A generic multi-level framework for microscopic traffic simulation

A generic multi-level framework for microscopic traffic simulation

Theory and an example case in modelling driver distraction

van Lint, J. W.C.; Calvert, S. C.

DOI

10.1016/j.trb.2018.08.009

Publication date

2018

Document Version

Final published version

Published in

Transportation Research Part B: Methodological

Citation (APA)

van Lint, J. W. C., & Calvert, S. C. (2018). A generic multi-level framework for microscopic traffic simulation:

Theory and an example case in modelling driver distraction. Transportation Research Part B:

Methodological, 117, 63-86. https://doi.org/10.1016/j.trb.2018.08.009

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ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

A

generic

multi-level

framework

for

microscopic

traffic

simulation—Theory

and

an

example

case

in

modelling

driver

distraction

J.W.C.

van

Lint

∗

,

S.C.

Calvert

Transport and Planning Department, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 27 March 2018 Revised 19 June 2018 Accepted 10 August 2018 Available online 8 September 2018

Keywords:

Human factors

Traffic simulation framework Workload

Task-Capacity-Interface model Distraction

a

b

s

t

r

a

c

t

Incorporationofmoresophisticatedhumanfactors(HF)inmathematicalmodelsfor driv-ingbehaviorhasbecomeanincreasinglypopularandimportantresearchdirectioninthe lastfewyears.Suchmodelsenableustosimulateunderwhichconditionsperception er-rorsand risk-taking leadtointeractionsthatresultinunsafetrafficconditionsand ulti-matelyaccidents.Inthispaper,wepresentagenericmulti-levelmicroscopictraffic mod-ellingand simulation framework that supports this important line of research. In this framework,thedriving taskis modeledinamulti-layeredfashion.Atthe highest level, wehaveidealized(collision-free)modelsforcarfollowingandotherdrivingtasks.These modelstypicallycontainHFparametersthatexogenously“governthehumanfactor”,such asreactiontime,sensitivitiestostimuli,desiredspeed,etc.Atthelowestlevel,wedefine HFvariables(taskdemandandcapacity,awareness)withwhichwemaintainwhatthe in-formationprocessingcostsareofperformingdrivingtasksaswellasnon-drivingrelated taskssuchasdistractions.Wemodelthesecostsusingso-calledfundamentaldiagramsof taskdemand.Inbetween,wedefinefunctionsthatgovernthedynamicsofthehigh-level HFparameterswiththeseHFvariablesasinputs.Whentotaltaskdemandincreases be-yondtaskcapacity, firstawareness may deteriorate,where weuseEndsley’sthree-level awarenessconstructtodifferentiatebetweeneffectsonperception,comprehension, antic-ipationandreactiontime.Secondly,driversmayadapttheirresponseinlinewithFullers riskallostasistheorytoreducerisktoacceptablelevels.Thisframeworkcanbeviewedas ametamodel,thatprovidestheanalystpossibilitiestocombineandmixawidevariety ofmicroscopicmodelsfordrivingbehavioratdifferentlevelsofsophistication,depending onwhichHFarestudied,andwhichphenomenaneedtobereproduced.Weillustratethe frameworkwithadistraction(rubbernecking)case.Ourresultsshow thattheframework resultsinendogenousmechanismsforinter-andintra-driverdifferencesindriving behav-iorandcan generatemultipleplausibleHFmechanismsto explainthe sameobservable trafficphenomenaand congestionpatternsthatariseduetothe distraction.We believe ourframeworkcanserveasavaluabletoolintestinghypothesesrelatedtotheeffectsof HFontrafficefficiencyandtrafficsafetyinasystematicwayforboththetrafficflowand HFcommunity.

© 2018TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

∗ _{Corresponding author.}

E-mail addresses: j.w.c.vanlint@tudelft.nl (J.W.C. van Lint), s.c.calvert@tudelft.nl (S.C. Calvert).

https://doi.org/10.1016/j.trb.2018.08.009

0191-2615/© 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

1. Introduction

Incorporation ofhumanfactors (HF)inmathematical modelsfordriving behavior hasbecomean increasingly popular researchdirectioninthetrafficflowtheorycommunitythelastdecade.Onemayargue,however,thatHFhavealwaysbeen atthecoreoftrafficflowmodelling.Sincethepioneeringworkof(Greenshields,1934,LighthillandWhitham,1955;Richards, 1956andmanyothers)onthefundamentalrelationandthefluiddynamicaldescriptionoftraffic,manyschoolsofthought haveemerged,eachcharacterizedbydifferentbehavioralassumptionsonhowdriversrespondtostimuli—andwhichstimuli they respondto; andby differentrangesofdescriptive and(partial) explanatorypowerfortheresulting phenomena. For example,safe-distancecarfollowing(CF)models(LavalandLeclercq,2010a;Newell,2002;Pipes,1953)assumethatdrivers maintaina large enough distance headway incase theleader brakesat maximumdeceleration; optimalvelocity models (Bandoet al., 1998; Davis, 2003) assume that drivers accelerate to their optimal velocity as a function of the distance headway;whereasapproachesinthemoregeneralgroupofstimulus-responsemodels(Gazisetal.,1961;KernerandKlenov, 2006; Treiber et al., 2000) make assumptions on how drivers adapt their response (acceleration)to a range of different stimuli(distanceheadway,speeddifferences).Overtheyears,manyapproachestoincorporatemore(HF)sophisticationhave beenproposed.So-calledpsycho-spacing(oraction point)models (Fritzsche,1994; Wiedemann,1974)incorporatedrivers’ inertiato observeandrespondtosmallchangesinstimuli;whereasforexamplemulti-anticipatorymodels (Hoogendoorn et al., 2006,2007;Treiber et al., 2006) include termsfor anticipation ofdrivers to traffic conditions furtherdownstream. What“classic” (orasSaifuzzamanandZheng,2014)putit:“engineering” models)forcarfollowing(CF)haveincommonis thattheyare—bydesign—collision-free.Thisisnolongerguaranteed,however,ifweincorporatereactiontimes,i.e.delayed stimuli, and/or perception errors in these stimuli (headways, relative speeds) or both (Hamdar and Mahmassani, 2008; Treiberetal.,2006).Anevenwiderdiversityofbehavioralassumptions andmodellingapproachescanbefoundforlateral drivingbehaviorthatgovernswhendriverschangelanes,diverge,andmerge(Choudhury,2008;Cohen,2004;Kestingetal., 2007; Laval andDaganzo, 2006; Schakelet al., 2012;Wei etal., 2000; Zheng, 2014). In mostcases here, the underlying theory isbasedonconditional decision-making.The correspondingmodels usuallyincorporatedecisiontrees,andmodels assessing the conditions(availabilityof gaps) andthe appropriate response (intentionand executionofcrossingsor lane changes). Also models for lateraldriving are—in principle—collision free. Like CF models, the inclusion ofreaction times and/orperceptionerrorsinlanechanging(LC)modelsrelaxesthatassumption.

Thereare severalgoodreasonswhyresearch hasacceleratedintomoresophisticatedandsystematicapproachesto in-corporateHFinmicroscopictrafficflowmodels.First,therestillaremanyphenomenaincurrenttrafficthatwedonotfully understand,such asthecapacitydrop,traffic hysteresis,andmanyphenomenarelatedto lateralmovement (Saifuzzaman andZheng,2014;Zheng,2014).Second,we areatthestartofamajor transitiontowards higherlevels ofvehicle automa-tion(VA).Paradoxically,trafficsimulationmodelshavealwaysbeencapableofsimulatingautomatedvehicles;nowthatVA becomesa reality,weneedtoincreasetheHFsophisticationinourhumandrivermodels.Sincetraffic flowoperationsare governed by interactionprocesses, we cannot predictthe changes inthose interactions andtheir consequences basedon knowledgeofthebehaviorofjustoneofthe‘players’(theautomated vehicle)—thehumanplayermayalsofundamentally changeinwaysnotcateredfor(sufficiently)by existingmodels.Third,whereasmostemphasisofmicroscopictraffic mod-elling hasbeen onreproducing safe traffic operationsandthe corresponding emerging phenomena(e.g. capacities, wave patterns),an increasedneedemerges tousethesemodelsto realisticallypredictalsopotentially unsafetraffic operations, andthe correspondingindicators (statisticsofaccidentsandsurrogatesafetymeasures) (Hamdar andMahmassani, 2009; Hamdaretal.,2015b).Theseconditionsarerelevantnotjustinstudyingvehicleautomation, butalsointhehereandnow. Toassesswhethersafety isatrisk, explanatorypsychologicalconstructs areneededthat can endogenouslypredictunder whichcircumstancesdriverstakerisksand/ormakeperceptionandjudgementerrorsthatmayleadtounsafesituationsand ultimatelyaccidents.Severalapproacheshavealreadybeenproposed inthisdirection,e.g.usingprospecttheory(inwhich driverswayfastertraveltimeagainsttherisk ofrear-endcrashes(Hamdaretal.,2015a, 2008);andusingFullers’Risk Al-lostasis Theory(Fuller,2011) (inwhichrisk takinganddriverresponseisconsidereda resultofcomparingsubjectivetask demand andtask capacityusingthe so-called Task-Capacity-Interfacemodel (e.g.Hoogendoorn et al., 2013;Saifuzzaman etal.,2015,2017).However,morebehavioralsophisticationcomesatamethodologicalandcomputationalprice,intermsof modelidentification,calibrationandvalidationefforts;andcomputationalefficiency.Therefore,thechallengeforour com-munityin thecomingyears,isto augmentexistingCF andLC modelswitha rangeofexplanatory(HF)mechanismsthat (a) endogenouslypredictwhereandunder whichcircumstances drivers e.g.make errors,take more(or less)risks,suffer fromlongerreactiontimes;using(b)mathematicsandsimulationlogicthatistractable andsimpleenough sothat large-scale simulation is (still) possible; while (c)still reproducing plausible vehicle trajectoriesand (byimplication) plausible macroscopictrafficpatterns.Thereisanadditionalpractical,butnonethelessimportantdesigncriterionthatrelatestothe (software)developmentoftraffic simulation models.Such newadditions tothe alreadybroadfamilyofmicro-simulation modelsneedtofindtheirwayintobothcommercial(closed-source)(e.g.Casasetal.,2010;FellendorfandVortisch,2009; MahutandFlorian,2010;Sykes,2010) andopen-source (Krajzewiczetal.,2012;TreiberandKesting,2010;vanLintetal., 2016) trafficmicrosimulation packages. Thisrequires ageneric modellingframework that allows combinationofdifferent modellingapproachesandimplementationsthataremodularandmaintainable.

The centralcontributionofthispaperissuch ageneric multi-levelmodellingandsimulation frameworkthat supports thisresearch challengeandthatgeneralizesexistingapproachestoincorporatehumanfactorsinmodelsfordriving behav-ior. Inthispaperwefocuson carfollowing(CF)only, however,theframeworkcan benaturallyextendedto supportlane

Fig. 1. Driving as a control task. For clarity, the many external factors affecting perception, response and driver characteristics (e.g. traffic conditions, environment, control, etc.) are not drawn.

changing(LC)orother drivingbehaviors(crossing, merging,etc).Thepaperisoutlinedasfollows.InSection2wediscuss humanfactorsinCFmodelsalongtwodimensions:thewhat(HFprocessismodelled)andthehow(thisisdone).Onthe ba-sisthereofwethenpresentourconceptualframeworkandthetheorybehinditInSection3.InSection4weoperationalize thisframeworkinasimulationcase,andinsection5wepresenttheresults.InSections4and7,werespectivelysynthesize andcriticallydiscusstheseresultsandclosewithconclusionsandanoutlookforfurtherresearch.

2. Humanfactorsincarfollowingmodels

There are many excellent reviewsand taxonomies of longitudinal driving available in the literature, e.g. (Brackstone and McDonald, 1999; Saifuzzaman and Zheng, 2014; van Wageningen-Kessels et al., 2014), of which Saifuzzaman and Zheng(2014) specificallydiscussHF incarfollowing models.Thissection is notintended asan additionalreview,butas a motivation andunderpinning forthe framework we presentin the next section. Tothis end, we review theliterature alongtwodimensionsthatweinformallydepictasthewhat(processismodeled)andthehow(thisisdone).

2.1. What(HF)processesaremodeled

Toexplainthe“what” dimensionofhumanfactorsmodelingintraffic,considerFig.1,inwhichthedrivingtaskisstylized asacontroltask.Thefigurehighlightsthetwomainprocessestoperformthiscontroltask,andHFplaysafundamentalrole inboth.First,thereisaperception process,inwhichthe(observed)environmentisrecognized,understood andtranslated into(possiblypredicted)stimuli,suchasdistancegapsandspeeddifferences.Thisprocessissubjecttodrivertraits,which encompass all relevant mentalstates, attitudes, preferences,skills, etc., andto the mechanical characteristics (inertia) of thevehicle.Second,thereisaresponseprocess,inwhichdriversactbasedontheperceivedandpossiblypredictedstimuli. Clearly,alsotheresponseprocessissubjecttodrivertraits.Inthispaper,weconsidertheperceptionandresponseprocesses atthetactical(maneuver)andoperational(control)levelonly,asdescribedbyMichon(1985)asillustratedinFig.2.

Weprefer to usetheterms tacticalandoperational fortheselevels, sincealso maneuversandstrategicbehaviors are controlprocessesthatinvolveperceptionandresponseaswell—albeitwithlongercontroltime-stepsand-spans.Perception andresponseonthetacticalandoperationallevelsarebothinfluencedbystrategical(route-andpath-planning)behaviors— theserelationshipsare beyond the scope ofthis paper.Fig. 2 further highlights that tacticaland operational driving are affectedbythedrivingenvironment.Theseincludetrafficconditions,roadlay-out,trafficrulesandcontrol,weather,andalso distractionsandtechnology(Farahetal., 2018). Finally,driversalsoperform othertasks thandriving(distractions), which mayaffecttheirperceptions,responsesandultimatelydrivingperformance.Notethatsomebehaviorsmaybecategorizedat morethanone level,e.g. pathchoice mayemergefrommaneuvering(ratherthanfromactivedecisionmaking)andspeed choicemaysimilarlyemergefromcontrolactions.

InTable1,wehaveadaptedthelistofHFusedinthereviewbySaifuzzamanandZheng(2014)by structuringitalong the“what” (perception andresponse)dimension. Notethatcontraryto othermore detailedpaths ofthoughtincognitive science,thisclassificationsimplifies thegeneralflowofinformationprocessingtothesetwocentralaspects,whichcanbe associatedwithinputandoutput.The authorsarewell awareofthe existenceofintermediate constructs;however,these arenotsufficientlyunderstoodtoallowdetailedapplicationinthetrafficmodellingatthistimeandarethereforeomitted.

Fig. 2. The hierarchical structure of the road user task. Performance is structured at three intertwined levels. (Source: Michon, 1985 , yellow “post-its” are added).

2.2. How(HF)processesaremodelled

Theseconddimension relatesto“how” HFareincorporatedinCFmodelsmathematically.Tostreamlinethediscussion, weframeCFmodelsusingthefollowinggeneralfunctional:

ai

(

t+

τ

i

(

t

)

)

=f

(

Si

(

t

)

,

θ

i

(

t

)

,

ω

i

(

t

)

)

(1)

inwhicha_i (t)denotesacceleration;

τ

_i (t)reactiontime;S_i (t)asetofavailablestimulie.g.speeds,speeddifferences,distance gapswithrespecttotheego-vehicleianditsleader(s);

θ

iasetofdriverpreferences(e.g.carfollowingparameters),amongst

whichhis/hersensitivitytothesestimuli;

ω

_i (t)asetofcharacteristicsofthe(perceivable)worldPW_i (t)fordriverithatmay affecttheresponse(e.g.control,visibility,etc).Inthesevariables,thesubscriptidenotes driver(i) specifity,andtdenotes (continuous)time.

We candistinguishbetweentwo mainapproachesofhow HFprocessesare modelled.The first,andmostcommon, is thatHFprocessesareincorporatedexogenouslybymeansoffixedparametersinthe(core)stimulus-responsecarfollowing logic,whichinthesimplestcasearedeterministic(i.e.meanvalues),ordrawnfromsome(known)distribution,thatis

θ

i ∼ gθ

(

μ

θ,

σ

θ

)

(2)

Thesameholds forreactiontime, e.g.,

τ

_i ∼ gτ(

μ

_τ,

σ

_τ). Thisapproachallowsone tovarybothresponse (behavior)and reactiontime overdrivers,butthisapproachcannotexplain thedynamicsofthatbehavior overtimeforthesamedriver.

OssenandHoogendoorn(2011)showthatbothdistributions(inter-driverandintra-driver,i.e.dynamicsovertime)arewide. The differences between drivers pertain not just to parameter distributions (i.e. trajectories of differentdrivers are best describedwith differentsets ofparameters fitted forthe same car followingmodels); butalso to (car following) model distributions (i.e.trajectoriesofdifferentdrivers arebestdescribedwithdifferentcarfollowinglogic; Ossenand Hoogen-doorn,2011).Perceptionerrors,i.e.errorsinstimuliinput tocarfollowingmodels,can alsobe modelledexogenously.The challengehereisthatsucherrorsaretypicallyauto-correlated.Treiberetal.(2006)suggestaWienerprocess(withknown parameters)tosimulateconsecutiveperceptionerrors.Thesametechniqueforexogenousmodelingofperceptionerrorsis usedin VanLint etal.(2018).Exogenousmodellingofperception errorsallowsone tostudythe robustnessto erroneous inputsoftheassumedcontrollawswithwhichdriversfollowleadingvehicles,butdonotrevealthemechanismsthatmay causetheseerrors.Insum,exogenous modellingofHFfactorscanbe usedtocater forinter-driverdifferences;andallow for“what-if” typeanalysis, buttheydonotprovideinsightinwhatcausesintra-driverdifferences(dynamicsovertime)in theHFmechanismsgoverningperception,anticipationorresponse.

Thesecond “how” approachovercomesthislimitationandincorporatesHFendogenously,by meansof(dynamic) func-tionsoralgorithmsthatexplainbothdynamicsinreactiontimeandresponseparameters,aswellasinter-driverdifferences. Thesedynamicscanbeformulatedingeneralas

d

dt

γ

i

(

t

)

=h

(

γ

i

(

t

)

,Si

(

t

)

,

ω

i

(

t

)

)

, with

γ

i =[

τ

i ,

θ

i ] (3)

in which the change in driver state (i.e. his/her reaction time and response parameters) is a function of their cur-rent state, the stimuli at hand and the environment. The perception threshold mechanism in the Wiedemann model (Wiedemann,1974), theprospect theorybased risk-taking mechanisminHamdar andMahmassani(2008), Hamdaret al. (2015a) and the taskcapability interface model implementations in Hoogendoornet al.(2013),Saifuzzaman et al.(2017, 2015) are examples of such an approach (we discussthe TCI model of Fuller (2011) in more detail in Section 3.3.2). In

Table 1

HF modelling in car following along the “what” (process is modelled) dimension, adapted from ( Saifuzzaman and Zheng, 2014 ).

What (process) Aspects and driver traits ∗ _Examples

Perception How drivers translate signals from the environment to (anticipated) stimuli.

Reaction time. This involves the physical delay between observing (a braking light) and responding (braking), and the delay caused by diverted attention.

Most CF models have been extended with reaction time, e.g. ( Davis, 2003; Gazis et al., 1961; Treiber et al., 2006 ). By and large, only for small reaction times do car following laws result in stable dynamics.

Estimation errors. Humans observe stimuli with a limited accuracy as a function of distance and visibility conditions.

There is abundant evidence that drivers have systematic biases in judging both distances and speeds ( Castro et al., 2005; Nilsson, 2000; Thiffault and Bergeron, 2003 ), and several CF models have been extensively tested under such errors, which are generally modelled as (auto-correlated) noise processes ( Hamdar and Mahmassani, 2008; Treiber et al., 2006; Van Lint et al., 2018 ). No attempts have been made to our knowledge in modelling these errors endogenously (i.e. as the result of a HF process).

Perception thresholds. Humans cannot perceive small changes in stimuli.

Wiedemann was among the first to formalize driver inertia to small stimuli in CF models ( Wiedemann, 1974 ), after which various adaptations followed ( Fritzsche, 1994 ). See also driver inertia

Spatial anticipation. Drivers look ahead (downstream).

Spatial adaptation is usually modelled by incorporating gaps between leader-follower pairs further downstream (i.e. downstream density), e.g. ( Hoogendoorn et al., 2006; Ossen and Hoogendoorn, 2011; Treiber et al., 2006, 2007; Van Lint et al., 2018 ). In general, spatial anticipation counter-effects the destabilizing effects of reaction times.

Temporal anticipation. Drivers extrapolate conditions (over space and time).

Typically, some form of dead-reckoning (using constant speed or acceleration) is adopted ( Treiber et al., 2006; Van Lint et al., 2018 ). Temporal anticipation also counter-effects reaction times.

Distractions. Competing information processing activities affect perception

Distractions, and particularly visual distractions are detrimental for the driving task ( Precht et al., 2017a , b; Rebecca et al., 2008 ). Various researchers have experimented with distractions in terms of consequences for the car following task ( Chan and Singhal, 2015; Hansen et al., 2017; Hoogendoorn et al., 2010; Hoogendoorn et al., 2011; Kaber et al., 2012b; Saifuzzaman et al., 2015; Schömig and Metz, 2013 ) and the modelling thereof.

Response How drivers dynamically and context-specifically respond to these stimuli.

Sensitivity to stimuli (relative speed,

position, etc) Every CF model contains parameters that govern the degree in which drivers respond to stimuli. These may dynamically change due to circumstances (see context sensitivity)

Preferences. Drivers’ desired speed, spacing, headway, etc.

Similarly, most CF models contain one or more parameters describing driver preferences. In exploring inter-driver heterogeneity, these are typically modelled as distributions over drivers ( Montanino and Punzo, 2015; Ossen and Hoogendoorn, 2011 ).

Context sensitivity. Different contexts may (dynamically) affect driving response.

It is widely recognized that parameters in CF models are both driver and context dependent, e.g. ( Laval and Leclercq, 2010a; Ossen and Hoogendoorn, 2011; Zheng et al., 2013 ). Clearly, as contexts dynamically change, so should the parameters.

Inertia. Even if drivers perceive (small) stimuli, they may not respond to these.

Wiedemann-type models ( Fritzsche, 1994; Wiedemann, 1974 ) are also termed action point models, since drivers are assumed to change their responses at discrete time instants (and continue according to the last response in between) rather than continuously.

Aggressiveness or risk-taking propensity

Laval and Leclercq (2010b) show how even in very simple CF models differentiating between timid and aggressive drivers complex but realistic stop-and-go patterns can be reproduced. More generally, there is rich literature on the driver- and context specific role of risk taking in various driving tasks ( Farah et al., 2008; Jamson et al., 2012; Michaels et al., 2017; Precht et al., 2017b ), and a few attempts to explicitly model this endogenously in CF models ( Hamdar et al., 2015a, 2008 ).

Saifuzzamanetal.(2015),forexample,theGibbsandIDMCFmodelsareaugmentedwithadynamictermfortaskdifficulty toadapttheaccelerationresponse.Theideaisthatdriversincreasetheirdesiredheadwayunderconditionswherethe driv-ingtaskbecomestoodifficult,whichtheyassumeisthecasewhentheiractualheadwaybecomestoosmall.Taskdifficulty

TD_i isconsideredproportionaltotheratioofadriver’sdesiredtimeheadwayT_i andtheactualheadway



v_i (t)/s_i (t),i.e.

TD_i

(

t+

τ

i

)

=





v

i

(

t

)

Ti

(

1−

δ

i

)

si

(

t

)



γ (4) Inwhich

δ

_i isa driverspecificrisk factor(the largeritis,the highertaskdifficulty) and

γ

ascaling factor(again,the largeritis,thehighertaskdifficultyundersimilarcircumstances).Saifuzzamanetal.(2015) estimatethe scalingfactor

δ

_i alongwithother modelparametersusing drivingsimulatordataandshow howincorporationofthisdynamictermgives a (potential) explanatorymechanism for trajectories of aggressive (small or negative

δ

i) and timid (large

δ

i) drivers. In

Saifuzzamanetal.(2017)thesameidea isusedasanexplanatorymechanismfortraffichysteresis andtraffic oscillations. Aswewillshowfurtherbelow,therearemanyotherpossibleHFmechanismsthatcouldexplainsuchdynamics.

2.3. Synthesis

Asarguedintheintroduction,thereisalreadyahugediversityinmodelsfordrivingbehaviorcharacterizedbydifferent assumptions on how drivers respond to stimuli—and which stimulithey respondto. Table 1 illustrates that the amount ofassumptions andmodellingapproacheswillmostlikelysteeply increaseoncewe startto incorporatemorerealisticHF processes into these different models. We propose a framework, withwhich it becomes possible to do this ina highly systematicway.Thisrequiresustogoonestepfurtherthantheendogenousmodellingexamplesabove,inwhichaselection ofHFparameters(e.g.taskdifficultyorriskpropensity)arefirstendogenouslycomputedandthenembeddedinthecorecar followinglogic. Wegeneralize theseapproaches, by decouplingidealizedandHFdriving completely.Wepropose a multi-levelrepresentationofthedrivingtask,witha“coarser level” inwhichidealized(collision-free) behaviorismodeled;and anexplanatory(HF)layerunderneaththatgovernsthedynamicsoftheinputsandparametersofthis“idealized” level,and hencerelaxesthecollision-free assumption.Byseparatingtheselayerscompletely, weallow analyststomix awiderange modelsfordrivingbehaviorwithawiderangeofmodelsfortheunderlyingHFprocesses.Intheremainderofthispaper, wefurtherdetailthisframeworkandexploreitspropertiesinacasewheredriversaredistracted.

3. Multi-levelmodellingandsimulationframework

3.1. Scopeandoverallidea

Weproposeamodelingandsimulationframework foroperationaldriving(contingenttotacticalandstrategicaldriving asdepictedinFig.1) inwhichwemaintainthetasksa driverexecutesandtheeffectsthesetaskshaveonthetwomain HF processesconsidered while driving:perception andresponse. Mostimportantly,there isthe driving taskitself,which fornowwerestricttojustcarfollowingandfreedriving.Wediscusspossiblegeneralizationtootherdrivingtaskssuchas lateral movementandconflict negotiationinthe discussion section.Secondly, thereare secondary tasks,that donot (di-rectly)contribute to driving, butthat mayaffectperception and response,forexample in-vehicleor outsidedistractions. Thiseffectisduetothefact thatall tasksconsumeinformationprocessingcapacitythrough theperceptionand interpre-tationofthedrivingenvironmentaswell asrespondingto these(carfollowing,lane changing,etc.). Toallowtheireffects on driving to be quantified, we propose to maintaina (minimal) set of key mental state variables, which are based on psychological constructs that are found inHF literature (e.g. (Endsley,2000; Fuller,2005,2011; Kaber et al., 2012a; Teh etal., 2014))and are usedto describe operational perception andresponse processesin the context ofdriving behavior. ThesearesituationalawarenessSA(t),whichencapsulatesmultipledimensionsofperception(includingfocus/distraction) (Endsley,1995;Wickens,2008);anddrivertaskdemandTD(t), whichisusedinmanystudiesas(oneofthe) explanatory concepts whenit comesto explainingdriving performance(Prechtetal., 2017b; Teh etal.,2014) underabroadrange of conditions.In simpleterms, TD(t) describesthe cumulative workloadofeach cognitive taskthat a driveris subjectedto, whereas SA(t) describeshowwell adriverisawareoftheir environment,particularlyofthosestimuliintheenvironment that adriverneedstosafelyandefficientlyperformthedrivingtask.Themannerinwhichinformationisprocessedfrom perceptionthroughtothecognitivedecisionprocessiscaughtwithintheSAconstructanddependsonadrivers’traitsand currentstate butgoesbeyondthescopeofthiscontribution.Inourframework,bothTD(t)andSA(t)aredynamic(i.e.they change over time andspace); andthey affectdriving parameters (reaction time, frequency andmagnitudeof perception errors)andtheresponseofdrivers(sensitivitiestoe.g.distancegaps).Wedescribethisinmoredetailbelow.Clearly,many more social/psychological/physiologicalfactors (e.g.interaction withpassengers, emotionalstate, fatigue, etc)are relevant indescribing(andsimulating)perceptionandresponseprocesses.However,itisimportanttostress thatweneedtostrike balancebetweenadescriptionthatis

• sufficientlyaccurate,sothatvalidinferenceispossibleofboththeefficiencyandsafetyeffectsofthetraffic operations resultingfrominteractionsbetweendrivers;

• sufficientlygeneric,sothatmanydifferentapproachestomodellingHFprocessesinmodelsfordrivingbehaviorcanbe castintheframework(directlyorvianaturalextensions);

• sufficientlysimple (mathematically/ computationally), sothat simulation inlargecongested networksisstill possible. Simplicityalsorelatestothelevelofdetailwithwhichnon-drivingtasksaremodelled.Forourpurposes,wedonotcare aboutthedetailsoftheHFprocessesinvolvedinnon-drivingtasks(inthispaperweconsidere.g.distractions);wecare onlyforthenetresultintermsofhowmuchinformationprocessingthesetasksconsumeandhowthataffectsdriving behavior.

Wemake twomoreoverall points.Firstly, theworddescriptionhereaboveis initalicsbecausethemodels castinthis frameworkmaynotprovidecausalexplanationsastoWHY(orhow)drivingperformancedeteriorates.Whatwe wantisa descriptivemodelthat—usingvariablesavailableinthesimulationonly—isabletopredictTHATthishappensundercertain conditions,analogouslytohowinamacroscopictrafficflowmodelthefundamentaldiagramisnotacausalmodelrelating

Fig. 3. Conceptual modelling & simulation framework.

speed todensity, buta descriptive (statistical)modelthat describes that speed dropswith increasing density.1 _Secondly, theframework outlined in thispaperis a theoreticalframework. We provide some face-validationin this(and a follow-up)paper,butvirtuallyeverycomponentandrelationshipinit(Fig.3) constitutesofmanyhypothesesthatrequiretesting withelaborateexperimental methods(drivingsimulatororfieldresearch,mathematicalanalysis, simulation,etc.)toresult invalidmathematicalandsimulationmodels.However,duetothemulti-levelstructure,theframework allowsresearchers todothisinasystematicway.

Fig. 3outlines the main mechanisms in our framework, using the functional in Eq. (1). First total taskdemands are computedusing(a)so-calledfundamentaldiagramsof taskdemandand(b)taskdemand aggregation.Thenthe effectof thoseaccumulatedtaskdemandsiscomputedon(c)personaltraits(desiredspeed,headway,etc.);on(d)situational aware-nessandasaconsequenceon(e) perceptionerrors and(f)reaction timedynamics.Belowwe elaborateon eachofthese components.

3.2.Statevariables&basicrelationships

3.2.1. Taskdemandandtaskcapacity

Inoursimulationframeworkwedefinetaskdemandastheamountofinformationprocessingeffort(perunittime)needed

tofulfillatask(i.e.toreachanobjectivesuchasnotcollidingintotheleadingvehicle)2_{Wedefinethefollowingvariables:}

TC Nominal Task Capacity Information processing capacity a nominal (standard) driver has available to execute tasks safely and efficiently. TC = 1 ( or 100%).

TCi ( t ) Driver Task Capacity Information processing capacity for driver i in units of TC

T D a

i(t) Driver Task Demand Variable that describes how much information processing effort driver i requires performing a

particular task a (safely and/or satisfactorily) in units of TC

TDi ( t ) Total Driver Task demand Sum of all task demands for driver i , that is,

TDi

(

t

)

=



a

TDa _i

(

t

)

. (5)

TSi ( t ) Driver Task Saturation Variable that expresses total driver task demand TD i ( t ) relative to TC i ( t ), that is,

TS_i

(

t

)

=TDi

(

t

)

TC_i

(

t

)

. (6)

Clearly,incaseswhendrivertasksaturationTS_i (t)iscloseto(orlargerthan)1theperformanceofadriverdeteriorates. Thisperformancedeteriorationmaytaketheformofchangesinawareness(largerperceptionerrors,longerreactiontimes);

1 In free flow drivers may reduce their speed based on decreasing distance gaps, whereas in congestion, drivers may adjust their distance gaps based

on decreasing speeds. The direction of causality does not matter for the validity of the predicted traffic conditions with macroscopic traffic flow theory. A descriptive (statistical) model suffices.

2 We realize this definition differs from what is commonly used in the HF field; e.g. _{De Waard (2002) defines task demand in terms of goals that have}

to be reached and (mental) workload as the proportion of a drivers mental processing capacity that is allocated for task performance (such that those task demands are met). However, for simplicity reasons, and because of the intuitive analogy with traffic flow modelling, we prefer to define driver task demand as a variable that expresses how much mental processing is demanded by a task versus driver task capacity that expresses how much processing “power” a driver has available for it.

changesinresponses(smallerorlargersensitivities)anddriverstate(changesinotherdrivertraits).Theseeffectsandhow tomodelthemarediscussedinthefollowingsub-sections.Wefirstfocusonhowtomaintaintotaldrivertaskdemandas astatevariable,whichrequiresthatitcanbeendogenouslycomputedwithinthesimulation,usingwhateverinformationis availableinthesimulation

3.2.2. Fundamentaldiagramsoftaskdemand

ConsideracarfollowingtaskasinFig.3(a).Inthelimit,withveryshorttimeheadwaysunderdensenearcapacity condi-tions,thistaskwillrequirevirtuallyall(driverspecific)informationprocessingcapacity,i.e.TDCF _i

(

t

)

≈ TC_i

(

t

)

,whereaswith longerheadwaysunderlight(er)conditions,thistaskmayconsumenotonlyafractionofadrivers’informationprocessing capacity,withsomesteep transitionbeyondacriticaltime headwayvalue.Thisidea,a so-called“fundamentaldiagramof taskdemand” (FDTD) for carfollowing, isillustrated in theexample inFig. 3(a),inwhich the relationship betweentask demandandtimeheadwayofadriverisdepictedundertwodifferentcircumstances—sayrainyanddryweather.Notethat inthefigure,drivertaskcapacityTC_i (t) isslightlybelownominaltaskcapacityTC(t),implyingthisdriverhascarfollowing skillsslightlybelowaverage.WeproposethefollowingrequirementsforFDTD’s:

Req. 1. Task demand is expressed in units of “nominal” task capacity—this is an arbitrary “red line of workload” (Rebecca etal., 2008) beyondwhich additionaltaskdemand resultsinperformance degradation (wereturnto this furtherbelow)

Req.2.Taskdemandmustbeexpressedasafunctionofvariables(made)availableinthesimulation.

Forcarfollowingtimeheadway seemsa logicalchoice,forlanechangemaneuversacombinationofheadway, den-sity,andavailable gapsmaywork whereasformergingadditionallydistance/ timeto divergepoint mayprovidea goodbasis. Possibly,variables (constructs)derived fromtheseprimaryvariables maybe usedsuch asmeasures for complexityofthedrivingtask(e.g.Hoogendoornetal.,2013;Tehetal.,2014).SeealsoReq.4.

Fornon-driving tasks, FDTD’scould be expressed asfunction of thelocation, time duration or severityof the dis-traction (see the example below). It is important to note that we exclusively consider secondary tasks that affect information processing capacityrequired for safe andefficient driving. Precht et al. (2017a) for exampleconclude thatparticularlyhigh-riskvisually/visual-manually distractingsecondarytasks(lookingawaybecause ofdistractions outsideorinsidethevehicle)resultinaberrantdrivingbehavior.Theyalsopointoutthatsomedistractions(e.g. con-versations)maydistractthedriverundersomeconditions(e.g.atdecisionpoints);butmayactually supportsafe(r) drivingunderotherconditions(incaseoffatigue).Bothsuchtaskscouldfitintheframework,however,thedirection inwhichasecondarytaskinterfereswiththedrivingtask(positiveornegative)isconsideredinputtoourframework. Although we restrict the discussion in thispaper to the car following taskonly, we believe that in principle, driver trait andcircumstance-specificFDTD’scanbe formulated fora widerselection ofrelevantoperationalandtacticaldriving tasks(e.g.overtaking, weaving,respondingto signaling,etc.),inwhichtaskdemandandtasksaturationplay aroleinthe performance of executing such tasks. We return to this claim in the discussion and synthesis section. To cater for that discussion,wedoproposeathirdrequirement:

Req.3.FDTD’sthatexpresstaskdemandatthetactical(maneuver)levelprevailover(areconsideredtosubsume)task demandsattheoperational(control)level.Forexample,aFDTDforexecutingalanechangemaneuvershould incor-poratetaskdemandfortheinherentcarfollowingsubtaskswithinthatmaneuver.

Req. 3 is importantfor two reasons. First,the amount of informationprocessing fora task is context sensitive.Very short headways may be comfortable duringa lane change maneuver, but highlyuncomfortable when following a truck on anarrowfreewaylane. Second,it seems apriorivery difficult(if notimpossible) toempiricallyvalidate a modelthat disentanglesalanechangemoveintoallitsconstituentsubtasksandtoquantifyseparatetaskdemandsforeachalongthe maneuver.ItmakesmoresensetokeepitsimpleandformulateaFDTDfordifferentmaneuvers(freedriving,carfollowing, lanechanging,merging,etc)intermsoftheirtotaldrivertaskdemand.

There isgoodreason, however,toconsider separationbetweensecondarytasks suchasdistractions, particularlythose that require visual perception (Precht et al., 2017a, b; Rebecca et al., 2008), since theseall utilizethe same information processingchannel(vision).Whetherornotmultipledistractionshaveanadditiveeffect(asinEq.(5))isahypothesisthat requirestesting.Forlackofevidenceotherwise,inthispaperweassumeanadditiveeffect,sothatgivenastackoftasksone can—asoftenasdeemednecessary(inthelimit atevery simulationtimestep)—computeadrivers’totaltaskdemandand tasksaturation(Fig.3(b))andconsequently,theresultingperformancedeterioration(ifany).Thisdeteriorationmayinvolve twothings:

• Deteriorationinawarenessintermsofincreasedreactiontimeandincreasedperceptionerrors(Fig.3(d)–(f))

• Responseadaptationintermsofchangesinpreferences(desiredspeed,headway,etc)andotherpersonaltraits(Fig.3(c)) Beforediscussingbothwithanillustrativeexample,wefirstfurtherdetailaconceptualmodelforawareness.

3.2.3. Conceptualmodelforsituationalawareness

FollowingEndsleys’dynamicsituationalawarenessmodel(Endsley,1995;Wickens,2008),weconsiderthreelevelsofSA. Theseare(1)sensingtherelevantobjectsandinformation;(2)comprehension(i.e.correctlyinterpretingthisinformation);

Fig. 4. Conceptual model for awareness based on Endsley ( Endsley, 1995; Wickens, 2008 ).

and(3)anticipation(makingshorttermpredictionsfordecisionmaking).Thesethreelevelsofawarenessconstitute three stagesintheperceptionprocess,asschematicallyoutlinedinFig.4.InSAstep1(sensing),adriverperceivesaselectionof theavailablepieces ofinformationfromthetotalperceivable world(PW_i ,i.e.everything adrivercould physicallyobserve) neededtoperformthedrivingtask(othervehicles,trafficsigns,controlsignals,geometry,etc)andstorestheseinitsmental

world(MW_i ). In SA, step2 (comprehension), the driver derives fromthese objects the set of stimulis_i (t) withwhich it

can make decisions (overtake, brake, etc). Depending on the (CF, LC andother) models used,si(t) may include(relative

differencesin)gap,andspeed;trafficsignals,etc.InSAstep3(anticipation),finally,a(timeseries)predictionS_i (t)={s_i (t),…,

s_i (t+T)}ofthesestimuliismade.Wethusdefinethreelevels: SAn

i(t) Situational Awareness Level

1 (sensing)

Variable that describes how aware a driver i is of all the objects (other traffic participants and information sources) required for performing the driving task.

SAc

i(t) Situational Awareness Level

2 (comprehension) Variable that describes how well (in terms of accuracy and efficiency) a driver is able to translate this information into stimuli.

SAa

i(t) Situational Awareness Level

3 (anticipation, prediction)

Variable that describes how well (in terms of accuracy and efficiency) a driver is able to anticipate (predict) the future evolution of these stimuli.

Fig.4alsoillustrates that inourframework we distinguishbetweenaphysical reactiontime

τ

_i p ,which istheresultof thethree-stageperceptionprocess;andan attentiontimelag

τ

a

i ,whichistheresultofcompeting(secondary)information

processingactivitieswhiledriving.Thetotalreactiontimethatisultimatelyusedintheupper-levelmodelsforCF(LC,GA, etc),equalsthesumofbothcomponents,i.e.

τ

i =

τ

i a +

τ

i p (7)

Like taskdemand,theSA variables maybe chosen ascontinuousvalues(e.g. between0and1), butone mayequally argue for categorical, ordinal or fuzzy values (e.g. “bad”, “moderate”, “good”), or whatever parameterization works in a particularcase. Thesethreeaspectsaffectdrivingperformanceindifferentwaysandmayalsobe(positivelyornegatively) influenced in different ways. For example, drivers behind a large truck have limited sensory awareness: they will miss relevantdownstream informationbutmaystill havean excellentcomprehensionandpredictiononthebasis ofwhatthey

cannotice(and perhapshavenoticed inthe past). Underadverse weather conditionsdrivers maystill noticeall relevant aspectsoftheenvironment,buttheconditionsmayaffectcomprehension(level2awareness),becauseit’smoredifficultto judgedistancesinheavyrain.Theiranticipation/predictionskillsmaynotsufferdirectly,althoughindirectpredictionsbased onerroneousinputs maybe lessreliable.Adverse weather mayalso increase physicalreaction time (the durationof the perceptionprocess)becauseittakesmore“processingtime” underlimitedvisibilitytojudgedistancesandrelativespeeds.

3.3.Conceptualframeworkusinganexample

We now discuss the conceptual framework of Fig. 3 on the basis of the illustrative example in Fig. 5, in which we follow(throughasequenceofevents) aparticularnominaldriveri(TC_i (t)₌TC)whoiscarfollowing.Attime t1,avehicle mergesinfrontofvehiclei,significantlydecreasingtheheadway.Alittlelateratt2,driverireceivesatelephonecall,which (s)hefinishes at time t6.First note that we propose two FDTD’s, one for the car following task (Fig. 5(a)-left) and one foradistraction: making a telephonecall (Fig. 5(a)-right).Forthe former,we considera simple linearfunction inwhich

Fig. 5. Illustrative example dynamics task demand and the use of fundamental diagrams of task demand.

TDCF _i

(

t

)

=TC_i

(

t

)

forheadwaysnearzeroseconds(h

↓

0)andTDCF _i

(

t

)

=0forsayh>4seconds.Forthelatter,weconsidera functionwithashortpeakatthestart(hearingandacceptingthecall)andaconstanttaskdemandduringtherestofthe conversation.Atevery stageinthesequenceofevents,thetaskdemandlevelforbothactivitiesareindicatedwithathick circleineachFDTDgraph.UnderFig.5(b),totaltaskdemandisdrawnateachtimeinstant,computedaccordingtoEq.(5); underFig.5(c),thevehiclepositionsareschematicallydrawnandunderFig.5(d)anarrativetotheexampleisprovided.

3.3.1. Initialstate(s)ofthedriver

Att0,thedriverfollowstheleaderatacomfortabletimeheadway(e.g.h=3s).Thisconsumessomecognitive informa-tionprocessingcapacitybuthasnodetrimentaleffectoneitherperceptionorresponse—thedriveroperatesaccordingtohis baselevelparameters (reaction time,sensitivities,SA levels,etc).Intheperiod[t0,t1],a vehiclemergesonto theroadway andbecomesthenewleader.Timeheadway nowdecreases (h=2s)andthecarfollowingtaskdemandincreases accord-ingly.Possibly,weseeasmalleffect(increasedperceptionerrors), althoughthesemaybe countereffectedby appropriate anticipationstrategies(Treiberetal.,2006;VanLintetal.,2018).

3.3.2. Effectsofadistraction

At t2, the driverreceives a telephone call.This results in a steep increase in total taskdemand dueto an additional taskTDcall _i ,such thatTD_i (t)_> TC_i (t)(i.e.TS_i (t)_> 1).Thedriverisnowoversaturated,whichwill(immediately) resultina deteriorationofperception/awarenesslevelsonanoperationallevel:

• Deteriorationinsensing:thedrivermaymiss relevantinformation(e.g.avehicleonan adjacentlane).Thisina sense istheworstpossibleeffect(overlookingvehiclesorothersafetycriticalinformation),whichinthissimplecarfollowing casewillnotoccur.

• Deterioration in comprehension: the driver will increasingly misjudge relative distances and speeds. There is much evidence in terms of which factors cause drivers to make errors, and visual distractions form an important category (Prechtetal.,2017a,b;Wickensetal.,2008).Thereisalsoevidenceintermsofthedirectionofspecificperceptionbiases that affectdriving.Forexample,thefindings inNilsson(2000) suggest thatgaps (perceivedwhiledriving at80km/h) are generallyunderestimated,andthat the front gapismore underestimated thanthe rear gap.Additionally,humans typicallyfinditevenmoredifficulttojudgespeeddifferencesthandistancegaps(Huntetal.,2011),andalsohere,the biasis towards underestimation. There isalso some evidence fordistanceoverestimation under specific circumstances andconditions(e.g.gapassessment,nighttimedriving(Castroetal.,2005;LeeandSheppard,2017)).Wewillvarywith bothbiasesinthesimulations.

• Deterioration in anticipation, either directly (the driver resorts to simpler more erroneousanticipation strategies) or indirectly(sincetheinputtoadrivers’“anticipationalgorithm” isnowmoreunreliable)

• Apossibleincreaseinoverallreactiontime.Onemayarguethisincrease relatestoattentiontime-lag(thedriverlooks atthephoneregularly)ortophysicalreactiontime (theconversationeatsupcognitiveprocessing,soparticularly com-prehensionandanticipationtakemoretime),ortoacombinationofthetwo.Thenetresultisthesame.

Fig. 6. Central ideas of Fullers’ task capability interface model (based on Fuller, 2011 ) for operational driving. Note that in this paper we use the term driver task capacity (instead of capability)—the reader may interpret these as synonyms (see running text).

Asecond effect takesplaceon a tacticallevel: the driverwill adapthis/her drivingbehavior to accommodate forthe increase in taskdemand. This behavioral adaptation is the key principle in Fuller’s taskcapability interface (TCI) model (Fuller, 2005, 2011) and is illustrated in Fig. 6. Note that in thispaper we use the termdriver task capacity instead of

capability,the readermayinterpret theseassynonyms(the Cambridgedictionary definescapability as“the abilityto do

something”,andcapacityas“theabilitytodoaparticularthing”). Wepreferthe termcapacityforitssecond connotation asaquantitativemeasure(forhowmuchabilitysomeonehas“todoaparticularthing”).WhendiscussingFuller,weusehis term(capability),butintheensuingwewillsticktoourterm(capacity).Fullerconceptualizesdrivingasacontroltask,in whichdriversattheoperationallevelattheveryleastavoidcollisions.Accordingtohistheorydriversmonitorthedifference between(perceived)taskdemandandtaskcapability.Thisresultsinaperceivedsafetymargin:thedifferencebetweenwhat adriverbelieves(s)heiscapableofhandlingandtheperceiveddemandsofaparticulartask(Fig.6right).Thesmallerthis safetymargin,thehighertheriskandlevelofarousaldriversexperience.Intheoriginalpaper(Fuller,2005),thetheoretical backgroundisthat oftaskdifficulty homeostasis,i.e.driversseekforandreturntoaconstantlevelofrisk (arousal).Later (Fuller,2011) Fuller relaxesthe theory witha risk allostasisprinciple, inwhich drivers dynamically adapt their “risk set-point”. Either way, whereas a driver reacts on a perceived risk, the actual consequences follow fromthe objective safety margin,whichistypicallysmaller,since driverstypicallyoverestimate theircapabilitiesandunderestimate theactualtask demand.Inourframework,thedifferencebetweenperceivedandobjectivetaskdemandandcapacitycanbeeasilycatered for,butevenifwegrantdriversanobjectivejudgementaboutboth,theirdecisionswillbebasedonperceivedstimuli(L2 awareness)andderivedanticipationthereof(L3awareness),whichcanbewrongasoutlinedabove.

ReturningtotheexampleofFig.5,thedriverinthissimplecarfollowingcasehasonechoicetoreturntoanacceptable risk level(safety margin),andthat is (betweent3 andt4) to decrease speed and(asa result) toincrease time headway. Bothcanbe achievedinvariouswaysindifferentcarfollowingmodels (e.g.by reducingdesiredspeed,increasingdesired headway,etc.). Despitethis adaptation,the driver inthis examplestill operates in an oversaturated state, whichimplies considerableperceptionerrorsand—dependingonthecharacteristicsofthedistraction—increasedreactiontime.When the callfinishesatt5 weassume thedriverrespondsagainbyreturningto(e.g.)adesiredspeedpreferenceslightlyabove his baselevelresultinginalevelofriskslightlyhigherthanjustbeforethecall.

3.4.Summary

Theproposed conceptual framework modelsthe drivingtask ina multi-layeredfashion. Atthe highestlevel,we have ideal (in principle collision-free) models for car following and other driving tasks. These models typically have reaction timesand a setof other high-level HF parameters that exogenously “govern the humanfactor” (typically sensitivities to stimuli,desiredspeed,etc).Atthelowestlevel,wedefinestatevariablesthatmaintainhowmanytasksdriversexecuteand whattheinformationprocessingcosts areofperformingthesetasks—thiswemodelusingso-calledfundamentaldiagrams oftaskdemand(Fig.3(a)).Inbetweenthesetwo,wedefinefunctionsthatgovernthedynamicsofhigh-levelHFparameters

withthesestatevariablesasinputs.Whentotaltaskdemand(Fig.3(b))increasesbeyondtaskcapacitytwoprocessestake place:

• Firstly,differentlevelsofawareness(basedonEndsley)maydeteriorate(Fig.3(d)):reactiontimemayincrease(Fig.3(e)) andperceptionerrorsmaybecomemoresevere(Fig.3(f))

• Secondly, drivers adapt their response in line with Fullers risk allostasis theory to reduce risk to acceptable levels (Fig. 3(c)), where “acceptable” may be driver, circumstance and maneuver specific. During a lane change manoeuver temporarilyveryshortheadwayswillnotresultinsuchhightasksaturations(recallReq4inSection3.2.2)

• Thirdly,over timeother driverspecific traitsmayexperience a (temporary)temporal adjustment.Inthe example,the driver increaseddesired speed after finishing a phone call to make up for lost time. Any hypothesis on longer term feedbackbetweenincreasedlevelsoftaskdemandandbehavioralresponsecouldbeincorporated.

Finally note that whereas very high tasksaturation may reduce awareness, the same maybe truefor very low task saturation (e.g. Thiffault and Bergeron, 2003), which is whyrelationship (d) in Fig. 3 has the characteristic “reverse U” shape.Inthiscase, wewouldmostlikelyobservesensing(SAlevel1)errorsandlong(er)attentiontime-lags.Wewillnot furtherelaborateonthisissue,otherthanthatitispossibletoincorporatethisbehaviorintheframework.

4. Case:simulatingdriverdistraction

The conceptual framework that we presentedin the previous sectionsis demonstratedin a simulationcase inwhich we apply differenttaskdemandcomponents andsensitivitiesto thesecomponents.Ouraimis twofold.Firstwe wantto verifythatimplementationofthemechanismsinFig.3indeedresultinplausiblechanges(deterioration)ofperformancein drivingability,whichalignstothatfoundinliterature(e.g.Hoogendoornetal.,2010;Saifuzzamanetal.,2017).Second,and related,wewanttoexplorethesensitivityofthoseresultswithrespecttoour(many)assumptions.

4.1. Casedescriptionandappliedtrafficmodel

Weconsider car-followingonlyandusetheIDM+Schakeletal.,2012) forthispurpose,whichisan adaptationofthe IntelligentDriver Model (IDM) (Treiber etal., 2000). The IDM+separatesthe free andcar-followingterms andtakesthe minimum, rather than superimposing the terms. This allows more realistic capacity values under reasonable parameter values.Thecar-followingaccelerationisdeterminedusingEqs.(8)and((9),where

τ

_i denotesthereactiontime;parameter

ai _max isthemaximumacceleration;bi _{com f} themaximumcomfortabledeceleration(expressed asapositive number);

v

i ₀ the desiredspeed;T_i thedesiredheadway,andsi ₀ isthestoppingdistance.Furthermore,wehavethreestimuli,thesearethe prevailing speed v_i (t) of driveri;speed difference with theleader



v_i (t)₌v_{i − 1}(t)_{− vi} (t) and(net distance) gap withthe leader s_i

(

t

)

=x_{i −1}

(

t

)

+si ₀−1− xi

(

t

)

.Forthebasecasewechoosethefollowingvalues:

τ

i =0;ai max =3m/s2;bi com f =3m/s2;

ν

i

0=35m/s; si 0=8m andTi =1.2s. Finally,for parameter

δ

we use a standard value of4, which reduces the maximum accelerationasspeedincreases.

a_i

(

t+

τ

i

)

=ai max min



1−



v

i

(

t

)

v

i 0



δ ,1−



s_i

(

t

)

s∗ i

(

t

)



2



(8) s∗_i

(

t

)

=si ₀+

v

i

(

t

)

· Ti +

v

i

(

t

)

·



v

i

(

t

)

2



ai _max bi _{com f} (9)

WeconsideranarbitraryhomogeneoussinglelaneroadcorridorofL₌3km,withoutrampsoranyotherinfrastructural disturbances.Atacertainlocationxacc anincidentispresumedontheoppositecarriagewaythatcausesadistraction (rub-bernecking)asschematicallysketchedinFig.7(a).Trafficisgeneratedduring15minutes(900seconds)upstreamaccording toa demandprofile witha pulseofhigh(nearcapacity)demandafter 100seconds(Fig.7(c)),whichleadstothe supply patternasinFig.7(b)(dashedlineindicates incidentlocation).The totaltimespent (TTS)inthiscaseequals451minutes (seefurtherbelow).

4.2. Specificationrelationshipsfortaskdemandandawareness

Fig.7(d)and(e)depictthefundamentaldiagramsoftaskdemandforthecarfollowinganddistractiontasks,respectively. Our(arbitrary)assumptionisthatadriverwhoiscarfollowingwithheadwayssmallerthanhi _min requiressomemaximum level of informationprocessing capacity(TDCF _i =TDi _max,CF ) to driveas an “ideal driver” Eqs. (8)and ((9)); whereas from headwayslargerthan hi ₀ seconds,(s)herequiresa muchlower level(TDCF _i =TDi _0,CF ). Inbetween, we assumea linear de-creaseintaskdemand.Specifically, wespecifytheFDTDforcarfollowingasa functionoftimeheadway h=v_i (t)



s_i (t)as

0 1 2 3 4 5 h e a d w a y (s ) 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 ta s kd e ma nd ( -)

FD task demand car following

- 4 0 0 - 2 0 0 0 2 0 0 4 0 0 d is ta n c e to d is tra c tio n (m ) 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 ta s k d e m a n d ( -)

FD task demand distraction

0 0 .5 1 1 .5 2 2 .5 ta s k s a tu ra ti o n (-) 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 aw a renes s (-)

Awareness (level 2: understanding)

(a) (c) (d) (e) (b) (f) 0 2 0 0 4 0 0 6 0 0 8 0 0 tim e (s ) 0 50 0 1 0 00 1 5 00 2 0 00 2 5 00 d e ma n d( ve h/ h ) Demand pattern

Fig. 7. Base case lay-out, base demand and supply pattern and HF functions for drivers with nominal task capacity. In this case the total time spent (TTS) by all vehicles in the simulation is 456 min.

follows(seeFig.7(c))

TDCF _i

(

h

)

=

⎧

⎪

⎨

⎪

⎩

TDi _max,CF h≤ hi min

(

a

)

TDi _max,CF − h− hi 0 hi _min

(

a

)

− hi 0



TDi _max,CF − TDi _0,CF

hi _min

(

a

)

<h≤ hi 0 TDi _0,CF h>hi ₀ , (10)

InwhichTDi _0,CF =0.5; TDi _max,CF =1(100%)aretheminimumandmaximumtaskdemandlevelwhilecarfollowingand

hi ₀=3(inseconds)andhi _min

(

a

)

theassociatedmaximumandminimumthresholdheadwayvaluesrespectively.Weexpress theminimumheadwayasafunctionofdecelerationa=a_i (t),toaccountfortheeffectthattaskdemandspecificallyincreases incaseofstrongdecelerations(emergencybraking).Wepropose

hi _min

(

a

)

=

⎧

⎨

⎩



1+ a+b i com f bi max − bicom f



hi _min a<−bi com f hi _min otherwise , (11)

Inwhich bi _max =8m/s2 _{representsthe maximumbraking accelerationwe assume inthispaper. Recall that b}i com f is a

decelerationexpressedasapositivevalue(inourcase3m/s2₎

Forthedistractiontask(Fig.7(e)),the assumptionisthat thisdistraction“eats up” task capacityalongthe samelines asintheexampleinthe previous paragraph.We(again arbitrarily)assume alinearincrease intaskdemand from400m towardsthedistractionuntilsomemaximumlevel(TDACC _i =TDi _max,ACC ), afterwhichadrivermaintainsthislevelfor200m

andthenrecuperates again. WespecifytheFDTDforthedistraction taskthus asafunction ofdistancetothedistraction

d=x_i (t)− xacc asfollows(seeFig.7(e))

TDacc _i

(

d

)

=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

TDi _max,ACC



max



0, 1− d di _min



d<0 TDi _max,ACC 0≤ d<di _med TDi _max,ACC



1− min



1, d− di med di _max − di med



d_med i ≤ d<d_max i , (12)

In whichTDi _max =0.8(80%) isthe maximumtaskdemand level; andd_min i =−400; di _med=200;and d_max i =400 (allin meters)arethethreedistanceparametersintheFDTDfunctionrespectively.Apossiblewaytointerpretthismaximumtask demand of 80% whilepassing thedistraction is that the driver takesprolonged glances atthe accident withbrief looks straight-aheadwitharatioofabout4:1(80/20).Sincebothvisualtasksusethesameinformationchannel(vision)theyare mutuallyexclusive.The distancevaluesareareasonableestimateofwhere(athighspeed)adrivermayfeelcompelledto lookattheaccident.

FillinginEqs.(10)and(12)in(5)and(6)givesustotaltaskdemandTD_i (t)andtasksaturationTS_i (t) (whichaslongas taskcapacityTC_i (t)=1 areequal). A thirdrelation betweenawarenessSA_i (t) andtasksaturationts=TS_i (t) (Fig.7(f)) now governstheeffectonawareness.Wespecifythisrelationasfollows

SA

(

ts

)

=

⎧

⎪

⎨

⎪

⎩

SAi _max ts<TSi _crit SAi _max − ts− TSi crit TSi _max − TSi _crit



SAi _max − SAi min

TSi _crit ≤ ts<TS_max i SAi _min ts≥ TSi _max , (13) InwhichSAi _max =1; SAi

min =0.5arethemaximumandminimumSAlevels;TSi crit =0.8thecriticaltasksaturationabove

which awarenessdecreases;and TSi _max =2.0 the maximumtasksaturation levelbeyond whichawareness nofurther de-teriorates.Clearly,both theshapeandtheparameters valuesinEqs.(10),(12),and(13)arearbitrary;we choosethemto accommodatethemainassumedtendencies.

4.3. Scenarios,hypothesesandspecificationofeffectsonperception&response

Weconsiderfourbehavioralscenarios,inwhichweexploreincreasinglycomplexcombinationsofHFeffectsasaresult ofthedistraction:

I. Distractionwitheffectsonperceptionerrors,i.e.errorsindistancegaps,speeddifferencesandboth II. Distractionwitheffectsonperceptionerrorsandreactiontime

III. Distractionwitheffectsonperceptionerrors,reactiontimeandresponseadaptationindesiredspeed,desiredheadway andboth

IV. Sameas3,nowwithdriverheterogeneity(varyingtaskcapacities)

Wedescribe theminthesubsectionsfurtherbelow.Ineachofthesefourscenarios,weconsiderfourphysicalreaction times:

τ

_i p =

{

0, 0.2, 0.5, 1

}

seconds.Clearly,settingphysicalreaction timeto zeroisan idealisation;however,one could interpretthisidealisationasfollows.Inthecase

τ

_i =0,weimplicitlyassumethatdriversareabletofullycompensatetheir physicalreactiontimeduetotheinformationprocessingforsensing,comprehensionandanticipation,withtheresultofthat verysameperceptionprocess:anadequateanticipationstrategy.Putdifferently,

τ

i=0reflectsanetresultoftheperception

process.Evenunderdensetrafficconditions,thisisareasonableassumptiontomake,inlinewithe.g.(Treiberetal.,2006; VanLintetal., 2018).Similarly,

τ

_i p =0.2, 0.5, 1representcasesinwhichdriversarenotabletofullycompensatephysical reactiontimewithanadequateanticipationstrategy.

4.3.1. ScenarioI:effectsonperceptionerrors

Inthisscenario,weconsidereffectsonlevel2awareness(comprehension)errors,whichresultinincorrect stimuli.The assumption isthat reducedawareness exacerbatesknown perceptionbiases, that is,either an under-or overestimationof bothdistancegapsand(relative)speeds(seeSection3.3.2).Wepropose

s_i percei ved

(

t

)

=



1+

δ

i

SA _i

(

t

)

si

(

t

)

(14)



v

_i percei ved

(

t

)

=



1+

δ

i

i SA

(

t

)



v

i

(

t

)

(15) inwhich

SA i

(

t

)

=SAi max − SAi

(

t

)

isafactorbetween0and

(

SAi _max − SAi

min

)

that determines the magnitude of the perceptionerror,and

T

δ

i =



−1 Driverisystematicallyunderestimatesgapsandspeeds 1 Driverisystematicallyoverestimates gaps andspeeds

representsafactorthatgovernsthedirectionoftheperceptionbias.Notethatwe assumethat asingledriverhasa fixed directiontowardeitherunder-oroverestimatingbothdistanceandspeeddifferences.Weconsiderthreedriverpopulations suchthat

δ

i =sign

(

D−

ν

)

with

ν

arandomvariabledrawnfromauniformdistributionover[0,1],and

D=



₀

_δ

i =−1,

∀

i.

(

alldriversunderestimate

)

0.5 50− 50mix

1

δ

_i =1,

∀

i.

(

alldriversoverrestimate

)

4.3.2. ScenarioII:effectsonperceptionandreactiontime

Inthisscenario,wedoconsideranincreasein(net)reactiontimewithanattentiontime-lag,againproportionaltothe decreaseinawareness,thatis,

τ

a

i

(

t

)

=

i SA

(

t

)

τ

i a,max

inwhich

τ

_i a,max ₌2sdepictsthe maximumattentiontime lagwe consider forthiscase.Clearly the(arbitrary) settingof

τ

a,max

i determinesthemagnitudeoftheeffect.Forthetotalreactiontime(Eq.(7))wethenhave

τ

i

(

t

)

=

i SA

(

t

)

τ

i a,max +

τ

i p (16)

With

τ

_i p=

{

0, 0.2, 0.5, 1

}

asdiscussed above.Note that sincewe donot implementspatialortemporal anticipation, driversthussimplybasetheirresponsesonstimuliof

τ

_i (t)secondsago,andtheydothiseverysimulationtimestep.

4.3.3. ScenarioIII:effectsonperception,reactiontime&response

Inthisscenario,weadditionallyconsidertwo kindsofresponseadaptationsthatboth resultinlarger gaps;butwhich mayhavedifferentconsequencesfortheresultingtraffic operations.First,weassume driversincrease theirdesiredspeed, andsecond,weassumeanincreaseindesiredtimeheadway,whichintheIDM+caseboilsdowntoanincreaseindesired distancegap(Eq.(9)).Tothisend,weproposeareductionfactorsimilartotheoneinEq.(16):

v

0 i

(

t

)

=



1−

β

v0 i

i T S

(

t

)

v

0 i (17)

inwhich

β

_i v0 isascalingparameterthatgovernsthemaximumreductioneffect(e.g.

β

_i v0 ₌0_.9impliesamaximumreduction of90%indesiredspeed),and

T S

i

(

t

)

=min



1,max



0, TS_i

(

t

)

− TSi _crit

isafactorbetween0and1thatinthiscasedependson theprevailingdrivertasksaturation.Thehighertasksaturation, thelargertheresponseadaptation.ThisrationaleisinlinewithFullersTCImodel(Fuller,2011);weessentiallyuse

_i T S

(

t

)

asa proxy forperceived risk. Thehigheritis,thestrongerthe behaviouraladaptationinthedirectionofa safergapand speeddifference.AnalogouslytoEq.(17)wehavefordesiredtimeheadway

Ti

(

t

)

=



1−

β

i T

i TS

(

t

)

T_i 0 (18)

4.3.4. ScenarioIV:driverheterogeneity(varyingtaskcapacities)

Inthefinalscenario,welookattheeffectsofdriverheterogeneity.ByvaryingdrivertaskcapacitiesTC_i (t)(andthereby tasksaturation) inunits of nominal taskcapacity (Section 3.2.1), we effectively vary driver skilllevel with justa single parameter. Thisparameter affects the fundamentaldiagrams oftask demands, leadingto lower (higher) tasksaturations formore(less) skilleddrivers(forwhateverreasons)undersimilarconditions.Tasksaturation,inturn,servesasinputfor theawarenessrelation(Eq.(13)),implyingalsoawareness deteriorateswithlowerdrivertaskcapacity.Atthestart ofthe simulationwegeneratetaskcapacityvaluesasfollows

TCi

(

t

)

=min

(

TCmax ,max

(

TCmin ,TC+

ψ

i

)

)

,

ψ

i ∼ N

(

0,0.1

)

(19)

InwhichTC_min andTCmax are setto 0.8and1.2respectively. Clearly,one could varywithmanymoreparameters and drivercharacteristics,andwereturntothisinthediscussionsection.