Transportation 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 1.3.2 Approaches to activity-based modelling 10 1.3.3 Towards elaborate agent-based simulations 11 In this paper, we focus on the different traffic flow models 1.4 Transportationeconomics. 12 that exist in literature. Due to our frequently encountered 1.4.1 Theeconomicalsetting . 13 confusion among traffic engineers and policy makers, this 1.4.2 Towardsroadpricingpolicies . 14 paper goes into more detail about transportation planning 2 Traffic flow propagation models 16 models on the one hand, and traffic flow models on the other hand. The former deal with households that make 2.1 Macroscopictrafficflowmodels . 17 certain decisions which lead to transportation and the use 2.1.1 Thecontinuumapproach . 17 of infrastructure, as opposed to the latter which explic- 2.1.2 Thefirst-orderLWRmodel. 18 itly describe the physical propagation of traffic flows in a 2.1.3 Analytical solutions of the LWR model 18 road network. Our goal is not to give a full account (as 2.1.4 Numerical solutions of the LWR model 20 that would be a dissertation of its own, given the broad- 2.1.5 FlavoursoftheLWRmodel . 22 ness of the field), but rather to impose upon the reader a 2.1.6 Higher-ordermodels . 23 thorough feeling for the differences between transporta- 2.1.7 Critiqueson higher-ordermodels 24 tion planningand traffic flow models. Because of the high 2.2 Mesoscopictrafficflowmodels . 26 course of progress over the last decade (or even during 2.2.1 The different meanings of ‘mesoscopic’ 26 the last five years), this paper tries to chronicle both past 2.2.2 Mesoscopic models considered from a gas-kinetic perspecti models, as well as some of the latest developments in this 2.2.3 Improvementsto the mesoscopic modelling approach 28 area. 2.3 Microscopictrafficflowmodels . 29 2.3.1 Classic car-following and lane-changing models 29 PACS numbers: 89.40.-a 2.3.2 Optimalvelocitymodels . 31 Keywords: land-use, trip-based, activity-based, trans- 2.3.3 Psycho-physiologicalspacing models 32 portation economics, macroscopic, mesoscopic, micro- 2.3.4 Trafficcellular automatamodels . 32 scopic 2.3.5 Modelsbasedonqueueingtheory 32 2.3.6 Microscopic traffic flow simulators 33 2.3.7 Calibration and validation issues . 34 Contents 2.4 Submicroscopictrafficflowmodels. 35 2.5 The debate between microscopic and macroscopic models 36 1 Transportation planning models 2 1.1 Landuse andsocio-economicalbehaviour 2 3 Conclusions 36 1.1.1 Classicland-usemodels . 2 1.1.2 Themodern approachto land-usemodels 3 AGlossaryofterms 37 1.2 Trip-basedtransportationmodels . 4 A.1 Acronymsandabbreviations . 37 arXiv:physics/0507127v1 [physics.soc-ph] 15 Jul 2005 1.2.1 Basicentitiesandassumptions . 4 A.2 Listofsymbols . 39 1.2.2 Thefoursteps. 4 Due to our frequently encountered confusion among traf- 1.2.3 Statictrafficassignment . 6 fic engineers and policy makers when it comes to trans- 1.2.4 Dynamictrafficassignment . 8 portation planning models and the role that traffic flow 1.2.5 Critiqueontrip-basedapproaches 9 models play therein, this paper strives to alleviate that 1.3 Activity-basedtransportationmodels . 9 bewilderment. The material elaborated upon in this pa- 1.3.1 Historicorigins . 10 per, spans a broad range going from transportation plan- ∗[email protected] ning models that operate on a high level and deal with †[email protected] households that make certain decisions which lead to 1 transportation and the use of infrastructure, to traffic flow strictly determined by economical activities alone, i.e., models that explicitly describe the physical propagation ethnic considerations et cetera can be taken into account), of traffic flows in a road network. Because transportation has spatial interactions with land use and vice versa, it can lead to a kind of chicken-and- egg problem [252]. For example, building a new road will 1 Transportation planning models attract some economical activity (e.g., shopping malls et cetera), which can lead to a possible increase of the travel Before going into detail about the possible mathematical demand. This in turn, can lead to an increase of extra models that describe the physical propagation of traffic economical activity (because of the well-suited location), flows, it is worthwhile to cast a glance at a higher level, and so on, resulting in a local reorganisation of the spa- where transportation planning models operate. The main tial structure. Resolving this chicken-and-egg paradox, is rationale behind transportation planning systems, is that typically done by means of feedback and iterations be- travellers within these systems are motivated by making tween land-use and transportation models, whereby the certain decisions about their wishes to participate in so- former provide the basic starting conditions for the latter cial, economical, and cultural activities. The ensemble models (with sometimes a reversal of the models’ roles). of these activities is called the activity system. Because In the following two sections, we first shed some light on these activities are spatially separated (e.g., a person’s liv- several of the archetypical land-use models, after which ing versus work area), the need for transportation arises. we take a look at some of the more modern models for In such a system, the so-called household activity pat- land use in the context of geosimulation. terns form the main explanation for what is seen in the transportation network. These models have as their primary intent the perform- 1.1.1 Classic land-use models ing of impact and evaluation studies, and conducting ‘before and after’ analyses. The fact that such trans- The discussion given in this section, talks about several portation studies are necessary, follows from a counter- kinds of land-use models that — at their time — were intuitive example whereby improving the transportation considered as landmark studies. That said, the models system (e.g., by making extra infrastructure available), presented here should be judged as being general in that they deal with (pre-)industrial American societies in the can result in an increase of the travel times. This phe- th nomenon, i.e., allowing more flexible routing that results first part of the 20 century. They are devised to gain in- in more congestion, is known as Braeß’ paradox,after Di- sight into the general patterns that govern the growth and etrich Braeß [33]. The underlying reason for this counter- evolution of a city. As such, they almost never ‘fit’ per- intuitive behaviour, is that people generally only selfishly fectly, leading to the obvious criticism that they are more try to minimise their own travel times, instead of consid- applicable to American cities than elsewhere. Notwith- ering the effects they have on other people’s travel times standing these objections, the models remain very use- as well [228]. ful as explanations for the mechanisms underpinning the socio-economical development of cities. As transportationis inherently a temporal and spatial phe- nomenon, we first take a look at the concept of land- One of the oldest known models describing the relation use models and their relation to the socio-economical be- between economic markets and spatial distances, is that haviour of individual people. In the two subsequent sec- of Johann Heinrich von Th¨unen [283]. As the model was tions, we consider two types of transportation planning published in 1826, it presents a rather ‘pre-industrial’ ap- models, i.e., the classic trip-based models, and the class proach: the main economical ingredients are based on of activity-based models, respectively. The section con- agricultural goods (e.g., tomatoes, apples, wheat, ...), cludes with a brief reflection on the economist’s view on whereas the transportation system is composed of roads transportation systems. on which carts pulled by horses, mules, or oxen ride. The spatial layout of the model, assumes an isolated state (self-sustaining and free of external influences), in 1.1 Land use and socio-economical be- which a central city location is surrounded by concentric haviour regions of respectively farmers, wilderness, field crops, and meadows for grazing animals. All farmers aim for maximum profits, with transportation costs proportion- As already stated, transportation demand arises because ally with distance, thus determining the land use around of the desire to participate in a set of activities (e.g., so- the city centre. cial, economical, cultural, ...). In order to deduce this derived transportation demand, it is necessary to map the Some 100 years later, inspired by von Th¨unen’s simple activity system and its spatial separations. This process and elegant model, Ernest W. Burgess developed what is is commonly referred to as land use, mainly playing the known as the concentric zone model [40]. It was based role of forging a relation between economical and geo- on observations of the city of Chicago at the beginning graphical sciences. In general, land-use models seek to of the 20th century. As can be seen in the left part of explain the growthand layout of urban areas (which is not Fig. 1, Burgess considered the city as growing around a 2 central business district (CBD), with concentric zones of tral location, surrounded by zones of pre- and post 1918 respectively the industrial factories and the low-, middle-, housing respectively. Dispersed around the outer con- and high-class residents. The outermost ring denotes the centric zone, the low-, middle-, and high-class residents commuter zone, connecting the CBD with other cities. would live.