A NEW MODEL FOR ESTIMATING CRITICAL GAP AND ITS DISTRIBUTION AT UNSIGNALIZED INTERSECTIONS BASED ON THE EQUILIBRIUM OF PROBABILITIES

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1 A NEW MODEL OR ESTIMATING CRITICAL GA AND ITS DISTRIBUTION AT UNSIGNALIZED INTERSECTIONS BASED ON THE EQUILIBRIUM O ROBABILITIES Ning Wu h.d. ofesso Insiue fo Tffic Engineeing Ruh Univesiy Bochum Gemny phone: fx: e-mil: ning.wu@ub.de hp://homepge.ub.de/ning.wu ublished in: In: oceeding of he 5h Inenionl Symposium on Highwy Cpciy nd Quliy of Sevice. Yokohm Jpn July ABSTRACT This ppe pesens new model fo esimion ciicl gps unsignlized inesecion. The heoeicl bckgound of he new model is he pobbiliy equilibium beween he ejeced nd cceped gps. The equilibium is esblished mcoscopiclly using he cumulive disibuion of he ejeced nd cceped gps. The model yields diecly he pobbiliy disibuion funcion of he ciicl gps. The new model hs he following posiive popeies: solid heoeicl bckgound equilibium of pobbiliies b obus esuls c independen of ny model ssumpions d possibiliy of king ino ccoun ll elevn gps no only he mximum ejeced gps s is he cse of he Toubeck model 992 e possibiliy of chieving he empiicl pobbiliy disibuion funcion of he ciicl gps diecly nd f simple clculion pocedue wihou ieion. The implemenion he new model is simple nd obus. I cn be cied ou using spedshee pogms e.g. EXCEL Quoo ec.. Thus wih he new model useful nd pomising ool cn be se up fo pofessionls in ffic engineeing. INTRODUCTION Ciicl gp is mjo pmee fo cpciy nlysis unsignlized inesecions. This pmee is sochsiclly disibued vlue nd i cnno be obined diecly by mesuemens. The esimion of ciicl gps unsignlized inesecions fom ffic obsevion is one of he mos difficul sks in ffic engineeing science. o esiming he ciicl gps sisicl models o pocedues e equied. Thee exis mny diffeen models fo esiming ciicl gps. Among hem he models of Siegloch 973 Rff e l. 95 Awoh 97 Hdes 968 Hewe 983 nd Toubeck 992 e he mos impon. Tody we cn find moe hn 2 o 3 models woldwide fo esiming ciicl gps. In pcice - fo unsued condiions - he mos common models e h of Rff e l. 95 nd Toubeck 992. Bilon nd König 997 gve n oveview of he mos impon models. Using micoscopic simulions hey lso conduced n ssessmen of hose models. I ws found h he model of Toubeck 992 gives he bes esuls. Thus his model is ecommended fo esiming he ciicl gps in mny sndd mnuls fo ffic engineeing e.g. HCM 2 HBS 2 ec..

2 The model of Toubeck 992 is micoscopic model. Th is he single vlues of he mesued gps e used in he model. The model is bsed on he heoy of Mximum Likelihood. In his model only he mximum ejeced gp nd he cceped gp of single vehicles e eed pi wise. In his model wo ssumpions e mde: log-noml disibuion fo he ciicl gps nd b he dive behviou is boh homogeneous nd consisen. Such pesumpions e disdvnges. uhemoe he model of Toubeck 992 is vey compliced o use nd is esuls e no vey obus. This model lso equies lge smple size fo esblishing sble esuls. To void such disdvnges in his ppe olly new model fo esiming he ciicl gps is pesened. The heoeicl foundion of his new model is he pobbiliy equilibium beween he ejeced nd he cceped gps. The equilibium is esblished mcoscopiclly fom he cumulive disibuion of he ejeced nd cceped gps. I uns ou h he model fom he mcoscopic equilibium is moe ppopie fo esiming ciicl gps. The new model yields simil esuls s h fom Toubeck's model. Moe imponly he new model yields diecly he empiicl disibuion of he ciicl gps. The new model does no equie ny pedefined ssumpions nd i is esy o use. 2 MODEL DESCRITION AN ALICATIONS 2.. The mehod of Rff nd Toubeck Le nd be he pobbiliy disibuion funcions Ds of ejeced nd cceped gps especively. Then nd cn be obined empiiclly by in siu mesuemens. Thus he obseved pobbiliy h gp of lengh is ejeced is nd h i is no ejeced is -. Moe hn foy yes go Rff 95 inoduced mcoscopic model fo esiming he ciicl gp. He defined he ciicl gp s he vlue of whee he funcions - nd inecep. Th is he vlue which is defined s he esimed ciicl gp c. Rff's mehod ws used in mny counies in elie imes. Becuse of is simpliciy i is sill being used ody in some esech pojecs. Toubeck 992 gve pocedue fo esiming ciicl gps bsed on he Mximum Likelihood echniques. This model is micoscopic model. In his model only he mximum ejeced gps e ken ino ccoun. Thus fo n cceped gp d hee is only coesponding ejeced gp d unde consideion. The likelihood h dive's ciicl gp is beween d nd d is given by d - d. The likelihood L * wih smple of n obseved dives is L * n [ d d ] i 2 If he D of he ciicl gps c is given he pmees of he D cn be obined by mximizing he likelihood L *. In he pcice he log-noml disibuion is ofen used s he D of he ciicl gps. uhemoe s model ssumpions he dive behviou hs o be boh homogeneous nd consisen. Nomlly he mximizion of he likelihood L * cn only be done using numeicl nd ieion echniques cf. Toubeck 992.

3 2.2. The new model bsed on he mcoscopic pobbiliy equilibium The mcoscopic pobbiliy equilibium of he cceped nd ejeced gps cn be esblished s follows. Accoding o he Ds of he cceped nd ejeced gps he obseved pobbiliy h gp of lengh is cceped is - nd h i is "no-cceped" is. And he obseved pobbiliy h gp of lengh is ejeced is nd h i is "no-ejeced" is -. In genel we hve - nd - becuse n cceped gp in he mjo sem my no hve he exc lengh of he cul ciicl gp. In fc n cceped gp is lwys gee hn he cul ciicl gp. Denoe he D of he ciicl gps o be esimed by c hen he pobbiliy c h gp of lengh in he mjo sem would be ejeced is c nd he pobbiliy c h i would be cceped is - c. Consideing he obseved pobbiliy of boh ccepnce nd ejecion we hve he pobbiliy equilibium + + c c c c c c 3 The equion 3 cn be ewien in he following mix fom: c c c c 4 Th is excly he descipion of he equilibium se of he pobbiliies c nd c s Mkov Chin. In his fomulion c c is he se veco nd he nsiion mix. The boundy condiion + c c holds. Wih c c nd c - c equion 4 yields c c c c 5 Solving equion 5 yields he D c of he ciicl gps: c The D c lwys lies beween nd see igue.

4 [-] c [s] igue Schemic elionship beween he D's fo he ejeced gps he cceped gps nd he esimed ciicl gps fom he new model I should be noed h his disibuion is only explicily defined beween fom he poin of view of ll vehicles he minimum cceped gp min nd he mximum ejeced gp mx. o c < min one hs c nd fo c > mx one hs c. Accoding o Rff's definiion fo he ciicl gp eq. we hve c This mens h he ciicl gp esimed fom Rff's mehod is he medin vlue bu no he men vlue of he ciicl gp. The new model hs solid heoeicl foundion in ems of he Mkov Chin nd equilibium of pobbiliies nd obus esuls. I is lso independen of ny model ssumpions. I equies neihe pedefined disibuion funcion of he ciicl gps no he consisency no he homogeneiy of dives. This model cn ke ino ccoun ll elevn gps no only he mximum ejeced gps s is he cse of he Toubeck model 992 nd yields he empiicl D of he ciicl gps diecly. The clculion pocedue of he model is simple nd wihou ieion. In picul he popey of he model h ll he ejeced no only he mximum ejeced gps cn be ken ino ccoun mkes he mjo diffeence beween he pesen new model nd he mos used model of Toubeck 992. If only he mximum ejeced gps e used fo esiming he ciicl gps he new model gives simil esuls deviions smlle hn.2s fo he men ciicl gps s h fom Toubeck 992. If ll ejeced gps e used he esimed men ciicl gps mus be shoe.

5 o implemening he poposed mcoscopic model useful clculion pocedue is ecommended. This pocedue cn be esily implemened ino Spedshee fo exmple EXCEL o Quoo. The pocedue is descibed s follows:. inse ll mesued nd elevn ccoding o whehe ll o only he mximum ejeced gps e ken ino ccoun gps in he mjo sem ino he column of he spedshee 2. mk he cceped gps wih "" nd he ejeced gps wih "" in column 2 of he spedshee especively 3. so ll gps ogehe wih hei mks "" nd "" in n scending ode 4. clcule he ccumule fequencies of he ejeced gps n j in column 3 of he spedshee h is: fo given ow j if mk"" hen n j n j + else n j n j wih n 5. clcule he ccumule fequencies of he cceped gps n j in column 4 of he spedshee h is: fo given ow j if mk"" hen n j n j + else n j n j wih n 6. clcule he D of he ejeced gps j in column 5 of he spedshee h is: fo given w j j n j /n mx wih n mx numbe of ll ejeced gps 7. clcule he D of he cceped gps j in column 6 of he spedshee h is: fo given w j j n j /n mx wih n mx numbe of ll cceped gps 8. clcule ccoding o equion 6 he D of he esimed ciicl gps c j in column 7 of he spedshee h is: fo given w j c j j /[ j +- j ] 9. clcule he fequencies of he esimed ciicl gps p c j beween he w j nd j- in column 8 of he spedshee h is: p c j c j - c j-. clcule he clss men dj beween he w j nd j- in column 9 of he spedshee h is: dj j + j- /2. clcule he vege vlue nd he vince of he esimed ciicl gps h is: cvege sum[p c j * dj ] nd σ 2 sum[p c j * dj 2 ]-sum[p c j * dj ] 2 This clculion pocedue ensues monoonous scending D fo he ciicl gps. In igue 2 n exmple of he pocedue fo esiming he ciicl gp wih spedshee is illused. The new pocedue sill hs limiion: in he mesued d he minimum cceped gp min hs o be smlle hn he mximum ejeced gp mx. Ohewise he pocedue yields no defined esul becuse in his cse he denomino of he equion 6 in he nge mx << min is no defined. This cse cn occu if he smple size is vey smll.

6 cceped o ejeced if 2"" nn+ if 2"" nn+ 3/nmx 4/nmx 6/[6+-5] 7_j-7_j- [_j-_j-]/2 index j gp n n c fc cl.m summe cmen 638 sigm igue 2 - Exmple of spedshee fo esiming he ciicl gp In igue 3 nd igue 4 he esuls of wo exmples e pesened d: Weine 2. In hese clculions only he mximum ejeced gps e used fo eson of compbiliy o he model of Toubeck 992. I cn be ecognised h he men vlues of he ciicl gps e simil fo boh models. Also he D esimed fom he new model e compble o he pedefined D lognoml fom Toubeck's model. This indices h he pedefined log-noml disibuion in Toubeck's model is suible fo descibing he disibuion of ciicl gps. In igue 5 nd igue 6 esuls fo he sme exmples bu using ll ejeced gps e pesened. I cn be seen h he men vlues of he ciicl gps e shoe comped o he esuls in igue 3 nd igue 4. The vege diffeence is bou 5%. To demonse his effec clely he esuled D fo boh cses e illused ogehe in igue 7 nd igue 8.

7 [-] cmcoc64 2 cmlc [s/] igue 3 - Exmple fo ciicl gp esimion. D of he mximum ejeced gps D of he cceped gps c MLD of he esimed ciicl gps fom Mximum Likelihood model of Toubeck c mcod of he esimed ciicl gps fom he new model fo mcoscopic equilibium D: Weine 2 Bd Nuheim 3 mino igh-un. [-] mcoc54 MLc [s/] igue 4 - Exmple fo ciicl gp esimion. D of he mximum ejeced gps D of he cceped gps c MLD of he esimed ciicl gps fom Mximum Likelihood model of Toubeck c mcod of he esimed ciicl gps fom he new model fo mcoscopic equilibium D: Weine 2 Köln mjo lef-un.

8 [-] _ll cmco_llc [s/] igue 5 - Exmple fo ciicl gp esimion. ll D of ll gps D of he cceped gps c mcod of he esimed ciicl gps fom he new model fo mcoscopic equilibium D: Weine 2 Bd Nuheim 3 mino igh-un. [-] _ll mco_llc [s/] igue 6 - Exmple fo ciicl gp esimion. D of ll ejeced gps D of he cceped gps c mcod of he esimed ciicl gps fom he new model fo mcoscopic equilibium D: Weine 2 Köln mjo lef-un.

9 [-] cmco_llc55 2 cmcoc [s/] igue 7 Compison of he esimed disibuions of ciicl gps. c mcod of he esimed ciicl gps fom he new model wih only he mximum ejeced gps c mco_lld of he esimed ciicl gps fom he new model wih ll ejeced gps D: Weine 2 Bd Nuheim 3 mino igh-un. [-] mco_llc5 mcoc [s/] igue 8 - Compison of he esimed disibuions of ciicl gps. c mcod of he esimed ciicl gps fom he new model wih only he mximum ejeced gps c mco_lld of he esimed ciicl gps fom he new model wih ll ejeced gps D: Weine 2 Köln mjo lef-un.

10 3 SUMMARY AND CONCLUSIONS Using he equilibium of pobbiliies fo ejeced nd cceped gps new model fo esiming he ciicl gp nd is disibuion cn be esblished. The new model does no equie ny pioi ssumpions nd he esuls e ccue. The poposed mcoscopic model equion 6 gives genelised pocedue fo esiming ciicl gps. Wih his pocedue he D of he ciicl gps cn be esimed empiiclly. The pocedue fo implemening he new model is simple nd obus. I cn be cied ou using spedshee pogms e.g. EXCEL Quoo ec. wihou ieion. Thus wih he new model useful nd pomising ool cn be se up fo pofessionls of ffic engineeing. o pcicl pplicions n implemened EXCEL-spedshee cn be obined fom he uho. 4 REERENCES BRILON W.; KÖNIG R.; TROUTBECK R Useful Esimion ocedues fo Ciicl Gps. In M. Kye ed.: oceeding of he Thid Inenionl Symposium on Inesecions Wihou Tffic Signls olnd Oegen USA. GSV 99. Mekbl zu Beechnung de Leisungsfähigkei von Knoenpunken ohne Lichsignlnlgen Guide fo clculion of cpciy unsignlized inesecions. oschungsgesellschf fü Sßen- und Vekehswesen Hsg. N. 27 Köln. GSV 2. Hndbuch fü die Bemessung von Sßenvekehsnlngen Gemn Highwy cpciy Mnul. oschungsgesellschf fü Sßen- und Vekehswesen Hsg. GSV 299 Köln. GROßMANN M. 99. Mehoden zu Beechnung und Beueilung von Leisungsfähigkei und Vekehsquliä n Knoenpunken ohne Lichsignlnlgen Mehod fo Clculion nd ssessmen of cpciy unsignlized inesecions. Schifeneihe Lehsuhl fü Vekehswesen Hef 9 Ruh-Univesiä Bochum. HARDERS J Die Leisungsfähigkei nich signlgeegele sädische Vekehsknoen Cpciy of unsignlized ubn inesecions. Sßenbu und Sßenvekehsechnik Hef 76. Hsg.: Bundesminise fü Vekeh Ab. Sßenbu Bonn. KYTE M.; TIAN Z.; MIR Z.; HAMEEDMANSOOR Z.; KITTELSON W.; VANDEHEY M.; ROBINSON B.; BRILON W.; BONDZIO L.; WU N.; TROUTBECK R Cpciy nd Level of Sevice Unsignlized Inesecions.. inl Repo: Volume Two Wy Sop-Conoled Inesecions. Nionl Coopeive Highwy Resech ogm RA M. S.; HART J. W. 95. A Volume Wn o Ubn Sop Sign. Tffic Engineeing n d Conol 5/983 pp SIEGLOCH W Die Leisungsemilung n Knoenpunken ohne Lichsignlseueung Cpciy esimion unsignlized inesecions. Sßenbu und Sßenvekehsechnik Hef 54. Hsg.: Bundesminise fü Vekeh Ab. Sßenbu Bonn 973. TRB 2. Highwy Cpciy Mnul HCM. Specil Repo 29. TRB Nionl Resech Council Wshingon D.C. TROUTBECK R Esiming he Ciicl Accepnce Gp fom Tffic Movemens. Resech Repo Qeenslnd Univesiy of Technology Bisbne. WEINERT A. 2. Genz- und olgezeilücken n Knoenpunken ohne Lichsignlnlgen. Schifeneihe Lehsuhl fü Vekehswesen Hef 23 Ruh-Univesiä Bochum.