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5Transportation Planning and Traffic Flow Models

Sven Maerivoet∗ and Bart De Moor†

Department of Electrical Engineering ESAT-SCD (SISTA)

Phone: +32 (0) 16 32 17 09 Fax: +32 (0) 16 32 19 70

URL:http://www.esat.kuleuven.be/scd

Katholieke Universiteit Leuven

Kasteelpark Arenberg 10, 3001 Leuven, Belgium

(Dated: February 2, 2008)

Abstract

In this paper, we focus on the different traffic flow modelsthat exist in literature. Due to our frequently encounteredconfusion among traffic engineers and policy makers, thispaper goes into more detail about transportation planningmodels on the one hand, and traffic flow models on theother hand. The former deal with households that makecertain decisions which lead to transportation and the useof infrastructure, as opposed to the latter which explic-itly describe the physical propagation of traffic flows in aroad network. Our goal is not to give a full account (asthat would be a dissertation of its own, given the broad-ness of the field), but rather to impose upon the reader athorough feeling for the differences between transporta-tion planning and traffic flow models. Because of the highcourse of progress over the last decade (or even duringthe last five years), this paper tries to chronicle both pastmodels, as well as some of the latest developments in thisarea.

PACS numbers: 89.40.-a

Keywords: land-use, trip-based, activity-based, trans-portation economics, macroscopic, mesoscopic, micro-scopic

Contents

1 Transportation planning models 21.1 Land use and socio-economical behaviour 2

1.1.1 Classic land-use models . . . . . 21.1.2 The modern approach to land-use models 3

1.2 Trip-based transportation models . . . . . 41.2.1 Basic entities and assumptions . . 41.2.2 The four steps . . . . . . . . . . . 41.2.3 Static traffic assignment . . . . . 61.2.4 Dynamic traffic assignment . . . 81.2.5 Critique on trip-based approaches 9

1.3 Activity-based transportation models . . . 91.3.1 Historic origins . . . . . . . . . . 10

∗sven.maerivoet@esat.kuleuven.be†bart.demoor@esat.kuleuven.be

1.3.2 Approaches to activity-based modelling 101.3.3 Towards elaborate agent-based simulations 11

1.4 Transportation economics . . . . . . . . . 121.4.1 The economical setting . . . . . . 131.4.2 Towards road pricing policies . . 14

2 Traffic flow propagation models 162.1 Macroscopic traffic flow models . . . . . 17

2.1.1 The continuum approach . . . . . 172.1.2 The first-order LWR model . . . . 182.1.3 Analytical solutions of the LWR model 182.1.4 Numerical solutions of the LWR model 202.1.5 Flavours of the LWR model . . . 222.1.6 Higher-order models . . . . . . . 232.1.7 Critiques on higher-order models 24

2.2 Mesoscopic traffic flow models . . . . . . 262.2.1 The different meanings of ‘mesoscopic’ 262.2.2 Mesoscopic models considered from a gas-kinetic perspecti2.2.3 Improvements to the mesoscopic modelling approach 28

2.3 Microscopic traffic flow models . . . . . 292.3.1 Classic car-following and lane-changing models 292.3.2 Optimal velocity models . . . . . 312.3.3 Psycho-physiological spacing models 322.3.4 Traffic cellular automata models . 322.3.5 Models based on queueing theory 322.3.6 Microscopic traffic flow simulators 332.3.7 Calibration and validation issues . 34

2.4 Submicroscopic traffic flow models . . . . 352.5 The debate between microscopic and macroscopic models 36

3 Conclusions 36

A Glossary of terms 37A.1 Acronyms and abbreviations . . . . . . . 37A.2 List of symbols . . . . . . . . . . . . . . 39

Due to our frequently encountered confusion among traf-fic engineers and policy makers when it comes to trans-portation planning models and the role that traffic flowmodels play therein, this paper strives to alleviate thatbewilderment. The material elaborated upon in this pa-per, spans a broad range going from transportation plan-ning models that operate on a high level and deal withhouseholds that make certain decisions which lead to

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transportation and the use of infrastructure, to traffic flowmodels that explicitly describe the physical propagationof traffic flows in a road network.

1 Transportation planning models

Before going into detail about the possible mathematicalmodels that describe the physical propagation of trafficflows, it is worthwhile to cast a glance at a higher level,where transportation planning models operate. The mainrationale behind transportation planning systems, is thattravellers within these systems are motivated by makingcertain decisions about their wishes to participate in so-cial, economical, and cultural activities. The ensembleof these activities is called theactivity system. Becausethese activities are spatially separated (e.g., a person’sliv-ing versus work area), the need for transportation arises.In such a system, the so-calledhousehold activity pat-terns form the main explanation for what is seen in thetransportation network.

These models have as their primary intent the perform-ing of impact and evaluation studies, and conducting‘before and after’ analyses. The fact that such trans-portation studies are necessary, follows from a counter-intuitive example whereby improving the transportationsystem (e.g., by making extra infrastructure available),can result in anincreaseof the travel times. This phe-nomenon, i.e., allowing more flexible routing that resultsin more congestion, is known as Braeß’ paradox, after Di-etrich Braeß [33]. The underlying reason for this counter-intuitive behaviour, is that people generally onlyselfishlytry to minimise their own travel times, instead of consid-ering the effects they have on other people’s travel timesas well [228].

As transportation is inherently a temporal and spatial phe-nomenon, we first take a look at the concept of land-use models and their relation to the socio-economical be-haviour of individual people. In the two subsequent sec-tions, we consider two types of transportation planningmodels, i.e., the classic trip-based models, and the classof activity-based models, respectively. The section con-cludes with a brief reflection on the economist’s view ontransportation systems.

1.1 Land use and socio-economical be-haviour

As already stated, transportation demand arises becauseof the desire to participate in a set of activities (e.g., so-cial, economical, cultural, . . . ). In order to deduce thisderivedtransportation demand, it is necessary to map theactivity system and its spatial separations. This processis commonly referred to asland use, mainly playing therole of forging a relation between economical and geo-graphical sciences. In general, land-use models seek toexplain the growth and layout of urban areas (which is not

strictly determined by economical activities alone, i.e.,ethnic considerations et cetera can be taken into account),

Because transportation has spatial interactions with landuse and vice versa, it can lead to a kind of chicken-and-egg problem [252]. For example, building a new road willattract some economical activity (e.g., shopping malls etcetera), which can lead to a possible increase of the traveldemand. This in turn, can lead to an increase of extraeconomical activity (because of the well-suited location),and so on, resulting in a local reorganisation of the spa-tial structure. Resolving this chicken-and-egg paradox, istypically done by means of feedback and iterations be-tween land-use and transportation models, whereby theformer provide the basic starting conditions for the lattermodels (with sometimes a reversal of the models’ roles).

In the following two sections, we first shed some light onseveral of the archetypical land-use models, after whichwe take a look at some of the more modern models forland use in the context of geosimulation.

1.1.1 Classic land-use models

The discussion given in this section, talks about severalkinds of land-use models that — at their time — wereconsidered as landmark studies. That said, the modelspresented here should be judged as being general in thatthey deal with (pre-)industrial American societies in thefirst part of the 20th century. They are devised to gain in-sight into the general patterns that govern the growth andevolution of a city. As such, they almost never ‘fit’ per-fectly, leading to the obvious criticism that they are moreapplicable to American cities than elsewhere. Notwith-standing these objections, the models remain very use-ful as explanations for the mechanisms underpinning thesocio-economical development of cities.

One of the oldest known models describing the relationbetween economic markets and spatial distances, is thatof Johann Heinrich von Thunen [283]. As the model waspublished in 1826, it presents a rather ‘pre-industrial’ ap-proach: the main economical ingredients are based onagricultural goods (e.g., tomatoes, apples, wheat, . . . ),whereas the transportation system is composed of roadson which carts pulled by horses, mules, or oxen ride.The spatial layout of the model, assumes anisolatedstate(self-sustaining and free of external influences), inwhich a central city location is surrounded by concentricregions of respectively farmers, wilderness, field crops,and meadows for grazing animals. All farmers aim formaximum profits, with transportation costs proportion-ally with distance, thus determining the land use aroundthe city centre.

Some 100 years later, inspired by von Thunen’s simpleand elegant model, Ernest W. Burgess developed what isknown as theconcentric zone model[40]. It was basedon observations of the city of Chicago at the beginningof the 20th century. As can be seen in the left part ofFig. 1, Burgess considered the city as growing around a

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central business district (CBD), with concentric zones ofrespectively the industrial factories and the low-, middle-,and high-class residents. The outermost ring denotes thecommuter zone, connecting the CBD with other cities.As time progresses, the city develops and the radii ofthese concentric zones would grow by processes of ‘in-vasion’ and ‘succession’: an inner ring will expand, in-vading an outer ring that in turn has to grow, in order tomake space.

Fifteen years after Burgess’ theory, Homer Hoyt intro-duced refinements, resulting in thesector model[139].One of the main incentives, was the observation thatlow-income residents were typically located in the vicin-ity of railroads. His model accommodates this kind ofobservation, in that it assumes that a city expands aroundmajor transportation lines, resulting in wedge-shapedpatterns (i.e., sectors), stretching outward from the CBD.A typical example of this development, can be seen inthe right part of Fig. 1.

CBDCBD

I

I

I

L

L

L

L

M

MM

H

HC

Figure 1: Typical examples of two models relaying theevolution of land use.Left: the concentric zone model ofBurgess.Right: the sector model of Hoyt. In both figures,CBD corresponds to the central business district,I to theindustrial factories,L, M, andH to the low-, middle-, andhigh-class residents respectively. In the Burgess model atthe left,C denotes the commuter zone.

Halfway the previous century, Chauncy D. Harris and Ed-ward L. Ullman were convinced that the previous typesof models did not correspond to many of the encoun-tered cities. The main reason for this discrepancy, wasto be found in the stringent condition of a central areabeing surrounded by different zones. As a solution tothis shortcoming, Harris and Ullman presented theirmul-tiple nuclei model[108]. Their theory assumed that inlarger cities, small suburban areas could develop intofully fledged business districts. And although Harris andUllman did not dispose of the CBD as the most importantcity centre, their smaller ‘nuclei’ would take on roles ofbeing areas for specialised socio-economical activities.

To end our discussion of classic land-use models, wehighlight the work of Peter Mann in 1965 [187], whoconsidered ahybrid modelfor land-use representation.He combined both Burgess’s and Hoyt’s models, whenderiving a model that described a typical British city. Inhis model that studied the cities of Huddersfield , Not-tingham, and Sheffield, the CBD still remained the cen-

tral location, surrounded by zones of pre- and post 1918housing respectively. Dispersed around the outer con-centric zone, the low-, middle-, and high-class residentswould live. A most notable feature of Mann’s model, isthe fact that he considered the industrial factories to beon one side of the city, with the high-class residents di-ametrically opposed (the rationale being that high-classresidents would prefer to stay upwind of the factories’smoke plumes).

1.1.2 The modern approach to land-use models

In the current time of living, most modern citizens have adifferent behaviour than their former counterparts at thebeginning of the 20th century. It seems there is an in-creased trend towards expansion, as people are feelingmore comfortable about covering larger distances, e.g.,working in a busy city centre or at a remote industrialfacility, coupled with living on the countryside). The ac-tivities related to working, living, and recreation appearto occur at substantially different spatial locations. Fur-thermore, several urban regions are composed of uniqueethnic concentrations, among other things leading to theconclusion that the emphasis on the geographical aspectof a city gets less important during its evolution.

Recognising these radical changes in the development,modern land-use models approach the integration of anactivity system from a completely different perspective.The growth of a city is represented as the evolution of amulti-agent system, in which a whole population of in-dividual households is simulated. Due to the tremen-dous increase in computational power over the last twodecades, these large-scale simulations are now possible.As an example, it is feasible to consider residential segre-gation in urban environments: within these environments(e.g., the city and housing market), individual agents(i.e., households) interact locally in a well-defined man-ner, leading to emergent structures, i.e., the evolving city.Besides data surveys that try to capture the households’behaviour, the basic landscape and mapping data is fedinto geographical information systems(GIS) that is cou-pled with a computer aided design(CAD) representa-tional model of the real world (although the differencebetween the traditional GIS and CAD concepts is slowlyfading away) [285]. A recent example of such an all-encompassing approach, is the work related to the Urban-Sim project, where researchers try to interface existingtravel models with new land use forecasting and analysiscapabilities [284]. It is being developed and improved bythe Center for Urban Simulation and Policy Analysis atthe University of Washington.

To conclude this section, we refer to the work of Benen-son and Torrens, who adopted the terminology ofgeosim-ulation [25]. Their methodology is based on what theycall the ‘collective dynamics of interacting objects’. Assuch, geosimulation hinges on the representation of whatwe would call asocio-economythat is simulated, tak-ing into account hitherto neglected dynamic effects (e.g.,

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demographic changes, shifts of the economic activities,. . . ).

1.2 Trip-based transportation models

The relation between activity patterns and the transporta-tion system has a long history, starting around 1954 withthe seminal work of Robert B. Mitchell and ChesterRapkin [196]. They provided the first integrated study,establishing a link that introduced a framework for trans-portation analysis, primarily intended for studying largescale infrastructure projects [192]. Their methodologywas based on four consecutive steps (i.e., submodels),collectively called thefour step model(4SM). In 1979,Manheim casted the model’s structure into a largerframework of transportation systems analysis, encap-sulating both activity and transportation systems [186].Central to this framework, was the notion of ‘demandand performance procedures’, which we can validly calldemandandsupply procedures. In a typical setup, theyrespectively represent the traffic that wants to use thisinfrastructure and the road infrastructure. For a morehistorically tinted recollection of the trip-based approach,we refer the reader to the outstanding overview of Boyce[30].

With respect the 4SM’s history, a subtle — almostforgotten — fact is that the classic four step modelwas actually conceived independently from the in-tegrated network equilibrium model proposed byBeckmann, McGuire, and Winsten in the mid-fifties;the 4SM can actually be perceived as a trimmed-down version of this latter model [30]. Intriguingly,over the years, the work of the ‘BMW trio’ has hadprofound impacts on the mathematical aspects of de-termining network equilibria, optimal toll policies,algorithms for variational inequalities, stability anal-yses, supply chains, . . . [5, 207, 30, 31].

In the next four sections, we consider the basic entitiesand assumptions of the four step model, followed bya brief overview of the four individual submodels withsome more detail on the fourth step (traffic assignment),concluding with some remarks on the criticisms often ex-pressed against the four step model. For a more extensivesurvey of the four step model, we refer the reader to thebooks of Sheffi [255] and Ortuzar and Willumsen [226].

1.2.1 Basic entities and assumptions

The basic ingredients on which the four step model isrooted, are thetrips. These trips are typically consid-ered at the household level, and relate to aggregate in-formation (individuals are no longer explicitly consid-ered). This level of detail, essentially collapses the wholetempo-spatial structure of transportation planning basedon individual travellers into bundles of trips, going fromone point in the transportation network to another.

In the four step model, one of the most rigid assump-tions is that all trips describe departure and arrival withinthe planning period (e.g., the morning commute). Fur-thermore, the usage of the model’s structure is intendedfor large-scale planning purposes, excluding small infras-tructural studies at e.g., a single intersection of urbanroads. Another assumption is based on the fact that anentity within the four step model has to make certain de-cisions, e.g., what is the departure time, which destinationis picked, what kind of transportation (private or public)will be used, which route will be followed, . . . In manycases, these decisions are considered concurrently, butthe four step model assumes they are made independentlyof each other. And finally, as each submodel needs input,most of the data is aggregated intospatial zones(oftenpresumed to be distinguished by socio-economic char-acteristics) in order to make the model computationallyfeasible. These zones are typically represented by theircentrally located points, calledcentroids.

1.2.2 The four steps

Within the four step model, the first three steps (I)– (III) can collectively be seen as a methodology forsetting up the travel demand, based e.g., on land use andother socio-economical activities. This travel demandis expressed asorigin-destination(OD) pairs (by somerespectively called ‘sources’ and ‘sinks’), reflecting theamount of traffic thatwants to travel from a certainorigin to a certain destination (these are typically thezones mentioned in the previous section). The last step(IV) then consists of loading this travel demand onto thenetwork, thereby assigning theroutesthat correspond tothe trips.

(I) Trip generationIn an essential first step, transportation engineers lookat all the trips that on the one hand originate in certainzones, and on the other hand arrive in these zones.As such, the first step comprises what are called theproductionsandattractions. Central to the notion of atrip, is themotive that instigated the trip. An exampleof such a motive is a home-based work trip, i.e., a tripthat originates in a household’s residential area, andarrives in that household’s work area. Other examplesinclude recreational and social motives, shopping, . . . andthe chaining of activities. Based on these intentions,productions and attractions consist of absolute counts,denoting the number of trips that depart from andarrive in each zone. Because of this, productions andattractions are in facttrip ends. Both of them are derivedusing techniques based on regression analysis, categoryanalysis, or even logit models. As different models canbe used for the derivations of the number of productionsand attractions, an a posteriori balancing is performedthat equalises both results. In the end, step (I) gives themagnitude of the total travel demand on the network.Note that all activities (i.e., the original motives) are at

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this point in effect transformed andaggregatedinto trips.More importantly, these trips areonly considered for aspecific timeperiod (e.g., the morning rush hour).

(II) Trip distributionOnce the total number of productions and attractions forall zones in the transportation network is known, the nextstep then consists of deriving how many trips, originat-ing in a certain zone, arrive at another zone. In otherwords, step (II)connectstrip origins to theirdestinationsby distributing the trips. The result of step (II) is then theconstruction of a completeorigin-destinationtable (ODtable). In such an OD table (orOD matrixas some peo-ple say), an element at a rowi and a columnj denotesthe total number of trips departing from origin zoneOi

and arriving in destination zoneDj . Diagonal elementsdenote intra-zonal trips. Note that step (II) does not stateanything about the differentroutesthat can be taken be-tween two such zones; this is something that is derivedin the final step (IV). Because of the implicit assump-tion in step (I), namely that all trips are considered for aspecific time period, the same premise holds for all thederived OD tables. Consequently, the four step model isapplied for different time periods, e.g., during rush hoursor off-peak periods. In this context, we advise to use thenomenclature oftime-dependentor dynamicOD tables,denoting OD tables that are specified for a certain period,e.g., from 07:00 until 08:00 (or even tables given for con-secutive quarter-hours).

Considering the fact that an OD table contains a largeamount of unknown variables (it is a considerably underdetermined system of equations), several techniques havebeen introduced to deal with this problem by introducingadditional constraints. If an OD table for a previous pe-riod (called abase table) is known, then a new OD tablecan be derived by using a so-calledgrowth factor model.Another method is by usinggravity models(also knownasentropy models, see e.g., the discussion in by Helbingand Nagel [122]), which are based ontravel impedancefunctions. These functions reflect the relative attractive-ness of a certain trip e.g., based on information retrievedfrom household travel surveys. In most cases, they arecalibrated as power or exponential functions. One of theharder problems that still remains to be solved, is howto deal with so-calledthrough trips, i.e., trips that origi-nate or end outside of the study area. Horowitz and Patelfor example, directly incorporate rudimentary geograph-ical information and measured link flows into a modelthat allows to derive through-trip tables, using a notionof external stations located in an external territory. Ap-plication of his methodology to regions in Wisonsin andFlorida, result in reasonable estimates of link flows thatare comparable with empirically obtained data [137].

Besides using results from productions and attractions,gathering the necessary information for construction ofOD tables can also be done using other techniques. Anequivalent methodology is based on the considerationof turning fractionsat intersections. The process can be

largely automated when using video cameras coupledwith image recognition software. Furthermore, thereliterally exist thousands of papers devoted to the estima-tion of origin-destination matrices, mostly applicable tosmall-scale vehicular transportation networks and localroad intersections. Some past methodologies used arethe work of Nihan and Davis who developed a recursiveestimation scheme [223], the review Cascetta andNguyen who casted most earlier methods into a unifiedframework [44], and Bell who estimated OD tablesbased on constrained generalised least squares [21]. Anexample of a more recent technique is the work of Li andDe Moor who deal with incomplete observations [167].

(III) Mode choice / modal splitOnce the origin-destination table for the given networkand time period is available, the next step deals with thedifferentmodes of transportationthat people choose be-tween. Typical examples are the distinction between pri-vate and public transportation (both vehicular and rail-road traffic). The ‘split’ in this step, refers to the fact thatthe OD table obtained from step (II), is now divided overthe supported transportation modes. To this end,discretechoice theoryis a popular tool that allows a disaggrega-tion based on the choice of individual travellers, e.g., byusing utility theory based on a nested logit model [23]. Amodern trend in this context is to work with fullymulti-modal transportation networks; these multi-layered net-works provide access points for changing from one layer(i.e., mode of transportation) to another [273].

Historically, steps (I) — production and attraction — and(III) were executed simultaneously, but nowadays theyare considered separate from step (I): the main reasonis the fact that the modal choice is not only dependenton e.g., a household’s income, but also on the type oftrip to be undertaken, as well as the trip’s destination.As a result, the modal split can be intertwined with step(II), trip distribution, or it can be executed subsequentlyafter step (II). In the former case, the same kind of travelimpedance functions are used in combination with anadjusted gravity model, whereas in the latter case, ahierarchic logit model can be used.

(IV) Traffic assignmentAt this point in the four step model, the total amount oftrips undertaken by the travellers is known. The fourthand final step then consists of finding outwhich routesthese travellers follow when going from their origins totheir destinations, i.e., which sequence of consecutivelinks they will follow ? In a more general setting, this pro-cess is known astraffic assignment, because now the totaltravel demand (i.e., the trips) are assigned to routes in thetransportation network. Note that in some approaches,an iteration is done between the four steps, e.g, using thetraffic assignment procedure to calculate link travel timesthat are fed back as input to steps (II) and (III).

It stands to reason that all travellers will endeavour to takethe shortest routebetween their respective origins and

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destinations. To this end, a suitable measure of distanceshould be defined, after which a shortest path algorithm,e.g., Dijkstra’s algorithm [81], can calculate the possi-ble routes. Such a notion of distance typically includesboth spatial and temporal components, e.g., the physicallength of an individual link and the travel time on thislink, respectively. The use of the travel time is one ofthe most essential and tangible components in travellers’route choice behaviour. Note that in a more general set-ting, the distance can be considered as acost, wherebytravellers then choose thecheapest route(i.e., the quick-est route when time is interpreted as a cost). Daganzocalls these formulations theforward shortest path prob-lem, as opposed to thebackward shortest path problemthat tries to find the cheapest route for a given arrival time[73].

The basic principles that underlay route choice behaviourof individual travellers, were developed by Wardrop in1952, and are still used today. In his famous paper, relat-ing space- to time-mean speed, Wardrop also stated twopossible criteria governing the distribution of traffic overalternative routes [289]:

User equilibrium (W1): “The journey timeson all the routes actually used are equal, andless than those which would be experienced bya single vehicle on any unused route.”

System optimum (W2): “The average journeytime is a minimum.”

The above two criteria are based on what is called theNash equilibriumin game theory [209], albeit that nowa very large number of individuals are considered1. Inthe first criterion (W1), it is assumed that all individu-als’ decisions have a negligible effect on the performanceof others. Two, more important, fundamental principleshere are the fact that in the equilibrium situation, there isno cooperation between individualsassumed, and that allindividuals make their decisions in anegoistic and ratio-nal way [107]. In real-life traffic, everybody is expectedto follow the first criterion (W1), such that the whole sys-tem can settle in an equilibrium in which no one is betteroff by choosing an alternative route. In this respect, thework of Roughgarden is interesting because it providesa mathematical basis for the quantification of the worstloss of social welfare due toselfish routing, and the man-agement of networks that limit these effects in order toobtain a socially desirable outcome [254]. In contrast tothis user equilibrium situation, the second criterion (W2)

1The difference between a Wardrop and a Nash equilibrium is asubtle but important one. In the Wardrop case, an infinite number ofindividuals is considered, each seeking their own optimum.Note thatthe concept ‘infinite number of individuals’ can practically be approxi-mated by ’a large amount of individuals’. The Nash case also considersan infinite number of individuals, but they are now grouped into a fi-nite number of classes, with each class seeking its own optimum. Ifin this latter case the number of classes goes to infinity, then the Nashequilibrium converges to a Wardrop equilibrium [112].

is unlikely to occur spontaneously. However, when theperceived cost of a route by a traveller is changed to ageneralisedor marginal cost(i.e., including the costs ofthe effects brought on by adding an extra vehicle to thetravel demand), then a system optimum is achieved withrespect to these latter costs. In any case, as some peoplewill be better off, others will be worse off, but the trans-portation system as a whole will be best off.

The above two principles, are a bit idealistic, in thesense that there are many exceptions to these behaviouralguidelines. For example, in urban city centres, a signifi-cant part of the congestion can be brought on by vehicleslooking for parking space. Furthermore, many driversjust follow theirusual route, because this is the route theyknow best, and they know what to expect with respect totravel time. In a broader setting, this make these ‘stan-dard’ routes moreappealingto road users than other un-familiar alternative routes. In some cases however, trav-ellers will opt for these less known routes, thereby possi-bly entering therisk of experiencing a higher travel timeas has been concluded in the work of Chen and Recker[50]. Another fact that we expect to have a non-negligibleeffect on the distribution of traffic flows, is that nowadaysmore and more people useintelligent route plannerstoreach their destinations. These planners take into accountcongestion effects, as the trip gets planned both spatiallyand temporally. This will result in a certain percentageof the population that is informed eitherpre-routeor en-route, and these people can consequently change theirdeparture time or actual route (e.g., through route guid-ance), respectively. Another interesting research prob-lem arises because transportation infrastructure managersshould then be able to adapt their policies to the changingtravel patterns. For example, how should a policy makeroptimally control the traffic when only 20% of the popu-lation will follow the proposed route guidance ?

Due to the importance of the subject, we have devotedtwo separate sections in this dissertation to the concept oftraffic assignment. In these sections, we discuss the traf-fic assignment procedure in a bit more detail, consideringtwo prominent methodologies from a historic perspec-tive, namelystaticversusdynamictraffic assignment.

1.2.3 Static traffic assignment

The classic approach for assigning traffic to a transporta-tion network, assumes that all traffic flows on the networkare in equilibrium. In this context, thestatic traffic as-signment(STA) procedure can be more correctly consid-ered as dealing withstationaryof steady-stateflows: thetravel demand and road infrastructure (i.e., the supply)are supposed to be time-independent, meaning that thecalculated link flows are the result of a constant demand.In a typical setup, this entails the assignment of an hourly(or even daily) OD table to the network (e.g., during on-and off-peak periods), resulting inaverageflows for thespecified observation period. Because the STA methodol-ogy neglects time varying congestion effects (it assumes

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constant link flows and travel times), various importantphenomena such as queue spill back effects are not takeninto account.

In general, several possible techniques exist for achiev-ing an STA. The first one assumes (i) that all drivers willchoose the same cheapest route between a pair of originsand destinations, (ii) that they all have the sameperfectinformation about the links’ impedances, and (iii) thatthese impedances are considered to be constant, i.e., inde-pendent of a link’s traffic load (so no congestion buildupis taken into account). As the methodology implies, thisis called anall-or-nothing assignment(AON). A secondtechnique refines this notion, whereby differences amongdrivers are introduced (i.e., giving rise to imperfect in-formation), resulting in astochastic assignment. In thismethodology, the link travel impedances are assumed tobe probabilistically distributed: for each link in the net-work, an impedance is drawn from the distribution afterwhich an AON assignment is performed on the resultingnetwork. This Monte Carlo process is repeated until acertain termination criterion is met.

Both previous methods carry a significant drawback withrespect to link capacities, that is to say, no effects aretaken into account due to the fact that an increased flowon a link will generally result in an increase of the traveltime (i.e., the link’s impedance). To this end, a thirdmethod introducescapacity restraintssuch that an in-crease of the travel demand on a link, will result in ahigher cost (thereby possibly changing the route withthe cheapest cost). This method is called anequilib-rium assignment, and just like as in the second method,a stochastic equilibrium assignmentversion can be de-rived, taking into account travellers’ imperfect knowl-edge. The underlying assumption is that all travellersbehave according to Wardrop’s user equilibrium (W1).Furthermore, the capacity restraints are included in thetravel impedance functions, as they are now synony-mously calledtravel time (loss) functions, congestionfunctions, volume delay functions, link impedance func-tions, or evenlink performance functions. A popular formof these functions that express the travel timeT in func-tion of the flowq on a link, is theBureau of Public Roads(BPR) power function [38]:

T = Tff

(1 + α

(q

qpc

)β)

. (1)

In this BPR relation, the coefficientsα andβ determinethe shape of the function. An example of such a functionis depicted in Fig. 2. For low flows, the BPR functionis rather flat and the travel time corresponds to the traveltime Tff under free-flow conditions. When higher flowsoccur on the link, the coefficientβ determines the thresh-old at which the BPR function rises significantly (in someformulations it asymptoticallyapproaches the capacityflow). The travel time will increase with the ratio of theflow q and the so-calledpractical capacityqpc. This lattercharacteristic is derived from the value of the travel time

under congested conditions. As a result, the practical ca-pacity is different from the maximum capacity of a linkas defined by a fundamental diagram. Finally, note thata serious disadvantage associated with these BPR func-tions in combination with static traffic assignment, is thefact that the travel demand on the network at a certaintime does not always correspond to the actual physicalflows that can be sustained. Under congested conditions,this implies that the flows in the STA approach can behigher than the physically possible link capacities (whichare different from the previously mentioned practical ca-pacities), leading to an incorrect assignment with faultyoversaturated links.

T

Tff

qqpc0

Figure 2: The Bureau of Public Roads (BPR) function,relating the travel time to the flow. It is based on the traveltime Tff under free-flow conditions and the practical ca-pacityqpc of the link under consideration.

Once the travel time of a link can be related to its cur-rent flow using e.g., a BPR function, an iterative schemeis adopted to calculate the equilibrium traffic assignment.Popular implementations are theFrank-Wolfe algorithm[91], and themethod of successive averages(MSA). Theformer method is based on principles of optimisation the-ory, as demonstrated by Beckmann et al. [20, 31] who re-formulated the Wardrop equilibrium as a convex optimi-sation problem, consisting of a single objective functionwith linear inequality constraints based on the Karush-Kuhn-Tucker (KKT) conditions, thereby resulting in aglobal minimum. Because travellers do not have perfectinformation, Daganzo and Sheffi formulated a variationon Wardrop’s first criterion (W1), whereby all traffic dis-tributes itself over the network with respect to aperceivedtravel timeof the individual drivers [77]. The resultingstate of flows on the network is called astochastic userequilibrium (SUE), as opposed to thedeterministic userequilibrium (DUE). Note that a further discrimination ispossible, as proposed by the work of Chen and Recker,who also make a distinction between travellers’ percep-tion errors on the one hand, and network uncertainty (i.e.,stochasticity of the travel times) on the other hand [50].For a thorough overview of the STA approach, we referthe reader to the work of Patriksson [229].

Although, as mentioned earlier, time varying congestioneffects are not taken into account, the STA approach doesfit nicely into the concept of long-term transportation

7

planning. For short-term analyses however, these effectscan have a significant impact on the end results, thusrequiring a more detailed approach to traffic assignment.

1.2.4 Dynamic traffic assignment

As explained in the previous paragraphs, the static trafficassignment heavily relies on simple travel time functions(e.g., BPR). An associated problem with these is thedifficulty in capturing the concept of ‘capacity of aroad’. In reality, congestion is a dynamic phenomenon,whereby its temporal character is not to be neglected.To tackle theses problems inherent to the STA approach,a more dynamic treatment of traffic assignment is nec-essary [180]. A fundamentally important aspect in thisdynamic traffic assignment(DTA) procedure, is the factthat congestion has a temporal character, meaning thatits buildup and dissolution play an important role: thehistory of the transportation systemshould be taken intoaccount (e.g., congestion that occurs due to queue spillback) [66]. Neglecting this time dependency by assum-ing that the entry of a vehicle to a link instantaneouslychanges the flow on that link, results in what is calledSmeed’s paradox. This leads to incorrect behaviour asa result of a violation of the so-called ‘first-in, first-out’(FIFO) property, because now a vehicle can leave linkearlier then a vehicle that enters it later (i.e., arrivingearlier by departing later) [260]. The methodology ofdynamic traffic assignment was now designed to dealwith all these particular aspects. The DTA technique iscomposed of two fundamental components:

Route choiceJust as in the STA approach, each traveller in the trans-portation network follows a certain route based on aninstinctive criterion such as e.g., Wardrop’s (W1). Theassociated component that takes care of travellers’ routechoices, can be complemented to allow for imperfectinformation. Another, more important, aspect related tothe route choice, is a traveller’schoice of departure time.An STA approach assumes that all traffic of a given ODtable is simultaneously assigned to the network, whereasDTA coupled with departure time choice can spread thedepartures in time (leading to e.g., realistically spreadingof the morning peak’s rush hour).

Dynamic network loading (DNL)Instead of using simple travel time functions, a DTA ap-proach typically has a component that loads the trafficonto the network. In fact, this step resembles thephys-ical propagationof all traffic in the network. In or-der to achieve reliable and credible results, a good de-scription of the network’s links is necessary, as well asthe behaviour of traffic at the nodes connecting the linkswithin this network (i.e., this is a mandatory requirementto achieve correct modelling of queue spill back). The

DNL component in the DTA approach has been an ac-tive field of research during the last decade, and it stillcontinues to improve the state-of-the-art. Testimonies in-clude the use ofanalytic modelsthat give correct repre-sentations of queueing behaviour, as well as detailed sim-ulations that describe the propagation of individual vehi-cles in the transportation network. Note that in the caseof simulation-based(also calledheuristic) traffic assign-ment, the route choice and DNL components can be iter-atively executed, whereby the former establishes a set ofroutes to follow, and the latter step feedbacks informationto the route choice model until a certain termination cri-terion is met (e.g., a relaxation procedure). Furthermore,using simulation-based traffic assignment with very largeroad networks is not always computationally feasible tocalculate all shortest paths. As a result, it might be ben-eficial to resort to simplifications of either the simulationmodel (e.g., using faster queueing models), or the num-ber of paths to consider (e.g., based on the hierarchy in-herently present in the road network) [253]. Finally, wemention the work of Astarita who provides an interestingclassification of DNL models, based on the discretisationwith respect to the spatial and temporal dimensions, aswell as with respect to the modelling of the traffic de-mand [10].

Despite the appealing nature of simulation-based DTA,there is in contrast to the STA approach, no unified frame-work that deals with the convergence and stability issues[94, 93, 233].

Some examples of these DTA mechanisms are: Gawronwho uses a queueing model to develop a simulation-based assignment technique that is able to deal withlarge-scale networks and is proven to be empirically sta-ble [94, 93], Bliemer who developed an analytical DTAapproach (with different user-classes) based on a vari-ational inequality approach [26], Bliemer’s work fur-thermore culminated in the development ofINteractiveDYnamic traffic assignment(INDY) [185] which — incombination with the OmniTRANS2 commercial trans-portation planning software — can be used as a fullyfledged DTA analysis tool [280], Lo and Szeto who de-veloped a DTA formulation based on a variational in-equality approach leading to a dynamic user equilib-rium [174], the group of Mahmassani who is actively en-gaged in the DTA scene with the development of the DY-NASMART (DYnamic Network Assignment-SimulationModel for Advanced Roadway Telematics) simulationsuite [189], . . . An excellent comprehensive overview ofseveral traditional DTA techniques is given by Peeta andZiliaskopoulos [234].

Another important field of research, is how individualroad travellers react to the route guidance they are given.In his research, Bottom considered this type ofdynamictraffic management(DTM), providing route guidance totravellers whilst taking into account theiranticipated be-haviourduring e.g., incidents [28]. Taking this idea one

2http://www.omnitrans-international.com

8

step further, it is possible to study the interactions be-tween the behaviour of travellers in a road network, andthe management of all the traffic controls (e.g., traffic sig-nal lights) within this network. An example of such adynamic traffic control(DTC) and DTA framework, isthe work of Chen who considers the management froma theoretic perspective based on a non-cooperative gamebetween road users and the traffic authority [51].

1.2.5 Critique on trip-based approaches

Considering its obvious track record of the past severaldecades, theconventionaluse of the trip-based approachis — to our feeling — running on its last legs. By ‘con-ventional’ we denote here the fact that the currentstate-of-the-practiceis still firmly based on the paradigm ofstatic traffic assignment, despite the recent (academic)progress on the front of dynamic traffic assignment tech-niques. The four step model still largely dominates thecommercial business of transportation planning, althoughits structure remained largely unchanged since its originalinception. As mentioned earlier, in the case of STA, alltrips are assumed to depart and arrive within the speci-fied planning period. This leads to an unnatural discrep-ancy between models and reality in congested areas dur-ing e.g., a morning rush hour: some travellerswant tomake a trip and, in the former case, are perfectly allowedto achieve this trip, whereas in the latter case they are infact physicallyunableto make the trip due to dynamicalcongestion effects.

In order to facilitate this disagreement between the bal-ancing of travel demand versus supply (i.e., the trans-portation infrastructure), the DTA approach is gainingimportance as more features are provided. An exam-ple of such a feature includes the framework of conges-tion pricing, where we have an incorporation of depar-ture time choice models coupled with the derivation ofoptimal road tolls. Some noteworthy studies that havebeen carried out in this respect, are the work of de Palmaand Marchal who present the METROPOLIS toolbox, al-lowing the simulation of large-scale transportation net-works [78], the work of Lago and Daganzo who com-bined a departure time equilibrium theory with a fluid-dynamic model in order to assess congestion policy mea-sures [159], the work of Szeto and Lo who coupled routechoice and departure time choice with the goal of nu-merically handling large-scale transportation networks[261, 175]. Closely related to Lago’s and Daganzo’swork is that of Yperman et al., who determined an op-timal pricing policy, describing the demand side with abottleneck model and an analytical fluid-dynamic modelas the DNL component [298].

At this point, we should mention some of the compli-cations associated with the traditional method of mod-elling traffic flow propagation using queue-based analo-gies. Historically, there have been two different queueingtechniques with FIFO discipline that describe this aspectin a trip-based assignment procedure:

• Point queues (also calledvertical queues): this typeof queue has no spatial extent. Because vehiclescan always enter the queue, and leave it after acertain delay time, congestion is incorrectly mod-elled. A well-known example of a model based onthis queueing policy is Vickrey’s bottleneck model[281].

• Spatial/physical queues (also calledhorizontalqueues): a queue of this type has an associatedfinitecapacity, i.e., a buffer storage. Vehicles can onlyenter the queue when there is enough space for themavailable.

The correct modelling of congestion effects such asqueue spill back, is of fundamental importance when as-sessing certain policy measures like e.g., road pricingschemes. To this end, the use of vertical queues shouldbe abandoned, in favour of horizontal queues. How-ever, even horizontal queues have problems associatedwith them: the buildup and dissolution of congestion in atransportation network are flawed, e.g., vehicles that areleaving the front of a queueinstantly open up a space atthe back of this queue, thus allowing an upstream vehi-cle to enter. This leads to shorter queue lengths, becausethe physical queueing effect of individual vehicles (i.e.,the upstream propagation of the empty spot) is absent[93, 256, 47, 159]. In order to alleviate this latter issue,a more realistic velocity should be adopted for the back-ward propagating kinematic wave, thus calling for moreadvanced modelling techniques thatexplicitly describethe propagation of traffic (e.g., fluid-dynamic approaches,models with dynamical vehicle interactions, . . . ).

As often is the case, a model’s criticisms can be found inits underlying assumptions. In the case of the four stepapproach, it is obvious that all information regarding in-dividual households is lost because of its aggregation toa trip level. As was already recognised from the start, theindividual itself loses value during this conversion pro-cess. This opened the door towards another approachto transportation planning, more preciselyactivity-basedmodelling(ABM) which is discussed in the next section.

A final complaint that is more common around many ofthese grotesque models, is their requirement of a vastamount of specific data. In many cases, a diverse range ofnational studies are carried out, having the goal of gath-ering as much data as possible. Regardless of this op-timism, some of the key problems remain, e.g., it is stillnot always straightforward to properly interpret and adaptthis data so it can be used as input to a transportation plan-ning tool.

1.3 Activity-based transportation models

As it was widely accepted that the rationale for travel de-mand can be found in people’s motives for participatingin social, economical, and cultural activities, the classictrip-based approach nevertheless kept a strong foothold

9

in the transportation planning community. Instead of fo-cussing attention elsewhere, the typical institutional pol-icy was to ameliorate the existing four step models [191].However, some problems persistently evaded a solutionwith the trip-based approach, e.g., shops that remain openlate, employers who introduce flexible working hours, theconsideration of joint activities by members of a house-hold, . . .

In the next few sections, we illustrate how all this changedwith the upcoming field of activity-based transportationplanning. We first describe its historic origins, afterwhich we move on to several of the approaches taken inactivity-based modelling. The concluding section gives aconcise overview of some of the next-generation mod-elling techniques, i.e., large-scale agent-based simula-tions.

1.3.1 Historic origins

The historic roots of the activity-based approach canprobably be traced back to 1970, with the querulouswork of Torsten Hagerstrand [106]. He asserted that re-searchers in regional sciences should focus more on theintertwining of both disaggregate spatial and temporal as-pects of human activities, as opposed to the more aggre-gate models in which the temporal dimension was ne-glected. This scientific field got commonly termed astime geography; it encompasses all time scales (i.e., fromdaily operations to lifetime goals), and focusses on theconstraints that individuals face rather than predictingtheir choices [194].

Central to Hagerstrand’s work was the notion of so-calledspace-time pathsof individuals’ activity and travel be-haviour. In a three-dimensional space-time volume, twospatial dimensions make up the physical world plane,with the temporal dimension as the vertical axis. Thejourney of an individual is now the path traced out inthis space-time volume: consecutive visits to certain lo-cations are joined by a curve, with vertical segments de-noting places where the individual remained stationaryduring a certain time period. The complete chain of ac-tivities (called atour) is thus joined by individualtriplegs. In this respect, the space-time path represents therevealed outcomeof an unrevealed behavioural process[191]. An example of such a path can be seen in Fig. 3:we can see a woman going from her home in Boulder(Colorado, USA), to the university’s campus, followedby a visit to the post office and grocery store, and finallyreturning home [80]. Note that Hagerstrand extended hisnotion in the space-time volume to includespace-timeprismsthat encapsulate and effectivelyconstrainall of aperson’s reachable points (i.e., all his/her possible space-time paths), given a certain maximum travel speed as wellas both starting and ending points within the volume [61].This environment is sometimes also referred to as a per-son’saction space, enveloping that person’stime budget.

Contrary to the belief that the field of activity-based trans-portation planning found its crux with the dissatisfac-

Figure 3: An example of a space-time path showing anindividuals’ activity and travel behaviour in the space-time volume: the two spatial dimensions make up thephysical world plane, with the vertical axis denoting thetemporal dimension. In this case, we can see a womangoing from her home in Boulder (Colorado, USA), to theuniversity’s campus, followed by a visit to the post of-fice and grocery store, and finally returning home (imagereproduced after [80]).

tion of trip-based modelling, it grew and emerged spon-taneously as a separate research study into human be-haviour [191]. The underlying idea however remained thesame as in the trip-based approach, namely that travel de-cisions arise from a need to participate in social, econom-ical, and cultural activities. But as opposed to the moreaggregated trip-based view, the basic units here are indi-vidual activity patterns, commonly referred to ashouse-hold activity patterns. In this context, the activity-basedapproach then studies the interactions between membersof a household, and the relation to their induced travelbehaviour [12].

1.3.2 Approaches to activity-based modelling

Departing from Hagerstrand’s initial comments, activity-based research progress has been slowly but steadily. Incontrast with the development of the trip-based approachthat culminated in the four step model,there is no ex-plicit general framework that encapsulates the activity-based modelling scheme. There were however early com-prehensive studies into human activities and their relatedtravel behaviour, e.g., the synopsis provided by Jones etal. [145]. As the field began to mature, certain ingredi-ents could be recognised, e.g. [12]:

• the generation of activities, which can be regardedas the equivalent of the production/attraction step intrip-based modelling,

• the modelling ofhousehold choices, i.e., with re-spect to their activity chains; this includes choosingstarting time and duration of the activity, its locationas well as a modal choice,

10

• thescheduling of activities, outlining how a house-hold plans and executes the tasks of its members forlong-, middle-, and short-term activities, going fromyear- and lifetime-long commitments, to daily oper-ations.

During the last three decades, many research models thatencompass activity scheduling behaviour have been de-veloped. An excellent overview is given by Timmer-mans, who makes a distinction betweensimultaneousandsequential models[265]. The former class is based onfull activity patterns (e.g., for one whole day), whereasthe latter is based on an explicit modelling of the activ-ity scheduling process. Simultaneous models compriseutility-maximisation models and mathematical program-ming models (e.g., Recker’s household activity patternproblem – HAPP, [247]). Sequential models are fre-quently implemented as so-called computational processmodels (CPM), acknowledging the belief that individu-als do not arrive at optimal choices, but rather employcontext-dependent heuristics.

As an example, we illustrate the seminal STARCHILDmodel, which was originally a simultaneous model basedon the maximisation of individuals’ utilities. Based onHagerstrand’s notion and derivatives thereof, i.e., thecentral idea that an individual’s travel behaviour is con-strained by its space-time prism, Recker et al developedthe STARCHILD research tool addressing activity-basedmodelling [248, 249]. The model hinges on three inter-dependent consecutive steps: (i) the generation of house-hold activities, (ii) constructing choice sets for these ac-tivities, as well as scheduling them, and (iii) constrain-ing these choices within the boundaries of the space-time prism [191]. Note that the principal critique to themodel’s operation, was — and today still is — its need foran extensive amount of specifically tailored data that en-compasses Hagerstrand’s concepts. Just as with the fourstep model, these data are arduous to come by. In short,most of the data are based on and transformed from e.g.,conventional trip-based surveys,travel diaries(e.g., theMOBEL (Belgium) and MOBIDRIVE (Germany) sur-veys of Cirillo and Axhausen [59]) and the like, althoughmore recently passive GPS based information is collected[12, 191, 194].

In the future, a complete integration of activity genera-tion, scheduling, and route choice (DTA) is expected totake place, on the condition that suitable empirical datawill become available. We must however be careful notto be too optimistic, as e.g., Axhausen states that depend-ing on the ‘research-political’ adoption of the activity-based approach,“both a virtuous circle of progress ora vicious circle of stagnation are a possibility for thefuture” [12]. An even more harsh argumentation wasvoiced by Timmermans, who looked back at the devel-opment of the integration between land-use models andtransportation planning [266]. In his overview, he iden-tified three waves, i.e., (i) aggregate spatial interaction-based models, (ii) utility-maximising multinomial logit-

based models, and (iii) activity-based detailed microsim-ulation models. His final conclusion states that, despitethe advances in finer levels of spatial detail, the scien-tific field has not undergone any significant theoreticalprogress. And although there exists a pronounced needfor better behavioural models, the critique remains thatthis implies a tremendous complexity, hence the insinu-ation that many of the approaches are in fact based onblack-box models.

1.3.3 Towards elaborate agent-based simulations

One of the most notable critiques often expressed againstclassic trip-based approaches such as the four step model,is the fact that all eye for detail at the level of the in-dividual traveller is lost in the trip aggregation process.Activity-based modelling schemes try to circumvent thisdisadvantage by starting from a fundamentally differentbasis, namely individual household activity patterns. Tothis end, it is necessary to retain all information regardingthese individual households during the planning process.

As hinted at earlier, an upcoming technique that fitsnicely in this concept, is the methodology ofmulti-agentsimulations. In such models, the individual householdsare represented asagents; the models then allow theseagents to make independent decisions about their actions.These actions span from long-term lifetime residentialhousing decisions, the mid-term planning of daily activ-ities, to even short-term decisions about an individual’sdriving behaviour in traffic. The following descriptionof such a simulation system is based on the work of thegroup of Nagel et al. [199, 13, 14, 15, 202, 122]:

• As a first step, asynthetic populationof agents isgenerated. There is a close relation with the com-mon land-use models, as these agents come frompopulations that should be correctly seeded, i.e.,they should entail a correct demographic representa-tion of the real world. Once the synthetic populationis available to the model, the next step is togener-ate activity patterns(i.e., activity chains), generatethese activities’locations, and finally theschedulingof the activities, as described in the previous section.Finally, modeandroute choiceform the bridge be-tween the activity-based model and the transporta-tion layer. As a consequence, it is beneficial to dealwith agents’ plans directly, rather than to rely on theinformation contained in OD tables [13].

• The component that represents thephysical propa-gation of agentsthroughout e.g., the road network,sits at the lowest level of the model. In this case, thenecessary ingredients constitute the physical propa-gation of individual vehicles in the traffic streams.Popular models aretraffic cellular automataand/orqueueing models, allowing a fast and efficient sim-ulation of individual agents in a network. Higherlevel models such as e.g., pure fluid-dynamic mod-els are inherently not suitable because they operate

11

on a more aggregated basis and consequently ignorethe individuality of each agent in the system. Notethat this latter type of model can be deemed appro-priate, on the condition that they can incorporate thetracking of individual particles by e.g., a smoothedparticle hydrodynamics method [14].

• An important issue that revolves around the two pre-vious aspects, is theclear absence of a rigidly de-fined direction of causality, i.e., when exactly dopeople choose their travel mode, is it before theplanning of activities, or is it rather a result from theplanning process ? This problem can be dealt within a broader context, wherein agents make certainplans about their activities, anditeratively learnbyreplanning and rescheduling (either on aday-to-dayor within-daybasis). This process ofsystematic re-laxationcontinues until e.g., a Wardrop equilibrium(W1) is reached (see step (IV) in section 1.2.2 formore details). However, note that the question ofwhether or not people in reality strive to reach suchan equilibrium, and whether or not such an equilib-rium is even reached, remains an open debate. Atthis stage in the model, we are clearly dealing withaspects from evolutionary game theory, be it cooper-ative or non-cooperative. In this context, the conceptof within-day replanning by agents is getting moreattention, as it constitutes a necessary prerequisitefor intelligent transportation systems, i.e., when andhow do travellers react (e.g.,en-route choice) tochanges (e.g., control signals, incidents, . . . ) in theirenvironment [14] ?

The above description of a multi-agent activity-basedsimulation system may seem straightforward, neverthe-less, no complete practical implementation exists today.The model suite that comes the closest to reaching thepreviously stated goals, is theTRansportation ANaly-sis and SIMulation System(TRANSIMS3) project. Thisproject is one part of the multi-track Travel Model Im-provement Program of the U.S. Department of Trans-portation, the Environmental Protection Agency, and theDepartment of Energy in the context of the IntermodalSurface Transportation Efficiency Act and the Clean AirAct Amendments of 1990. Its original developmentstarted at Los Alamos National Laboratory, but a com-mercial implementation was provided by IBM BusinessConsulting.

Since its original inception, TRANSIMS has been ap-plied to a various range of case studies. One of the mostnotable examples, is the truly country-wide agent-baseddetailed microsimulation of travel behaviour in Switzer-land (see also Fig. 4) [282, 245]. A similar study encom-passing the iteration and feedback between a simulationmodel and a route planner, was carried out for the regionof Dallas [198, 204]. In the context of large-scale agent-based simulations, queueing models were employed as aTRANSIMS component by the work of Simon et al. for

3http://www.transims.net

the city of Portland [256], as well as the work of Gawron[93, 94] and Cetin et al. [47]. Because of the computa-tional complexity involved in dealing with the enormousamount of agents in real-world scenarios, a popular ap-proach is the use of parallel computations, as describedin the work of Nagel and Rickert [203]. Another exam-ple of this last type of simulations, is the work of Chopardand Dupuis who applied the methodology of large-scalesimulations to the city of Geneva [53, 52, 83].

Figure 4: An example of a multi-agent simulation of theroad network of Switzerland, around 08:00 in the morn-ing: each vehicle is indicated by a single grey pixel, withlow-speed vehicles coloured black. The image clearly re-veals more vehicular activity (and congestion) in the citycentres than elsewhere in the country (image reproducedafter [282]).

As a final remark, we would like to draw attention tosome morecontrol-oriented aspects of multi-agent sim-ulations. In this respect, the transportation system is con-sidered as a whole, whereby the agents are now the lo-cal controllers within the system (e.g., traffic lights, vari-able message signs, . . . ), instead of individual house-holds as was previously assumed. Using a coordinatedcontrol approach, is it then possible to achieve a sys-tem optimum. An example of such a control system isthe Advanced Multi-agent Information and Control forIntegrated multi-class traffic networks(AMICI 4) projectfrom The Netherlands. As one of its goals, it strives toprovide routing information to different classes of roadusers, as well as controlling them by means of computersimulated agents who operate locally but can be steeredhierarchically.

1.4 Transportation economics

Most of the work related to traffic flow theory hasbeen considered by researchers with roots in engineer-ing, physics, mathematics et cetera. With respect to trans-portation planning, the scene has shifted somewhat dur-ing the last couple of decades towards policy makers who

4http://www.amici.tudelft.nl

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test and implement certain strategies, based on e.g., thefour step modelling approach. Around 1960 however,another branch of scientists entered the field of trans-portation planning: economists developed standard mod-els that viewed transportation as a market exchange be-tween demand and supply.

Generally stated, the economics of transportation doesnot exclusively focus on traffic as a purely physical phe-nomenon, but also takes into account the fact thattrans-portation incurs certain costs, both to the individual aswell as to the society as a whole [172].

In the following sections, we describe the setting in whicheconomists view transportation, after which we discussthe concept of road pricing.

1.4.1 The economical setting

In the context of economic theory, a transportation sys-tem can be seen as an interaction betweendemand(prof-its) andsupply(costs). In a static setting, both demandand supply are frequently described by means of func-tions: they are expressed as theprice for a good asso-ciated with thequantityof that good. In transportationeconomics, quantity is frequently identified as thenum-ber of trips made(e.g., by the macroscopic characteristicof traffic flow) [20]. In the remainder of this section, weuse the termtravel demandto denote the demand side,as opposed to the supply side which is composed of thetransportation infrastructure (including changes due to in-cidents, . . . ). In a more broader context, travel demandis typically described as the amount of traffic volumethat wants to use a certain infrastructure (i.e., the sup-ply): when demand thus exceeds the infrastructure’s ca-pacity under congested conditions (implying queueing),this supply effectively acts as a constraint to the presentvolume of traffic flow.

According to the aforementioned conventions, a demandside function is expressed as a certain cost associated witha level of flow (i.e., number of trips). We call such acurve atravel demand function(TDF), and it is typicallydecreasing with increasing flow; an example of such afunction can be seen as the dotted curve in Fig. 5. Notethat, to be correct, the depicted curve actually representsamarginal travel demand: it gives the additional profit thatis received with the obtaining of one extra unit (the totalamount of profit is just the area under (i.e., the integral of)the demand curve). Translated to a transportation system,this means that the benefits of a traveller tend to decreasewith increasing travel demand (i.e., congestion).

In similar spirit, we can consider a supply side curve, i.e.,price (costs) versus quantity (flow). One of the most usedapproaches for describing traffic flow operations at thesupply side from an economical point of view, is the useof anaverage cost function(ACF) [277] as proposed byA.A. Walters in 1961 [288]. The theory was based on thedescription of traffic flow by means of fundamental dia-grams. Consider for example Greenshields’ simple lin-

ear relation between densityk and space-mean speedvs

[105, 182]. The correspondingvse(q) fundamental dia-

gram, consists of a tilted parabola, lying on its side [182].As the travel timeT is inversely proportional to the space-mean speedvs, Walters’ idea was to assume certaincostsrelated to the travel demand. Some examples of thesecosts are those associated with [27, 110, 111]:

• (i) the construction of the transportation infrastruc-ture,

• (ii) vehicle ownership and use,

• (iii) taxes,

• (iv) travel time, i.e.,value of time(VOT),

• and (v) environmental and social costs.

Based on these costs, and using the relation betweentravel time and travel demand, Walters derived a func-tional relationship for the economical costC associatedwith the travel demandq. This relationship (i.e., theACF) denotes the supply side of transportation eco-nomics; an example is the thick solid curve in Fig. 5.

Once both travel demand and average cost functionsare known, they can be used to determine theequilib-rium pointsthat occur at their intersection(s): givenboth curves, the transportation system is assumed tosettle itself at these equilibria, with a certain travelcost associated with the equilibrium traffic demand.Note that because of the nature of the analysis proce-dure, i.e., using static (stationary) curves, the result-ing travel costs areaveragecosts, hence the name ofthe average cost function.

There are some distinct features noticeable in the relationdescribed by the average cost function. For starters, thecurve does not go through the origin, i.e., at low travel de-mands there is already afixed costincurred. The depictedcost then typically increases with increasing travel de-mand, mainly due to the contribution of the value of timeassociated with the travel time. The most striking featurehowever, is the fact that the curve contains abackward-bendingupper branch [86]. This peculiar branch has anasymptotic behaviour, i.e.,

limq→qcap→0

C(q) = +∞, (2)

where we have denoted the path taken by the limit, i.e.,passing through the capacity flowqcap towards the upperbranch, which in fact corresponds to an increase towardsthe jam densitykjam. Also note the presence of an in-flection point (for concaveqe(k) fundamental diagrams),which can be located analytically by differentiating thefunctional relation twice, and solving it with a right handside equal to zero.

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ACF

TDF

congested

hypercongestedC

qqcap0

Figure 5: A graphical illustration of the economics oftransportation operations: the dotted curve represents thedemand side, i.e., the travel demand function (TDF),whereas the thick solid curve represents the supply side,i.e., the average cost function (ACF). Both curves expressthe costC associated with the number of trips made (e.g.,level of traffic flowq). The latter curve is said to have twostates, namely the congested and the hypercongested area(identified as the backward-bending part of the curve).Points where both demand and supply curves intersecteach other denote equilibrium points: given both curves,the transportation system as assumed to settle itself attheir intersection(s), with a certain travel cost associatedwith the equilibrium traffic demand.

In contrast to the nomenclature adopted by the traf-fic engineering community and in this dissertation,economists typically refer to the lower branch ofFig. 5 as thecongestedstate, and to the upper branchas thehypercongestedstate. Their line of reasoningbeing the conviction that in a certain sense, conges-tion also occurs when the speed drops below the free-flow speed on the free-flow branch [277, 172, 259].

With respect to the relevance of this hypercongested state,there has been some debate in literature. Among mosteconomists there seems to be a consensus, in the sensethat the hypercongestedbranch is actually a transient phe-nomenon [296, 172]. Walters thought of the branch asjust a collection of inefficient equilibria, but it was shownby Verhoef that all equilibria obtained on the hypercon-gested branch are inherently unstable [277, 259]. An-other argument, that discards the use of the branch, goesas explained by Yang and Huang [296]: many traditionaleconomical models of transportation assume a static (sta-tionary) model of congestion, similarly as in classic statictraffic assignment described in section 1.2.3. Under thispremise, the relations as described by the fundamental di-agrams, should be considered for complete links, and notonly — as is usual in traffic flow theory — at local pointsin space and time. Therefore, a link may contain twodifferent states: a free-flow state and a congested state.Hence, the average cost function should only describe theproperties that are satisfied on the whole link, and as a re-sult this excludes the global hypercongested regime.

Several ad hoc solutions exist for dealing with this prob-

lem, which is a consequence of using cost functions basedon stationary equilibria: some of these solutions typi-cally entail the use of vertical segments near the capac-ity flow in Fig. 5, resulting in finite queueing delays onheavily congested links [277, 296, 259, 278]. Anothermuch used solution that ignores the backward-bendingbranch, is to directly specify the average cost functionbased on a link’s observed capacity, instead of derivingit through the fundamental diagram of space-mean speedversus flow [172]. Note that in most cases,the economicinterpretation of capacity is different from the one in traf-fic flow theory: the capacity considered by economistshas a lower value as opposed to its engineering counter-part. A similar example that specifies the relation be-tween travel demand and travel time (e.g., VOT), is theBPR travel impedance function as described in section1.2.3. Notwithstanding these proposed specific solutions,the mainstream tendency nowadays seems to imply theuse of traffic flow models that explicitly describe the dy-namics of congestion, either by using queueing models,or more elaborate models based on fluid dynamics or de-tailed simulations of individual vehicles [296].

1.4.2 Towards road pricing policies

In an economical treatment of transportation, road usersin general only take into account their ownprivate costs,such as (ii) vehicle ownership and use, (iii) taxes, and(iv) costs related to the travel time. Note that because,as mentioned earlier, we are working withmarginal costfunctions, the cost (i) related to the transportation infras-tructure is not taken into account (as this is only a one-time initial cost).

To this end, we consider the average cost function fromtwo different points of view: on the one hand, we have theprivate costsborne by an individual traveller, and on theother hand, we have theexternal coststhat the travellerbears to the rest of society. In accordance with economicliterature, we call the former associated cost function themarginal private cost function(MPCF), and the latter themarginal social cost function(MSCF). The extra coststo society brought on by individual travellers, are callednegative externalities.

In Fig. 6, we have depicted the resulting equilibria thatarise from the intersections of the travel demand functionwith both marginal private and social cost functions (notethat we disregard the upper backward-bending branch ofthe average cost function as was shown in Fig. 5). Inan unmanaged society, i.e., where no measures are takento change individual travellers’ behaviour, the resultingequilibrium will be found atque, which is in fact auserequilibrium corresponding to a cost as dictated by themarginal private cost function (MPCF) [9]. As travellershandle selfishly, not considering the costs inflicted uponother travellers (e.g., more road users imply more conges-tion for everybody), this pricing method is termedaver-age cost pricing. At this equilibrium, the unpaid external

14

cost to society equals the difference between the MSCFand MPCF curves at a demand level ofque.

TDF

MPCF

MSCFC

qqueqso

optimaltoll

welfare benefit

0

Figure 6: An economical equilibrium analysis basedon a travel demand function (TDF) represented by thedotted curve, and marginal private and social cost func-tions (MPCF and MSCF) represented by the thick solidcurves. The user equilibrium is located at a demand ofque, whereas the system optimum is located at a lower de-mand ofqso. The welfare benefit (indicated as the greytriangular region) can be gained by levying a congestiontoll equal to the difference between the marginal socialand private cost function defined at the system optimumdemand levelqso.

As early as in 1920, Arthur Cecil Pigou noted that roadusers do not take into account the costs they inflict uponother travellers. In order to rectify this situation, he pro-posed to levy governmental taxes on road use. Pigou ac-tually discussed his idea in a broader economic setting,by making a distinction between the private and the so-cial costs. Charging of a suitable governmental tax couldchange the balance so the negative externalities would beincluded, resulting in a new equilibrium [237]. This pro-cess is calledinternalising the external costs.

Some years later in 1924, Frank Hyneman Knight furtherexplored Pigou’s ideas5: Knight fully acknowledged thefact that congestion justified the levying of taxes. In con-trast to Pigou however, Knight raised some criticism inthe sense that not governmental taxes were necessary, butinstead private ownership of the roads would take care oftax levying and consequently resulting in a reduction ofcongestion [150].

In 1927, Frank Plumpton Ramsey cast this methodology— calledmarginal cost pricing— in the light of socialwelfare economics. This latter type employs techniquesfrom a branch that is called micro-economics, which is aneconomical treatment of society based on the behaviourof individual producers and consumers. Welfare eco-nomics embraces two important concepts:

Efficiency: a measure for assessing how much bene-fit society gains from a certain policy rule. It can

5Note that Knight apparently was clued in his research by an errormade on Pigou’s behalf in his study of a two-route road network [30].Even more intriguing, is the fact that this type of problem was alreadyconsidered as far back as 1841, with the work of the German economistJohann Georg Kohl [151].

be considered with the(strict) Pareto criterion(in-vented by Vilfredo Pareto), which states that effi-ciency improves if a policy rule implies an increaseof welfare for at least one individual, but no other in-dividual of society is worse off. Nicholas Kaldor andJohn Hicks restated Pareto’s criterion, but this timefrom the point of view of those who gain and thosewho lose, respectively. TheirKaldor-Hicks criterionstates that society gains welfare, but not everybodyreceives personal gain, i.e., there will be winners andlosers. The crucial assumption on the Kaldor-Hickscriterion is that the winners could fully compensatethe losers, in theory; whether or not this happens atall, is not the issue.

Equity: if society benefits from a certain policy rule,then its efficiency can be measured using e.g., thePareto criterion as stated earlier. However, nothingis said about the size of the benefit each individualsof society receives. This is were the concept of eq-uity enters the picture: it refers to a fair distributionof the total benefits over all individuals in society(note that in this case, there typically is a strong cor-relation with theincome distribution).

In this context, Ramsey thus stated a policy rule, imply-ing a maximisation of the social welfare[244]. In thefield of transportation, this can be done by marginal costpricing, also calledroad pricing, congestion tolls, . . . Thenature of the measure is that it signifies a demand-sidestrategy, with the goal of inducing a change in travellers’behaviour. Road pricing typically entails a shift from on-peak to off-peak periods, switching mode (e.g., from pri-vate to public transportation), car pooling, route change,. . . Considering again Fig. 6, we can see that if users wereto consider the marginal social cost function, instead ofonly their marginal private cost function, this would shiftthe resulting equilibrium fromque to qso, which is aso-cial optimum. As said at the beginning of this section,travellers do not take into account the negative external-ities they cause to the rest of society, and as such, theycan be charged with anoptimal toll that is defined as thedifference between both marginal social and private costfunctions. Levying the correct congestion toll, would re-move the original market failure, resulting in a net socialwelfare benefit that is visualised as the grey triangular re-gion in Fig. 6. Note that in an ideal world, congestiontolls exactly match the caused negative externalities. Inpractice however, this can not be accomplished, result-ing in so-calledsecond-best pricingschemes. Practicalreal-life examples of this are tolling the beltway arounda city upon entering it (e.g., London’s recent congestioncharge), using step-tolls, tolling at fixed time periods in-stead of based on traffic conditions, . . . [173]

Reconsidering the work of Wardrop with respect to thecriteria (W1) and (W2) highlighted in section 1.2.2,Beckmann, McGuire, and Winsten found that the systemoptimumqso can be reached if the standard cost (i.e., jour-ney time) is replaced with ageneralised cost, which is

15

just the marginal social cost function as described earlier[20]. Consequently, the total travel time in the systemcan be minimised (from an engineering perspective), bylevying a so-called efficiency toll, which corresponds toRamsey’s optimal toll.

One of the most notable extensions in the economic treat-ment of transportation and congestion tolls, is the semi-nal work of the late Nobel prize winner William Vickrey[281]. As already stated in section 1.2.5, correct mod-elling of e.g., queue spill back, is of fundamental im-portance when assessing the effectiveness of road pric-ing schemes for example. Vickrey’sbottleneck modelisone step in this direction: it is based on the behaviourof morning commuters, whereby the model takes into ac-count the departure times of all travellers. As everybody’sdesire is to arrive at work at the same time, some willarrive earlier, others later. Besides the traditional traveltime costs, travellers therefore also experience so-calledschedule delay costs. Levying suitable tolls that dependon the travellers’ arrival times, allows to reach a systemoptimum [8, 172]. It is important to realise here that thelevied toll should vary over time and space, in order tocorrespond to the governing traffic conditions.

To most people in society road pricing is a highly unpop-ular measure, as well as a controversial political issue,whereby public acceptance is everything [188, 109, 110,111]. Alternatives to road pricing can include upgradingexisting roads and/or public transportation services, strictcontrol-oriented regulation by means ofadvanced trafficmanagement systems(ATMS), issuing elaborate parkingsystems, fuel taxes, et cetera [7]. In spirit of second-bestpricing schemes, it was a wise thing in the UK to con-nect London’s congestion charge to the simultaneous im-provement of public transportation [258]. Similarly, thecordon toll system in the city of Oslo, Norway, quicklyfound acceptance among the population [109]. Neverthe-less, road pricing is considered anunfair policy measureto most people: households (and firms) with higher in-comes, can more easily afford to pay the charge, hencethey will keep the luxury of travelling at their own discre-tion, whilst others might not be able to pay the requiredtoll. As a consequence, an inconsiderate internalisationof the external costs, does not lead to an equitable Paretooptimal scenario. Despite this resistance, there does seemto be a general consensus among members of society thatcongestion caused by traffic induced by recreational ac-tivities, is not tolerated during peak periods; congestiontolling for these travellers is deemed appropriate.

Despite the advances in the methodology underlyingroad pricing, there is still one major gap that has yet tobe filled in, i.e., the equity of the principle, or otherwisestated: where do the gained social welfare benefits(i.e., tax revenues) go in the redistribution ? As Smallstates, road pricing is more acceptable to the broadpublic, if it is presented as a complete financial package[257]. As welfare economists debate on whether ornot the revenues should go back to the transportationsector or rather elsewhere, Small asserts that inclusion

of the former is mandatory for achieving substantialsupport from both the political side and the investors.Complementary, in order to satisfy the global population,it is advisory to use the collected charges in order todiminish e.g., labour taxes, as they are perceived as beingtoo high [103, 110, 111].

In the end, we should note that both economistsand traffic engineers are essentially talking aboutthe same subject, although by approaching it fromdifferent angles. In the field of economics, roadpricing policies are introduced based on averagecost functions, allowing an optimisation of thesocial welfare. This effectively corresponds to theengineers’ idea of static traffic assignment, based ona system optimum using travel impedance functions(see e.g., section 1.2.3). The validity of usingthese average cost functions (with or without theirbackward-bending parts as explained in section1.4.1), has instigated several debates in road pricingliterature, most notably between Else and Nash[87, 208], Evans and Hills [88, 89, 130], and Ohtaand Verhoef [224, 225, 279].

In continuation, the approach followed by Vickrey’sbottleneck model provides a nice, first alternative,using schedule delay costs (see sections 1.2.5 and1.4.2). Although Vickrey’s idea introduces a hithertoabsent time dependence, it has the disadvantage thatspatial extents are neglected through the assumptionof point-queues (see section 1.2.5). Lo and Szetohave rigourously shown that hypercongestion is es-sentially a spatial phenomenon, and that by neglect-ing this facet, a road pricing policy might actuallyworsen traffic conditions [176]. The correct way outof this problem, is by explicitly taking the tempo-spatial characteristics of traffic flows into account.As an engineering analogy, this can be accomplishedby introducing dynamic traffic assignment (see sec-tion 1.2.4) which uses physical propagation modelsto describe the buildup and dissolution of congestion(see also some of the models presented in section1.2.5, e.g., the work of Lago and Daganzo [159]).

2 Traffic flow propagation models

In contrast to the previous section, which dealt with highlevel transportation planning models, the current sectionconsiders traffic flow models that explicitly describe thephysical propagationof traffic flows. In a sense, thesemodels can be seen as being directly applicable for thephysical description oftraffic streams. There exist sev-eral methods for discriminating between the families ofmodels that describe traffic flow propagation, i.e., basedon whether they operate in continuous or discrete time (oreven event-based), whether they are purely deterministicor stochastic, or depending on thelevel of detail(LOD)

16

that is assumed, . . . More detailed explanations can foundin the overview of Hoogendoorn and Bovy [134]. In thisdissertation, we present an overview that is based on thelatter method of discriminating between the level of de-tail. We believe that this classification most satisfactorilydescribes the discrepancies between the different trafficflow models. Thus, depending on the level of aggrega-tion, we can classify the propagation models into the fol-lowing four categories:

• macroscopic(highest level of aggregation, lowestlevel of detail, based on continuum mechanics, typ-ically entailing fluid-dynamic models),

• mesoscopic(high level of aggregation, low level ofdetail, typically based on a gas-kinetic analogy inwhich driver behaviour is explicitly considered),

• microscopic (low level of aggregation, high levelof detail, typically based on models that describethe detailed interactions between vehicles in a trafficstream),

• and submicroscopic (lowest level of aggregation,highest level of detail, like microscopic models butextended with detailed descriptions of a vehicles’ in-ner workings).

Note that some people regard macroscopic modelsmore from the angle ofnetwork models. In this case,the focus lies onperformance characteristicssuchas total travel times (a measure for the quality ofservice), number of trips, . . . [92] To this end, sev-eral quantitative models were introduced, such as Za-havi’s so-calledα relation between traffic flow, roaddensity, and space-mean speed [299], and Prigogineand Herman’stwo-fluid theoryof town traffic [128].

2.1 Macroscopic traffic flow models

In this section, we take a look at the class of trafficflow models that describe traffic streams at an aggregatedlevel. We first introduce the concept behind the models(i.e., the continuum approach), after which we discuss theclassic first-order LWR model. Because of its historicalimportance, we devote several sections to the model’s an-alytical and numerical solutions, as well as to some exten-sions that have been proposed by researchers. We con-clude our discussion of macroscopic models with a de-scription of several higher-order models, and shed somelight on the problems associated with both first-order andhigher-order models.

2.1.1 The continuum approach

Among the physics disciplines, exists the field ofcontin-uum mechanicsthat is concerned with the behaviour ofsolids and fluids (both liquids and gasses). Consideringthe class offluid dynamics, it has spawned a rich variety

of branches such as aerodynamics, hydrodynamics, hy-draulics, . . .

Underlying these scientific fields, is thecontinuity as-sumption(also called thecontinuum hypothesis) that theyall have in common. In a nutshell, this assumption statesthat fluids are to be treated as continuous media (in con-trast to e.g., molecular gasses, which consist of distinctparticles). Stated more rigourously, the macroscopic spa-tial (i.e., the length) and temporal scales are considerablylarger than the largest molecular corresponding scales[62]. As a consequence, all quantities can be treated asbeing continuous (in the infinitesimal limit). The deci-sion to use either a liquid-like or a gas-like treatment, isbased on theKnudsen numberof the fluid: a low value(i.e., smaller than unity) indicates a fluid-dynamic treat-ment, whereas a high value is indicative of a more gran-ular medium. In this section, we consider the former ap-proach. In the latter case, we enter the realm of statisticalmechanics that deals with e.g., kinetic gasses, requiringthe use of the Boltzmann equation (as will be explainedin section 2.2 on mesoscopic traffic flow models).

Historically, the fluid-dynamic approach found its rootsin the seminal work of Claude Louis Navier (1822),Adhemar de Saint-Venant (1843), and George GabrielStokes (1845) [101]. This gave rise to what we know asthe Navier-Stokes equations(NSE), formulated as a setof non-linear partial differential equations(PDEs). Forour overview, the most relevant equation is actually thelocalconservation law, stating that the net flux is accom-panied by an increase or decrease of material (i.e., fluid).In general, we can discern two subtypes:compressibleorincompressiblefluids, andviscousor inviscid fluids. In-compressibility assumes a constant density (and a highMach number), whereas inviscid fluids have a zero vis-cosity (with a corresponding highReynolds number) andare typically represented as theEuler equations.

Note that the NSE are still not fully understood. The factof the matter is that theClay Mathematics Institutehasdevised a list ofMillennium Problems6, among which adeeper fundamental understanding of the NSE holds a re-ward of one million dollar. Because the original Navier-Stokes equations are too complex to solve, scientists de-veloped solutions to specific subproblems, e.g., Euler’sversion of inviscid fluids. As an example, we give theBurgers equation, as derived by Johannes Martinus Burg-ers [39], for a one-dimensional fluid in the form of ahy-perbolic conservation law:

∂u

∂t+ u

∂u

∂x= ν

∂2u

∂x2, (3)

in which theu ∈ R typically represents the velocity, andν is the viscosity coefficient. For inviscid fluids,ν = 0,such that equation (3) corresponds to afirst-order par-tial differential equation. This type ofhyperbolic PDEis very important, as its solution can develop discontinu-ities, or more clearly stated, it can containshock waves

6http://www.claymath.org/millennium

17

which are of course directly relevant to the modelling oftraffic flows. The inviscid Burgers PDE can be solvedusing the standardmethod of characteristics, as will beexplained in further detail in the next three sections.

2.1.2 The first-order LWR model

Continuing the previous train of thought, we can considertraffic as an inviscid but compressible fluid. From thisassumption, it follows that densities, mean speeds, andflows are defined as continuous variables, in each point intime and space, hence leading to the names ofcontinuummodels, fluid-dynamic models, or macroscopic models.

The first aspect of such a fluid-dynamic description oftraffic flow, consists of ascalar conservation law(‘scalar’because it is a first-order PDE). A typical derivation canbe found in [92] and [147]: the derivation is based on con-sidering a road segment with a finite length on which novehicles appear or disappear other than the ones that en-ter and exit it. After taking the infinitesimal limit (i.e., thecontinuum hypothesis), this will result in an equation thatexpresses the interplay between continuous densities andflows on a local scale. Another way of deriving the con-servation law, is based on the use of a differentiable cu-mulative count functionN(t, x) that represents the num-ber of vehicles that have passed a certain location [215]:

k(t, x) = −∂N(t, x)/∂x and q(t, x) = ∂N(t, x)/∂t,

⇓

∂k(t, x)

∂t= −

∂2N(t, x)

∂t ∂xand

∂q(t, x)

∂x=

∂2N(t, x)

∂t ∂x,

⇓

∂k(t, x)

∂t+

∂q(t, x)

∂x= 0, (4)

with the densityk and flowq dynamically (i.e., time vary-ing) defined over a single spatial dimension. Lighthill andWhitham were among the first to develop such a trafficflow model [169] (note that in the same year, Newell hadconstructed a theory of traffic flow at low densities [218]).Crucial to their approach, was the so-called fundamentalhypothesis, essentially stating that flow is a function ofdensity, i.e., there exists aqe(k(t, x)) equilibrium rela-tionship [182]. Essentially to their theory, Lighthill andWhitham assumed that the fundamental hypothesis holdsat all traffic densities, not just for light-density traffic butalso for congested traffic conditions. Using this trick withthe fundamental diagram, relates the two dependent vari-ables in equation (4) to each other, thereby making it pos-sible to solve the partial differential equation.

One year later, in 1956, Richards independently derivedthe same fluid-dynamic model [251], albeit in a slightlydifferent form. The key difference, is that Richardsfocusses on the derivation of shock waves with respectto the density, whereas Lighthill and Whitham considerthis more from the perspective of disruptions of the flow

[239]. Another difference between both derivations, isthat Richards fixed the equilibrium relation, whereasLighthill and Whitham did not restrict themselves to ana priori definition; in Richards’ paper, we can find theequationV = a(b − D), with V the space-mean speed,D the density, anda and b fitting parameters [251].Note that all three authors share the following samecomment: because of the continuity assumption, thetheory only holds for a large number of vehicles, hencethe description of“long crowded roads”in Lighthill andWhitham’s original article.

Because of the nearly simultaneous and independentdevelopment of the theory, the model has becomeknown as theLWR model, after the initials of its in-ventors who receive the credit. In some texts, themodel is also referred to as thehydrodynamic model,or thekinematic wave model(KWM), attributed tothe fact that the model’s solution is based on char-acteristics, which are called kinematic waves (e.g.,shock waves).

2.1.3 Analytical solutions of the LWR model

Reconsidering equation (4), taking into account the fun-damental diagram, the conservation law is now expressedas:

kt + qe(k)x = 0, (5)

in which we introduced the standard differential calculusnotation for PDEs. Recognising the fundamental relationof traffic flow theory, i.e.,q = k vs, equation (5) [182]then becomes:

kt + (k vse(k))x = kt +

(vse

(k) + kdvse

(k)

dk

)kx = 0.

(6)

The above hyperbolic PDE, can be translated into theBurgers equation (3), using a suitable transformation toa dimensionless form as explained in the rigourous math-ematical treatment provided by Jungel [147]. The con-servation law (5) can also be cast in a non-linear waveequation, using the chain rule for differentiation [92]:

kt +dqe(k)

dkkx = 0. (7)

Analytically solving the previous equation using themethod of characteristics, results in shock waves thattravel with speeds equal to:

w =dqe(k)

dk, (8)

18

i.e., the tangent to theqe(k) fundamental diagram. Thistangent corresponds to the speedw of the backward prop-agatingkinematic wave[182]. As a consequence, solu-tions, being the characteristics, of equation (7) have thefollowing form:

k(t, x) = k(x − wt), (9)

with the observation that the density is constant alongsuch a characteristic. Note that in order to obtain shockwaves that are only decelerating, the usedqe(k) funda-mental diagram should be concave (a property that is of-ten neglected) [79]:

d2qe(k)

dk2≤ 0. (10)

Starting from an initial condition, the problem of find-ing the solution to the conservation PDE, is also calledan initial value problem(IVP), whereby the solution de-scribes how the density evolves with increasing time. Theproblem is called ageneralised Riemann problem(GRP)when we consider an infinitely long road with given con-stant initial densities up- and downstream of a disconti-nuity.

Because the method of characteristics can result innon-unique solutions, a trick is used to select the correct,i.e., physically relevant, one. Recall from equation (3)that the right-hand side of the Burgers PDE contained aviscosity termν. The general principle that is adopted forselection of the correct solution, is based on theOleinikentropy condition, which regards the conservation law asa diffuse equation. In this context, the viscosity coeffi-cient is multiplied with a small diffusion constantǫ. Inthevanishing viscosity limitǫ → 0, the method returns aunique, smooth, and physically relevant solution insteadof infinitely many (weak) solutions [164, 201]. For moredetails with respect to the application of traffic flows, werefer to the excellent treatment given by Jungel [147].

Ansorge, Bui et al., Velan and Florian later reinter-preted this entropy condition, stating that it is equiv-alent to adriver’s ride impulse[6, 37, 276]. Driversgoing from free-flow to congested traffic encountera sharp shock wave, whereas drivers going in the re-verse direction essentially pass through all interme-diate points on the fundamental diagrams, i.e., thesolution generates a fan of waves. It is for this lattercase that the ‘ride impulse’ is relevant: it denotes thefact that stopped drivers prefer to start riding again,resulting in a fan of waves.

A more intuitive explanation can also be given based onthe anticipation that drivers adopt when they approacha shock wave: their equilibrium speedvse

(k) is also afunction of the change in density, e.g.:

vse(k)

.= vse

(k) −ν

k

∂k

∂x. (11)

Substituting this new equilibrium relation in equation (6),results in a right-hand side equal toν ∂2k

∂x2 . Applying thesame methodology based on the vanishing viscosity limitof the entropy solution, results in the same unique solu-tion. Because the shock waves are in fact mathematicaldiscontinuities, and as such, infinitesimally small, theyare typically ‘smeared out’ by numerical schemes. Infact, this is just the equivalent of introducing an artificialviscosity (as explained earlier), which allows diffusion(i.e., the combined effect of dissipation and dispersion)of the shock waves. Note that this diffusion is a con-sequence of the numerical solution, and not necessarilycorresponding to the real diffusion processes in a viscousfluid. This numerical smoothing helps to retain numericalstability of the final solution.

Whenever in the solution of the conservation equation,two of its characteristics intersect, the density takes ontwo different values (each one belonging to a single char-acteristic). As this mathematical quirk is physically im-possible, the entropy solution states that both character-istics terminate and breed ashock wave; as such, theseshock waves form boundaries that discontinuously sepa-rate densities, flows, and space-mean speeds [92]. Thespeed of such a shock wave is related to the followingratio [239]:

wshock =∆q

∆k, (12)

with ∆q = qu − qd and∆k = ku − kd the relative differ-ence in flows, respectively densities, up- and downstreamof the shock wave.

Note that going from a low to a high density regime typ-ically results in a shock wave, whereas the reverse tran-sition is accompanied by an emanation of afan of char-acteristics(also calledexpansion, acceleration, or rar-efaction waves). In shock wave theory, the densities oneither side of a shock are well defined (i.e., unique solu-tions exist); along the shock wave however, the densityjumps discontinuously from one value to another. In thislatter respect, equation (12) is said to satisfy theRankine-Hugoniot jump condition.

The previous remarks with respect to the entropy con-dition, are closely related to the concavity of theqe(k)fundamental diagram, as defined by equation (10): forconcaveqe(k) fundamental diagrams, all shock wavesarecompression wavesgoing from lower to higher densi-ties. However, forqe(k) fundamental diagrams that con-tain convex regions, application of the entropy conditioncan return the wrong solution [165]. Although the mathe-matics of using these kinds of fundamental diagrams hasbeen worked out, see for example the work of Li [168], aunified physical interpretation is still lacking [181, 201]:instead of only deceleration shock waves and accelera-tion fans, we now also have acceleration shock waves anddeceleration fans. Finally, it is important to realise thatfor non-smoothqe(k) fundamental diagrams, the entropycondition is not applicable and no fans occur because thecorrect unique solution is automatically obtained [276].

19

In Fig. 7, we have depicted a classic example that is of-ten used when illustrating the tempo-spatial evolution ofa traffic flow at a traffic light (left part), based on the LWRfirst-order macroscopic traffic flow model with a triangu-lar qe(k) fundamental diagram (right part). The applica-tion of the traffic flow model is visible in the time-spacediagram to the left. A traffic light is located at positionxlight; it is initially green, and attred it turns red untiltgreen

when it switches back to green. The initial conditions atthe road segment are located at point1© on the fundamen-tal diagram. Because all characteristics of the solution aretangential to the fundamental diagram, the results can beelegantly visualised when using a triangular diagram: ex-cept for the fan of rarefaction waves (we approximate thenon-differentiable tip of the triangle with a smooth one,such that we can show the fan4© for all didactical intentsand purposes), only two kinematic wave speeds are pos-sible. When the traffic light turns red, a queue of stoppedvehicles develops. Inside this queue, the jam density statekj holds, corresponding to point2© on the fundamentaldiagram. The upstream boundary of the queue is demar-cated by the shock wave3© that is formed by the intersec-tions of the characteristics1© and 2©. Downstream of thejam, there are no vehicles: because we are working witha triangular fundamental diagram, the characteristics areparallel to the vehicle trajectories (their speeds are equalto the slopes of points on the free-flow branch). The ini-tial regime at state1© and the ‘empty’ regime downstreamof the queue are separated from each other by acontactdiscontinuityor slip. When the traffic light turns greenagain, the queue starts to dissipate, whereby the solutionof characteristics becomes a fan of rarefaction waves4©,taking on all speeds between states2© and 1© on the fun-damental diagram. A final important aspect that can beseen from Fig. 7, is the fact that in the LWR model theoutflow from a jam, i.e., going from a high to a low den-sity regime, always proceeds via the capacity-flow regimeatqcap: sothere is no capacity drop in the LWR modelbe-cause the outflow is always optimal.

To conclude our summary of analytical derivations, wepoint the reader to the significant work of Newell, whoin 1993 cast the LWR theory in an elegant form. Thekey ideas he introduced were on the one hand the useof cumulative curves for deriving the conservation law,and on the other hand the use of a triangularqe(k) funda-mental diagram [215]. Due to Newell’s work, traffic flowanalysis in this respect gets very simplified, as it is nowpossible to give an exactgraphicalsolution to the LWRmodel for both free-flow and congested conditions [216].To complete his theory, Newell also provided us witha means to include multi-destination flows, i.e., specifi-cations of which off-ramp vehicles will use to exit themotorway [217]. Note that for the LWR model with aparabolicqe(k) fundamental diagram and piece-wise lin-ear and piece-wise constant space and time boundaries,respectively, Wong and Wong recently devised an exactanalytical solution scheme. Their method is based on theefficient tracking and fitting of generated and dispersedshock waves within a time-space diagram [295].

2.1.4 Numerical solutions of the LWR model

Besides the previousanalytic derivationof a solution tothe conservation law expressed as a PDE, it is also pos-sible to treat the problemnumerically. By trying to finda numerical solution to the PDEs, we enter the field ofcomputational fluid dynamics(CFD). In a typical setup,the ‘fluid domain’ is first discretised into adjacent cells(called a one-dimensionalmesh) of size∆X (note that allcells need not to be equal in size), after which aniterativeschemeis used to update the cells’ states (e.g., the densityk in each fluid cell) at discrete time stepsm ∆T with m ∈

N0. Typically, this entailsfinite difference schemes(or ina broader context,finite element methodsor finite volumemethods), which replace the continuous partial derivativewith adifference operator, thereby transforming the con-servation equation into afinite difference equation(FDE).Examples of these difference operators are theforwarddifference operator∆f(x) = f(x + 1) − f(x) and thebackward difference operator∇f(x) = f(x)−f(x−1),which is not to be confused with the gradient vector off(x). Examples of finite difference schemes are thecen-tral scheme, the Lax-Friedrichs scheme, the downwindscheme, the upwind scheme, the MacCormack scheme,theLax-Wendroff scheme, theSteger-Warming Flux Split-ter scheme, theRieman-based Harten-van Leer-Lax andEinfeldt scheme, . . . For a more complete overview ofthese schemes, we refer the reader to the work of Hel-bing and Treiber [124], Jungel [147], and Ngoduy et al.[222]. A practical software implementation of amoving-mesh finite-volume solverfor the previously mentionedhyperbolic PDEs, can be found in van Dam’s TraFlow-PACK software [272]. LeVeque also developed a numer-ical solver, called CLAWPACK, that is designed to com-pute numerical solutions to hyperbolic partial differentialequations using a wave propagation approach [166]. Acentral precaution for all these schemes, is the so-calledCourant-Friedrichs-Lewy (CFL) conditionwhich guar-antees numerical stability of the algorithms; for trafficflows, it has the physical interpretation that no vehiclesare allowed to ‘skip’ cells between consecutive time steps(i.e., all physical information that has an influence on thesystem’s behaviour should be included):

∆T ≤∆X

vff. (13)

Just over a decade ago, Daganzo constructed a numericalscheme based on finite difference equations. It is knownas thecell transmission model(CTM), which solves theLWR model using a trapezoidalqe(k) fundamental dia-gram [63]. At the heart of his model lies a discretisa-tion of the road into finite cells of width∆X , each con-taining a certain number of vehicles (i.e., an average celldensity). When time advances, these vehicles are trans-mitted from upstream to downstream cells, taking intoaccount the capacity constraints imposed by the down-stream cells. The CTM converges to the LWR modelin the limit when ∆X → 0. In 1995, Daganzo ex-

20

ttred tgreen

x

xlight

kkc kj

q

qcap

1

1

2

2

33

4

4

Figure 7: Example of an analytical solution based on the LWR first-order macroscopic traffic flow model with a triangularqe(k) fundamental diagram.Left: a time-space diagram with a traffic light located at positionxlight. It is green, exceptduring the period fromtred until tgreen. The solution is visually sketched by means of the characteristics that evolve duringthe tempo-spatial evolution of the traffic flow.Right: a triangular fundamental diagram, with the initial conditions at state1©. When the traffic light is red, a queue develops in which the jam density state at point2© holds. Its upstream boundary

is demarcated by the shock wave3©. When the queue starts to dissipate, the solution of characteristics generally becomesa fan of rarefaction waves4©.

tended the model to include network traffic, i.e., two-way merges and diverges, thereby allowing for the cor-rect modelling of dynamic queue spill backs [64]. Healso cast the model in the context ofGodunov FDE meth-ods7, allowing for arbitraryqe(k) fundamental diagrams.The exchange of vehicles between neighbouring cells isthen governed by so-calledsendingand receiving func-tions[65]. Lebacque derived a similar numerical schemethat performed the same functions as Daganzo’s CTM,at approximately the same time (the debate on whomeverwas first is still not resolved) [161, 162]. In his derivation,he employed the termsdemandandsupply functionstodenote the exchange of vehicles between cells. He alsoprovided the means to handle general (i.e., multi-way)merges and diverges. Both the original cell transmissionmodel and an implementation of the Godunov scheme forthe LWR model with an arbitraryqe(k) fundamental di-agram, were provided by Daganzo et al. in the form ofa software package called NETCELL [46]. Note that, asmentioned earlier, numerical methods tend to smear outthe shock waves; this diffusion is therefore a consequenceof the solution methodology and not of the LWR modelitself [177].

Daganzo also developed another methodology for numer-ically solving the LWR equations, based on avariationalformulation. Rather than extending the existing conceptof a conservation equation coupled with a vanishing vis-cosity limit, he derived a solution based on the principlesbehind cumulative curves. The initial value problem be-comes well-posed, and the methodology is able to handlecomplex boundary conditions. In short, the problem is

7Sergei K. Godunov’s numerical solution of PDEs is considered asa breakthrough in computational fluid dynamics: it providesa uniquesolution based on a stable Riemann problem [102, 162, 70].

transformed into finding shortest paths in a network ofarcs that comprise the kinematic waves; as a surplus, themethod is computationally more efficient than traditionalsolutions based on conservation laws [74, 75].

Traditional cell-based numerical methods are fairly com-putationally intensive, because they have to discretise theroad entirely (even in regions where there is no variationin density), resulting in a solution that is composed oflinear shock waves and continuous fans (i.e., the rarefac-tion waves). In order to derive a solution that is com-putationally more efficient, Henn proposed to replace thecontinuous fans of rarefaction waves with a discrete setof angular sectors (i.e., the density now varies with dis-crete steps). The efficiency now stems from the fact that,instead of a whole array of cells, only list structures needto be maintained [126].

Only recently, a combination of Daganzo’s CTM with atriangularqe(k) fundamental diagram and Newell’s cu-mulative curves was constructed by Yperman et al, result-ing in thelink transmission model(LTM). Because wholelinks can be treated at once, the LTM’s computational ef-ficiency is much higher than that of classic numerical so-lution schemes for the LWR model, whilst retaining thesame accuracy [?].

With respect to the applicability of the LWR model toreal-life traffic flows, we refer the reader to two studies:the first was done by Lin and Ahanotu in the course of theCalifornia Partners for Advanced Transit and Highways(PATH) programme (formerly known as theProgram onAdvanced Technology for the Highway). In their work,they performed a validation for the CTM with respectto the formation and dissipation of queues, concludingthat the most important first-order characteristics (corre-

21

lations in measurements of free-flow traffic at successivedetector stations, and the speed of the backward propagat-ing wave under congested conditions) perform reasonablewell when comparing them to field data [171].

A second, more thorough and critical study was done byNagel and Nelson. In it, they scrutinise the LWR model,both with concaveqe(k) fundamental diagrams and thosewith convex regions. Their main conclusion states thatit remains difficult to judge the model’s capabilities ona fair basis, largely due to the fact that there do not ex-ist many real-world data sets which also contain a geo-metrical description of the local infrastructural road lay-out. This latter ingredient is a requirement for assessingwhether or not an observed traffic breakdown is eitherspontaneously induced or due to the presence of an activebottleneck (because the LWR model constitutes a strictlydeterministic model) [201].

2.1.5 Flavours of the LWR model

Considering this elegant first-order traffic flow model,its main advantages are that it is simple, and in a sensereproduces the most important features of traffic flows(i.e., shock waves and rarefaction waves). However, be-cause of its restriction to a first-order partial differentialequation, certain other effects, such as stop-and-go traf-fic waves, capacity drop and hysteresis, traffic flow in-stabilities, finite acceleration capabilities, . . . can notberepresented [169]. In many cases, these ‘deficiencies’can be tackled by switching to higher-order models, aswill be elaborated upon in section 2.1.6. Interestingly, intheir original paper, Lighthill and Whitham recognisedthe fact that drivers tend to anticipate on downstreamconditions, changing their speed gradually when crossingshock waves. This in fact necessitates a diffusion term inthe conservation equation that captures a density gradi-ent.

Instead of using a higher-order model, traffic flow engi-neers can also resort to more sophisticated approaches,such as extensions of the first-order model. To concludethis section, let us give a concise overview of some ofthe model flavours that have been proposed as straight-forward extensions to the seminal LWR model.

An interesting set of extensions launched, was created byDaganzo, dealing with two classes of vehicles, of whichone class can use all lanes of a motorway, whereas theother class is restricted to a right-hand subset of theselanes. When the capacity of the latter vehicles in regularlanes is exceeded, a queue will develop in those lanes, butthe former vehicles will still be able to use the other lanes;this is called a2-piperegime. Similarly, if the capacity ofthe yet freely flowing vehicles is exceeded, all lanes en-ter a queued state, which is called a1-pipe regime. Inshort, interactions between vehicles in this and the fol-lowing models are nearly always considered from a userequilibrium perspective [68]. Daganzo et al. applied thetheory to a case where there is a set of special lanes onwhich only priority vehicles can drive. The theory was

also suited to describe congestion on a motorway diverge,such that the motorway itself can still be in the free-flowregime [69]. For the special case of queue spill back ata motorway’s off-ramp, Newell also provided a graphi-cal solution that is based on the use of cumulative curves[220].

Continuing the previous train of thought, Daganzo pro-vided a logical extension: he again considered differ-ent lanes, but now introduced two different types ofdrivers: aggressive ones (calledrabbits) and timid ones(calledslugs). Daganzo himself states that a correct traf-fic flow theory should involve both human psychologyand lane-changing aspects, leading him to such abe-havioural description [71]. The theory was also usedto explain the phenomenon of a capacity funnel [182]:according to the theory, once a capacity drop occurs,the recovery to the capacity-flow regime can not occurspontaneously, thereby requiring an exogeneous mech-anism. Daganzo provides an explanation, called thepumping phenomenon: drivers temporarily accept shortertime headways downstream of an on-ramp, leading toa ‘pumped state’ of high-density and high-speed traffic,or in other words, a capacity-flow regime [72]. Chungand Cassidy later provided a validation of the theory, byapplying it to describe merge bottlenecks on multi-lanemotorways in Toronto (Canada) and Berkeley (Califor-nia). In their study, they introduced the concept ofsemi-congestion, denoting a regime in which on vehicle classenters a state with a reduced mean speed, whereas theother vehicle class can still travel unimpeded. Their find-ings indicated an agreement between both shock wavespeeds empirically observed and predicted by the model[58].

An interesting case to which the LWR theory can be ap-plied, is the problem ofmoving bottlenecksas stated byGazis and Herman [96]. Examples of such bottlenecksare slower trucks on the right shoulder lanes, which canimpede upstream traffic. Newell was among the first totry to give a satisfactory consistent treatment of this typeof bottlenecks. The trick he used was to translate theproblem into a moving coordinate system that is travel-ling at the bottleneck’s velocity. This resulted in a de-scription of a stationary bottleneck, after which the clas-sic LWR theory can be applied [219]. Although the the-ory is sound, there exist some serious drawbacks, mainlydue to its underlying assumptions. For example, the mov-ing bottlenecks are assumed to be long convoys, and otherdrivers’ behaviours are not affected by the bottlenecks’speeds; even more serious is the fact that the theory isnot valid for very light traffic conditions, and that severalstrange effects are predicted by the theory (e.g., a bottle-neck with increasing speed can result in a lower upstreamcapacity). To this end, Munoz and Daganzo applied thepreviously mentioned behavioural model with rabbits andslugs to the problem of a moving bottleneck. Their the-ory performs satisfactorily and agrees well with empiri-cally observed motorway features. However, because ofthe fact that it relies on the LWR model, it is not entirely

22

valid for bottlenecks that travel at high speeds under lighttraffic conditions. In this latter case, they state that driverdifferences are much more important than the dynamicsdictated by the kinematic model [197].

Another theory that deals with the problem of movingbottlenecks, is the one proposed by Daganzo and Laval:they treat moving bottlenecks as a sequence of consec-utive fixed obstructionsthat have the same capacity re-straining effects. Despite the fact that the previous the-ory of Munoz and Daganzo has a good performance, itdoes not easily lend itself to discretisation schemes thatallow numerical solutions. In contrast to this, the hybridtheory (fixed obstructions coupled with the LWR modeldynamics) of Daganzo and Laval holds high promiseas they have shown that it can be discretised in a nu-merically stable fashion [76]. As a continuation of thiswork, Lavel furthermore investigated the power of thesefixed obstructions, allowing him to capture lane-changesas random events modelled by moving bottlenecks in aLWR 1-pipe regime. It is suggested that (disruptive) lanechanges form the main cause for instabilities in a trafficstream. This leads the ‘Berkeley school’ to the statementthat incorporating lane-change capabilities into multi-lane macroscopic models seems a prerequisite for ob-serving effects such as capacity drops, kinematic wavesof fast vehicles, . . . [160] In this respect, Jin provides atheory that explicitly takes into account to effects of lanechanges [144]. The starting point in this model, is thepresence of certain road areas in which traffic streamsmix. The underlying assumption here is that all lanechanges lead to the same traffic conditions in each lane:the crucial element in the model is that vehicles perform-ing lane changes are temporarily counted twice in thedensity total. This new ‘effective density’, is then usedto transform theqe(k) fundamental diagram, leading toa reversed lambda shape. However, because the currentversion of the theory employs a small artificial constantto introduce the lane changes, we question its practicalapplicability when it comes to calibration and validation.

To conclude this overview of the first-order models, wehighlight two other successful attempts at increasing thecapabilities of the classic LWR model. A first importantextension was made by Logghe, who derived amulti-class formulation8 that allows for the correct modellingof heterogeneous traffic streams (e.g., preserving theFIFO property for interacting classes). Classes aredistinguished by their maximum speed, vehicle length,and capacity (all intended for a triangularqe(k) funda-mental diagram). A central ingredient to his theory, isthe interactions between different user classes that resideon a road: in this respect, each class acts selfishly, withslower vehicles taking on the role of moving bottlenecks.Besides being able to construct analytical and graphicalsolutions, Logghe also provided a stable numericalscheme, as well as a complete network version with

8At approximately the same time, Chanut and Buisson constructeda first-order model that incorporates vehicles with different lengths andfree-flow speeds [49]. Their model can be considered as a trimmed-down version of Logghe’s multi-class formulation.

road inhomogeneities, and two-way merges and diverges[177]. A second extension was made by Jin; it actuallydeals with a whole plethora of extensions, in particularfor inhomogeneous links (e.g., lane drops), merge anddiverge zones, and mixed-type vehicles (i.e., havingdifferent qe(k) fundamental diagram characteristics).All these models are then combined in a description fora multi-commodity kinematic wave model for networktraffic, whereby the commodities signify vehicles takingdifferent paths [143].

A finally important aspect that is mainly related tolane changes, is theanisotropy propertyof a trafficstream. This property basically states that driversare not influenced by the presence upstream vehi-cles. In a sense, most models describing the acceler-ation behaviour of a vehicle, only take into accountthe state of the vehicle directly in front. For mostmacroscopic traffic flow models, this anisotropy con-stitutes a necessary ingredient. However, in his orig-inal paper, Richards very subtly points out that thefact of whether or not drivers only react to the con-ditions ahead, remains an open question [251]. Incontrast to this, Newell states that a driver is only in-fluenced by downstream conditions, leading to a nat-ural cause-and-effect relation, making the problemmathematically well-posed [216]. Recently, Zhangstated that the anisotropy property is likely to be vi-olated in multi-lane traffic flows. His explanationis closely tied to the concavity character of aqe(k)fundamental diagram (non-concave regions can leadto characteristics that travel faster than the space-mean speed of the traffic stream). He also providesan intuitive reasoning based on Daganzo’s rabbitsand slugs, wherebytailgatingvehicles induce slowerdownstream vehicles to ‘make way’. Note that forsingle-lane traffic flows, the anisotropy property isexpected to hold because of the FIFO property (ve-hicles can not pass each other), although there areexceptions in the case of some higher-order macro-scopic traffic flow models [305].

2.1.6 Higher-order models

The development of higher-order macroscopic modelscame as a response to the apparent shortcomings of thefirst-order LWR model. Harold Payne was among thefirst in 1971 to develop such a higher-order model [231].In those days, ramp metering9 control strategies were ba-sically an all-empirical occasion. Payne recognised thenecessity to include dynamic models in the control of on-ramps; the celebrated LWR model however, was found toperform unsatisfactorily with respect to the modelling ofreal-life traffic flows. One of these shortcomings, was the

9Ramp metering is an ATMS whereby a traffic light is placed atan on-ramp, such that traffic enters the highway from the on-ramp bydrops. We refer the reader to the work of Bellemans [22] and Hegyi[114] for an overview and some recent advancements.

23

model’s inability to generate stop-and-go waves. Zhanglater traced this to be a consequence of the model’s persis-tent reliance on a single equilibrium curve (i.e., the fun-damental diagrams) [306]. In the LWR model, driversare assumed to adapt their vehicle speedinstantaneouslyaccording to the fundamental diagram when crossing ashock wave, a phenomenon termed theno-memory effect(i.e., they encounter infinite accelerations and decelera-tions [300]). One option that leads to a solution of thepreviously mentioned problems, is to introduce differentfundamental diagrams for vehicles driving under differ-ent traffic conditions; this avenue was explored by Newell[213] and Zhang [301]. Another, more popular type ofsolution was proposed by Payne (as well as by Whithamsome years later [291]): they suggested to add an equa-tion to the LWR conservation law (6) and its fundamentaldiagram10. This newdynamic speed equationwas de-rived from the classical car-following theories of Gaziset al. [97] (see also section 2.3.1 for more details). Animportant aspect is this derivation, is the fact that the car-following model includes areaction time, resulting in amomentum equationthat relates the space-mean speed ofa vehicle stream to its density. As a result, vehicles nolonger instantaneously change their speed when crossinga shock wave. Payne’s second-order macroscopic traf-fic flow model is now described by the following pair ofPDEs, i.e., a conservation law and a momentum equation:

kt + (k vs)x = 0, (14)

dvs = vst+ vsvsx︸ ︷︷ ︸

convection

=vse

(k) − vs

τ︸ ︷︷ ︸relaxation

−c2(k)

kkx

︸ ︷︷ ︸anticipation

, (15)

with vstandvsx

denoting the partial derivatives of thespace-mean speed with respect to time and space, re-spectively,vse

the traditional fundamental diagram, andτ thereaction time. The functionc(k) corresponds to themodel-dependentsound speed of traffic(i.e., the typicalspeed of a backward propagating kinematic shock wave);examples ofc(k) are [307]:

−

√−

12τ

dvse(k)

dk(Payne) (16)

−

√v

τ(Whitham) (17)

kdvse

(k)

dk(Zhang) (18)

with v being a parameter in equation (17).

In equation (15), the left hand side corresponds to thederivative of the speed, i.e., the acceleration of vehicles.

10Note that Lighthill and Whitham originally proposed to extend theconservation law in their model with relaxation and diffusion terms, butthe idea did not receive much thought at the time [169].

As can be seen from the formulation, Payne identifiedthree different aspects for the momentum equation: aconvectionterm describing how the space-mean speedchanges due to the arrival and departure of vehicles at thetime-space location(t, x), a relaxation term describinghow vehicles adapt their speed to the conditions dictatedby the fundamental diagram, but with respect to a certainreaction time (as opposed to the instantaneous adaptionin the LWR model), and finally ananticipationterm de-scribing how vehicles react to downstream traffic condi-tions.

In continuation of the above derivation, many otherhigher-order models have been based on the Payne-Whitham (PW) second-order traffic flow model. An ex-ample is the work of Phillips, who changed the reactiontimeτ in the relaxation term of equation (15) from a con-stant to a value that is dependent on the current density[236]. Another example is due to Kuhne, who artificiallyintroduced a viscosity term into equation (15), in orderto smooth the shock waves [157]. The physical role thatviscosity plays in a vehicular traffic stream is howevernot entirely understood: according to Zhang, the viscos-ity reflects the resistance of drivers against sharp changesin speeds [306]. For a rather complete overview of exten-sions to the PW model, we refer the reader to the work ofHelbing [121].

2.1.7 Critiques on higher-order models

Higher-order models have been successfully applied invarious computer simulations of traffic flows, e.g., theoriginal FREFLO implementation by Payne [232], thework of Kwon and Machalopoulos who developed KRO-NOS which is an FDE solver for a motorway corridor[158], the METANET model of Messmer and Papageor-giou [193], . . . Despite their success, it was Daganzo whoin 1995 published their final requiem, which stood outas an obituary for all higher-order models [67]. From atheoretical perspective, there were some seriousphysicalflaws that littered these second-order models. Most no-tably was the fact that there exist two families of charac-teristics (calledMach lines) in the Payne-Whitham typemodels. On the one hand, there are characteristics thatimply a diffusion-like behaviour, which under certain cir-cumstances can lead to negative speeds at the end of aqueue, i.e., vehicles travelling backwards. On the otherhand, there are characteristics that have the property oftravelling faster than the propagation of traffic flow. Thislatter gas-like behaviour means that vehicles can get in-fluenced by upstream conditions (because information issent along the characteristics), which is a clear violationof the anisotropy property for single-lane traffic as ex-plained in the previous section. From a physical point ofview, the relaxation term in equation (15) may even intro-duce a ‘suction process’ because slower vehicles can getsucked along by leading faster ones [113].

Several years after these critiques, Papageorgiou re-sponded directly to the comments stated in Daganzo’s

24

article [227]. In his response, Papageorgiou put a lotof emphasis at the incapabilities of first-order trafficflow models for use in a traffic control strategy (e.g.,ramp metering). He very briefly reacts to the anisotropyviolation, by mentioning that in multi-lane traffic flows,the space-mean speeds of the different lanes are notall the same, leading to characteristics that areallowedto travel faster than the space-mean speed of all lanescombined. With respect to negative speeds (and hence,negative flows), he proposes to simply include an aposteriori check that allows to set the negative flowsequal to zero. One year later, in 1999, Heidemannreconsidered these higher-order models, but this timefrom the perspective ofmathematical flaws. His mainargument was the fact that the models led to an internalinconsistency, because they ignored some aspects relatedto the conservation law [115]. However, after carefulscrutiny, Zhang later refuted Heidemann’s claims: theinconsistencies that plague the models are a result of theinsistence on the universality of a conservation law andthe imposing of arbitrary solutions. As a consequence,the Payne-Whitham type of models are mathematicallyconsistent theories, although they may suffer from theaforementioned physical quirks [307].

Note that the dynamic speed equation (15), can alsobe cast in another form that is more closely relatedto a gas-kinetic analogy. With this in mind, we canrewrite the momentum equation as follows [134]:

dvs = vst+ vsvsx︸ ︷︷ ︸

transport

=vse

(k) − vs

τ︸ ︷︷ ︸relaxation

−Px

k︸︷︷︸pressure

+ν

kvsxx

︸ ︷︷ ︸viscosity

, (19)

with now P denoting thetraffic pressureand νthe kinematic traffic viscosity(as introduced byKuhne [157]). The convection term has been re-labelled atransport term, describing the propaga-tion of the speed profile with the speed of the vehi-cles. Thepressure termreflects the change in space-mean speed due to arriving vehicles having differentspeeds, and theviscosity termreflects changes dueto the ‘friction’ between different successive vehi-cles. The classic Payne model is obtained if we setP = kc2(k) andν = 0.

In contrast to Papageorgiou’s response which did not pro-vide a definite answer, Aw and Rascle carefully examinedthe reason why the PW model exhibited the strange phe-nomena indicated in Daganzo’s requiem [11]. The rootcause of this behaviour can be traced back to the spatialderivativePx of the pressure term in equation (19). Theirsolution suggests to abandon the transport and relaxationterms, and replace the spatial derivative of the pressureP(which is a function of the densityk) with a convective(Lagrangian) derivative, i.e.,D/Dt = ∂t + (vs · ∇) =

∂t + vs∂x, with (vs · ∇) called theadvective derivativeterm [240]:

(vs + P (k))t + vs (vs + P (k))x = 0. (20)

This new formulation allows to remedy all Daganzo’sstated problems [11]. Because of the somewhat limitedcharacter of their derivation of equation (20), Rascle adda relaxation term to the equation’s right-hand side, anddeveloped a numerically stable discretisation scheme, aswell as showing convergence to the classic LWR modelwhen the relaxation tends towards zero [246].

To end our overview of higher-order models, we illustratetwo other types. The first model is actually athird-ordermodelcreated by Helbing. It is based on the two PDEs ofthe Payne-Whitham type models, but is extended with athird equation that describes the change in thevariance ofthe speed, denoted byΘ [118]. Helbing derived his equa-tions using a gas-kinetic analogy, resulting in the follow-ing Navier-Stokes-like equation (it is typically encoun-tered in the pressure term forP ):

Θt + vsΘx =2(Θe(k) − Θ)

τ︸ ︷︷ ︸relaxation

+

2k

vsx(ν vsx

− P )︸ ︷︷ ︸

pressure

+

ν

kvsxx

+κ

kΘxx

︸ ︷︷ ︸viscosity

, (21)

with now the equilibrium relationvse(k, Θ) of equation

(19) also depending on the speed varianceΘ. In additionto the viscosityν, the dynamic speed variance equation(21) also contains an equilibrium relationΘe(k, vs) forthe variance of the speed, andκ which is akinetic coef-ficient that is related to the reaction timeτ , the densityk, and the speed varianceΘ. For ν = κ = 0, Hel-bing’s model reduces to an inviscid Euler type model asexplained in section 2.1.1 [149]. Whereas in the LWRmodel there is only one family of characteristics, and inthe PW model there are two families, the Helbing modelgenerates three different families of characteristics; thisimplies that small perturbations in the traffic flow propa-gate both with the traffic flow itself, as well as in upstreamand downstream direction relative to this flow [134].

The second model we illustrate, is thenon-equilibriummodel of Zhang. Because of the relaxation terms inthe Payne-Whitham equations, drivers initially tend to‘overshoot’ the equilibrium speed as dictated by thevse

(k) fundamental diagram. It takes a certain amountof time for them to adapt to their speed to the newtraffic conditions (i.e., a change in density is accom-panied by asmoothchange in space-mean speed), af-ter which they converge on the diagram. This latter as-pect gives rise to the empirically observed scatter in the

25

(k,q) phase space, leading Zhang to the terminology of‘non-equilibrium models’ because of the deviation fromthe one-dimensional equilibrium fundamental diagram[300].

In his model, Zhang considers equilibrium traffic to bea state in whichdvs/dt = ∂k/∂x = 0. In similarspirit of Payne’s theory, Zhang constructs his model usingan equilibrium relation between density and space-meanspeed (i.e., the fundamental diagram), a reaction time thatallows relaxation, and an anticipation term that adjuststhe space-mean speed to downstream traffic conditions.This results in a macroscopic model that contains equa-tion (14) as the conservation law, as well as the followingmomentum equation:

dvs = vst+ vsvsx

=vse

(k) − vs

τ− k

(dvse

(k)

dk

)2

kx, (22)

with the last anticipation term showing the dependenceon the spatial change of the density. Zhang also com-plements the theory with a finite difference scheme thatallows to solve the equations in a numerically stable fash-ion, based on an extension of the Godunov scheme thatsatisfies the entropy condition referred to in section 2.1.3[303].

Just as with the improved PW model of Aw and Ras-cle, this model alleviates Daganzo’s stated problemof wrong-way travel, even though there are also twofamilies of characteristics, travelling slower, respectivelyfaster, than the space-mean speed of traffic. An impor-tant fact here is that for the slower characteristics, theassociated shock waves and fans correspond perfectlyto those of the first-order LWR model. However, theshock waves and fans associated with the faster familyof characteristics can still violate the anisotropy propertyof traffic (although they decay exponentially), but inthe end, Zhang questions its universal validity, statingthat traffic might occasionally violate this principle dueto the heterogeneity of a traffic stream [302, 305]. Theviolation of anisotropy, i.e., drivers get influenced byupstream traffic, is sometimes referred to as gas-likebehaviour, because in contrast to fluid-dynamics, gasparticles are not anisotropic. In an attempt to removethis faulty behaviour, Zhang developed yet anothernon-equilibrium model that removed the gas-like be-haviour, thereby respecting the anisotropy property.Moreover, both families of characteristics in his modelsatisfy the condition of travelling at a lower speed thanthe space-mean speed of the traffic stream, but stillkeeping the one-to-one correspondence between theslower characteristics and those of the first-order LWRmodel. At present, it is however unclear if this newmodel can generate stop-and-go waves, although thereare indications that it can because of the non-equilibriumtransitions that can occur [304].

Despite the significant progress that has been madeon the front of higher-order macroscopic traffic flowmodels, the Berkeley school firmly holds its faithin first-order models and their extensions. Its mainreason is because of the numerical solution schemesthat are well developed and understood. This is notthe case for higher-order models, as these containother characteristics that complicate the finitedifference schemes (because information is nowcarried both up- and downstream, and because theirnumerical schemes initially were flawed [303, 160]).Related to this critique, is the fact that in contrast tothe first-order model, no analytical solutions existfor the higher-order models.

Another reason for sticking with first-order models,is because the school believes that first-order char-acteristics are sufficient for the description of trafficflows [45]. Using a triangularqe(k) fundamental di-agram that captures the most important traffic flowcharacteristics (i.e., the free-flow speedvff , the ca-pacity qcap, the jam densitykj , and the backwardkinematic wave speedw), results in a further ele-gance of the models.

2.2 Mesoscopic traffic flow models

The previous section dealt with macroscopic models thatdescribed traffic streams at an aggregated level, derivedfrom a fluid-dynamic analogy. This section describeshow traffic can be modelled at this aggregate level, butwith special consideration for microscopic characteristics(e.g., driver behaviour). Because of the ambiguity thatsurrounds mesoscopic models, we first elucidate what ismeant by the term mesoscopic (i.e., it is something be-tween a macroscopic and a microscopic approach). In thesections thereafter, we zoom in on a derivation of meso-scopic models based on a gas-kinetic analogy. For anoutstanding overview of gas-kinetic models, we refer thereader to the work of Tampere [262].

2.2.1 The different meanings of ‘mesoscopic’

Considering the amount of literature that has been gener-ated during the last few decades, it seems to us that thereexists no unanimous consensus as to what exactly consti-tutes mesoscopic traffic flow models. In general, there arethree popular approaches when it comes to mesoscopicmodels [134]:

• Cluster models

When considering vehicles driving on a road, a pop-ular method is to group nearby vehicles togetherwith respect to one of their traffic flow characteris-tics, e.g., their space-mean speed. Instead of havingto perform detailed updates of all vehicles’ speedsand positions, the cluster approach allows to treatthese vehicles as a set of groups (calledclusters,

26

cells, packets, or macroparticles); these groups arethen propagated downstream without the need forexplicit lane-changing manoeuvres (leading to thecoalescing and splitting of colliding and separatinggroups).

Examples of this kind of models, are theCONtin-uous TRaffic Assignment Model(CONTRAM) ofLeonard et al. [163], the work of Ben-Akiva et al,called Dynamic network assignment for the Man-agement of Information to Travellers(DynaMIT),which is based on a cell transmission model with acell of a link containing a set of vehicles with iden-tical speeds [24], theMesoscopic Traffic Simulator(MesoTS) of Yang, which allows fast predictions offuture traffic states [297], . . .

• Headway distribution models

This rather unknown and somewhat outdated classof models, places the emphasis on the probabil-ity distributions of time headways of successive ve-hicles (this aggregation makes them mesoscopic).Two popular examples are Buckley’s semi-Poissonmodel [36], and Branston’s generalised queueingmodel [34]. As clarified in the summary of Hoogen-doorn and Bovy, the original versions of these head-way distribution models assume homogeneous traf-fic flows and they are inadequate at describing theproper dynamics of traffic flows [134].

• Gas-kinetic models

The third and most important characterisation ofmesoscopic models comes from a gas-kinetic anal-ogy. Because macroscopic models aim towardsobeying the fundamental diagram (either instanta-neously as in the first-order LWR model or througha relaxation process as in higher-order models), thefocus there lies on the generation and dissipation ofshock and rarefaction waves. As a consequence,more complex and non-linear dynamics can not bereproduced. To remedy this, gas-kinetic models im-plicitly bridge the gap between microscopic driverbehaviour and the aggregated macroscopic mod-elling approach.

In the next sections, we will first give an overview of theoriginal gas-kinetic model as derived by Prigogine andHerman, after which we discuss some of the recent suc-cessful modifications that allow for heterogeneity in thetraffic stream (i.e., multi-class modelling), as well as theinclusion of more specific driver behavioural characteris-tics.

2.2.2 Mesoscopic models considered from a gas-kinetic perspective

As opposed to the macroscopic traffic flows models thatare derived from a conservation equation based on the

Navier-Stokes equations, mesoscopic models can be de-rived from a gas-kinetic analogy. From individual driv-ing behaviour (termed a microscopic approach), a macro-scopic model is derived. The earliest model can be tracedback to the work of the late Nobel laureate Ilya Prigogine,in cooperation with Frank Andrews and Robert Herman[241, 242]. A central component in their theory, is theconcept of a phase-space density (PSD):

k(t, x, vs) = k(t, x) P (t, x, vs), (23)

in whichP (t, x, vs) denotes thedistributionof the vehi-cles with space-mean speedvs at locationx and timet;the concept of this distribution originated in Boltzmann’stheory of gas dynamics. For the above density function, akinetic conservation equation can be derived, looking asfollows [121]:

dk

dt= kt + vs kx =

(kt

)

acc+(kt

)

int, (24)

with now the two terms on the right hand side denotingthe accelerationsof and interactionsbetween the vehi-cles; they are also calledgainsandlosses, relaxationandslowing down, or continuousanddiscreteterms, respec-tively [149, 262]. Equation (24) is called thePrigogine-Herman kinetic modeland it actually describes three pro-cesses:

1. Similar to the macroscopic conservation equation,the termvs kx describes a convective behaviour: ar-riving and departing vehicles cause a change in thedistributionk of vehicle speeds.

2. The first term on the equation’s right hand side,(kt

)

acc, describes the acceleration behaviour of ve-

hicles, which is assumed to be adensity-dependentrelaxation process of the speed distributionP ofequation (23) towards some pre-specified targetspeed distributionP0 (typically based on an equi-librium speed).

3. The second term on the equation’s right hand side,(kt

)

int, describes the interactions between vehicles,

as fast vehicles either must slow down or overtakeslower ones (hence implying inherently multi-lanetraffic). The decision on when to either slow downor to overtake (which is assumed to be adiscreteevent), is governed by the probabilities(1− π) andπ, respectively. The interaction term is called acol-lision equation, in analogy with the physics of theBoltzmann equation (where the collision term de-scribes the scattering of the gas molecules). Becausethere occur joint distributions in this latter equation(i.e., the probability of a faster vehicle encounteringa slower one), a common assumption calledvehicu-lar chaosis used, which states that vehicles’ speedsare uncorrelated, hence allowing to split the jointdistribution.

27

More than a decade later, Paveri-Fontana criticisedthe assumption of vehicular chaos in the interactionterm [230]. He subsequently proposed an improvedgas-kinetic model, in which he extended the phase-spacedensity of equation (23) with a dependence on thedesired speedvdes, i.e., k(t, x, vs, vdes); in Prigogine’soriginal derivation, this desired speed was incorrectlyconsidered to be a property of the road, instead of beinga driver-related property [121].

An interesting property of the gas-kinetic modellingapproach instigated by the seminal work of Pri-gogine, is that for densities beyond a certain criticaldensity, Nelson and Sopasakis found that the modelsolutions split into two distinct families. The currenthypothesis surrounding this phenomenon states thatthis corresponds to the widely observed data scat-ter in the empirically obtained (k,q) fundamental di-agrams [211].

2.2.3 Improvements to the mesoscopic modelling ap-proach

Significant contributions to the gas-kinetic mesoscopicmodel have been sporadic; after the work of Paveri-Fontana, Nelson was among the first to tackle the compu-tational complexity associated with the four-dimensionalphase-space densityk(t, x, vs, vdes) [210]. In his deriva-tion, he reformulated the relaxation and interaction termsboth as discrete events, based on a bimodal distributionof the vehicles’ speeds (i.e., corresponding to stoppedand moving vehicles). In contrast to the classic modelwhich uses a relaxation process in the acceleration term,Nelson furthermore based his derivation on a microscopicbehavioural model [121, 134, 262].

Building on the work of Nelson (which is, as he de-scribes, just a first initial step towards constructing a suit-able kinetic model), Wegener and Klar derived a kineticmodel in similar spirit, based on a microscopic descrip-tion of individual driver behaviour with respect to accel-erations and lane changes. Attractive to their work, is thefact that they also pay attention to the numerical solutionsof their model, with respect to the description of homo-geneous traffic flows [290].

Noting that the correct reproducing of traffic flowbehaviour at moderate to higher densities still troubledthe existing mesoscopic models, Helbing et al. exploredan interesting avenue. Not only did they capture theeffect that vehicles require a certain finite space (leadingto an Enskog- instead of a Boltzmann-equation), theyalso generalised the interaction term of equation (24).This last method allowed them to dismiss the traditionalassumption of vehicular chaos, i.e., they were now ableto treat correlations between vehicles’ speeds (whichhave a substantial influence at higher densities). Thetrick to obtain this behaviour, was to assume that driversreact to the downstream traffic conditions. This leads

to the inclusion of non-local interaction (braking) term,and hence their model is referred to as thenon-localgas-kinetic traffic flow model[119, 117]. Interestingly,this non-locality can generate effects that are similar tothe ones induced by viscosity/diffusion terms in macro-scopic traffic flows models, causing smooth behaviour atdensity jumps [121]. The power of their model is alsodemonstrated as it is able to reproduce all traffic regimesencountered in Kerner’s three-phase traffic theory [148].

Central to some of the recently proposed models,is the step process that transforms one model classinto another. Starting from microscopic driver be-havioural principles (e.g., accelerating, braking, . . . ),a mesoscopic model is deduced. This mesoscopicmodel can then be translated into an equivalentmacroscopic one by applying themethod of mo-ments. This allows to obtain PDEs that describethe dynamic evolution of the densityk, space-meanspeedvs, and its varianceΘ (an exception to thismethodology is the previously mentioned model ofWegener and Klar that obtains dynamic solutions di-rectly [290]). As an example, Helbing et al. alsodevised a numerical scheme for their previously dis-cussed model. It was implemented in a simulationpackage called MASTER [120].

Important progress was made by the work of Hoogen-doorn et al., who extended the gas-kinetic traffic flowmodels withmultiple user classes, in the sense that dif-ferent classes of drivers have different desired speeds.In order to achieve this, they replaced the traditionalphase-space density with amulti-class phase-space den-sity (MUC-PSD). The kinetic conservation equation thusdescribes the tempo-spatial evolution of this MUC-PSD(i.e., the interactions between different user classes), afterwhich an equivalent system of macroscopic model equa-tions is derived. The generalisation power of their modelis exemplified as the previously mentioned model of Hel-bing et al., which is just a special case, having only oneclass [132, 133]. The developed multiclass gas-kineticmodel is currently being integrated in a macroscopic sim-ulation model for complete road networks, called HE-LENA, which allows prediction of future traffic states,and hence to assess the effectiveness of policy measures[135].

Recently, Waldeer derived a kinetic model that is basedon the description of a driver’s acceleration behaviour (asopposed to hisobservedspeed behaviour). This novelapproach attempts to alleviate the unrealistic jumps inspeeds that are typically encountered in kinetic models.To this end, Waldeer extends the phase-space densityeven further, including a vehicle’s acceleration in addi-tion to its position, speed, and desired speed (leading toan even more complex system). Because now the acceler-ation is updated discretely, the speed will change contin-uously as a result [287]. Furthermore, Waldeer provideda numerical scheme for solving his model, by employinga Monte Carlo technique that is frequently used in non-

28

equilibrium gas-kinetic theory [286].

To end this overview of gas-kinetic models, we men-tion the important work of Tampere, who significantlyextended the previous modelling approaches [262, 264].In his work, he used thegeneralised phase-space density(as derived by Hoogendoorn [132]), which incorporatesa dependency on thetraffic stateS (e.g., encompassingvehicles’ speeds and their desired speeds). As it is anincreasingly recognised fact that a complete traffic flowmodel should contain elements which describe the humanbehaviour (see for example the comments made by Da-ganzo [71, 72]), Tampere proposes to include a driver’sactivation level. His human-kinetic model(HKM) is, justlike that of Helbing et al. and Hoogendoorn, able to re-produce all known traffic regimes. Because of the de-pendency of the PSD on a behavioural parameter (i.e.,the activation level that describes a driver’s awareness ofthe governing traffic conditions), the model is well-suitedto evaluate the applicability ofadvanced driver assis-tance systems(ADAS). As another illustrating example,the phenomenon of a capacity funnel can be realisticallyexplained and reproduced [263]. However, despite theprogress related to incorporating human behaviour intomathematical models for traffic flows, Tampere arguesthat most of the work can currently not be validated be-cause there is no appropriate data yet available.

2.3 Microscopic traffic flow models

Having discussed both mesoscopic and macroscopic traf-fic flows models, we now arrive at the other end of thespectrum where the microscopic models reside. Whereasthe former describe traffic operations on an aggregatescale, the latter kind is based on the explicit considerationof the interactions between individual vehicleswithin atraffic stream. The models typically employ character-istics such as vehicle lengths, speeds, accelerations, andtime and space headways, vehicle and engine capabili-ties, as well as some rudimentary human characteristicsthat describe the driving behaviour.

The material in this section is organised as follows:we first introduce the classic car-following (and lane-changing) models as well as some of their modern succes-sors, after which we discuss the optimal velocity model,then introduce the more human behaviourally psycho-physiological spacing models, which are subsequentlyfollowed by a brief description of traffic cellular automatamodels. After some words on models based on queueingtheory, the section concludes with a concise overview ofsome of the (commercially) available microscopic traf-fic flow simulators, as well as some of the issues that arerelated to the calibration and validation of microscopictraffic flow models.

More detailed information with respect to microscopicmodels (more specifically, car-following models), can befound in the book of May [190], the overview of Rothery[92], the work of Ahmed [2], and the overview of Brack-stone and McDonald [32].

2.3.1 Classic car-following and lane-changing mod-els

Probably the most widely known class of microscopictraffic flow models is the so-called family ofcar-following or follow-the-leadermodels. One of the oldest‘models’ in this case, is the one due to Reuschel [250],Pipes [238], and Forbes et al. [90]. It is probably bestknown as the“two-second rule”taught in driving schoolseverywhere11 . An earlier example of this line of reason-ing is the work of Herrey and Herrey, who specified asafedriving distancethat also included the distance needed tocome to a full stop [129].

It still remains astonishing that the seemingly dauntingand complex task that encompasses driving a vehicle,can be executed with such relative ease and little exer-cise, as is testified by the many millions of kilometresthat are driven each year. In spite of this remark, the firstmathematical car-following models that have been devel-oped, were based on a description of the interaction be-tween two neighbouring vehicles in a traffic stream, i.e.,a follower and its leader. In this section, we historicallysketch the development of car-following theories, as theyevolved from conclusions about early experiments intomore sophisticated models.

The above mentioned model was originally formulatedas the followingordinary differential equation(ODE) forsingle-lane traffic:

dvi(t)

dt=

vi+1(t) − vi(t)

T, (25)

with vi(t) and vi+1(t) the speeds of the following, re-spectively leading, vehicle at timet, andT a relaxationparameter. For the above case, the underlying assump-tion/justification is that vehiclei (the follower) tries toachieve the speedvi+1(t) of vehicle i + 1 (its leader),whilst taking a certain relaxation timeT into account.

As equation (25) describes a stable system, Chandler etal. were among the first to include an explicitreactiontime τ into the model (e.g.,τ =1.5 s), leading to desta-bilisation of vehicle platoons [48]. This reaction encom-passes both aperception-reaction time(PRT), i.e., thedriver sees an event occurring (for example the brakelights of the leading vehicle), as well as amovementtime (MT), i.e., the driver needs to take action by apply-ing pressure to the vehicle’s brake pedal [92]. Introduc-ing this behaviour, resulted in what is called astimulus-response model, whereby the right-hand side of equation(25) describes the stimulus and the left-hand side the re-sponse (the response is frequently identified as the ac-celeration, i.e., the actions a driver takes by pushing theacceleration or brake pedal). The relaxation parameteris then reciprocally reformulated as the sensitivity to thestimulus, i.e.,λ = T−1, resulting in the following ex-pression:

11Note that, in his article, Pipes actually stated his safe-distance ruleas keeping at least a space gap equal to a vehicle length for every 15km/h of speed you are travelling at [238, 134].

29

response = sensitivity× stimulus

dvi(t + τ)

dt= λ (vi+1(t) − vi(t)). (26)

Additional to this theoretical work, there were also someearly controlled car-following experiments, e.g., the onesdone by Kometani and Sasaki, who add a non-zero ac-celeration term to the right-hand side of the stimulus-response relation, in order to describe collision-free driv-ing based on a safety distance [152, 153].

Equation (26) is called adelayed differential equation(DDE), which, in this case, is known to behave in an un-stable manner, even resulting in collisions under certaininitial conditions. Gazis et al. remedied this situation bymaking the stimulusλ dependent on the distance, i.e., thespace gapgsi

between both vehicles [97]:

dvi(t + τ)

dt= λ

vi+1(t) − vi(t)

xi+1(t) − xi(t). (27)

Further advancements to this car-following model weremade by Edie, who introduced the current speed of thefollowing vehicle [84]. Gazis et al, forming the club ofpeople working at General Motors’ research laboratories,generalised the above set of models into what is called theGeneral Motors non-linear modelor theGazis-Herman-Rothery (GHR) model[95]:

dvi(t + τ)

dt= λ vm

i (t)vi+1(t) − vi(t)

(xi+1(t) − xi(t))l, (28)

with nowλ, l, andm model parameters (in the early days,the model was also called theL&M model [98]). For agood overview of the different combinations of parame-ters attributed to the resulting models, we refer the readerto the book of May [190], and the work of Ahmed [2].

A recent extension to the classic car-following theory, isthe work of Treiber and Helbing, who developed thein-telligent driver model(IDM). Its governing equation isthe following [268, 267, 269]:

dvi

dt= amax

1−

(vi

vdes

)δ

︸ ︷︷ ︸acceleration

−

(g∗s(vi, ∆vi)

gsi

)2

︸ ︷︷ ︸deceleration

, (29)

with amax the maximum acceleration,vdes the vehicles’desired speed, and∆vi the speed difference with the lead-ing vehicle (we have dropped the dependencies on timet for the sake of visual clarity). The first terms withinthe brackets denote the tendency of a vehicle to acceler-ate on a free road, whereas the last term is used to al-low braking in order to avoid a collision (the effectivedesired space gapg∗s(vi, ∆vi) is based on the vehicle’s

speed, its relative speed with respect to its leader, a com-fortable maximum deceleration, a desired time headway,and a jam space gap). The finer qualities of the IDM arethat it elegantly generalises most existing car-followingmodels, and that it has an explicit link with the non-localgas-kinetic mesoscopic model discussed in section 2.2.3[267]. It is furthermore quite capable of generating allknown traffic regimes. Based on the IDM, Treiber et al.also constructed thehuman driver model(HDM), whichincludes a finite reaction time, estimation errors, tempo-ral and spatial anticipation, and adaptation to the globaltraffic situation [270].

Similar to the work of Kometani and Sasaki, Gipps pro-posed a car-following model based on a safe brakingdistance, leading to collision-free dynamics [100]. Themodel is interesting because no differential equations areinvolved (i.e., the speeds are computed directly from onediscrete time step to another), and because it can captureunderestimation and overreactions of drivers, which canlead to traffic flow instabilities. In similar spirit of Gipps’work, Krauß developed a model that is based on assump-tions about general properties of traffic flows, as well astypical acceleration and deceleration capabilities of ve-hicles. Fundamental to his approach, is that all vehiclesstrive for collision-free driving, resulting in a model thathas the ability to generalise most known car-followingmodels [155, 156].

Another example of a recently proposed car-followingmodel, is the ‘simple’ model of Newell, who formulateshis theory in terms of vehicle trajectories whereby thetrajectory of a following vehicle is essentially the sameas that of its leader12. Remarkable properties are thatthe model has no driver reaction time, and that it cor-responds to the first-order macroscopic LWR traffic flowmodel with a triangularqe(k) fundamental diagram (seesection 2.1.3) [221]. The model furthermore also agreesquite well with empirical observations made at a signal-lised intersection, which support the model and conse-quently also the first-order macroscopic LWR model [3].

As a final example, we briefly illustrate Zhang’s car-following theory which is based on a multi-phase vehicu-lar traffic flow. This means that the model is able to repro-duce both the capacity drop and hysteresis phenomena,because his theory is based on the asymmetry betweenacceleration and deceleration characteristics of vehicles[182]. The model also holds a generalisation strength,as it is possible to derive all other classic car-followingmodels [308].

With respect to the stability of the car-following models,there exist two criteria, i.e.,local andasymptotic stability(also calledstring stability). The former describes howinitial disturbances in the behaviour of a leading vehicleaffect a following vehicle, whereas the latter is used todenote the stability of aplatoon of following vehicles. Bysuch a stable platoon it is then meant that initial finite dis-turbances exponentially die out along the platoon. Early

12A similar model was proposed earlier by Helly [125, 99, 221].

30

experiments by Herman et al. already considered thesecriteria for both real-life as for the developed mathemati-cal car-following models [127].

As an example, we graphically illustrate in Fig. 8 theasymptotic stability of a platoon of some 10 identical ve-hicles. We have used the simple car-following model ofGazis et al. of equation (27) to describe how a followingvehicle changes his acceleration, based on the speed dif-ference and space gap with its direct leader in the platoon(the sensitivityλ was set to 5000 m/s2 with a reactiontimeτ = 1 s). The left part of the figure shows all the ve-hicles’ positions, whereas the middle and right parts showthe speeds and accelerations of the 2nd , the 5th , and the10th vehicle respectively. We can see that all vehicles areinitially at rest (homogeneously spaced), after which theleading vehicle applies an acceleration of 1 m/s2, decel-erates with -1 m/s2, and then comes to a full stop. As canbe seen, the first 4 following vehicles mimic the leader’sbehaviour rather well, but from the 5th following vehicleon, an instability starts to form (note that all followingvehicles suffer from oscillations in their acceleration be-haviour). This instability grows and leads to very largeaccelerations for the last vehicle, which even momen-tarily reaches a negative speed of some -150 km/h; thisis clearly unrealistic (the vehicles shouldn’t be drivingbackwards on the road), indicating that the specified car-following model is unsuitable to capture the realistic be-haviour of drivers under these circumstances.

In continuation of this small excerpt on stability, we referthe reader to the work of Zhang and Jarrett who analyt-ically and numerically derive the general stability condi-tions (in function of the reaction time and the sensitiv-ity to the stimulus) for the previously mentioned classiccar-following models [309], the work of Holland who de-rives general stability conditions and validates them withempirical data containing non-identical drivers (i.e., ag-gressive and timid ones); central to Holland’s work isthe source for instability with respect to a breakdownof a traffic flow. He relates this event to a so-calledanticipation time that describes the duration for a wavecontaining an instability to travel to the current driver[131]. Finally, we mention that stability analysis is ofparamount importance for e.g., automated vehicle tech-nologies (‘smart cars’) such asintelligent or adaptivecruise control(ICC/ACC), as in for example the platoon-ing experiments in the PATH project where a platoon ofvehicles autonomously drives close to each other at highspeeds.

To conclude this section, we shed some light on the typ-ical mechanisms behind lane-changing models. With re-spect to microscopic models for multi-lane traffic, it isa frequent approximation to only take lateral movementsbetweenneighbouring lanes into account (as opposed tothe within-lane lateral dynamics of a vehicle). In suchcases, a vehicle changes a lane based on an incentive:these lane changes can then be classified as beingdiscre-tionary (e.g., to overtake a slower vehicle), ormandatory(e.g., to take an off-ramp). When a vehicle (i.e., driver)

has decided to perform a lane change, a check is madeon whether or not it is physically possible to merge in tothe adjacent lane (note this lane changing process also de-scribes vehicles turning at street intersections). This latterprocess is called thegap acceptancebehaviour: if thereis no such possibility (as it is frequently the case in densetraffic), a driver may initiate atforced merging, in whichcase the following vehicle in the target lane might have toyield. This interaction between forced merging and yield-ing can be frequently observed at on-ramps where heavyduty vehicles enter the motorway. Although it seems in-tuitive that there is an asymmetry between the frontal andbackward space gaps in the target lane (i.e., the former isusually smaller than the latter due to the human behaviourassociated with forced merging and yielding), there is inour opinion nevertheless not enough empirical data avail-able to calibrate the microscopic models that describelane-changing (see for example the work of Ahmed [2]).One way to obtain a correct behaviour is to use a kind ofa black box approach, in which for example the down-stream capacity of a motorway section is used as a mea-sure for calibrating the interactions (i.e., lane changes)between vehicles in a traffic stream. Note that as tech-nology advances, new detailed data sets are constructed.An example is the work of Hoogendoorn et al. who usea remote sensingtechnique to capture vehicle trajectoriesbased on aerial filming of driving behaviour under con-gested conditions [136].

2.3.2 Optimal velocity models

Closely related to the previously discussed classic car-following models, are the so-calledoptimal velocity mod-els(OVM) of Newell and Bando et al. Whereas the previ-ous car-following models mostly describe the behaviourof a vehicle that is following a leader, the OVMs modifythe acceleration mechanism, such that a vehicle’s desiredspeed is selected on the basis of its space headway, in-stead of only considering the speed of the leading vehicle[121]. Newell was the first to suggest such an approach,using an equilibrium relation for the desired speed as afunction of its space headway (e.g., thevs(hs) fundamen-tal diagram [212].

Bando et al. later improved this model, resulting in thefollowing equation that describes a vehicle’s accelerationbehaviour [16]:

dvi(t)

dt= α (V (hsi

(t)) − vi(t)) , (30)

in which V () is called theoptimal velocity function(OVF). The difference between this desired speed, asso-ciated with the driver’s current space headway, and thevehicle’s current speed, is corrected with an accelerationαV (), with nowα a coefficient expressing the sensitivityof a driver. Specification of the optimal velocity func-tion (typically a sigmoid function such astanh) is donesuch that it is zero forhsi

→ 0, and bounded tovmax forhsi

→ +∞; this latter condition means that the model is

31

0 10 20 30 40 50 60 70 80 90−1000

−800

−600

−400

−200

0

200

400

Time [s]

Pos

ition

[m]

0 10 20 30 40 50 60 70 80 90−200

−150

−100

−50

0

50

100

Time [s]

Spe

ed [k

m/h

]

0 10 20 30 40 50 60 70 80 90−4

−3

−2

−1

0

1

2

3

Time [s]

Acc

eler

atio

n [m

/s2 ]

Figure 8: An example based on Gazis et al.’s car-following model of equation (27), indicating an asymptotically unstableplatoon of 10 vehicles.Left: the time-space trajectories of all ten vehicles (the leading vehicle is shown with a thick solidline). We can see an instability occurring at the 6th (⋆) vehicle, growing severely such that the last vehicle even has to drivebackwards at a high speed of -150 km/h.Middle: the speeds of the first, the 2nd (◦), the 6th (⋆), and the last vehicle (×).Note the oscillations and negative values for the speeds of the vehicles at the end of the platoon.Right: the accelerationsof the first, the 2nd (◦), and the 6th (⋆) vehicle (example based on [141]).

able to describe the acceleration of vehicles without theexplicit need for a leader as in the previous car-followingmodels.

Interestingly, the OVM requires, in contrast to the clas-sic car-following models, no need for a reaction time inorder to obtain spontaneous clustering of vehicles [156].Unfortunately, the model is not always free of collisions,and can result in unrealistically large accelerations [206].

2.3.3 Psycho-physiological spacing models

Instead of using continuous changes in space gaps andrelative speeds, it was already recognised in the early six-ties that drivers respond to certainperception thresholds[32]. For example, a leading vehicle that is looming infront of a follower, will be perceived as having approxi-mately the same small size for a large duration, but oncethe space gap has shrunk to a certain size, the size of thelooming vehicle will suddenly seem a lot bigger (i.e., likecrossing a threshold), inducing the following vehicle toeither slow down or overtake.

The underlying thresholds with respect to speeds, speeddifferences, and space gaps, were cast into a model bythe work of Wiedemann et al. [292]. In this respect, themodels are calledpsycho-physiological spacing models,and although they seem quite successful in explaining thetraffic dynamics from a behavioural point of view (evenlane-change dynamics can be included based on suitableperception thresholds), calibration of the models has nev-ertheless been a difficult issue [32].

2.3.4 Traffic cellular automata models

In the field of traffic flow modelling, microscopic traf-fic simulation has always been regarded as a time con-suming, complex process involving detailed models thatdescribe the behaviour of individual vehicles. Approxi-mately a decade ago, however, new microscopic models

were being developed, based on thecellular automataprogramming paradigm fromstatistical physics. Themain advantage was anefficient and fast performancewhen used in computer simulations, due to their ratherlow accuracy on a microscopic scale. These so-calledtraffic cellular automata(TCA) are dynamical systemsthat are discrete in nature, in the sense that time advanceswith discrete steps and space is coarse-grained (e.g., theroad is discretised into cells of 7.5 metres wide, eachcell being empty or containing a vehicle). This coarse-graininess is fundamentally different from the usual mi-croscopic models, which adopt a semi-continuous space,formed by the usage of IEEE floating-point numbers.TCA models are very flexible and powerful, in that theyare also able to capture all previously mentioned basicphenomena that occur in traffic flows [18, 56]. In a largersetting, these models describeself-driven, many-particlesystems, operating far from equilibrium. And in contrastto strictly gaseous analogies, the particles in these sys-tems are intelligent and able to learn from past experi-ence, thereby opening the door to the incorporation ofbehavioural and psychological aspects [55, 294, 121].

Not only in the field of vehicular traffic flow mod-elling, but also in other fields such aspedestrian be-haviour, escapeandpanic dynamics, . . . the cellular au-tomata approach proved to be quite useful. It is nowfeasible to simulate large systems containing many ‘in-telligent particles’, such that is it possible to observetheir interactions, collective behaviour, self-organisation,. . . [140, 274, 123, 121, 199, 200, 54]

2.3.5 Models based on queueing theory

In this final section dealing with types of microscopictraffic flow models, we briefly summarise some of themodels that are based on the paradigm of queueing the-ory. Early applications of queueing theory to the field oftransportation engineering are mostly related to descrip-tions of the behaviour signallised and unsignallised in-

32

tersections, overtaking on two-lane roads with opposingtraffic, . . . [60]. Another more theoretically oriented ap-plication can be traced back to the work of Newell, whogives a nice summary of the mathematical details relatedto thepractical application of the methodology. Newellwas one of the few people who directly questioned theusefulness of cleverly devising a lot of methods and solu-tions, whereby corresponding problems remained absent[214]. In his later work, Newell reintroduced the conceptof arrival and departure functions (i.e., the cumulativecurves as briefly described in section 2.1.2), giving an an-alytical but still highly intuitive method for solving traf-fic flow problems, and drawing parallels with the well-known and studied first-order macroscopic LWR model[215, 216, 217].

During the mid-nineties, Heidemann developed severalqueueing-based traffic flow models, of which the mostpowerful version deals with non-stationary conditionsand is able to model the capacity drop and hysteresisphenomena, as well as providing an explanation for thewide scatter observed in empirical fundamental diagrams[116, 182].

Central to the approach in this field, is the partitioningof a road into equal pieces of width 1/kjam. Each ofthese pieces is then considered as aservice stationop-erating with aservice rateµ = kjam · vff . Equivalently,vehicles arrive at each service station with anarrival rateλ = k · vff , with the assumption thatk is the prevailingdensity and that traffic can flow unimpeded in the free-flow traffic regime. When vehicles enter the motorway,they can get stuck inside the queues, thereby reducingthe space-mean speed in the system. Differentqueueingpoliciescan be specified in the form of service and arrivaldistributions. In queueing theory, theKendall notationisadopted, whereby a system is described asA/S/m withA the arrival distribution,S the service distribution, andm the number of servers (i.e., service stations). Typicalforms are theM/M/1 queues that have an exponentiallydistributed arrival time, exponentially distributed servicetime, and one server (with an infinite buffer).

Recently, Van Woensel extended the existing queueingmodels for traffic flows, leading to e.g., analytical deriva-tions of fundamental diagrams based onG/G/m queuesthat have general distributions for the arrival and servicerates with multiple servers [293]. The methodology alsoincludes queues with finite buffers, and has been appliedto the estimation of emissions, although we question thevalidity of this latter approach (which is essentially basedon a one-dimensional fundamental diagram) as we be-lieve dynamic models are necessary, e.g., to capture tran-sients in traffic flows [275].

Queue-based models were also used to describe large-scale traffic systems, e.g., complete countries, as wasmentioned in section 1.2.5 [47]. In that section, we al-ready mentioned that queues with finite buffer capacitiesare to be preferred in order to correctly model queue spillback. However, with respect to a proper description oftraffic flow phenomena, some of the problems can not

be so easily solved, e.g., the speed of a backward prop-agating kinematic shock wave. Take for example vehi-cles queued behind each other at a traffic light: once thelight turns green, the first-order macroscopic LWR modelcorrectly shows the dispersal of this queue. In a queue-based model however, once a vehicle exits the front of thequeue, all vehicles simultaneously and instantly move upone place, thus the kinematic wave propagates backwardsat an infinite speed !

To conclude this short summary on queueing models, wemention the work of Julvez and Boel, who present a simi-lar approach, based on the use ofPetri nets13. Their workallows them to construct complete urban networks, basedon the joining together of short sections, with continuousPetri nets for the propagation of traffic flows, and discretePetri nets for the description of the traffic lights [19, 146].

2.3.6 Microscopic traffic flow simulators

In continuation of the previous sections that gave anoverview of the different types of existing microscopictraffic flow models, this section introduces some of thecomputer implementations that have been built aroundthese models. In most cases, the computer simulators in-corporate the car-following and lane-changing processesas submodels, as opposed to strategic and operationalmodules that work at a higher-level layer (i.e., routechoice, . . . ).

Whereas most microscopic traffic simulators allow tobuild a road network, specify travel demands (e.g., bymeans of OD tables), there was quite some effort spentover the last decade, in order to achieve a qualitative visu-alisation (e.g., complete virtual environments with trees,buildings, pedestrians, bicycles, . . . An example of such avirtual environment is shown in Fig. 9, which is based onVISSIM’s visualisation module. Note that in our opinion,the usefulness of these virtual scenes should not be un-derestimated, as in some cases a project’s approval mighthinge on a good visual representation of the results. It isone thing for policy makers to judge the effects of replac-ing a signallised intersection with a roundabout, based ona report of the observed downstream flows of each inter-section arm, but it gives a whole other feeling when theyare able toseehow the traffic streams will interact ! Evenin the early sixties, it was recognised that a visual rep-resentation of the underlying traffic flow process, was anundeniable fact for promoting its acceptance among traf-fic engineers [99]. With respect to this latest comment,Lieberman even states that“There is a need to view vehi-cle animation displays, to gain an understanding of howthe system is behaving, in order to explain why the result-ing statistics were produced”[92].

13Petri nets (invented in the sixties by Carl Adam Petri) are a for-malism for describing discrete systems [235]; they consistof directedgraphs of ‘transitions’ and ‘places’, with arcs forming theconnectionsbetween them. Places can contain ‘tokens’, which can be ‘consumed’when a transition ‘fires’.

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Figure 9: A screenshot of the VISSIM microscopic traf-fic flow simulator, showing a detailed virtual environmentcontaining trees, buildings, pedestrians, . . . (image repro-duced after [243]).

Quite a large amount of microscopic traffic flow mod-els have been developed, in most cases starting from aresearch tool, and — by the law of profit — naturallyevolving into full-blown commercial packages, includinge.g., dynamic traffic assignment and other transportationplanning features. Note the sad observation that this com-mercialisation inherently tends to obscure the underlyingmodels. In such cases, privacy concerns, company poli-cies, and project contracts and agreements prohibit a totaldisclosure of the mathematical details involved. Some ofthese computer models are listed here. For starters, theGeneric Environment for TRaffic Analysis and Modeling(GETRAM) couples the multi-modal traffic assignmentmodel EMME/2 to theAdvanced Interactive MicroscopicSimulator for Urban and Non-Urban Networks(AIM-SUN2) model [17]. Next, theParallel microscopic traf-fic simulator(Paramics), initially developed at the Edin-burgh Parallel Computing Centre, but afterwards boughtby Quadstone [43, 170]. Subsequently, Yang developeda MIcroscopic Traffic flow SIMulator(MITSIM) [297],and Maerivoet constructed aMIcroscopic TRAffic flowSIMulator (Mitrasim 2000) which was mostly based onand influenced by MITSIM’s and Paramics’ dynamic be-haviour [179]. Two further examples are the Open SourceSoftware (OSS) package calledSimulation of Urban MO-bility (SUMO), developed at the Deutsches Zentrum furLuft- und Raumfahrt [154], and the VISSIM programmedeveloped by the German PTV group [243]. In addi-tion, there is theINteractive DYnamic traffic assignment(INDY) model [185], and the INTEGRATION softwarepackage developed by Van Aerde et al. This latter sim-ulator deserves a special mention: it is microscopic innature, but the speeds of the vehicles that are propagatedthrough the network, are based on a macroscopicvse

(hs)fundamental diagram for each link [1]. Finally, we men-tion theTRansportation ANalysis and SIMulation System(TRANSIMS) project [205], . . .

An extensive overview of all existing microscopic traffic

flow simulators until 1998 is provided by theSimulationModelling Applied to Road Transport European SchemeTests, or better known as the SMARTEST report [4].

When using one of these microscopic simulators, itis important to understand the assumptions and limi-tations inherent to the implemented models, in orderto judge the results objectively. Indeed, as with anymodel, the question on whether some observed be-haviour arises due to the implemented model, or asa result of the imposed boundary conditions, shouldalways be asked, understood, and answered.

2.3.7 Calibration and validation issues

Due to the sometimes large amount of parameters typi-cally involved in microscopic traffic flow models, theircomputational complexity is often a significant disadvan-tage when compared to meso- or macroscopic models (al-though there are some exceptions, e.g., the traffic cellu-lar automata models of section 2.3.4). From the point ofview of model calibration and validation, this poses aninteresting conundrum, as in many cases not all param-eters are equally influential on the results (thus requir-ing some sensitivity analyses). In this sense, microscopicmodels contain a real danger of purporting to convey asort of fake accuracy. Different parameter combinationscan lead to the same phenomenological effects, leavingus pondering as to what exactly is causing the observedbehaviour [262]. As there is no clear road map on how tocalibrate microscopic traffic flow models, we here give asmall sample of some of the numerous attempts that havebeen made.

There is the work of Jayakrishan and Sahraoui who dis-tinguish between calibration in the conceptual (i.e., atthe level of the underlying mathematical model) and op-erational (i.e., within the global context of the study)phases; they apply their operational methodology to bothPARAMICS (micro) and DYNASMART (macro), usingthe California Freeway Performance Measurement Sys-tem(PeMS) database from the PATH project to feed andcouple both models [142].

Based on a publicly available data set of a one-lay roadcorridor of six kilometres long (the data contained de-tailed cumulative curves), Brockfeld et al. systemati-cally tested the predicted travel times of some ten well-known microscopic traffic flow models. As a result of anon-linear optimisation process to calibrate the models,they found that the intelligent driver model and the cell-transmission model perform the best (i.e., below an errorrate of 17%), due to the fact that these models require theleast amount of parameters (there were even some mod-els such as the Gipps-based ones that hadhiddenparam-eters). Their final conclusion is noteworthy, as they statethat “creating a new model is often done, however cali-brating this model to reality is a formidable task, whichexplains why there currently are more models than resultsabout them”[35].

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Related to the previous study, Hourdakis et al. present anautomated systematic calibration methodology based onan optimisation process, applied to the AIMSUN2 sim-ulator. The data used for the calibration procedure stemfrom a twenty kilometres long motorway in Minneapo-lis, Minnesota. The process first involves a calibrationof the global model parameters (i.e., to get the macro-scopic flows and speeds correct), after which the localparameters are dealt with (i.e., ramp metering setups, etcetera). In their results, Hourdakis et al. state an obtainedaverage correlation coefficient of 0.961 for manual cali-bration (the results for the automated calibration are sim-ilar), which is quite high (they mostly explain this due tothe data’s high level of detail, as well as the quality of thesimulator software) [138].

Recently, Chu et al. extended the systematic, multi-stagecalibration approach for the PARAMICS simulator.Based on data of a highly congested six kilometres longcorridor network in the city of Irvine, Orange County,California, they first calibrate the driving behaviourmodels, then the route choice model, after which estima-tion and fine-tuning of the OD tables is done. Despitethe good reproduction of travel times, their calibrationmethodology was done manually, and an automatedoptimisation procedure remains future work [57].

Other examples of calibration of microscopic traffic flowmodels, include the work of Dowling et al., who givean extensive account on the application of commerciallyavailable simulation tools to typically encountered trafficengineering problems [82], and the work of Mahantiwhich is primarily based on the correct representation ofOD tables [184].

To end this section, we state some important princi-ples that are — in our opinion — related to a correctcalibration methodology. First and foremost, we be-lieve that all traffic flow models (whether they aremacro-, meso- or microscopic in nature), should beable to accurately reproduce and predict the encoun-tered delays, queue lengths, and other macroscopicfirst-order characteristics(i.e., the kinematic wavespeed, a correct and realistic road capacity, . . . ). Oneway to test this is the use of cumulative curves, asthey provide an elegant way to automatically per-form a good calibration. It is for example possible toconsider the difference between observed and sim-ulated curves, and then use a Kolmogorov-Smirnovgoodness-of-fit statistical test to decide on whetherthe difference is statistically significant, or if it is justa Brownian motion with a zero mean. Only whenthese first-order effects can be correctly reproduced,the next step can be to considersecond-order effectssuch as waves of stop-and-go traffic, oscillations, . . .

Furthermore, it is important to take into account thespa-tial natureof the study area, i.e., a detailed description ofthe road infrastructure, with bottleneck locations as wellas up- and downstream boundary conditions. With re-

spect to the model that is created within the computer, itis paramount to know how the model behaves on both thelink as well as the node level. Because the models aremost of the time working with fairly homogeneous roadlinks (e.g., constant elevations, no road curvature, . . . ),itmight be necessary to allow for small deviations from (orfixes to) reality (e.g. inserting extra intermediate nodes inthe network in order to artificially obtain bottlenecks).

2.4 Submicroscopic traffic flow models

As the level of modelling detail is increased, we enterthe realm of submicroscopic models. Traditional mi-croscopic models describe vehicles as single operatingunits, putting emphasis on the interactions between dif-ferent (successive) vehicles. In addition to this, submi-croscopic models push the boundaries even further, giv-ing detailed descriptions of a vehicle’s inner workings.This typically entails modelling of thephysical charac-teristics such as engine performance, detailed gearboxoperations, acceleration, braking, and steering manoeu-vres, . . . Complementary to the functioning of a vehi-cle’s physical components, submicroscopic models canalso describe ahuman driver’s decision taking processinmuch more detail than is usually done. Some examplesof submicroscopic models are:

• van Arem’s Microscopic model for Simulation ofIntelligent Cruise ControlMIXIC: it contains adriver model (for deciding on and executing of lanechanges, car-following behaviour, and the appli-cation of intelligent (or adaptive) cruise control –ICC/ACC) and a vehicle model (dealing with the en-gine, the transmission, road friction, aerodynamic,rolling, and slope resistance) [271].

• In similar spirit, Minderhoud has developed theSim-ulation model of Motorways with Next generationvehicles(Simone); this model focusses on intel-ligent driver support systems, such as ICC/ACC,platoon driving, centralised control of vehicles, etcetera. In contrast to most other (sub)microscopicmodels, Simone explicitly allows for rear-end col-lisions to occur under certain parameter combina-tions. As there is a close coupling between driverbehaviour related parameters and those of the simu-lation, these collision dynamics enable the modellerto find realistic values (or ranges) for these parame-ters [195].

• Ludmann’sProgram for the dEvelopment of Lon-gitudinal micrOscopic traffic Processes in a Sys-temrelevant environment(PELOPS), is akin to theprevious two models. It is however more techno-logically oriented with respect to the car-followingbehaviour of vehicles, aiming at merging both adriver’s perceptions and decisions, the car’s han-dling, and the surrounding traffic conditions. Atthe core of the model, there are four modules that

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respectively describe vehicle routing in a road net-work, human decision taking (i.e., car-following,tactical decisions with respect to lane-changing,. . . ), vehicle handling (i.e., a driver’s physical acts ofsteering, accelerating and braking, . . . ), and finallya module that describes physical vehicle character-istics (traction on elevations, engine capabilities, ex-haust gas modelling, . . . ) [178, 85].

To conclude this section, we like to mention an often sci-entifically neglected area of research, namely the popularfield of simulation in thecomputer gaming industry. Overthe last couple of decades, numerous arcade-stylerac-ing simulationshave been developed, allowing aplayerto be completely immersed in a three-dimensional vir-tual world in which racing at high speeds is paramount.Examples of these kinds of programmes are the highlyaddictive world ofFormula 1 racing, street racing incity environments, off-road rally races, . . . The underly-ing submicroscopic models in these games, have overthe course of several years been evolved to incorporateall sorts of physical effects. Friction characteristics (e.g.,pavement versus asphalt), road elevation, wet conditions,air drag and wind resistance (including effects such asslip streaming and downforce), car weight depending onfuel consumption, tyre wear, . . . have had influences onwhat we commonly refer to ascar handling, i.e., realisticbehaviour with respect to car acceleration, braking, andsteering. Thanks to the increasing computational powerof desktop computers, graphics cards, as well as dedi-cated gaming consoles (e.g., Microsoft’sXbox, Sony’sPlayStation, Nintendo’sGameCube, . . . ), the path to awhole plethora of extra realistic effects has been paved:skidding, under- and oversteering, sun glare, overly real-istic collision dynamics (in our opinion, this is where thearcade sensation plays a major role), . . .

2.5 The debate between microscopic andmacroscopic models

Deciding which class of models, i.e., microscopic, ormacroscopic (and we also include the mesoscopic mod-els), is the correct one to formulate traffic flow problems,has been a debate among traffic engineers ever since thelate fifties. Although the debate was not as intense assay, the one between first- and higher-order macroscopictraffic flow models (see section 2.1.7 for more details),it nevertheless sparkled some interesting issues. As isnearly always the case, the true answer to the above ques-tion depends on the kind of problem one is interested insolving [98].

In the beginning years of traffic flow engineering, abridge was formed between the microscopic General Mo-tors car-following model of equation (28), and the Green-berg macroscopic model [104, 95]. This proved to bequite a significant breakthrough, as it was now possible toobtain all known steady-state macroscopic fundamentaldiagrams, by integrating the car-following equation with

suitably chosen parameter values [98]. A recent exampleof this kind of linking, was done by Treiber and Helbing,who provided amicro-macro linkbetween their non-localgas-kinetic mesoscopic model (see section 2.2.3) and theintelligent driver model (see section 2.3.1) [119, 117].

Besides this explicit translating of microscopic intomacroscopic (mesoscopic) models and vice versa, itis also possible to develophybrid modelsthat couplemacroscopically modelled road links to microscopicallymodelled ones. Examples include the work of Magneet al., who develop a hybrid simulator that couples aMETANET-like second-order macroscopic traffic flowmodel with theSImulation TRAfic(SITRA-B+) micro-scopic traffic flow model. Special attention is given tothe interfaces between macroscopically and microscop-ically modelled road segments; each macroscopic timeiteration in the simulator, is accompanied by a number ofmicroscopic iterations [183]. In similar spirit, the workof Bourrel and Henn links macroscopic representationsof traffic flows to microscopic ones, using interfaces thatdescribe the transitions between them. As an applicationof their methodology, they describe the translation be-tween the first-order macroscopic LWR model and a ve-hicle representation of this model (based on trajectories)[29]. Another avenue was pursued by the Wilco [41] andWilco et al. [42], who developed an integration frame-work between the MITSIMLab microscopic model andthe Mezzo mesoscopic model. By building upon a meso-scopic approach, the strength of their work lies in the factthat no aggregation and disaggregation of flows needs tobe performed.

3 Conclusions

The material elaborated upon in this paper, spanned abroad range going from transportation planning modelsthat operate on a high level, to traffic flow models that ex-plicitly describe the physical propagation of traffic flows.

As explained in the introduction, we feel there is a fre-quent confusion among traffic engineers and policy mak-ers when it comes to transportation planning models andthe role that traffic flow models play therein. To thisday, many transportation planning bureaus continue touse static tools for evaluating policy decisions, whereasthe need for dynamic models is getting more and morepronounced [180].

Even after more than sixty years of traffic flow modelling,the debate on what is the correct modelling approachremains highly active. On the transportation planningside, many agencies still primarily focus on the tradi-tional four-step model (4SM), because it is the best intu-itively understood approach. In contrast to this, activity-based modelling (ABM) is gaining momentum, althoughit remains a rather obscure discipline to many people. Atthe basis of this scrutiny towards the ABM, lies the ab-sence of a generally accepted framework such as the oneof the 4SM. It is tempting to translate the ABM approach

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to the 4SM, by which e.g., the ABM’s synthetic popula-tion generation (including activity generation, householdchoices and scheduling) corresponds to the 4SM’s pro-duction and attraction, distribution, and modal split (orto discrete choice theory in a broader setting), therebygenerating (time dependent) OD tables. Similarly, theABM’s agent simulation can be seen as an implementa-tion of the 4SM’s traffic assignment. However, it remainsdifficult to gain insight into this kind of direct translationand the resulting travel behaviour, although the ABM’sscientific field is continuously in a state of flux thanks tothe increasing computational power.

On the traffic flow modelling side, the debate on whetheror not to use macro-/meso- or microscopic models stillcontinues to spawn many intriguing discussions. De-spite the respective criticisms, it is widely agreed uponthat modelling driver behaviour entails complex human-human, human-vehicle, and vehicle-vehicle interactions.These call for interdisciplinary research, drawing fromfields such as mathematics, physics, and engineering, aswell as sociology and psychology (see e.g., the overviewof Helbing and Nagel [122]).

A Glossary of terms

A.1 Acronyms and abbreviations

4SM four step modelAADT annual average daily trafficABM activity-based modellingACC adaptive cruise controlACF average cost function

ADAS advanced driver assistance systemsAIMSUN2 Advanced Interactive Microscopic

Simulator for Urban and Non-UrbanNetworks

AMICI Advanced Multi-agent Information andControl for Integrated multi-class trafficnetworks

AON all-or-nothingASDA Automatische StauDynamikAnalyseASEP asymmetric simple exclusion processATIS advanced traveller information systemsATMS advanced traffic management systemsBCA Burgers cellular automatonBJH Benjamin, Johnso, and HuiBJH-TCA Benjamin-Johnson-Hui traffic cellular

automatonBL-TCA brake-light traffic cellular automatonBML Biham, Middleton, and LevineBML-TCA Biham-Middleton-Levine traffic cellular

automatonBMW Beckmann, McGuire, and WinstenBPR Bureau of Public RoadsCA cellular automatonCA-184 Wolfram’s cellular automaton rule 184CAD computer aided designCBD central business districtCFD computational fluid dynamicsCFL Courant-Friedrichs-LewyChSch-TCA Chowdhury-Schadschneider traffic

cellular automatonCLO camera LinkeroeverCML coupled map latticeCONTRAM CONtinuous TRaffic Assignment

ModelCOMF car-oriented mean-field theoryCPM computational process modelsCTM cell transmission modelDDE delayed differential equationDFI-TCA deterministic Fukui-Ishibashi traffic

cellular automatonDGP dissolving general patternDLC discretionary lane changeDLD double inductive loop detectorDNL dynamic network loadingDRIP dynamic route information panelDTA dynamic traffic assignmentDTC dynamic traffic controlDTM dynamic traffic managementDUE deterministic user equilibriumDynaMIT Dynamic network assignment for the

Management of Information toTravellers

DYNASMART DYnamic Network Assignment-Simulation Model for AdvancedRoadway Telematics

ECA elementary cellular automatonEP expanded congested patternER-TCA Emmerich-Rank traffic cellular

automatonFCD floating car data

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FDE finite difference equationFIFO first-in, first-outFOTO Forecasting of Traffic ObjectsGETRAM Generic Environment for TRaffic

Analysis and ModelingGHR Gazis-Herman-RotheryGIS geographical information systemsGNSS Global Navigation Satellite System

(e.g., Europe’s Galileo)GoE Garden of Eden stateGP general patternGPRS General Packet Radio ServiceGPS Global Positioning System

(e.g., USA’s NAVSTAR)GRP generalised Riemann problemGSM Groupe Speciale MobileGSMC Global System for Mobile

CommunicationsHAPP household activity pattern problemHCM Highway Capacity ManualHCT homogeneously congested trafficHDM human driver modelHKM human-kinetic modelHRB Highway Research BoardHS-TCA Helbing-Schreckenberg traffic cellular

automatonICC intelligent cruise controlIDM intelligent driver modelINDY INteractive DYnamic traffic assignmentITS intelligent transportation systemsIVP initial value problemJDK JavaTM Development KitKKT Karush-Kuhn-TuckerKKW-TCA Kerner-Klenov-Wolf traffic cellular

automatonKWM kinematic wave modelLGA lattice gas automatonLOD level of detailLOS level of serviceLSP localised synchronised-flow patternLTM link transmission modelLWR Lighthill, Whitham, and RichardsMADT monthly average daily trafficMC-STCA multi-cell stochastic traffic cellular

automatonMesoTS Mesoscopic Traffic SimulatorMFT mean-field theoryMITRASIM MIcroscopic TRAffic flow SIMulatorMITSIM MIcroscopic Traffic flow SIMulatorMIXIC Microscopic model for Simulation of

Intelligent Cruise ControlMLC mandatory lane change

moving localised clusterMOE measure of effectivenessMPA matrix-product ansatzMPCF marginal private cost functionMSA method of successive averagesMSCF marginal social cost functionMSP moving synchronised-flow pattern

MT movement timeMUC-PSD multi-class phase-space densityNaSch Nagel and SchreckenbergNAVSTAR Navigation Satellite Timing and RangingNCCA number conserving cellular automatonNSE Navier-Stokes equationsOCT oscillatory congested trafficOD origin-destinationODE ordinary differential equationOSS Open Source SoftwareOVF optimal velocity functionOVM optimal velocity modelParamics Parallel microscopic traffic simulatorPATH California Partners for Advanced Transit

and HighwaysProgram on Advanced Technology forthe Highway

PCE passenger car equivalentPCU passenger car unitPDE partial differential equationPELOPS Program for the dEvelopment of

Longitudinal micrOscopic trafficProcesses in a Systemrelevantenvironment

PeMS California Freeway PerformanceMeasurement System

PHF peak hour factorPLC pinned localised clusterpMFT paradisiacal mean-field theoryPRT perception-reaction timePSD phase-space densityPW Payne-WhithamQoS quality of serviceSFI-TCA stochastic Fukui-Ishibashi traffic

cellular automatonSimone Simulation model of Motorways with

Next generation vehiclesSLD single inductive loop detectorSMARTEST Simulation Modelling Applied to Road

Transport European Scheme TestsSMS space-mean speedSOC self-organised criticalitySOMF site-oriented mean-field theorySP synchronised-flow patternSSEP symmetric simple exclusion processSTA static traffic assignmentSTCA stochastic traffic cellular automatonSTCA-CC stochastic traffic cellular automaton

with cruise controlSUE stochastic user equilibriumSUMO Simulation of Urban MObilityT2-TCA Takayasu-Takayasu traffic cellular

automatonTASEP totally asymmetric simple exclusion

processTCA traffic cellular automatonTDF travel demand functionTMC Traffic Message ChannelTMS time-mean speed

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TOCA time-oriented traffic cellularautomaton

TRANSIMS TRansportation ANalysis and SIMulationSystem

TRB Transportation Research BoardTSG triggered stop-and-go trafficUDM ultra-discretisation methodUMTS Universal Mobile Telecommunications

SystemVDR-TCA velocity-dependent randomisation traffic

cellular automatonVDT total vehicle distance travelledVHT total vehicle hours travelledVMS variable message signVMT total vehicle miles travelledVOT value of timeWSP widening synchronised-flow patternWYA whole year analysis

A.2 List of symbols

amax the maximum acceleration in the IDM

c(k) the sound speed of traffic

C(q) the economical cost associated with the

travel demandq

∆f(x) the forward difference operator applied

to f(x)

∆k the difference in density up- and

downstream of a shock wave

∆q the difference in flow up- and

downstream of a shock wave

∆X the width of a cell in a numerical

discretisation scheme

∆T the size of a time step in a numerical

discretisation scheme

Dj a destination zonej

ǫ a small diffusion constant for the

viscosityν

g∗s(vi, ∆vi) the effective desired space gap in

the IDM

κ a kinetic coefficient related toτ , k,

andΘ

kt the partial derivative ofk(t, x) with

respect to time

kx the partial derivative ofk(t, x) with

respect to space

k(t, x, vs) the phase-space density at(t, x)

associated with SMSvs

kt the partial derivative ofk(t, x) with

respect to time

kx the partial derivative ofk(t, x) with

respect to space

λ the sensitivity to the stimulus in a

car-following model

the arrival rate at a server in queueing

theory

µ the service rate of a server in queueing

theory

∇f(x) the backward difference operator applied

to f(x)

the gradient vector off(x)

ν the kinematic traffic viscosity coefficient

Oi an origin zonei

π the probability of overtaking (as opposed

to slowing down)

P the traffic pressure

Px the partial derivative of the traffic pressure

with respect to space

P (t, x, vs) the distribution of the vehicles with SMS

vs at (t, x)

qpc the practical capacity

qso travel demand associated with a system

optimum

que travel demand associated with a user

equilibrium

S a traffic state in the human-kinetic model

τ a driver’s reaction time

Θ the variance of the speed

Θe(k, vs) an equilibrium relation between the speed

variance, the density, and the SMS

T a travel time

a relaxation parameter (in Pipes’ car-

following model)

Tff a travel time under free-flow conditions

u the velocity (in the context of a Navier-

Stokes fluid)

vdes the desired speed of drivers

vstthe partial derivative of the space-mean

speed with respect to time

vsxthe partial derivative of the space-mean

speed with respect to space

vse(k, Θ) an equilibrium relation between the SMS,

the density, and the speed variance

V () the optimal velocity function

wshock the speed of a shock wave

39

Acknowledgements

Dr. Bart De Moor is a full professor at the KatholiekeUniversiteit Leuven, Belgium. Our research is supportedby: Research Council KUL: GOA AMBioRICS, sev-eral PhD/postdoc & fellow grants,Flemish Govern-ment: FWO: PhD/postdoc grants, projects, G.0407.02(support vector machines), G.0197.02 (power islands),G.0141.03 (identification and cryptography), G.0491.03(control for intensive care glycemia), G.0120.03 (QIT),G.0452.04 (new quantum algorithms), G.0499.04 (statis-tics), G.0211.05 (Nonlinear), research communities (IC-CoS, ANMMM, MLDM), IWT : PhD Grants, GBOU(McKnow), Belgian Federal Science Policy Office:IUAP P5/22 (‘Dynamical Systems and Control: Com-putation, Identification and Modelling’, 2002-2006),PODO-II (CP/40: TMS and Sustainability),EU: FP5-Quprodis, ERNSI, Contract Research/agreements:ISMC/IPCOS, Data4s,TML, Elia, LMS, Mastercard.

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