons. It also helps to assess the increase in theacceptability of public transport, when the comfort oftravel is improved, to forecast the change in travel behav-ior in an aging society, to predict the usage of new modesof transport, and to estimate potential market penetra-tions of new travel-related products. VII. ACTIVITY PATTERNS
Sometimes, the models outlined in the previous sec-tions are not sufficient to model demand generation fortraffic. The first obvious shortfall is that they do not sayanything about a time-of-day dynamics. This is some-times introduced by assuming that all trips from resi-dential to commercial locations represent morning traf-fic, and the afternoon/evening peak is then obtained bytransposing the trip matrix.Often, even that is not enough. This happens typicallywhen there are significant correlations between travelers’attributes and their decisions. For example, the numberof trips starting in a zone in general does not just de-pend on the number of residents, but also on their incomeor age; destination choice in general depends on vehicleavailablity; the acceptance of a new mode of transportin general depends on the question if people can reorga-nize their entire day plans so that the new facility can beefficiently integrated.A possible way out of this is to make the entireapproach completely microscopic, or agent-based. This means that throughout the whole simulation, all travelersare maintained as individuals, and individual behavioralmodels are applied at all steps of the decision-making.For example, synthetic (= simulated) travelers may plantheir activity patterns (e.g. home – work – shop – home– leisure – home), then the corresponding locations, thenthe modes of transportation, and finally the precise tim-ing of their activities. The advantage of this approachover the ones discussed so far is that, at each step of thedecision-making, all “internal” information of the agentis available.Such an approach draws from many areas of science.Clearly, psychology is involved because it describes thebehavior of human individuals. Yet, since the goal is notto describe each individual traveler correctly but to getcorrect aggregate results, economics and the social sci-ences are involved as well, and there is in fact a com-munity of “travel behavior research” [74] which appliesthese principles to traffic. However, the approach to goto microscopic principles when an analysis on the macro-scopic level fails is also deeply rooted in physics, and infact, both the computing methods to large-scale problemsand the general intuition that simple (and therefore indi-vidually wrong) microscopic models may lead to correctmacroscopic results come to a large extent from physics.Given its highly interdiscplinary nature plus the factthat only recently data and computing methods becameavailable to do activity-based demand generation forlarge urban areas, it is understandable that the area is inrapid flux. There is one method, discrete choice theory[75], that is arguably better established than others, andit will be described in a bit more detail in the following.However, there are some drawbacks to that theory, andit is questionable whether it will be possible to correctthem within the framework of that theory. For that rea-son, alternative, less well established approaches will alsobe discussed.
A. Discrete Choice Theory
Assume, for simplicity, a situation where an agent hasa choice between several alternatives, j , for example be-tween several locations for the same activity. Also assumethat there are utilities U j associated with those alterna-tives. Discrete choice theory starts from two assumptions: • It is possible for the observer/modeler to mea-sure/predict some part of the utilities. These mea-surable parts are called “systematic utilities” V j . • The differences between the V j and the U j arerandom and statistically independent. These dif-ferences are denoted as ε j .With these assumptions, one obtains essentially the fol-lowing algorithm:1. Compute the V j based on your observations. Forexample, one might have observed that one loca-3tion is accessible only via some steep stairs, andtherefore avoided by physically less fit persons.2. Add the random components: U j = V j + ε j .3. Select the alternative that now has the highest util-ity. Note that, because of the random component,it can well be that the alternative with a smaller V j in the end “wins”.There are at least two different interpretations wherethe random parts ε j come from: Economic theory statesthat economic agents always select the option withthe largest utility, and the difference between the ob-served/predicted utilities V j comes from so-called unob-served attributes of the agent which are thus unknownto the observer/analysis. Psychology, in contrast, mightsay that there is just a random component to people’sbehavior. B. Multinomial Logit Model (MNL)
The question now is how to obtain real-world numbersfor the V j and the ε j . The typical way to progress is in twosteps: (i) make some assumptions about the ε j ; (ii) as-sume that the V j are additive in their contributions anddo a maximum likelihood estimation of the parameters. a. The random components. A conceptually easyway to proceed is to assume that the ε j are normallydistributed. The result of this is called a (multinomial)probit model . Its solution, however, cannot be writtendown in terms of simple functions. Another path is toassume that the ε j follow so-called Gumbel-distributionswith zero mean, i.e. their generating function is F ( ǫ j ) =exp[ − exp( − µ j ǫ j )], where µ j is a width parameter. Inthis case, the result is that the probability for agent i toselect option j follows the Boltzmann distribution p i,j = e β ′ V i,j P k e β ′ V i,k . (37)This is known as the (multinomial) logit model , cf.Eq. (36). β ′ results from the “widths” µ j of the distribu-tions. It models how “rational” the agents behave withrespect to what is observed: a small β ′ makes the agentsnearly random with respect to V j ; for β ′ → ∞ one ob-tains an agent that always takes the option j with thebest V j . b. The systematic components. In order to makefurther progress, one has to make an assumption aboutthe functional form of the V j , and its free parameters.The most typical assumption is that the V j are linear intheir components. For example, the utility for agent i togo swimming at location j might be V i,j = β T T i,j + β C C i,j + β G G i (38)where T i,j is the travel time to get there, C i,j is the mon-etary cost, and G i is i ’s gender (e.g. 0 for male, and 1 for female). Normally, β T and β C will be negative. A posi-tive β G would express that, all other things being equal,females like to go swimming more than males. Note thatattributes of the alternative as well as attributes of theperson can occur, and that attributes can be continuousor discrete.One now proceeds by asking a large number of peo-ple about their individual T i,j , C i,j , G i , plus the decisionthat they actually made (revealed preference) or wouldmake (stated preference) in the given situation. Maxi-mum likelihood estimation then finds coefficients β X suchthat the probability to obtain the decisions from the sur-vey is maximized ( X stands for the different possible in-dices). More precisely, it finds products β ′ β X , meaningthat the β ′ from Eq. (37) is already included. Note that itis not necessary for Eq. (38) to be linear; something like β T T T i,j or β CI C i,j /I i is possible (where I i is i ’s income).In the past, the utility function V i,j had to be linear in the β X (because of the need for a partial derivative for themaximum likelihood estimation), but that requirement isnow lifted [76]. C. Extensions
Multinomial logit models (MNLs) have a propertycalled independence from irrelevant alternatives (IIA).It means that taking an alternative out of the choiceset does not change the relative probabilities: p i,j /p i,k does not depend on a third alternative m . This has some-times odd consequences, typically explained in terms ofthe so-called “red bus blue bus” paradoxon. Assume thata traveler has the option to take a car (option 1) or totake the bus, and that she takes the bus with 50% prob-ability. Now assume that the bus route is served by twobuses, one of them red (option 2) and one of them blue(option 3), and both of them depart at exactly the sametime, they have exactly the same service quality, and theyare always nearly empty. Estimation of an MNL wouldresult in a 25% probability for the red bus and a 25%probability for the blue bus.So far, this is all fine. However, now assume that theblue bus gets taken out of service. The model for ourtraveler now predicts that she uses the car with prob-ability 2 /
3, and the red bus with probability 1 /
3. Thisis due to the fact that the ratio of the probabilities ofoption 1 and option 2 has to remain unchanged.The problem sits in the fact that for this example, therandom components ε and ε are not independent. Oneway to deal with this problem is to make the model hi-erarchical (“nested”), where first the choice is betweencar and bus, and then about the type of bus. This re-duces computational complexity, but demands that theanalyst specifies the hierarchical structure. Another op-tion is to use models in which correlations between therandom variables are included as covariances. The resultsare more sophisticated than for the MNL model, but withincreasingly better computers they enter the domain of4being useful for practical cases.Another extension concerns the parameters β X , where X again stands for the different possible indices. Forexample, in Eq. (38), it would make sense to assumethat the ratio β T /β C , related to the “value of time”, de-pended on income. The two options in this situation are(i) to assume an explicit income-dependent model such as V ij = β T T i,j + β CI C i,j /I i ... (where once more β T and β CI would be expected to be negative) or (ii) to assume ran-dom taste variations across the population, which meansthat the coefficients β X become random variables, withaccordingly more free parameters to be estimated.Software for the estimation of the models describedin this section and pointers to relevant literature can befound at [76]. D. Discrete Choice Theory for Activities
Existing discrete choice models for the generation ofdaily activities use the nested MNL approach, whichtranslates into a hierarchical decision tree: First a travelerdecides on the daily pattern, then on activity locations,then on modes of transportation, and then on exact times[77, 78]. A problem is that decisions on the higher leveldepend on decisions on the lower levels; for example, thedecision about the inclusion of an activity depends onhow close its location is to the locations of other possibleactivities. The practical approach to that problem is tofirst compute the lower level options and then pass themon as “expected” utilities to the higher levels. For exam-ple, the utility of doing a trip both by car and by bus iscomputed, and (roughly) the maximum of both is passedon to the location choice module.
E. Alternative Methods for the Generation ofActivity Patterns
In spite of the random term, discrete choice theory as-sumes extremely rational agents. In fact, a traveler mod-eled by a discrete choice model for activity planning willconsider the utilities of all different options before mak-ing a decision. A consequence of this is that the computa-tion is in fact much more expensive than finding the bestoption , since for a best option, suboptimal branches inthe search tree can be pruned, while for discrete choicetheory, all branches need to be followed to the leaves. Inaddition, discrete choice models for human activity plan-ning need of the order of several hundred free parameters,and it is questionable how well so many parameters canbe estimated from surveys.For those reasons, it makes sense to look at alterna-tive models. For example, one can make models wheretravelers do not look forward on the time axis [79]. Sucha model is much easier to calibrate, but it is not sensi-tive to changes in the time structure: For example, themodel will not make people get up later in the morning if opening times of shops are extended in the evening. Animprovement of this is the use of Q-learning, where agentslearn, by doing the same day over and over again, to back-propagate the temporal effects to the decision points [80].Again, other models are entirely based on rules [81] or ongenetic algorithms [82]. A newer development is the useof mental maps, where the agent remembers which partsof the system it knows [83–85].
VIII. LEARNING, FEEDBACK, ANDEVOLUTIONARY GAME THEORY
Neither entropy-based destination choice (Sec. VI) norcomplete activity patterns (Sec. VII) nor route choice(Sec. IX) can be sensibly modeled without including theeffect of learning or adaptation. The simple reason forthis is that there are nonlinear circular dependencies: Hu-man plans depend on (expected) congestion, but conges-tion is the result of (the execution of) plans.The obvious solution for this is any kind of relaxationmethod, i.e. some version of “make initial plans – com-pute congestion – adjust the plans – re-compute conges-tion” etc. Another interpretation of the same approach isthat it models human learning from one day to the next.It is interesting to note that there is actually sometheory available to describe this process. For example, ifall agents always play “best reply” (i.e. the best answerto the last iteration), and if the iterative dynamics con-verges to a fixed point attractor, than that fixed pointis also a
Nash Equilibrium (NE) [86]. For the traditionalmethod of transportation planning, called static assign-ment (similar to flows in electric networks except thatparticles know their destinations), one can actually showthat under some assumptions the NE is unique (in thelink flows), and therefore the iterative process does nothave to be explicitly modeled [87]. For the more compli-cated models discussed in this paper, no such mathemat-ical statement is available, and therefore multiple fixedpoint attractors could be possible, or the iterative processcould converge to a periodic or chaotic attractor. If themodels are stochastic, then under mild assumptions theyconverge to a fixed steady-state state-space density [88].In practice, only a small number of iterations is feasible,and effects such as broken ergodicity [89] need to be takeninto account. A very illustrative example of these differ-ent types of possible learning dynamics will be providedin Sec. IX.Real world scenarios are even more complicated. De-cisions of different agents may happen on different timescales, which depending on their hierarchical order leadsto different solutions [90, 91]. This is related to the issuesof sequential games in game theory. Agents may have lim-ited information, or they may not even use “best reply”at all, but rather some constraint satisficing method, inwhich they improve until they are satisfied. Technically,this means that some important real world results maylie in the transients rather than in the steady state be-5havior. A research agenda for the future needs to clarifythese questions, and needs to identify the limits of pre-dictability for such systems.
IX. ROUTE CHOICE
Once the activity pattern of an individual is selected(including the destination j and the mode of transport k [51–53]), one of the alternative routes m to the des-tination needs to be chosen. This is typically done bymaximizing the utility U k , which in most cases comesdown to minimizing the overall travel time. As a conse-quence, the fastest connection is filled with vehicles, untilthe traffic density has increased so much that their traveltimes become comparable with the travel time on an al-ternative road with greater length or smaller speed limit.In this way, a distribution of traffic is produced, in whichthe travel times of all alternative routes are the same.This distribution is called the Wardrop equilibrium [92].However, does this distribution really describe the routechoice behavior of drivers? The following paragraphs willshow that the Wardrop equilibrium systematically un-derestimates travel times, as drivers do not match theideal traffic distribution due to coordination problems.
FIG. 10: Schematic illustration of the day-to-day route choicescenario (from [44, 93, 94]). Each day, the drivers have todecide between two alternative routes, 1 and 2. Note that,due to the different number of lanes, route 1 has a highercapacity than route 2. The latter is, therefore, used by lesscars.
A. Decision Experiments
The efficient distribution of road capacities amongusers based on individual decisions is still a fundamentalproblem. As individuals normally have aggregate infor-mation (such as radio news) rather than complete infor-mation, one frequently observes far-from-optimal traveltimes. In order to learn more about this decision behav- ior, we have carried out a variety of route choice experi-ments [93, 94] (see Figs. 10 to 13).
FIG. 11: Schematic illustration of the decision experiment(from [93, 94]). Several test persons take decisions based onthe aggregate information their computer displays. The com-puters are connected and can, therefore, exchange informa-tion. However, a direct communication among players is sup-pressed.
In these experiments, N test persons had to repeatedlydecide between two alternative routes m ∈ { , } andshould try to maximize their resulting payoffs P m , whichwere chosen proportionally to the inverse travel times. Ifthe average vehicle speed V m on route m is approximatedby the linear relationship [95] V m ( N m ) = V m (cid:18) − N m ( t ) N max m (cid:19) , (39)the inverse travel times obey the payoff relations P m ( N m ) = P m − P m N m with P m = V m L m and P m = V m N max m L m . (40)Herein, V m is the maximum velocity (speed limit) and N m the number of drivers on route m , L m its length,and N max m its capacity, i.e. the maximum possible num-ber of vehicles on route m . For an improved approach todetermine the travel times in road networks see Ref. [96].Note that alternative routes can reach comparable pay-offs and travel times only when the total number N = N + N of vehicles is large enough to fulfil the rela-tions P ( N ) < P (0) = P and P ( N ) < P (0) = P .Our route choice experiment has focussed on this traf-fic regime. Furthermore, we have the capacity restriction6 N < N max1 + N max2 , as N = N max1 + N max2 is connectedwith a complete gridlock.The Wardrop or user equilibrium corresponds to equaltravel times and payoffs for both alternative decisions. Itis found for a fraction f eq1 = N N = P P + P + 1 N ( P − P )( P + P ) (41)of persons choosing alternative 1 and agrees with the sys-tem optimum only in the limit N ≫ “treatment” 1 , whereall players knew only their own (previously experienced)payoff, but also in treatment 2 , where the payoffs P ′ ( N )and P ′ ( N ) for both, 1- and treatment 3 , we managed to reveal the mysteri-ous persistence in the changing behavior and to achievemore than three times higher payoff increases. Every testperson was informed about the own payoff P ′ ( N ) [or P ′ ( N )] and the potential payoff P ′ ( N − N + ǫN ) = P ′ ( N ) − ǫN P [or P ′ ( N − N + ǫN ) = P ′ ( N ) − ǫN P ]he or she would have obtained, if a fraction ǫ of per-sons had additionally chosen the other alternative (here: ǫ = 1 /N = 1 / P ′ ( N ) = P ′ ( N ) every player knew that he or shewould not get the same , but a reduced payoff, if he or shewould change the decision. That explains why treatment3 could reach a better adaptation performance, reflectedby a low standard deviation and close-to-optimal aver-age payoffs. Moreover, even the smallest individual cu-mulative payoff exceeded the highest one in treatment 1.Therefore, treatment 3’s way of information presentationis much superior to the ones used today.Figure 13 clears up why players changed their deci-sion in the user equilibrium at all. We discovered inter-mittent behavior, i.e. quiescent periods without changes,followed by turbulent periods with many changes. Thisis reminiscent of volatility clustering in stock market in-dices, where individuals also react to aggregate informa-tion summarizing all decisions (the trading transactions).Single individuals seem to change their decision specu-lating for above-average payoffs. In fact, although thecumulative individual payoff is anticorrelated with theaverage changing rate, some individuals receive higherpayoffs with larger changing rates than others, i.e. theyprofit from the overreaction in the system: Once the sys-tem is out of equilibrium, all individuals respond in oneway or another. Typically, there are too many decision N o . o f - D e c i s i on s S t anda r d D e v i a t i on C hang i ng R a t e O v e rr ea c t i on -4-20246810 0 500 1000 1500 2000 2500 A v e r age P a y o ff s Iteration t All PlayersChange of Decision (a)(b)(c)(d)(e)
FIG. 12: Overview of treatments 1 to 5 (with N = 9 and pay-off parameters P = 28, P = 4, P = 6, and P = 34 for0 ≤ t ≤ P = 44and P = − ≤ t ≤ all players. In treatment 4,the changing rate and the standard deviation went up, sincethe user equilibrium changed in time. The user-specific rec-ommendations in treatment 5 could almost fully compensatefor this and managed to reach the minimum overreaction inthe system. The above conclusions are also supported by ad-ditional experiments with single treatments. (After [93, 94].) R an k i ng Iteration t0246810 0 100 200 300 400 500 N u m be r o f P l a y e r s Iteration tNo. of 1-DecisionsStandard DeviationChanging Rate (a)(b)
FIG. 13: Illustration of typical results for treatment 3 (from[93, 94]). (a) Decisions of all 9 players. Players are displayed inthe order of increasing changing rate. Although the rankingof the cumulative payoff and the changing rate are anticor-related, the relation is not monotonic. Note that turbulentor volatile periods characterized by many decision changesare usually triggered by individual changes after quiet pe-riods (dotted vertical lines). (b) The changing rate is oftenlarger than the (standard) deviation from the user equilib-rium N = f eq1 N = 6 (red horizontal line), indicating anoverreaction in the system (see also Fig. 12d). changes.In treatment 4 , we have tested the group performanceunder changing environmental conditions, when the par-ticipants got the same information as in treatment 3.That is, some of the payoff parameters were varied intime, implying a time-dependent user equilibrium. Evenwithout recommendations, the group managed to adaptsurprisingly well to the variable conditions, but the stan-dard deviation and changing rate were approximately ashigh as in treatment 2 (see Fig. 12). This adaptability(the collective ‘group intelligence’) is based on comple-mentary responses [94, 97]. That is, if some players donot react to the changing conditions, others will takethe chance to earn additional payoff. This experimentallysupports the behavior assumed in the theory of efficientmarkets, but here the efficiency is limited by overreac-tion.To avoid overreaction, in treatment 5 we have recom-mended a number f eq1 ( t +1) N − N ( t ) of players to changetheir decision and the other ones to keep it. These user-specific recommendations helped the players to reach thesmallest overreaction of all treatments and a very lowstandard deviation, although the payoffs were changingin time as in treatment 4 (see Fig. 12). X. MIGRATION AND THE LAW OF GRAVITY
Urban and regional evolution, in particular the changeof street infrastructures, housing, industrial and businessquarters, has a natural feedback on the spatial distribu-tion of people. The temporal development of the popula-tion distribution P ( x , t ) in space x can be described bymigration models. Pioneers in this field have been Weid-lich and Haag [98]. In the following, we will discuss avariant of their master equation model [99–101] dP ( x , t ) dt = X x ′ h w ( x | x ′ , t ) P ( x ′ , t ) | {z } Immigration into x − w ( x ′ | x , t ) P ( x , t ) | {z } Emigration out of x i . (42)According to this equation, the stream of immigrants intoplace x is given by the fraction P ( x ′ , t ) of the populationat place x ′ which could potentially move to place x . Theproportionality factor is the transition rate w ( x | x ′ , t )from x ′ to x , i.e. the migration probability to x per unittime, given one lives at place x ′ before. Analogously, thestream of emigrants leaving place x is proportional tothe fraction P ( x , t ) of the population living in x , andthe proportionality factor is the migration rate w ( x ′ | x , t )from x to x ′ . This migration rate is proportional to thefraction of people living at place x ′ , as the number ofacquaintances with friends, family members, or formercollegues increases linearly with the number of inhabi-tants of a town x . However, the migration rate decreasessignificantly with the distance y = k x − x ′ k between twoplaces x and x ′ (and/or with the transaction costs). Ac-cordingly, we assume [102] w ( x ′ | x , t ) P ( x , t ) = ν ( t )e U ( x ′ ,t ) e − U ( x ,t ) P ( x ′ , t ) P ( x , t ) D ( k x − x ′ k ) , (43)where D ( y ) is a monotonously growing function in y .The parameter ν ( t ) reflects the mobility of the popu-lation and U ( x , t ) the utility/attractiveness of living atplace x . Note that formula (43) reminds of the grav-ity law in physics [103], where P ( x ′ , t ) and P ( x , t ) havethe meaning of the masses of two celestial bodies x and x ′ and D ( y ) = y . Moreover, Eq. (43) is analogousto Eq. (32) with b xx ′ = P ( x ′ , t ) P ( x , t ) /D ( k x − x ′ k ), g x ( t ) = e − U ( x ,t ) , and h x ′ ( t ) = e U ( x ′ ,t ) . That is, the grav-ity law can also be motivated by entropy considerations,but this time P ( x , t ) is adapted rather than g x ( t ) and h x ′ ( t ). A great advantage of Eq. (42) is that it can de-scribe disequilibrium situations such as resulting migra-tion streams when borders between countries are opened. XI. URBAN AND REGIONAL EVOLUTION
Weidlich and Haag have modeled urban and regionalevolution with a master equation approach similar toEq. (42), but with additional birth- and death-terms de-scribing the generation or the removal of entities from8the system. Moreover, in their simulations, they distin-guished different kinds of entities (“agents”): inhabitants,who may migrate, streets which may be expanded, indus-trial areas which may grow or decay, building blocks inwhich people may live, and leisure areas such as parks.Positive and negative feedbacks between these differententities were represented by interaction terms in the util-ity functions determining the corresponding transitionrates w . In this way, they could describe the empiricallyobserved differentiation in spatial usage patterns and in-terdependencies between them. Moreover, they have ap-plied their approach to the evolution of cities in China[104].An advantage of models based on the master equa-tion is their huge flexibility. A disadvantage is that thereis a large number of coefficients that need to be defined.Therefore, there is also a search for models that naturallyinclude (usually geometric) constraints and by doing socan explain some aspects of urban dynamics with a muchsmaller number of free parameters. These models origi-nate from the observation that some properties of citiesare fractal. For example, the fractal dimension of the bulkof cities seems to be around 1.9, and the fractal dimen-sion of the (outer) perimeter of cities seems to be around1.3 [105, 106]. This implies that one might attempt to de-scribe city growth with local growth models. There areboth attempts to explain the fractal dimension by vari-ations of models originally from physics [107, 108] andattempts to re-create (and thus predict) the growth ofreal-world cities [109, 110]. Yet, not all aspects of urbangrowth can be explained by entirely local rules. For exam-ple, construction of a large mall or a golf course certainlydepends on the availability of a similar facility somewhere in the city. Unfortunately, this leads to models and simu-lations where everything depends on everything, whichmeans complicated models and long simulation times.Yet, plausibly, this is also not correct: Although largeand important facilities have a far-reaching impact, theeffect still decays with distance. In consequence, hierar-chical models, as they are known from particle dynamicswith long range interactions, are now also transfered tourban dynamics [111].As was the case at other places in this paper, the higherlevel dynamics of urban growth is not independent fromlower level dynamics such as traffic congestion. In con-sequence, the ultimate challenge of this area may be topresent a uniform view that bridges all scales from cardriving via human activity planning to urban evolution.In such a view, the models presented so far would be lim-iting cases for the situation where a separation of (time)scales is possible. There are three approaches to achievethis bridging of the scales: • Parameterization:
If a separation of (time) scales isnot possible, then lower level processes need to betaken into account at least via parameterizations,i.e. simplified aggregated models of the lower levelprocesses. As is well known, parameterization needsto be preceeded by scientific understanding. • Model coupling:
Instead of using a parameteriza-tion, i.e. a macroscopic description of the lower leveldynamics, a (computational) model of the lowerlevel dynamics could be used directly. Such modelcoupling is another big challenge, not only for com-putational reasons, but also because the combinedmodel systems are once more dynamical systemswith a behavior that is usually not well understood. • Microscopic across all scales:
A further alternativeis to construct a model that is microscopic or agent-based across all scales, that is, at all levels all de-cisions are made by synthetic versions of the re-spective actors, usually humans. This is similar toa molecular dynamics simulation in physics. It alsofaces the same challenge: It is difficult to simulatelarge scale systems in spite of all the computationalprogress we had in recent times [112]. Neverthe-less, the first partial systems of this approach areemerging [113–115], and validations of partial as-pects look rather promising [116, 117].Once such models are available and operational, theycould be applied to a rather wide range of questions,ranging from zoning via infrastructure planning to thedeliberate attraction of commercial companies and theeffects of migration. And although such models cannottake away the responsibility for decisions from politiciansor from society, they might help that such decisions arebased on a more informed understanding of the systemthan today.
XII. SUMMARY AND OUTLOOK
Physics is a science that has a tradition in derivinglarge scale emergent phenomena from microscopic rules.Sometimes, these derivations can be done analytically,and in such cases the thermodynamic limit is useful. Inother situations, the microscopic rules are too compli-cated to make analytical progress, and the computer isused. However, even with today’s computational power,only up to about 10 particles can be simulated directly.The situation in traffic and regional development issimilar in the sense that most effects are also causedby the combined behavior of many particles. In contrastto materials science, the number of particles in regionalsimulations, typically several millions, is well within therange that is computationally accessible. On the otherhand, the rules that each individual particle follows areconsiderably more complicated.In this situation, it makes sense to look at differentscenarios one by one. The first question is to identifythose areas where very simple models already lead togood levels of understanding and prediction. This refers,for example, to the area of traffic flow, where alreadymodels which just include limits on acceleration, braking,and excluded volume generate a wide range of dynamicalphenomena that can all be identified in the real world.9Similar models are tried in the areas of route choice, ac-tivity generation, migration, or urban growth.If these simple models fail, then more complicatedmodels are in order. These models typically contain morebehavioral parameters, and because the behavior of hu-mans is difficult to predict, these models are more diffi-cult to calibrate or validate. The challenge is, as so often,to keep the models as simple as possible, but no simpler.Also, one has to keep in mind that we are interested inmacroscopic, “emergent” phenomena such as traffic jams,or general urban patterns, and those should be easier topredict than the behavior of individual humans. Never-theless, one has to get used to validation error bars thatare much larger than in physics.With respect to real-world prediction, models thatcover a large range of scales would be useful. Most no-tably, there is a rather old dream for integrated land usetransport models, in which the location choices of in-habitants and commercial entities lead to traffic, whichin turn triggers revised location choices, etc. [118] Ithas, however, been difficult to put that dream into prac-tice [118], and one could argue that the main obstacle hasbeen that the interactions between the scales were notwell understood but rather included in an ad-hoc man-ner. Simultaneously, computing power in the past has notreally been sufficient for sound solutions.However, in our view the situation is now changing.Based on the combination of computational, analyti-cal, and experimental work, there has been considerableprogress with respect to understanding aspects at thedifferent levels, leading to better and faster submodels inthe future. At the same time, it is now possible to buildcomputational models that bridge large ranges of scaleswhile remaining microscopic, which, although computa-tionally demanding, again solves a large variety of con-ceptual problems.Finally, one should also mention that data availabil-ity in the field of geo-science has made a quantum jumpforward. Satellite-derived high resolution spatial data isnow available for nearly everywhere; decent census datais available in nearly every industrialized country; vec-tor data for the infrastructure is increasingly becomingavailable; and a whole range of telematics devices is capa-ble of collecting important data about spatial behavior,although aspects of privacy need to be valued against sci-entific interest. This also means that the traditional ap-proach of physics, where theory, computation, and real- world measurement enhance each other, can now be ap-plied to traffic and regional systems.Overall, it is our impression that the new computingand data collection technologies have had a huge impactin the area of traffic and regional systems, and that boththe computational and the theoretical work are strug-gling to keep up with the new possibilities. This, com-bined with the obvious societal relevance of the area,makes this a very exciting field. XIII. AUTHORS’ BIOGRAPHIES
Dirk Helbing (*19/Jan/1965) is the Managing Direc-tor of the Institute for Economics and Traffic at DresdenUniversity of Technology. Originally, he studied Physicsand Mathematics in G¨ottingen, Germany, but soon hegot fascinated in interdisciplinary problems. Therefore,his master thesis dealt with physical models of pedes-trian dynamics, while his Ph.D. thesis at the Univer-sity of Stuttgart focussed on modeling interactive deci-sions and behaviors with methods from statistical physicsand the theory of complex systems. After his habilitationon the physics of traffic flows, he received a Heisenbergscholarship and worked at Xerox PARC in Silicon Val-ley, the Weizmann Institute in Israel, and the CollegiumBudapest—Institute for Advanced Study in Hungary. Hegets excited when physics meets traffic or social science,economics, or biology, and if the results are potentiallyrelevant for everyday life.Kai Nagel (*17/Sep/1965) is Professor for TransportSystems Analysis and Transport Telematics at the Insti-tute of Land and Sea Transport Systems at the Techni-cal University Berlin in Germany. He studied physics andmeteorology at the University of Cologne and the Univer-sity of Paris, with one master’s thesis in the area of cel-lular automata models for cloud formation and anotherone in the area of large scale climate simulations. HisPh.D., in computer science at the University of Cologne,was about cellular automata models for large scale trafficsimulations. He then was postdoc, staff member, and re-search team leader at Los Alamos National Laboratory,working on the TRANSIMS (Transportation ANalysisand SIMulation System) project. In 1999–2004, he wasassistant professor for computer science at ETH Zurichin Switzerland. His interests are in large scale simulationand in the simulation and modeling of socio-economicsystems. [1] Helbing, D., 2001, Traffic and related self-driven many-particle systems.
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