Delay Variability at Signalized Intersections

Transcription

1 Trnsporttion Reserh Reord Pper No Dely Vribility t Signlized Intersetions Liping Fu nd Brue Helling Delys tht individul vehiles my experiene t signlized intersetion re usully subjet to lrge vrition beuse of the rndomness of trffi rrivls nd interruption used by trffi signl ontrols. Although suh vrition my hve importnt implitions for the plnning, design, nd nlysis of signl ontrols, urrently no nlytil model is vilble to quntify it. The development of n nlytil model for prediting the vrine of overll dely is desribed. The model is onstruted on the bsis of the dely evolution ptterns under two extreme trffi onditions: highly understurted nd highly oversturted onditions. A disrete yle-by-yle simultion model is used to generte dt for librting nd vlidting the proposed model. The prtil implitions of the model re demonstrted through its use in determining optiml yle times with respet to dely vribility nd in ssessing level of servie ording to the perentiles of overll dely. The bility to urtely quntify vehile delys t signlized intersetions is ritil omponent for the plnning, design, nd nlysis of signl ontrols. As result of rndom flututions in trffi flow nd interruptions used by trffi ontrols, delys tht individul vehiles experiene t signlized pproh re often subjet to highly stohsti nd time-dependent vrition. It hs been inresingly reognized tht the estimte of the vribility of delys is lso of importne for mny pplitions (1 3). For exmple, knowledge of the vribility of delys mkes it possible to estimte the onfidene limits bout the men delys nd thus provide more informtive omprison of lterntive signl plns in identifying optiml signl settings. By onsidering the vribility of dely, more relible signl ontrol strtegies my be generted, potentilly leding to improved levels of servie t signlized intersetions. The problems of estimting delys t signlized intersetions hve been extensively studied in the literture; however, the mjority of the work hs foused on developing models for estimting men dely point estimte of stohsti delys. Detiled disussions of these verge dely predition models hve been provided by Allsop (4), Newell (5), nd Hurdle (6). However, muh less work hs been done to quntify the vribility of dely t signlized pproh. Teply nd Evns (7) nlyzed the dely distribution t signlized pproh for the evlution of signl progression qulity. They observed tht most of the dely distributions re bimodl, nd point estimtor is not dequte to desribe these distributions. By onsidering the yli overflow dely over time s Mrkov hin, Kimber nd Hollis (8), Cronje (9), nd reently Olszewski (1, 10) developed numeril methods to lulte the verge dely nd time-dependent distribution of verge yli dely. This type of model, lthough pble of ompletely speifying the dely distribution, requires substntil omputtionl resoures for lulting nd storing stte nd trnsition probbilities nd therefore is not well suited for use in prtil situtions. The objetive of this pper is to develop n nlytil Deprtment of Civil Engineering, University of Wterloo, Wterloo, Ontrio NL 3G1, Cnd. model for the estimtion of the vribility of delys t signlized intersetions with speifi fous on prediting the vrine of delys of vehiles trversing signlized pproh during given time intervl. An pproximte model for prediting the vrine of delys is presented. The methodologies pplied to develop the pproximte model re outlined, followed by presenttion of the development of the pproximte model. Then the disrete yle-by-yle simultion model tht ws developed for librting nd vlidting the proposed model is desribed. This simultion model is then used to generte dt for librting nd vlidting the proposed model under vriety of signl operting onditions. Applitions of the developed model re demonstrted lst through its use in sensitivity nlysis nd in determining relibility-oriented optiml yle times nd levels of servie. Finlly, onlusions nd reommendtions re presented. ASSUMPTIONS AND NOTATION The dely tht prtiulr vehile experienes when it trvels through the pproh to signlized intersetion depends on number of ftors, inluding the probbilisti distribution of rrivl flow, signl timings, nd the time when the vehile rrives t the pproh. In rel pplition environment, mny of these ftors re rndom vribles, whih mkes urte estimtion of this dely very omplited proess. As n initil reserh effort, the following idelized rod trffi nd signl ontrol onditions re onsidered in this pper: 1. The intersetion pproh onsists of single through lne ontrolled by fixed-time signl. The pproh hs unlimited spe for queuing nd hs onstnt sturtion flow rte.. The vehile rrivl t the pproh is rndom vrible with known probbilisti distribution. The rte of vehile rrivls during the evlution time is ssumed to be onstnt. No initil queue is present t the beginning of the evlution time. The flow rte inreses bruptly from zero to the rte for the evlution time. The trffi strem onsists only of pssenger-r units (pu). Consider the umultive rrivl nd deprture of vehiles during the time intervl [0, T] t the stopline of signlized pproh s illustrted in Figure 1. The dely for prtiulr vehile rriving t time t, lled overll dely nd noted s D, is onsidered to inlude two omponents: uniform dely nd overflow dely, s follows: D = D1 + D ( 1) where the uniform dely omponent, D 1, is defined s tht portion of dely inurred by vehile when the pproh is understurted nd ll vehiles rrive uniformly. The overflow dely omponent, D, represents tht portion of dely used by temporry overflow queues resulting from the rndom nture of rrivls nd by ontinuous

2 16 Pper No Trnsporttion Reserh Reord 1710 rrive uniformly t onstnt rrivl rte less thn or equl to pity; Vr[D ] is the vrine of rndom dely or the differene between the vrine of overll dely nd the vrine of uniform dely. The vrine of uniform dely n be derived theoretilly nd the vrine of overflow dely n be diretly librted from simultion dt. A detiled desription of the development of these models is provided next. APPROXIMATE MODEL FOR VARIANCE OF OVERALL DELAY FIGURE 1 Queuing digrm illustrting omponents of dely. overflow when the rrivl rte during the time period [0, T] exeeds the pity. The estimtion of the overflow dely omponent in Eqution 1 is omplited s result of the omplex time-dependent stohsti nture of the queuing proess, nd urrently no theory is vilble for the development of single nlytil model suitble ross ll sturtion levels. Pst reserh hs minly foused on developing pproximte models for estimting the verge overll dely using simultion dt s mehnism to obtin dt for librtion (8, 11 15). A number of similr dely models re vilble to provide estimtes of this mesure. For exmple, the Cndin dely model uses Eqution to estimte the verge overll dely (13, 16), in whih the units of some prmeters hve been hnged for use in this pper: ED k y( 1 λ) x [ ] = f T x x ( x ) + ( ) + ( ) T ( ) λ 1 where E[D] = verge overll dely (s); T = evlution time (s); y = yle time (s); λ=g e / y ; g e = effetive green intervl durtion (s); k f = djustment ftor for effet of qulity of progression, defined s k f = (1 p) f p /(1 λ) (p is the proportion of vehiles rriving during the green intervl, nd f p is supplementl djustment ftor for pltoon rrivl type; note tht this study does not onsider the effet of signl progression; i.e., k f = 1.0); = pity (pu/s), determined by sλ, where s is sturtion flow rte (pu/s); x = degree of sturtion, defined s q/ ; q = verge rrivl flow rte from time 0 to time T (pu/s); nd x 1 = minimum of (1.0, x). The development of model for the vrine of overll dely, Vr[D], is onsidered, whih is defined s the summtion of the vrine of uniform dely nd the vrine of rndom dely: Vr[ D] = Vr[ D] + Vr[ D ] 1 3 where Vr[D 1 ] is the vrine of uniform dely, defined s the vrine of dely tht would be experiened by vehiles when ll vehiles ( ) The vrine of uniform dely, Vr[D 1 ], represents the vrition of uniform dely tht would be experiened by vehiles rriving during time intervl [0, T]. This vrition results from the unertinty of the vehile s rrivl time during eh yle of the intervl. The vehile n rrive t ny moment within yle nd thus experiene vrible delys s result of the signl ontrol. An estimte of this vrine omponent n be obtined theoretilly on the bsis of deterministi queuing model with vehiles rriving uniformly during the yle (): Vr[ D ] = 1 In order to estblish model for the vrine of dely used by n overflow queue, two extreme trffi onditions re first investigted: understurted onditions (x < 1.0) nd oversturted onditions (x > 1.0). For understurted onditions, overflow dely experiened by vehile rriving during the time intervl [0, T] is minly used by osionl overflows of trffi from eh yle. The reltionship between the vrine of this dely nd the degree of sturtion n be pproximted from the well-known Pollzek-Khinthine formul for n M/G/1 system (for the generl formul nd derivtion) (17) by supposing tht the signl is ting s server with onstnt servie time 1/, s follows: x ( 4 x) Vr[ D ] 1 ( 1 x) ( 1 λ) ( 1 + 3λ 4λx1) 1( 1 λx ) 3 y 1 ( 4) ( 5) It should be emphsized tht the foregoing model is merely n pproximte estimte of the vrine beuse stedy stte my not be rehble during time intervl [0, T]. Nevertheless, the eqution n be used to illustrte the qulittive reltionship between the vrine of dely nd the degree of sturtion. With this ssumption, the vrine is time-independent nd n infinite vrine would be predited s the degree of sturtion (x) pprohes unity. In relity, t high degrees of sturtion, the system is not likely to settle into stedy stte by time T. Consequently, it n be expeted tht Eqution 5 provides resonble pproximtion of the vrine only under light trffi onditions (x << 1.0). If the intersetion pproh is highly oversturted during time period [0, T], there is high probbility tht n overflow queue lwys exists during the period from time 0 to time T. Consider vehile rriving t time t during time period [0, T]. The overflow queue for vehile rriving t time t, Q t, n be determined s the totl rrivls minus the totl deprtures: Qt = Nt t ( 6) The number of rrivls, N t, is rndom vrible with men equl to qt. The dely experiened by the vehile n then be simply determined on the bsis of the overflow queue:

3 Fu nd Helling Pper No D Nt t = 7 On the bsis of Eqution 7, the vrine of dely for vehiles rriving during time intervl [0, T] n be obtined by ssuming tht the rrivl time t is rndom vrible with known distribution: ( ) Vr[ D ] = E[ Vr[ Nt tt] ] + Vr[ E[ Nt tt] ] = = E[ Vr[ Nt t] 0] + Vr[ E[ Nt t] t] E[ Vr[ Nt t] ] + Vr[ qt t] = E[ Vr[ Nt t] ] + ( q ) Vr[] t () 8 FIGURE Models for vrine of overflow dely. If the rtio of the vrine to the men of the vehile rrivls, denoted s I, is ssumed to be onstnt during the time intervl [0, T] nd given, then Vr[ Nt t ] = IENt [ t ] = IqT t ( 9) Note tht if the vehile rrivls follow Poisson distribution, I is equl to 1. In this study, Poisson rrivl is ssumed, but the prmeter I is still used for the onveniene of future extension. With Eqution 9, Eqution 8 n be further expressed s Vr[ D ] = If it is ssumed tht the rrivl time is uniformly distributed during the time intervl [0, T ], Eqution 10 n be further expressed s ITx T ( 1 x) Vr[ D ] = + 1 It must be emphsized tht Eqution 11 is vlid only when n overflow queue is present during the period from time 0 to time t. In relity, however, it is possible tht no overflow queue exists t time t, nd onsequently no overflow dely is experiened. Therefore, it n be onluded tht Eqution 11 represents n upper-bound estimte of the vrine of overflow dely. The tul vrine would be lower thn tht predited by Eqution 11, but the predition error should beome smller s the degree of sturtion inreses nd the ssoited likelihood of overflow queuing inreses. Figure shows the reltionships between the vrines of overflow dely s funtions of the degree of sturtion represented by Equtions 5 nd 11. Both urves re only pproprite within ertin flow domins: either highly understurted or highly oversturted trffi onditions. Consequently, it is hypothesized tht the true reltionship between the vrine nd the degree of sturtion follows the dshed urve in Figure. It n be observed tht it is diffiult, if not impossible, to derive the funtionl reltionship for the trnsitionl urve diretly from Equtions 5 nd 11 through the trditionl oordinte trnsformtion tehnique. Therefore, the nonliner funtion, expressed in Eqution 1, is proposed to model the vrine: ITx T ( 1 x1) Vr[ D ( t) ] = + 1 where x 1 = mx{1, x}. IqEt []+ ( q ) Vr[] t ( 10) e x0 β x ( 11) ( 1) The prmeters x 0 nd β determine the shpe of the dely urve, nd their vlues need to be librted. It n be observed tht the proposed funtion hs two desired ttributes. First, the funtion is symptoti to the model for oversturted onditions (Eqution 11). Seond, similr to the understurted model (Eqution 5), the funtion goes to zero s x pprohes zero. However, lthough these hrteristis re neessry, they do not of themselves demonstrte tht the proposed funtion is relisti. Therefore, dt from simultion model were used to librte pproprite vlues for x 0 nd β nd to vlidte the librted model, s disussed in the next setion. Expressions for the vrines of uniform dely nd overflow dely hving been developed, the vrine ssoited with the overll dely (Eqution 7) n be expressed s Vr[ Dt ( )] = SIMULATION MODEL In order to obtin dt to librte nd vlidte the proposed models, disrete yle-by-yle simultion system ws developed. Logi ( 1 λ) ( 1 + 3λ 4λx1) 1( 1 λx ) 3 y ITx T ( 1 x1) e x β 0 x The simultion model expliitly models the dely tht vehile experienes when it trverses signlized intersetion pproh. The pproh is used exlusively for through trffi nd is ontrolled by pretimed trffi signl. The vehile rrivls re rndomly distributed with the vehile hedwy following negtive exponentil distribution with minimum hedwy equl to 1 s. The vehile dishrge pttern during the green intervl depends on the queue sttus t the pproh. If no queue is present when vehile rrives, it n immeditely be dishrged without dely. Otherwise, the vehile must wit until the dishrge of the queued vehiles hed of it. Vehile dishrge hedwy is determined on the bsis of sturtion flow rte. The simultion strts with no queue present nd resets the queue size to zero whenever the elpsed lok time rehes prespeified evlution time. The simultion termintes one the required totl ( 13)

4 18 Pper No Trnsporttion Reserh Reord 1710 number of yles hs been simulted. The rrivl time nd dely ssoited with eh vehile re reorded for use in the nlysis stge. Informtion suh s the men nd vrine of delys experiened by vehiles rriving during the evlution time n then be derived. Verifition Before the simultion model ws used to generte dt for librting nd testing the proposed models, it ws verified ginst results from other vilble models. Two omprisons were mde. First, the verge overll delys obtined from the simultion model for given evlution period under different sturtion rtios were ompred with the results from the Austrlin (1), Cndin (13), Highwy Cpity Mnul (HCM) (18), nd Mrkov hin models (1). For onveniene, the senrio used in this omprison is the sme s tht used by Olszewski (1) for similr purpose. The evlution period durtion ws 15 min. The signl timing onsisted of yle time of 60 s, n effetive green intervl of 4 s, nd sturtion flow of 1,800 pu/h. A totl of 6,000 yles, orresponding to 100 h of trffi flow, were simulted for eh degree of sturtion. It ws estimted tht this number of simultions would result in n estimtion error of less thn 0.5 s t signifine level of 95 perent. Figure 3 illustrtes the verge overll dely obtined from the simultion model nd the four other methods. It should be noted tht the overll delys ssoited with the HCM model were obtined by multiplying the stopped delys from the HCM formul by 1.3 to onvert stopped dely to overll dely. The Mrkov hin model ssumes Poisson rrivls nd onstnt deprture during the green intervl. As would be expeted, the simultion results re lmost identil to those of the Mrkov hin model. Among the three other models, the Austrlin model shows the best greement with the simultion model under ll levels of sturtion, nd the Cndin model provides the best greement with the simultion for oversturted onditions. It should be noted tht the differenes mong the HCM, Cndin, nd Austrlin dely equtions were expeted nd hve been ddressed by Akelik (19). The objetive of the seond omprison ws to provide n indition of the vlidity of the simultion model in estimting the vrine of delys. The simultion results were ompred with those reported by Olszewski (1) in whih the ext mens nd vrines of delys under vrious levels of sturtion were obtined for given se from Mrkov hin model. The system prmeters re the sme s those for the previous omprison exept tht the evlution time ws 30 min insted of 15 min. In this omprison, the number of yles to be simulted ws estimted on the bsis of n nlysis of the onfidene intervl for the vrine. It ws estimted tht totl of 6,000 yles for eh degree of sturtion would yield n estimtion error for the stndrd devition of less thn s t signifine level of 95 perent. Figure 4 shows the stndrd devitions of dely estimted by the simultion model nd those provided by Olszewski (1) from the Mrkov hin model. It n be observed tht the estimtes of the stndrd devition of dely from the simultion model re quite onsistent with those obtined from the Mrkov hin model. The overestimtion of the stndrd devition of dely by the simultion model, espeilly in the rnge x < 1.0, ws expeted beuse the Mrkov hin model does not onsider the vrition of trvel time within the yle s quntified by Eqution 4. MODEL CALIBRATION AND VERIFICATION Clibrtion To determine the pproprite prmeter vlues for the overflow dely vrine model shown in Eqution 3, two-step sequentil librtion proedure ws performed. The first step is to find the x 0 - nd β-vlues tht would produe the best fit between the estimtes of the vrine of overflow dely from Eqution 1 nd the estimtes from the simultion model (representing the true vlues) for given yle time (), effetive green intervl (g e ), nd evlution time (T ). Following the definition of Eqution 3, the vrine of rndom dely from the simultion is obtined s the differene between the vrine of overll dely lulted from the dely of simulted vehiles nd the vrine of uniform dely from Eqution 4. Beuse of the nonliner reltionship between the vribles, nonliner regression proess ws onduted by first trnsforming Eqution 1 into n equivlent liner eqution: Y = + bx ( 14) FIGURE 3 Averge overll dely estimted by Austrlin, Cndin, HCM, Mrkov hin, nd simultion models ( y = 60 s, g e = 4 s, s = 1,800 pu/h, nd t = t e = 15 min; simulted yles = 6,000). FIGURE 4 Stndrd devition of dely estimted by Mrkov hin model nd simultion model ( y = 60 s, g e = 4 s, s = 1,800 pu/h, nd t = t e = 30 min; simulted yles = 6,000).

5 Fu nd Helling Pper No where Y Iqx T ( 1 x) = ln ln + δ 1 D ln( Vr[ ] ) X = ln(x), =βln(x 0 ), nd b = β. The simultion model ws used to obtin the vlues of the vrine of overflow dely (Vr[D ]), whih is the differene between the vrine of overll dely nd the vrine of uniform dely lulted from Eqution 4, under vrious ombintions of, x, nd T. These dt were trnsformed to X- nd Y-vlues s in Eqution 14. For set of prespeified x-vlues, liner regressions were performed to determine the vlues of nd b, whih were subsequently trnsformed bk to vlues for x 0 nd β. The dt points used in regression were determined by simultion by fixing the vlues of x, g e, nd T nd vrying the degree of sturtion x from 0.8 to 1. with n inrement of Eh dt point results from simultion of 15,000 yles. The regressed x 0 nd β, together with ( y, g e, T ), form new dt point ( y, g e, t, x 0, β). By hnging the vlues of the prmeter set ( y, g e, T ) nd repeting the regression nlysis, number of suh dt points n be obtined. In this study, totl of 18 points were generted with the following ombintions of prmeters: y ={70, 90, 10}; λ =g e / y ={0., 0.5, 0.8}; T ={900, 3600}. It ws found tht the liner reltionship shown in Eqution 14 is sttistilly signifint for eh of the 18 ombintions with minimum R of 0.95, whih indites tht the proposed funtionl form is pproprite. In the seond step, series of orreltion nlyses of the reltionships between the prmeters (x 0, β) nd ( y, g e, T, λ, T/ ) were onduted nd the following best-fit equtions were obtined: x = T λ ( 15) 0 6 (R = 0.87, t 1 = 13.07, t = 3.84) 4 β= T ( 16) (R = 0.93, t 1 = 13.07) FIGURE 5 Correltion of stndrd devitions of overll dely estimted by nlytil model with simultion results (s = 1,800 pu/h; simulted yles = 15,000 per ombintion). The librted model ws further evluted using results from Olszewski (1) in whih the ext vrines of overflow delys under vrious levels of sturtion were obtined for given se from Mrkov hin model. The vlues of the system prmeters for the se s well s the results re shown in Figure 6. It n be observed tht the estimtes of the stndrd devition of dely from the simultion model re very onsistent with those obtined from the Mrkov hin model. APPLICATIONS OF VARIANCE MODEL Optiml Cyle Time Averge overll dely hs trditionlly been used s one riterion in determining optiml yle times. An exmintion is mde to see if there is n optiml yle time tht minimizes the vribility of dely tht individul vehiles experiene t signl-ontrolled intersetion. An idelized two-phse, four-pproh intersetion with equl flows The obtined high R -vlues indite tht both equtions explin lrge portion of the vrition in the simulted dt. All t-vlues re greter thn the ritil t-vlue t the 5 perent level of signifine, whih indites tht the inluded prmeters re sttistilly signifint. Evlution The simultion system is first used to estimte the vrine of overll dely orresponding to vrious evlution times nd trffi onditions. A totl of 10 ombintions were simulted with the following ombintions of prmeters: y ={50, 60, 80, 100, 10}, λ={0.3, 0.5, 0.7), t ={900, 3600}, nd x ={0.5,0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.}. Figure 5 shows the orreltion between the stndrd devition of the dely obtined by the nlytil model nd the simultion results. Eh point represents the result of simultion runs of 15,000 yles. The pproximte model exhibits no pprent bis nd hs high orreltion with the simulted estimtes (R = 96.1 perent). FIGURE 6 Stndrd devition of overflow dely estimted by Mrkov hin model (1) nd simultion model ( y = 60 s, g e = 4 s, s = 1,800 pu/h, nd T = 30 min).

6 0 Pper No Trnsporttion Reserh Reord 1710 on ll pprohes is onsidered. The sturtion flow rte is 1,800 pu/h nd the lost time t eh phse is 4 s. Figure 7 shows the reltionship between the vrine of overll dely nd the verge overll dely s funtion of yle time. It n be observed tht the verge overll dely nd the vrine of overll dely hve similr trends with respet to yle length. Furthermore, the rnge of optiml yle times with respet to verge overll dely (q = 600 pu/h: optiml y = 40 ~ 60 s, nd q = 800 pu/h: optiml y = 80 ~ 100 s) overlps with those determined on the bsis of minimizing the vrine of overll dely (q = 800 pu/h: optiml y = ~40 s nd q = 800 pu/h: optiml y = 90 ~ 10 s). This finding indites tht for the senrios exmined, the urrent prtie of determining optiml yle lengths on the bsis of minimizing verge overll dely is pproprite with respet to the objetive of minimizing the vrine in overll delys. Vribility of Level of Servie The possible use of dely vribility in quntifying level of servie for signlized intersetions is illustrted in this setion. In the HCM (18), level of servie for signlized intersetions is defined in terms of verge overll stopped dely. With the bility to estimte the vrine of overll dely, it is fesible to integrte the onept of relibility into design nd nlysis of signlized intersetions. For exmple, dely of ertin perentile, insted of verge vlue, n be used to define the level of servie. A 95th-perentile dely mens tht 95 perent of the vehiles would experiene dely less thn or equl to this dely. The perentile vlue n be pproximtely estimted using E[D] + z α (Vr[D]) 1/, where z α is sttisti for the norml distribution nd n be determined on the bsis of the prespeified relibility level. Figure 8 shows verge overll dely nd 90th-perentile dely (with z α 1.3) under different degrees of sturtion. It is ssumed tht rnges of dely vlues used in defining eh level of servie in the HCM re lso pplible to individul vehiles, s shown in Figure 8. It n be observed tht for the given se with degree of sturtion of 0.9, the verge overll dely is 0 s, whih would yield level-of-servie (LOS) C (point ). However, if the 90th-perentile dely is used, the FIGURE 8 Level of servie nd dely vribility. level of servie would be D (point b). On the other hnd, in order to gurntee tht 90 perent of the vehiles going though the intersetion pproh experiene LOS C or higher, the degree of sturtion needs to be redued to 0.8 (point ) by either inresing the pity or deresing the demnd. CONCLUSIONS AND FUTURE RESEARCH The development of n nlytil model for estimting the vrine of dely t signl-ontrolled pprohes is desribed. The model ws onstruted on the bsis of the dely evolution ptterns under two extreme trffi onditions: highly understurted nd highly oversturted. A disrete yle-by-yle simultion model ws developed nd used to generte dt for librting nd vlidting the proposed models. The results of orreltion nlysis indite remrkble greement between the model estimtes of the stndrd devition of dely nd simultion results (R = 96.1 perent). The developed model provides vluble tool for the plnning, design, nd nlysis of signl ontrols. Prtil pplitions hve been demonstrted through its use in determining optiml yle times with respet to dely vribility nd in ssessing level of servie ording to the perentiles of overll dely. The proposed nlytil models were librted nd vlidted with simultion results tht re bsed on severl importnt ssumptions, inluding rndom trffi rrivls with onstnt flow rte nd unlimited queuing spe. These ssumptions my be overly restritive nd re likely to be violted in prtie. The impt of these ssumptions on the vlidity of these models hs not yet been determined. It is reommended tht future reserh fous on the following spets: the potentil impts of the ssumptions pplied in this pper should be quntified, nd field dt should be used in onjuntion with simultion results to librte nd verify the proposed models. FIGURE 7 Reltionship between optiml yle times with respet to men nd stndrd devition of overll dely (two-phse, four-pproh intersetion with equl flows on ll pprohes; sturtion flow = 1,800 pu/h; lost time = 4 s/phse). ACKNOWLEDGMENT This reserh ws supported by the Nturl Sienes nd Engineering Reserh Counil of Cnd.

7 Fu nd Helling Pper No REFERENCES 1. Olszewski, P. Modeling Probbility Distribution of Dely t Signlized Intersetions. Journl of Advned Trnsporttion, Vol. 8, No. 3, 1994, pp Rouphil, M. N., nd N. Dutt. Estimting Trvel Time Distribution for Signlized Links: Model Development nd Potentil IVHS Applitions. Pro., Annul Meeting of ITS Ameri, Vol. 1, Mrh 15 17, Fu, L., nd S. Teply. Improving the Relibility of Prtrnsit Servie Using Better Trvel Time Informtion. Pro., Joint Conferene of the Cndin Institute of Trnsporttion Engineers, Pifi Northwest ITE Qud Setion nd Western Cnd Trffi Assoition, Vnouver, British Columbi, Allsop, R. E. Dely t Fixed Time Trffi Signl I: Theoretil Anlysis. Trnsporttion Siene, Vol. 6, 197, pp Newell, G. F. Applitions of Queueing Theory. Chpmn Hll, London, Hurdle, V. F. Signlized Intersetion Dely Models A Primer for the Uninitited. In Trnsporttion Reserh Reord 971, TRB, Ntionl Reserh Counil, Wshington, D.C., 1984, pp Teply, S., nd G. D. Evns. Evlution of the Qulity of Signl Progression by Dely Distributions. In Trnsporttion Reserh Reord 15, TRB, Ntionl Reserh Counil, Wshington, D.C., 1989, pp Kimber, R. M., nd E. M. Hollis. Trffi Queues nd Delys t Rod Juntions. LR 909. UK Trnsport nd Rod Reserh Lbortory, Crowthorne, Berkshire, Englnd, Cronje, W. B. Derivtion of Equtions for Queue Length, Stops, nd Dely for Fixed-Time Trffi Signls. In Trnsporttion Reserh Reord 905, TRB, Ntionl Reserh Counil, Wshington, D.C., 1983, pp Olszewski, P. Modeling of Queue Probbility Distribution t Trffi Signls. In Trnsporttion nd Trffi Theory (M. Koshi, ed.), Elsevier, New York, Webster, F. V. Trffi Signl Settings. Tehnil Pper 39. UK Trnsport nd Rod Reserh Lbortory, Crowthorne, Berkshire, Englnd, Akelik, R. Trffi Signls: Cpity nd Timing Anlysis. Reserh Report 13. Austrlin Rod nd Reserh Bord, Vermont South, Teply, S., D. I. Allinghm, D. B. Rihrdson, nd B. W. Stephenson. Cndin Cpity Guide for Signlized Intersetions, nd ed. (S. Teply, ed.), Institute of Trnsporttion Engineering, Distrit 7, Cnd, Rouphil, N. M. nd R. Akelik. Oversturtion Dely Estimtes with Considertion of Peking. In Trnsporttion Reserh Reord 1365, TRB, Ntionl Reserh Counil, Wshington, D.C., 199, pp Brilon, W., nd N. Wu. Delys t Fixed Time Trffi Signls Under Time-Dependent Trffi Condition. Trffi Engineering nd Control, Vol. 31, No. 1, 1990, pp Teply, S., D. I. Allinghm, D. B. Rihrdson, nd B. W. Stephenson. Cndin Cpity Guide for Signlized Intersetions, 1st ed. (S. Teply, ed.), Institute of Trnsporttion Engineering, Distrit 7, Cnd, Medhi, J. Stohsti Models in Queueing Theory. Hrourt Bre Jovnovih, Sn Diego, Clif., Speil Report 09: Highwy Cpity Mnul, 3rd ed. TRB, Ntionl Reserh Counil, Wshington, D.C., Akelik, R. The Highwy Cpity Mnul Dely Formul for Signlized Intersetions. ITE Journl, Vol. 58, No. 3, 1988, pp Publition of this pper sponsored by Committee on Highwy Cpity nd Qulity of Servie.