COMPARAIVE SUDY OF CAR-FOLLOWING MODELS FOR DESCRIBING BREAKDOWN PHENOMENA A SAGS akashi Oguchi * and Ryoichi Konuma okyo Meropolian Universiy, JAPAN, - Minamiosawa, Hachiouji, 9-97 JAPAN 8-6-77-78, oguchi@mu.ac.jp Meropolian Epressway Co., Ld., okyo, JAPAN ABSRAC he paper describes characerisics of capaciy boleneck phenomena a sag secions in JAPAN from a viewpoin of car-following behaviour. hrough observaion and evaluaion of drivers' behaviour and he parameer esimaion and comparaive evaluaion of several car-following models are inroduced. he model of gradual gradien change effecs are inroduced and compared beween driving behaviour a sag secions and ha a a consan gradien secion. he sysem design of ACC Adapive Cruise Conrol o preven breakdown a sag secions could become more realisic hrough he evaluaion using microscopic simulaion wih he beer car-following model evaluaed in his sudy. KEYWORDS boleneck a sags, car-following behaviour, effecs of gradien change, model comparison INRODUCION here are many capaciy bolenecks on Japanese Epressways where raffic queue congesion of long lengh and duraion is occurred very frequenly. Mos major boleneck sies are ordinary secions including verical alignmen sag curve secions and unnel enrances, oherwise oll gaes on he main rack and acciden or inciden sies. his paper deals wih characerisics of capaciy boleneck phenomena a ordinary secions, paricularly a sag secions, and shows analyical measures for proving he mechanism which causes he breakdown phenomena a sag secions from a viewpoin of car-following behaviour. he mechanism of he acivaion of bolenecks a sag secions proposed by Koshi e. al. [] becomes very popular, bu only in Japan. Some of he key facors are he eisence of dense plaoon of vehicles on median lane, he gradual speed decrease being sared from he beginning of a verical sag curve secion, drivers' unawareness of such gradual change of verical gradien, drivers' endency of keeping space clearances in he dense plaoon, and resuling in he oubreak of an upsream amplified propagaion of deceleraion shockwave. hese phenomena consis in each driver's car maneuvering behaviour including he car-following behaviour and he driver's response o such a gradual gradien change. Alhough several conribuions o he developmen of models o simulae such driving behaviours were carried ou, here are sill many problems.
Few of he eising car-following models are, firs, proved wih acual driving daa ecep for paricular sudies. here is no comprehensive comparaive sudy among many famous car-following models. he disribuional naure of he parameers of hese eising models are, hird, no eamined enough wih observaions or eperimens of drivers' behaviour. he appropriaeness of parameers is, fourh, almos always evaluaed wih he reproducibiliy correlaion coefficien or RMSE of acceleraion only, bu i should also be done wih ha of velociy and space clearance also. he effecs of gradual gradien change on drivers' behaviour are, las, no eamined wih enough care hough hese are mos imporan facors for eplaining he difference beween he sags which cause breakdown phenomena and hose which do no. By he way, here are many well-known bolenecks of sag secions in Japan bu hese are fied a some sag secions no a ohers, and here is no obvious difference of highway geomery or raffic characerisics beween hese wo groups of sag secions. here are hree objecives of he paper. he firs objecive is observaion and evaluaion of drivers' behaviour wih high accuracy a he ime of breakdown phenomena. he second one is parameer esimaion and comparaive evaluaion of several car-following models in consideraion of he appropriaeness concep. he las one is evaluaion of he gradual gradien change effecs hrough comparison beween driving behaviour a sag secions and ha a a consan gradien secion. NAURE OF CAPACIY BOLENECK PHENOMENA A SAGS Generally, raffic flow characerisics in basic segmens wih sags were already known Koshi e al. []. Before congesion occurs in a boleneck, he flow rae in he median lane is always higher han ha in he shoulder lane, and he flow rae of median lane is abou,8 o, vphpl while he maimum flow rae is from, o,5 vph/-lane. his shows ha he speed reducion causing he breakdown sars from he median lane. Afer his however, he flow raes for boh lanes become almos equal and he capaciy flow rae afer formaion of he queue is reduced o, o,7 vph/-lane. he fac ha a sag secion can be a boleneck does no necessarily mean ha his is because of a consan, seep and long up-grade secion downsream of he sag curve. Xing e al. [] observed successfully, for he firs ime in Japan, he rajecories of more han vehicles showing he amplifying shock wave propagaion o he upsream secion, only by an inernal deceleraion a a sag secion where he gradien changes from -.6 o -.%. he downsream secion is almos level, wih a sligh down-grade slope no upgrade. Unil he consrucion of an addiional lane, his par was a boleneck where breakdowns occurred mos frequenly among many bolenecks of basic secions. I is hough ha he cause of he boleneck phenomenon a sags is no a seep up-grade bu a mild gradien change Koshi e al. []. he mechanism of he boleneck acivaion is simulaed by sligh speed decrease wih he gradien change. he drivers going hrough a sag secion canno fully compensae he gradien change, bu sill hey ry o mainain heir space clearance wihin plaoon in he median lane; his naure for all drivers causes amplifying shock wave propagaion in he plaoon on he median lane. I is hough ha drivers, afer several seconds passed hrough his secion, are accusomed wih he downsream grade condiions, herefore, he effec from he grade or he gradien change around he sag secion is already vanished. Heavy vehicles do no necessarily affec on he acivaion of he
boleneck phenomena because of he slighness of gradien change, and he boleneck phenomena a many sags are acivaed only in weekends or holidays wih few heavy vehicles. EXPERIMEN MEHOD AND RESULS Ouline of he Eperimens An eperimenal vehicle, equipped wih several sensors o measure he behaviour of iself and surrounding vehicles Oguchi e al. [], is employed for es run in realiy. Several es runs, going hrough one of he mos popular boleneck named "Yamao sag" in Japan, were planed repeaedly. Figure shows he verical alignmen of he secion. he auhors ried o sar he vehicle seeking he good iming for he vehicle o go hrough he secion wihin a dense plaoon and o be encounered wih deceleraion shockwave. Because he breakdown phenomenon happens suddenly, i is no easy o make he vehicle mee he shockwave. One of he es runs, which relaively eperienced wih a shockwave, are seleced as a former vehicle of he firs virual es run for each virual highway condiion creaed as -D CG for a driving simulaor DS developed in Universiy of okyo. As a join research, he auhors are permied o use he DS. he DS sysem conains he 6-degree-of-freedom shaking insrumens wih urn-able, and he insrumens providing full-direcional visual image. he DS conneced wih KAKUMO, which is a microscopic raffic simulaor S as for each vehicle can move and change lanes considering spacing and/or gaps among surrounding vehicles, can provide he condiion in which subjecs can feel as o be in he virual raffic condiion. KAKUMO enables o reproduce he ime-space rajecory recorded by boh in realiy and in he DS sysem as a virual leading vehicle. Uilizing his funcion, subjecs, driving he DS, can drive in a virual siuaion wih he leading vehicle as a reproducion of cerain behaviour of oher drivers and wih he neighbour lane raffic condiion. Aliude m 9 8 Direcion oubound VLC 7 m "Yamao Sag" 7 6 5..... Posiion km Figure Verical alignmen of he "Yamao sag".9-.6kp, oubound. algebraic gradien change able Prepared highway geomeric condiions Lengh of Verical Curves 7m m wihou curve.% case case.6% case case in realiy.9% case case 5 no change case 6
Seven virual highways are creaed wih differen verical alignmen condiions using -D CAD sofware; combinaion of he same, a half of and one hird of he lengh of verical curve and he same, a half of and one hird of he algebraic gradien change of he "Yamao sag" secion in addiion o a condiion wih no gradien change. able summarizes hese seven cases. Oher geomeric condiions for he seven cases are always same as he "Yamao sag" such as horizonal alignmens, cross-secion specificaions, ec. Procedures and Resuls hiry-hree subjecs were asked o follow he former vehicle. he former vehicle of he firs subjec is he reproducion of he seleced rajecory obained by he eperimenal vehicle. Ecep for firs subjec, he former vehicle is he reproducion of he behaviour of he former subjec in he order of hem, for seven virual highway condiions wih he DS. Seven es runs of car-following behaviour by he hiry-hree subjecs are recorded. Figure shows one of he eamples of ime-space rajecories of hiry-hree subjecs obained by he DS. CAR-FOLLOWING MODELS A CONSAN GRADIEN CONDIION Oguchi [] already colleced many eising ypes of car-following models. he GM model [5], he eponenial epression model [6], OV model [7], polynomial epression models including he models proposed by Koshi [5] and Ozaki [8], [9] and spiral curve model [] are seleced wih small modificaions for comprehensive comparaive sudy here. Seleced Car-following Models he noaions of variables used here are as below; : elapsed ime[s],,, : posiion[m], speed[m/s], acceleraion[m/s ] of he leader, posiion[m], speed[m/s], acceleraion[m/s ] of he follower, θ n : verical grade [rad] a ime of he n-h vehicle's posiion, : reacion delay ime [s],,,,,,, β : coefficien parameers, and l, m, n : raising parameers. posiion [km] Figure Eample of ime-space rajecories of subjecs case
5 Model- Linear Monomial Model LM Model: his is he simples model proposed by Chandler e al. [] which consiss of linear differenial equaion. Model- Non-Linear Monomial Model NLM Model: his is he model wih small simplificaion derived from he one developed by Gazis e al. [5] as shown in equaion. Model- Monomial Elemenary Funcion Model GM Model: his is he mos famous model, so called "GM Model", developed by Gazis e al. [5]. his Model includes he ones in equaion and as specific cases. l m { { Model- Eponenial Funcion Model Newell Model: Newell [6] proposed a model which allows space headway o be given as an eponenial funcion of speed. o ake differenial of speed, he model shown equaion is inroduced. e { Model-5 Eponenial Funcion Model ype Ceder Model: Ceder [] proposed a model which allows he reacion srengh of GM Model o be an eponenial funcion of space headway. / e { { 5 Model-6 Linear Polynomial Model KS Model: Komeani and Sasaki [] proposed a model adding he effec of acceleraion of he leading vehicle o he LM Model. 6 Model-7 Hyperbolic angen Funcion Model OV Model: Bando e al. [7] proposed a model which allows he acceleraion is given by he difference beween Opimal Velociy OV and he real speed, and he OV is given by he hyperbolic angen funcion of space headway. heir model includes he consan values of "" and "anh ", hough hese consans would be arbirary. he original model does no conain he physical dimensional adjusmen facors before aking hyperbolic angen funcion and afer doing so. So he auhors add four coefficien parameers o he original one. he original model does no also consider he effec of reacion delay ime, herefore i is included in he model here. ] - [ - anh { 7 Model-8 Linear Polynomial Model ype Helley Model: Helley [] added anoher erm including he effecs of he difference beween he opimal space clearance given by linear funcion of speed and acceleraion and real clearance o he LM Model. - β 8 Model-9 Non-Linear Polynomial Model Spiral Model: Nakayama e al. [] paid aenion o he fac ha he rajecory on he plane made of space headway and relaive speed become spiral. aking approimaion of he leader's speed be consan, he model shown in he equaion 9 is derived; he consan L is se o be a consan as an average space clearance. { { Y L Y 9 where, / L Y Model- Non-Linear Polynomial Model Koshi Model: Koshi [5] proposed a model which consiss of four erms as shown below.
f f f sin{ θ where, f l { f g n { f { V f β β β β g In his sudy, he hird and forh erms are omied like equaion. he model is applied only a he consan gradien condiion and he effec of gradien change is separaely deal. he reacion ime delay is pu ogeher ino one and he opimal space headway is se o be consan β. { { β l n { { hough GM, Newell, KS and Koshi Models include LM, NLM and Helley Model as special cases, every model parameers are se o be non-zero o make clear he difference beween each model. Simulaion Analyses a Consan Gradien Condiion he auhors creae a micro raffic simulaor for evaluaing he en ypes of car-following models seleced above. Simulaion analyses are conduced in 6 m secion from. km poin o.8 km poin wih consan gradien condiion in he virual highway alignmen case. he simulaor makes each run hrough he secion wih a cerain se of parameers for a cerain car-following model wih he iniial speed and space clearance condiions of he saring posiion in addiion o he oal rajecory of he leading vehicle. Reproducibiliy wih each model of any rajecories can be judged by roo mean square error RMSE of he simulaed calculaion o he observed daa recorded by DS, in addiion o he correlaion coefficien CC beween hem. hough evaluaion indices may no be only for acceleraion bu also speed and space clearance, Oguchi [6] found ha he space clearance, being he mos inegraed variable, is mos appropriae as an evaluaion inde. he simulaor can also simulaneously check he calculaed vehicle condiions, which are used for judging he local sabiliies. he sabiliy judging hresholds are as follow; avoidance of rear-end collision: >, no ecess of upper/lower limis of realisic acceleraion: 9.8 [m/s ] < <. [m/s ], avoidance of no follow-he-leader driving condiion: < 5 [m], and no sopping or being backed up siuaion: >. A simulaed calculaion should break off a he ime when one of he any condiions above does no mee, wih a cerain se of parameers. I means ha his se of parameers does no conain "Local Sabiliy". Moreover, one more vehicle he second follower is added o follow he simulaed firs follower. he "Asympoic Sabiliy" can be evaluaed parially by 6
able Range of he se of parameers for each model LM # of division.~..5~.5 NLM # of division.~..5~5. GM l m # of division.~. 5.~. 5.~..~. Newell # of division.~. 5.~. 5 5 Ceder.~. ~ # of division 5 6 KS.~..~.5 # of division 5 5 7 OV.~..~.7 # of division 5 8 Helly.~..~.5 # of division 5 5 9 Spiral.~.5 5.~. # of division 5 Koshi # of division.~. 5.~..5~5..~..5~5. 6~5.5.~..~. 5.5~..5~.5 5.~.5.5~..~.5 5 8 5 l n.~. 5.~.5.~..5~.8 checking he running condiion of he second follower hrough he simulaed calculaion as same way as he firs one. he parameers for each car-following model are idenified wih RMSE value of he simulaed space clearances o he recorded ones of a cerain subjec. he search mehod adoped here is he simple repeiion from fron o back so called "area bombing mehod" for every parameer wih cerain discree valuessee able, and he parameers wih he smalles RMSE value are se o be he idenified model parameers. If he simulaed calculaion for he firs follower successfully finished wih he smalles RMSE value, ha of he second follower sars and ry o find again he anoher smalles RMSE value, which is he fied one wih he finally idenified parameers. Reproducibiliy a Consan Gradien Condiion From he view poin of reproducibiliy, a car-following model should allow smaller RMSE value. On he oher hand, from he view poin of robusness, i should provide RMSE values for more number of subjecs. For eample, even if one model can allow very small RMSE value for some paricular subjecs, and canno provide RMSE value for oher many subjecs because of insufficien naure abou any of he four sabiliy judging hresholds, he evaluaion 7
RMSE [m] 5 LM NLM GM Newell Ceder KS OV Helley Spiral Koshi Figure 5 5 5 RMSE ascending order Smalles RMSE ranked ascending order for he en car-following models he second-follower simulaion resuls, oal number resul is no well. he desirable model should have small RMSE value for as many subjecs as possible and also should have as small number of subjecs as possible wihou RMSE value. Figure shows he smalles RMSE values, derived from he second-follower simulaed calculaion, ranked in ascending order for each of en car-following models. hose five models, Ceder, OV, Spiral, Newell and Helley Models, can reproduce only half of he subjecs' follow-he-leader behaviour. Koshi Model can reproduce he mos number of subjecs' behaviour, followed by GM Model, and NLM, KS, and LM Models, respecively. he ranking, sars from Koshi Model, means he order of desirable robusness. NLM, KS, and LM Models can be described by Koshi and GM Models as a paricular parameer condiion. here is no significan difference in he performance of reproducibiliy and robusness beween he Koshi Model and he GM Model, bu he GM Model can allow smaller values of RMSE han he Koshi Model can do. herefore, GM Model is found o be he bes model o reproduce he behaviour of hiry-hree subjecs wih DS in he consan gradien condiion; followed by Koshi Model, and he oher eigh models are significanly inferior o hese wo models. MODELS OF GRADIEN CHANGE EFFECS Proposed Models Because of he performance in reproducibiliy and robusness of he compared en car-following models, he effecs of he gradien change are invesigaed only combined wih GM model. In his sudy, he resuls of virual es runs on DS for si differen gradien change condiions a he 'Yamao Sag' secion shown in able are used for comparison among models of gradien change effecs proposed. he basic idea of he model of gradien change effecs is shown as below; GM Model γ g{sinθ sinθ u, where g : graviy acceleraion [m/s ], 8
θ : grade [radian] a elapsed ime a he posiion, θ u : grade [radian] in upsream secion of he sag secion, and γ : gradien change effec parameer. If a driver can compleely compensae he graviy change in a sag secion and he downsream consan grade secion, his or her car-following behaviour does no affec from he gradien change a all and he gradien change effec parameer should be se o zero γ. On he oher hand, if he driver never be able o noice and reac o he gradien change and canno compensae a all, he componen of graviy change oally affecs on he car-following behaviour, i.e. he gradien change effec parameer should be se o uniy γ. In general he driver's behaviour would be in he middle condiion beween hese wo γ. Because he drivers may gradually sar o noice he gradien change in a sag secion, he gradien change effec parameer migh change from zero o uniy wih he gradien condiion and elapsed ime. hrough he discussion above, five differen models are proposed as below. No Effec NE Model: I is a model wihou effec from he gradien change. γ oally Affeced A Model: Drivers never compensae he effec of gradien change. γ Consan Effec CE Model: he effec from gradien change is consan hrough oal ravel from upsream secion of a sag boleneck o downsream secion. γ consan < γ < Linear Funcion LF Model: he gradien change effec parameer is se o be a linear funcion of elapsed ime from he ime a driver sars o noice he gradien change a W o he ime he or her sars o compensae compleely. γ γ < a W W a γ a W γ < a W 5 W γ a W γ where a and W are he model parameers. he change of γ is illusraed in Figure. Non-Linear Funcion NL Model: he gradien change effec parameer is se o be a non-linear funcion of elapsed ime wih he model parameers of a and κ. he change of γ for NL Model is also illusraed in Figure. anh{ κ γ a 6 a W γ.8.6.. NE Model A Model CE Model LF Model NL Model a W a Figure Gradien change effec parameers of he five models a W elapsed ime 9
Simulaion Analyses wih Gradien Change Condiions Daa of he hree subjecs who go virual carsick and canno carry ou heir assigned ask for driving on seven differen highway alignmen condiions are eliminaed. Daa of oher hree subjecs for whom GM Model canno reproduce heir behaviour RMSE value canno be esimaed are also eliminaed. herefore, rajecories derived by weny-seven subjecs are uilized o evaluae he gradien change effecs. he evaluaion process should be done only under he condiion of he almos same parameers of GM Model can be applied for all of he si differen gradien change condiions. herefore, before analysing he gradien change effecs, one se of he parameers of GM Model is idenified checked wih he reproducibiliy and robusness for each subjecs on each of si alignmen change condiions, ecluding no gradien change case case 6, in he same secion of.~.9 km poin wih lile gradien change. As a resul, eleven subjecs' behaviour canno be reproduced by GM Model because any of he sabiliy judging condiions ~5 is no filled in some highway alignmen condiions RMSE value canno be esimaed. For oher welve subjecs, relaively differen ses of parameers of GM Model are appropriae o differen alignmen condiions. he ses of parameers for he oher only four subjecs are good for eplaining oal of si differen alignmen condiions. I is found ha he drivers, who drive differen si highways mainaining same driving behaviour wih he same se of parameers of GM Model, are very rare. Reproducibiliy in Gradien Change Condiions Among daa of weny-seven subjecs running on si alignmen condiions, he reproducibiliy and robusness of five gradien change effec models are eamined uilizing only he daa in he condiion boh wih fied ses of parameers of GM Model esimaed shown in Figure and wih he smalles RMSE value being smaller han meers. As a resul, niney runs are used in his evaluaion. hen he parameers of he five gradien change effec models are idenified wih a RMSE value of he simulaed space clearances o he recorded ones of a cerain subjec and wih a cerain alignmen condiion hrough he secion beween. km poin o. km poin. he search mehod adoped here is also "area bombing mehod", and RMSE [m] 5 NE Model A Model CE Model LF Model NL Model 6 8 RMSE accending order Figure 7 Smalles RMSE ranked ascending order for he five gradien effec models he second-follower simulaion resuls, oal number9
he parameers wih he smalles RMSE value are idenified. If he simulaed calculaion for he firs follower successfully finished wih he smalles RMSE value, ha of he second follower sars and he anoher smalles RMSE value, which is he fied one wih he finally idenified parameers, is ried o be found. Figure 7 shows he smalles RMSE values, derived from he second-follower simulaed calculaion, ranked in ascending order for each of he five gradien effec models combined wih GM Model. LF and NL Models, which have changing naure of he gradien effec parameer, can reproduce more number of behaviours han oher hree NE, A, and CE Models wih fied consans of he parameer. he figure shows he fac ha LF and NL Models have no only beer reproducibiliy bu also beer robusness. CONCLUDING REMARKS he idenificaion of he parameers of en car-following models, such as LM, NLM, GM, Newell, Ceder, KS, OV, Helley, Spiral and Koshi Models, are ried in he raffic condiion of boleneck acivaion around a cerain sag secion. GM Model is he bes model from he view poins boh of reproducibiliy and robusness, and Koshi Model has he bes performance on robusness only. he parameer idenificaion mehod used here is o find he smalles RMSE of simulaed space clearances o recorded daa of hiry-hree subjecs in a consan grade secion on DS. Reproducibiliy is evaluaed by he smallness of RMSE for each subjec; on he oher hand robusness is evaluaed by he sabiliy naure for all of he subjecs. he idenificaion of he parameers of five gradien change effec models, such as NE, A, CE, LF and NL Models, are ried wih combinaion of GM Model a he sag secion. LF and NL Models are he bes model from he view poins of boh reproducibiliy and robusness. Boh LF and NL Models have changing naure of he gradien change effec parameers, herefore, he gradien change effec is gradually vanishing as drivers pass hrough he sag secion. hese characerisics are inerpreed as he drivers' psychological behaviour. here are sill many issues lef o fuure sudy. he DS used in he sudy should be verified for reproducibiliy and feelings of drivers compared o hose of hem in realiy, paricularly in he driving condiion on an epressway wih many surrounding vehicles and wih gradien change. GM Model nor Koshi Model is no compleely enough for reproducibiliy and robusness for describing car-following behaviour observed by DS, because he smalles RMSE value is no sufficien enough. In addiion, he simulaed calculaion of acceleraion and speed is no fi o recorded one. he difference of driving behaviour beween each driver will be epeced o be described by he difference of parameers idenified for a proper model of car-following behaviour affeced by gradien change. From he view poins of reproducibiliy and robusness or sabiliy beween drivers he comprehensive and simulaneous mehod o idenify he model parameers should be buil up. Even if here are many issues lef o fuure sudy, he resuls can sugges he developmen of he beer car-following model o describe he breakdown phenomena a sag secions, and he sysem design of ACC Adapive Cruise Conrol o preven breakdown a sag secions could become more realisic hrough he evaluaion using microscopic simulaion wih he model.
ACKNOWLEDGEMEN his sudy was fully suppored by "Susainable IS Projec" in he Universiy of okyo. he auhors would like o remark special acknowledgmens o Professor M. Kuwahara and Professor Y. Suda for giving his join research opporuniies o hem. hey also would like o epress heir graiude o he Projec Members, especially o Mr. K. Honda, Mr. M. Onuki and Mr.. Shiraishi for heir useful and generous advice and suppor. REFERENCES [] Koshi, M, Kuwahara, M and Akahane, H, "Capaciy of sags and unnels on Japanese Moorways", IE Journal, 65, 99, pp.7-. [] Xing, J and Koshi, M, "A Sudy on he Boleneck Phenomena and Car-following Behaviour on Sags on Moorways", J. Infrasrucure Planning and Managemen JSCE, 5, 995, pp.69-78 in Japanese. [] Oguchi,, Akahane, H, Nishikawa, H, and Kuwahara, M "Developmen of an Eperimenal Vehicle for Evaluaing Highway raffic Composed of Auomoives wih and wihou Adapive Cruise Conrol Sysems", Proc. h FISIA,, on CD-ROM. [] Oguchi,, "Needs for Developing New Car-following Model for Evaluaing Cruise-assis Highway Sysems", Proc. 7h World Congress on IS,, CD-ROM. [5] Gazis, D, Herman, R, and B. Pos, "Car-Following heory of Seady-Sae raffic Flow", Oper. Res., 7, 959, pp.99-55. [6] Newell, G, "Nonlinear Effecs in he Dynamic of Car-following", Oper. Res., 9, 96, pp. 9-9. [7] Bando, M, Hasebe, K, Nakayama, A, Shibaa, A, and Sugiyama, Y, "Dynamical Model of raffic Congesion and Numerical Simulaion", J. of Phys. Rev. E, 5, 995, pp.5-. [8] Ozaki, H, "Reacion and Anicipaion in he Car-following Behaviour", Proc. h In'l Symp. on ransporaion and raffic heory, 99, pp.9-66. [9] Ozaki, H, "Assisance of Drivers o Miigae Highway Capaciy Problem", Proc. nd World Congress on IS, 995, pp.77-7. [] Nakayama, H, Wada, M, and Ichikawa, K, "Invesigaion of raffic Simulaion Model Using Spiral Curve", Proc. of raffic Eng. JSE,, 99, pp.5-8 in Japanese. [] Chandler, R, Herman, R, and Monroll, E, "raffic Dynamics: Sudies in Car Following", Oper. Res., 6, 958, pp.65-8. [] Ceder, A, "Deerminisic raffic Flow Model for he wo-regime Approach", ranspn. Res. Recrd., 567, 976, pp.6-. [] Komeani, E, and Sasaki,, "On he Sabiliy of raffic Flow Repor ", J. Oper. Res. Soc. Japan,, 958, pp.-6. [] Helley, W, "Simulaion of Bolenecks in Single-lane raffic Flow", heory of raffic Flow, 959, pp.7-8. [5] Koshi, M, "Capaciy of Moorway Bolenecks", J. Infrasrucure Planning and Managemen JSCE, 7, 986, pp.-7 in Japanese. [6] Oguchi,, "Observaion and Analysis Measure of Vehicle Moion for Evaluaion of raffic Capaciy", ebook of he One-day Seminor on Infrasrucure Planning JSCE,,, pp.5-56 in Japanese.