USE OF STATISTICAL TECHNIQUES FOR CRITICAL GAPS ESTIMATION

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1 Sessio 1. Sttistic Methods d Thei Applictios Poceedigs of the 12 th Itetiol Cofeece Relibility d Sttistics i Tspottio d Commuictio (RelStt 12, Octobe 2012, Rig, Ltvi, p ISBN Tspot d Telecommuictio Istitute, Lomoosov 1, LV-1019, Rig, Ltvi USE OF STATISTICAL TECHNIQUES FOR CRITICAL GAPS ESTIMATION Ade Gvulová Uivesity of Žili, Fculty of Civil Egieeig Deptmet of Highwy Egieeig Uivezitá 8251/1, SK Žili, Slovk Republic Ph.: E-mil: de.gvulov@fstv.uiz.sk The citicl gp is vey impott pmete fo cpcity clcultio of usiglized itesectios. This pmete is stochsticlly distibuted vlue d it cot be mesued diectly o the field. Oly ejected gps d ccepted gps of ech mio stem vehicle c be mesued t the itesectio. Cosequetly, some sttisticl model o pocedue fo estimtig citicl gps is eeded. Thee e my diffeet methods o pocedues fo citicl gp estimtio d my studies bout estimtig the citicl gp hve bee cied out. But i Slovki we hve oly little pcticl expeieces becuse simil studies hve ely bee doe i ou couty. This ppe pesets shot oveview of thee methods fo the estimtio of citicl gps t usiglized itesectios: method of Rff (1950, Mximum Likelihood Method (MLM of Toutbeck (1992 d method of Wu (2006. These methods e peseted o pcticl exmple. Iput dte obseved ccepted d mximum ejected gps of ech mio stem vehicle wee detemied o the bsis of video suvey t the usiglized itesectio i Žili. Citicl gps e estimted by usig these thee methods d esults e comped ech othe. Keywods: citicl gp estimtio, ejected d ccepted gp, mximum likelihood method, pobbility, usiglized itesectio 1. Itoductio Methods of cpcity clcultio fo usiglized itesectios e mostly bsed o Gp ccepe pocedue (GAP. I this pocedue the citicl gp is impott pmete fo cpcity clcultio. Thee e diffeet vlues of citicl gps fo dives i diffeet geomety d tffic coditios. The poblem is the citicl gps cot be mesued diectly. Oly eject gps d ccepted gps of sigle vehicles i the mio stem c be mesued t the itesectio. Cosequetly, some sttisticl model o pocedue fo estimtig citicl gps t usiglized itesectios is eeded. Thee e my diffeet methods o pocedues fo citicl gp estimtio (moe th 20 methods, such s method of Rff (1950, Hdes (1968, Awoth (1970, Siegloch (1973, Toutbeck (1992 d so o. The most commoly used methods fo estimtig the citicl gp e the method of Rff (1950 d Toutbeck (1992. The method of Rff (1950 is bsed o mcoscopic model d it is the eliest method fo estimtig the citicl gp which is used i my couties becuse of its simplicity. I Toutbeck's micoscopic model the pobbility of the citicl gp is clculted though the Mximum Likelihood Method (MLM.A itetio pocess is ecessy. The ouome of Bilo et l. compehesive lyses [1] is MLM of Toutbeck (1992 gives the best esults. This method is ecommeded fo estimtig the citicl gps i the best-kow stdd muls fo tffic egieeig HCM 2000 d HBS Ufotutely, the model of Toutbeck (1992 is vey complicted to commo use fo tffic egiees. Besides the metioed methods, Wu (2006 developed mcoscopic method bsed o equilibium of pobbilities betwee the ejected d ccepted gps [2]. Usig this method is simple; it c be cied out esy by spedsheet pogms without itetios. This ppe biefly itoduced method of Rff (1950, MLM of Toutbeck (1992 d method of Wu (2006. I Slovki we hve oly little pcticl expeieces bout estimtig the citicl gp becuse simil studies hve ely bee doe i ou couty. This ppe pesets pcticl exmple of citicl gps estimtio by usig these thee methods. The citicl gps wee clculted d comped ech othe. Iput dte (ccepted gps d mximum ejected gps wee detemied o the bsis of video suvey t the usiglized itesectio i Žili. 20

2 The 12 th Itetiol Cofeece RELIABILITY d STATISTICS i TRANSPORTATION d COMMUNICATION Methods fo Estimtig the Citicl Gp The citicl gp t c c be defied s the miimum time itevl betwee the mjo stem vehicles tht is ecessy fo oe mio stem vehicle to mke meuve (see Figue 1.Vlues of citicl gps e diffeet fo diffeet dives (some of them e too fst o isky, some of them e slow o ceful d thee e depedet o types of movemets, geomety pmetes of itesectios, tffic situtio. Due to this vibility gp ccepe pocess is coside s stochstic pocess d the citicl gps e dom vibles. The estimtio of citicl gps ties to fid out vlues fo the vibles d s well s fo the pmetes of thei distibutios, which epeset typicl dive behvio t the ivestigted itesectio. Figue 1. The citicl gp fo mio stem ight-tu vehicle I usiglized itesectio theoy it is geelly ssumed tht dives e both cosistet d homogeeous. Cosistet dives e expected to behve the sme wy evey time i ll simil situtios. This mes: dive with specific t c -vlue will eve ccept gp less th t c d he will ccept ech mjo stem gp lge th t c. Howeve, withi popultio of sevel dives, ech of which behves cosistet, diffeet dives could hve thei ow t c -vlue. These t c -vlues e the teted s dom vible with specil sttisticl desity fuctio f (td cumultive distibutio fuctio F (t. The popultio of dives is homogeeous if ech sub-goup of dives out of the popultio hs the sme fuctios f (td F (t. The poblem is tht the citicl gps cot be mesued diectly. Oly ejected gps d ccepted gps of ech mio stem vehicle c be mesued t the itesectio. The citicl gps c be estimted fom these iput dt usig some sttisticl method o pocedues. Fo the estimtio of citicl gps fom obsevtios thee methods e biefly descibed i this ticle d pplied i pcticl exmple method of Rff (1950, Mximum Likelihood Method (MLM of Toutbeck (1992 d method of Wu ( The Rff s Method The method of Rff (1950 is bsed o mcoscopic model d it is the eliest method fo estimtig the citicl gp which is used i my couties becuse of its simplicity. This method ivolves the empiicl distibutio fuctios of ccepted gps F (t d ejected gps F (t. Whe the sum of cumultive pobbilities of ccepted gps d ejected gps is equl to 1 the gp of legth tis equl to citicl gp t c. It mes the umbe of ejected gps lge th citicl gp is equl to the umbe of ccepted gps smlle th citicl gp. = 1. ( The Mximum Likelihood Method The model of Toutbeck (1992 fo estimtig citicl gps is bsed o the Mximum Likelihood Method (MLM. This micoscopic model ssumes the log-oml distibutio of ccepted d mximum ejected gps. I this model, oly the ccepted gp d the mximum ejected gp of ech vehicle e 21

3 Sessio 1. Sttistic Methods d Thei Applictios teted pi wise. Fo oe idividul mio steet dive i we hve obseved: oe ccepted gp i d oe coespodig the mximum ejected gp i. The mximum ejected gp is the lgest vlue of ll ejected gps fo oe mio steet dive. The MLM is bsed o the ssumptio tht mio stem dives behve cosistetly. It mes tht ech dive will eject evey gp smlle th his citicl gp d will ccept the fist gp lge th the citicl gp. Ude this ssumptio, the distibutio of the citicl gps lies betwee distibutios of lgest ejected d ccepted gps (see Figue 2. The pmetes of distibutio fuctio of the citicl gps, the me µ d vice σ 2, e obtied by mximizig the likelihood fuctio. The likelihood fuctio is defied s the pobbility tht the citicl gp distibutio lies betwee the obseved distibutio of the mximum ejected gps d the ccepted gps: [ F i F( i ] 1 (, (2 whee L mximum likelihood fuctio, i logithm of the ccepted gp of dive i, i logithm of the mximum ejected gp of dive i, F( i, F( i cumultive distibutio fuctios fo the oml distibutio. L = Figue 2. Theoeticl distibutio fuctio of ccepted gps F(t, mximum ejected gps F(t, d citicl gps F(t The logithm of fuctio (3 is s follows: 1 [ F( F( ] l. (3 i i Likelihood pmetes µ d σ 2 e solutios whe the ptil deivtive of equtio (4 is equl to 0. They c be simplified s follows: f ( i f ( i = 0 1 F( i F( i, (4 1 ( i µ. f ( i ( i µ. f ( i = 0 2 2σ 1 F( i F( i whee f( i, f( i pobbility desity fuctios fo the oml distibutio with pmetes µ d σ 2. Pmetes µ d σ 2 c be clculted by umeicl d itetio techiques [5]. Subsequetly the me E(t c d vice D(t c of citicl gp c be deived by equtios: 22

4 The 12 th Itetiol Cofeece RELIABILITY d STATISTICS i TRANSPORTATION d COMMUNICATION µ + 0,5σ E( t c = e. (5 2 2 σ D( = E(.( e 1. ( The Model Bsed o the Mcoscopic Pobbility Equilibium The theoeticl bckgoud of Wu s model [2] is the pobbility equilibium betwee the ejected d the ccepted gps. The equilibium is estblished mcoscopiclly fom the cumultive distibutio of the ejected d ccepted gps. The obseved pobbility tht gp of legth t is ccepted is F (t d tht it is ot-ccepted is 1 F (t. The obseved pobbility tht gp of legth t is ejected is F (t d tht it is ot-ejected is 1 F (t. Geelly F (t 1 F (t d 1 F (t F (t. Cosideig the obseved pobbility of both ccepe d ejectio, Wu s method is bsed o the followig pobbility equilibium: P, (,, t t = P ( t + P (. (7 Deote the distibutio fuctio of the citicl gps to be estimted by F (t, the the pobbility P, (t tht gp of legth t i the mjo stem would be ejected is F (t, d the pobbility tht it would be ccepted is 1 F (t. Substitutig P, (t=f (t d P, (t=1 F (t, fo Wu s model we hve the followig distibutio fuctio F (t of the citicl gps: 1 = = 1. (8 Fo implemetig the poposed mcoscopic model of Wu is detiled clcultio pocedue step by step give i [2]. This pocedue is esily implemeted ito Excel spedsheet. Accodig to Rff s defiitio fo the citicl gp (Equtio (1 it c be witte [2]: 1 = = 1. (9 It mes tht the citicl gp estimted fom Rff s method is the medi vlue but ot the me vlue of the citicl gp. 3. Pcticl Exmple of Estimtig the Citicl Gps 3.1. Results of Mesuemets Mesuemets wee cied out t the usiglized itesectios i Žili. All tffic stems t this itesectio wee video-ecoded. Subsequetly video-ecodigs wee pocessed by ow specific softwe (usig MATLAB.Time positio of ech vehicle o ceti poit (desiged lie coss the od ws ecoded i sped sheet MS Excel with ccucy 0,04 s (fme te of cme: 25 fmes pe sec.. Fo ech mio stem vehicle ws ecoded: ivl time, vehicle type, diectio d deptue time. Fo ech mjo stem vehicle ws ecoded: pssig time desiged lie coss the od, vehicle type d diectio. All dt wee pocessed. The ccepted d lgest ejected gps fo idividul mio stem vehicle wee detemied (fo ech mio stem septely. If dive fom mio stem ccepted the lg d did ot eject y time gp: lgest ejected time gp ws equl 0 s. Exmple of estimtig citicl gps of mio left-tu stem t usiglized itesectio i Žili (see Figue 3 by thee estimtig methods usig sttisticl techiques (Rff, MLM d Wu is give i this ticle. Tffic suvey ws cied out by video ecodig i duce oe hou (pek hou.all obseved iput dt ccepted d the mximum ejected gps fo ech mio stem left-tu vehicle e peseted o Figue 4. Two simples fo estimtig citicl gps hve bee give. I simple 1 thee e ot cosideed vehicles which hd the mximum ejected gp equl to zeo, so 80 couples of ccepted d mximum ejected gps wee deived fo estimtig citicl gp by ll thee methods. I simple 2 thee e lso cosideed vehicles which immeditely eteed to itesectio without witig d fo estimtig citicl gp wee deived 80 ejected gps d 103 elevt ccepted gps. Dt fom simple 2 wee used oly fo two mcoscopic methods of Rff d Wu, becuse of equiemet of MLM to hve couples of ccepted d mximum ejected gps fo estimtio of citicl gps. 23

5 Sessio 1. Sttistic Methods d Thei Applictios Figue 3. Moitoed usiglised itesectio i Žili Figue 4. Mesuemets of the mximum ejected gps d ccepted gps (Dt: Žili 2012, mio stem left-tu 3.2. Estimtio d Compiso of Citicl Gps Results of Rff s method The cuves of ccumultive pobbility of ccepted gps F (t d mximum ejected gps F (t e show o Figue 5. The citicl gp t c = 5,80 s is i coss poit of F (td 1 F (t. The citicl gp ws estimted by Rff s model lso cosideig ll ccepted gps fom iput dt (simple 2. Although distibutio fuctio F (t fo simple 2 is slightly diffeet to F (t fo simple 1, coss poit is i the sme gp time, the citicl gp t c = 5,80 s. Results of MLM of Toutbeck I ccodce with MLM of Toutbeck the pobbility desity fuctios of ccepted gps f( i d mximum ejected gps f( i, distibutio fuctios of ccepted gps F( i d mximum ejected gps F( i wee clculted fo ech logithm of ccepted gp i d logithm of mximum ejected gp i. The likelihood pmetes: the me µ d vice σ 2, wee obtied s solutio of two equtios (4 by itetio pocess. The itetio pocess ws pogmmed i Excel pogm. Vlues µ d σ 2 whee chged step by step util fuctios i equtio (4 ted to zeo. Filly the me E(t c = 6,04 s d vice D(t c = 5,38 s 2 of citicl gp wee deived by equtios (5 d (6. Figue 5. The citicl gp estimtio by Rff s method, distibutio fuctios of the mximum ejected gps F(t d ccepted gps F(t fo simple 1 d simple 2 24

6 The 12 th Itetiol Cofeece RELIABILITY d STATISTICS i TRANSPORTATION d COMMUNICATION Results of mcoscopic method of Wu The distibutio fuctios of ccepted gps F (t d mximum ejected gps F (t e show o Figue 6. The distibutio fuctio of citicl gp F (t ws clculted i ccodce with mcoscopic pobbility equilibium model of Wu by equtio (8 d the cuve of this fuctio lies betwee F (t d F (t. The me vlue of the citicl gp t c = 5,91 s d the vice σ 2 = 2,89 s 2 wee clculted. The medi vlue of the citicl gp ws lso clculted d its vlue is equl to Rff citicl gp (5,8 s. Fo compiso, distibutio fuctio of the estimted citicl gp fom MLM of Toutbeck F (t [MLM] is lso show o Figue 6. The citicl gp ws estimted by mcoscopic model of Wu lso cosideig ll ccepted gps fom iput dt (simple 2. The me vlue of the citicl gp t c = 5,60 s d the vice σ 2 = 3,34 s 2 wee clculted. Figue 6. Distibutio fuctios of the mximum ejected gps F(t, ccepted gps F(t, citicl gps fom the Wu s model F(t [Wu] d distibutio fuctio of the estimted citicl gp fom MLM of Toutbeck F(t [MLM] 4. Coclusios Mximum likelihood method of Toutbeck (1992 belogs to the most impott models fo estimtig the citicl gps d ccodig to [1] gives the best esults. The mcoscopic method of Wu (2006 gives simil esults fo the me citicl gps s tht fom ecogized MLM of Toutbeck (1992. I dditio, we c get ot oly the mi vlue of citicl gps but lso the medi vlue, which is the sme vlue s the esult of Rff s method. Also we c diectly get the pobbility distibutio fuctio of the citicl gps. Advtge of this method is simple clcultio pocedue without itetio s it is i the cse of the Toutbeck model (1992. It c be cied out usig spedsheet pogms, e.g. Excel, d theefoe it is vey useful fo pofessiols of tffic egieeig. I dditio, ot oly couples of ccepted d ejected gps e equied fo estimtio of citicl gps s i MLM of Toutbeck is equied. It c be used ll elevt ccepted d mximum ejected gps. I these cses the me citicl gp is shote. Refeeces 1. Bilo, W., Köig, R., Toutbeck, R. (1997. Useful Estimtio Pocedues fo Citicl Gps. I Poceedigs of the 3d Itetiol Symposium o Itesectios without Tffic Sigls, July 21-23, 1997 (pp Potld Oego, U.S.A: Uivesity of Idho, Ntiol Cete fo Advced Tspottio Techology. 2. Wu, N. (2006. A ew model fo estimtig citicl gp d its distibutio t usiglized itesectios bsed o the equilibium of pobbilities. I Poceedigs of the 5th Itetiol Symposium o Highwy Cpcity d Qulity of Sevices. July 25-29, Yokohm, Jp: Jp Society of Tffic Egiees. 3. Weiet, A. (2000. Estimtio of Citicl Gps d Follow-Up Times t Rul Usiglized Itesectios i Gemy. I Poceedigs of the 4th Itetiol Symposium o Highwy Cpcity. Jue 27 July 1, 2000 (pp Mui, Hwii: Tspot Resech Bod. 25

Sessio 1. Sttistic Methods d Thei Applictios 4. Guo, R. (2010. Estimtig Citicl Gp of Roudbouts by Diffeet Methods.

, Vdehey, M., Robiso, B., Kittelso, W., Kyte, M., Toutbeck, R., Bilo, W., Wu, N. (1999.

Ackowledgemets This cotibutio is esult of the poject s implemettio: Cete of Excellece fo Systems d Sevices of Itelliget Tspot II,

7 Sessio 1. Sttistic Methods d Thei Applictios 4. Guo, R. (2010. Estimtig Citicl Gp of Roudbouts by Diffeet Methods. I Poceedigs of the 6th Advced Foum o Tspottio of Chi Octobe 16, 2010 (pp Beijig, Chi: Cu Assocites, Ic. 5. Ti, Z., Vdehey, M., Robiso, B., Kittelso, W., Kyte, M., Toutbeck, R., Bilo, W., Wu, N. (1999. Implemetig the mximum likelihood methodology to mesue dive s citicl gp. Tspottio Resech, Pt A, 33, Ackowledgemets This cotibutio is esult of the poject s implemettio: Cete of Excellece fo Systems d Sevices of Itelliget Tspot II, ITMS suppoted by the Resech&Developmet Opetiol Pogmme fuded by the ERDF. "Podpoujeme výskumé ktivity Slovesku/Pojekt je spoluficový zozdojov EÚ" ("We suppot esech ctivities i Slovki / The Poject is co-fuded by the EU" 26

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