Ipleenting Atuated Signal-Controlled Intersetion Capaity Analysis with Pedestrians DingXin Cheng, Zong Z. Tian, and Honghao Liu For an atuated signalized intersetion, pedestrian alls are likely to affet the effetive greens serving the vehile oveents, whih affet the apaity and delay of the intersetion. However, the urrent proedure in the Highway Capaity Manual 2000 (HCM) for analyzing atuated signalized intersetions treats pedestrian rossing and tiing statially, with either pedestrian alls at all signal yles or no pedestrians at all. In reality, pedestrian arrivals are rando events with soe yles having ore pedestrians than others and other yles having no pedestrian all at all. This paper deonstrates that the urrent proedure an lead to erroneous results in apaity and delay estiations. A odel is introdued to overoe the shortoings in the urrent proedure. The odel takes into aount the stohasti nature of pedestrian rossings and their effets. The odel oputes the probability of having pedestrian alls in a yle and the orresponding apaities and delays for traffi oveents. An ipleentation fraework was developed to help pratitioners ondut apaity analyses using the odel. The odel s results on a seiatuated signalontrolled intersetion were oparable with the results fro the Si- Traffi irosiulation odel. The effets of pedestrians on intersetion apaity and delay were analyzed using the proposed odel. Depending on the pedestrian volue and traffi onditions, the urrent HCM proedure ould produe signifiant error, espeially when the pedestrian volue is low, beause it does not onsider the stohasti nature of pedestrian arrivals. For atuated signalized intersetions, during a signal yle in whih there is a pedestrian all, the pedestrian phase will be ativated with an indiation of the solid Walk interval followed by a Flashing Don t Walk (FDW) learane interval. Beause pedestrian arrivals are rando events, pedestrian alls ay our only in soe yles. When there is a pedestrian all, the orresponding green split needs to be at least the su of the Walk and the FDW intervals. However, the urrent proedure in the Highway Capaity Manual 2000 (HCM) for analyzing atuated signalized intersetions does not onsider the stohasti nature of pedestrian arrivals and tiing (). The urrent HCM proedure reoends that the phase split satisfy the pedestrian tiing regardless of the pedestrian volues even though there ay be D.-X. Cheng, Departent of Civil Engineering, California State University, Chio, CA 95929. Z. Z. Tian, Departent of Civil and Environental Engineering, University of Nevada, Reno, Mail Stop 258, Reno, NV 89557-052. H. Liu, Departent of Civil and Environental Engineering, M.S. 023, Texas Teh University, Lubbok, TX 79409. Corresponding author: D.-X. Cheng, dxheng@suhio.edu. Transportation Researh Reord: Journal of the Transportation Researh Board, No. 207, Transportation Researh Board of the National Aadeies, Washington, D.C., 2008, pp. 25 30. DOI: 0.34/207-5 no pedestrian all during soe yles. It is not a proble if the required vehile phase split is greater than the pedestrian Walk plus FDW tie. However, if the vehile deand is low and the pedestrian tie is high, the urrent HCM ethod ay not provide aurate apaity and delay estiates. The objetive of this paper is to inorporate the stohasti nature of pedestrian all events into the HCM apaity analysis proedure. Although the stohasti nature of pedestrian rossings has been entioned by researhers and pratitioners, not any analytial odels are available to address the pedestrian issues. In 2000, Tian et al. disussed the ipat of pedestrian rossing on signal oordination (2). They found that aoodating pedestrian tiing in the signal tiing is generally a better strategy fro the signal oordination point of view, beause the signal will reain oordinated even when pedestrian alls our. In 200, Tian et al. addressed the effet of pedestrians on signalized intersetion apaity on a split-phase operation (3). They developed a odel to deterine the onditions for whih the use of an exlusive pedestrian phase ight atually inrease the apaity opared with the typial split-phasing operation. In 2006 Tian and Xu developed a atheatial odel to apture the stohasti effet of pedestrians on the apaity and delay of a pedestrian oveent phase (4). The present study further extends the odel to evaluate the ipat on the intersetion apaity and delay. Furtherore, this paper provides a odeling fraework for researhers and pratitioners to ipleent the odel when analyzing atuated signalized intersetion operations. To illustrate the odel and its ipleentation, the paper inludes the following setions. First, a brief review of urrent HCM proedure to handle pedestrian all events is given. Then, an analytial odel is introdued that onsiders the stohasti nature of pedestrian alls. Next, a fraework to inorporate the proposed odel into the HCM apaity and delay analysis proedure is presented. Finally, the effet of pedestrians on apaity will be disussed and the onlusions reahed fro this study will be presented. HCM PROCEDURE AND CURRENT PRACTICE The urrent proedure in the HCM treats pedestrian alls as fixed events without onsidering their stohasti harateristis. When dealing with pedestrian rossings, the HCM reoends setting the green split to satisfy the pedestrian tiing (split) of the Walk plus FDW intervals if pedestrian phases and push buttons are provided. However, it is up to the analyst to deide whether the pedestrian split should be used to perfor the apaity analysis. A general pratie is that the vehile split is used only when the pedestrian volue is low and the pedestrian split is used when the pedestrian 25
26 Transportation Researh Reord 207 volue is high. However, traffi engineers do not have lear guidane on when to use pedestrian split and when not to. Most traffi analysis software that ipleents the HCM proedure does not provide ore pedestrian appliation beyond what is offered in the HCM. For exaple, Synhro (5) and the Highway Capaity Software fro MTrans (6) will display a warning essage if the input green split is less than the pedestrian tiing. The TRAFFIX software fro Dowling and Assoiates, In. (7), does not provide any warning if the vehile tiing setting violates the pedestrian tiing. Therefore, it is up to analysts to deide whether to onsider pedestrian effets or not. In suary, the urrent analytial proedure for analyzing signalized intersetions an produe either the result without pedestrian alls or the result with pedestrian alls. There are norally no differenes aong the results with different pedestrian volues, exept soe inor adjustents on the saturation flow rates based on the HCM proedure. When pedestrian volue is low and pedestrian rossings do not our in every yle, the urrent proedure overestiates the apaity for the pedestrian phase oveent and underestiates the apaity for the other oveents if the pedestrian phase split option is hosen. However, when the pedestrian split is not onsidered for a partiular phase (i.e., a onurrent phase), the urrent proedure ould underestiate the apaity for the phase and ay overestiate the apaity of other phase oveents. PROPOSED ANALYTICAL MODEL Model Developent This setion presents the developent of an analytial odel for onsidering the stohasti nature of pedestrian rossing. For easy illustration, the data in Figure are used to deonstrate the odel. This diagra will be used later in the paper to disuss the effets of pedestrians on intersetion apaity analysis based on the proposed odel. The geoetri layout and traffi deands for eah oveent are presented in Figure, in whih the north south street is a ajor street and the east west street is a inor street. To siplify the proble, only the east west pedestrian rossings are onsidered, and it is assued that the north south street has enough green tie to over pedestrian rossings on the north south approahes. The pedestrian oveent rossing the south leg is denoted as =, and the pedestrian oveent rossing the north leg is denoted as = 2. The eastbound through oveent is therefore the onurrent phase of =, and the westbound through oveent is the onurrent oveent or phase of = 2. The nubers following the turning oveent arrows are the traffi deands in vehiles per hour (veh/h), whih will be used in the exaples that follow. Beause the east west through oveent phases are the onurrent vehile phases for the two pedestrian oveents, the green splits for both through oveent phases ust satisfy the pedestrian Walk plus FDW intervals if both phases have pedestrian alls. The Walk interval is used to initiate pedestrians to enter the rosswalk, and the FDW is a learane interval that ust be long enough to allow pedestrians to safely ross the intersetion. Typially, in pratie the Walk interval is set between 4 and 7 s based on the FHWA Manual on Unifor Traffi Control Devies (8) and the FDW is deterined based on the rossing distane and an assued pedestrian walking speed, typially 4 ft/s. The HCM also inludes an equation to alulate the required pedestrian tie, whih onsiders the width of the rosswalk in the equation as well. In atuated signal operations, the aount of green tie needed for the east west inor street is governed totally by the pedestrian alls and ontroller paraeter settings, suh as the iniu green, passage tie, and axiu green. When there is no pedestrian all, the vehile phase will run the iniu green and then either gap out or extend to its axiu, depending on the vehile deands. When there is a pedestrian all, the vehile phase will run a iniu of Walk plus FDW and then either gap out or extend to the axiu. Pedestrian alls affet the length of the iniu green of a tiing phase with pedestrian push button. Furtherore, pedestrian alls affet the apaity of oveents orresponding to the previous tiing phase. Assue that pedestrian arrivals are rando and follow a Poisson distribution and x is a rando variable denoting the nuber of pedestrian alls during a yle on one approah. The average nuber of pedestrian alls per yle, λ, an be alulated fro the pedestrian volue of the oveent using Equation. 0 2500 80 20 =2 = 20 80 2500 0 FIGURE Saple intersetion layout and traffi deand.
Cheng, Tian, and Liu 27 λ V p, equals pedestrian volue of oveent (ped/h) and C equals yle length (s). The probability of having x nuber of pedestrian alls, p x,, is obtained by x λe px, = x! The probability of having no pedestrian alls, p 0,, is obtained by 0 λ λe p0, = = e 0! The probability of having at least one pedestrian all, p, is obtained by p = p = e λ 0, ( 4) For the ase of a dual entry ontroller setup for the two onurrent vehile phases on the inor street, any one approah that has a pedestrian all will trigger the pedestrian phase, thus the two through oveent phases will extend to at least Walk plus FDW. In this ase, the probability of having at least one pedestrian all during a yle, p, is alulated by λ λ2 λ+ λ2 p = p0, = ( e e ) = e () 5 The probability of having no pedestrian alls, p 0,, is then p0 p e 2, = = ( λ + λ ) () 6 The probabilities above would not be affeted by the fat that pedestrians ross in groups. Equation 7 gives the proposed for of a apaity odel, whih deterines the apaity of oveent based on the probabilities of two apaities: the apaity without pedestrians and the apaity with pedestrians. = p0,, + p2, = p0,, + p0, 2, ( 7) = apaity for the pedestrian onurrent through oveent (veh/h);, = apaity without pedestrians (veh/h); 2, = apaity with pedestrians (veh/h); p 0, = probability of having no pedestrian alls, or equivalent to the proportion of yles that have no pedestrian alls; and p = probability of having pedestrian alls, or equivalent to the proportion of yles that have pedestrian alls. Substituting Equations 5 and 6 into Equation 7, the following equations are obtained: λ+ λ2 λ λ2 = e + e 2 8, 2, Vp, = C 3, 600 2 = ( ) ( + ),, () g, = C s λ ( p) s ax g,, g = C λ ( ) ( ) () ( 2) () 3 () 9 ( 0) g, = green split without pedestrians, whih an be deterined on the basis of the ethodology developed by Webster (9); g p = green split with pedestrians, g p = walk + FDW or is alulated by the HCM equation; and s = saturation flow rate for oveent (veh/h). Delay an be alulated siilar to the way oveent apaities are alulated, as shown in Equations and 2. d = p0, d, + ( p0, ) d2, ( ) ( λ+ λ2) ( λ+ λ2) d = p0, d, + ( p0, ) d2, = e d, + [ e ]d2, ( 2) d = delay for pedestrian onurrent through oveent (s), d, = delay without pedestrians (s), and d 2, = delay with pedestrians (s). To obtain d, and d 2,, the HCM proedure or any software ipleenting the HCM proedure an be used. Model Validation with Mirosiulation To validate the analytial odel result with the atuated signal operation with pedestrian push button, the proposed odel was opared against the irosiulation results using SiTraffi by Hush and Albek (0). The SiTraffi software was hosen beause of the ease with whih stohasti pedestrian rossings with atuated signal ontrols ould be odeled. Certainly, other advaned irosiulation software an also work for this validation purpose. At first, the siulation odel was alibrated using the HCM proedure based on two ases that the HCM handles: (a) with pedestrian alls and (b) without pedestrian alls. The purpose of the alibration is to ensure that siulation results are onsistent with the two base ases above. During the alibration with the two base senarios using the HCM, adjustents were ade in SiTraffi to the following settings: detetor length, passage tie, and soe driver- and vehilerelated paraeters. After the siulation odel alibration, siulation runs were onduted using different pedestrian volues. Eah siulation was set up to have a 3-in seeding period and a 5-in siulation period, whih is onsistent with the 5-in analysis period used in the HCM. A total of 20 runs were onduted for eah pedestrian volue senario, and the average intersetion delays were seleted for oparison. The reason delay was seleted for aking oparisons is that SiTraffi does not expliitly produe apaity estiates. To opare the different odel results, Figure 2 illustrates the delays fro the SiTraffi siulation odel, the proposed odel, and the HCM results based on different pedestrian volues. Basially, the results based on the urrent HCM proedure are indiated by only two values: delay without pedestrians and delay with pedestrian alls. The HCM results with pedestrians are the sae regardless of the pedestrian volues. Figure 2 indiates that the results athed well between the SiTraffi siulation and the proposed odel. Fro a pratial point of view, the proposed odel and the siulation produed alost the sae delay results for all pedestrian volues. The HCM proedure has siilar results for only very high pedestrian volues and no pedestrian volues. For all other pedestrian volues, the HCM results are different fro the other two odels. The differene in delays
28 Transportation Researh Reord 207 Average Delay, se/veh 50 45 40 35 30 25 20 5 0 5 0 HCM 0 2 5 0 5 20 30 50 Max Pedestrian Volue, Ped/hr SiTraffi Proposed HCM FIGURE 2 Coparison of intersetion delays between SiTraffi and proposed odel. between the urrent HCM proedure and the SiTraffi and proposed odels an be treated as the errors of the urrent HCM proedure. As the figure indiates, the highest error is at the lower end of pedestrian volues. IMPLEMENTING THE ANALYTICAL MODEL To help pratitioners use the proposed odel to ondut apaity analysis, Figure 3 illustrates a proedure to estiate one vehile oveent with pedestrian phase when rando arrivals of pedestrians are onsidered. When dealing with a seiatuated signal operation using the proposed odel, norally only the dual entry through phase on the inor street needs to be onsidered beause typially the green split on a ajor street through oveent an aoodate the pedestrian rossing on a inor street. As indiated in the flowhart, an analyst needs to run the urrent HCM apaity proedure twie: with pedestrian all on rossing the ajor street and without pedestrian all on rossing the ajor street. Then the oveents and intersetion apaity and delay an be estiated using the proposed odel in Equations 7 and. For a fully atuated signal-ontrolled intersetion, the ase beoes ore oplex beause the yle length of the intersetion an hange with the pedestrian alls, and pedestrian alls fro both the ajor and inor streets need to be onsidered. However, the priniple of using the stohasti approah is siilar. It requires alulating the probabilities of no pedestrian alls, pedestrian alls on ajor street approahes only, pedestrian alls on inor street approahes only, and pedestrian alls on both ajor and inor streets. The su of the probabilities above ultiplied by the orresponding apaities will give the estiated intersetion apaity. Siilarly, one an obtain the delay for a fully atuated intersetion with pedestrian push buttons. SAMPLE CALCULATION A saple proble is presented in this setion to illustrate the appliations of the proposed odel. The saple proble is based on the data shown earlier in Figure. To siplify the proedure, all vehiles were Set Pedestrian Volue to Zero Calulate Delay/ Capaity Using HCM Software Copute Probability of No Pedestrian Cyle Start Obtain Intersetion Lane Configuration and Traffi Control Collet Traffi Counts and Pedestrian Counts Calulate Average Nuber of Peds Per Cyle Obtain Probability Weighted Average Delay and Capaity End Set Pedestrian Volue to Average Volue Calulate Delay/ Capaity Using HCM Software Copute Probability of Pedestrian Cyle FIGURE 3 Capaity and delay proedure onsidering stohasti pedestrian alls.
Cheng, Tian, and Liu 29 assued to be passenger ars and the default saturation flow rates of,900 vehiles per hour per lane (vphpl) for through oveents and,805 vphpl for left-turn oveents were used. No other adjustents on the saturation flow rate are ade. This exaple addresses the eastbound through oveent. The westbound through oveent an be analyzed using a siilar approah. In the exaple, the following paraeters are given: The intersetion uses seiatuated oordinated signal ontrol with yle length, C = 90 s; lost tie, L = 4.0 se; Walk = 6.0 s; FDW = 24.0 s; V p, = V p,2 = 20 ped/h; V p, and V p,2 are fro V p, defined previously; = for eastbound through oveent; and = 2 for westbound through oveent. Results without pedestrians: Capaity (EB), vph 900 800 700 600 500 400 300 200 C=60 C=80 C=00 C=20 C=40 Y = 06. X = 074. Y = su of intersetion ritial lane flow ratios, X CI = intersetion ritial volue apaity ratio, y = eastbound through oveent ritial lane flow ratio, g, = eastbound through oveent green split (s),, = eastbound through oveent apaity (veh/h), and d, = eastbound through oveent delay (s/veh). Results with pedestrians: Y = 054. X =. y = 0. g = 30. 0 s p p CI y = 0. g = 77. s CI = 63 veh h d = 55. = 633 veh h d = 2. 5 Based on Equations 5 and 6:, = p, = 0. 368 = p = 0. 632 0 02 2 sveh sveh p 0, and p 0,2 are defined by p 0, desribed previously with = and p and p 2 are defined by p desribed previously with =. Final apaity,, and delay, d, are alulated using Equations 7 and : ( ) = = 0. 368 63 + 0. 368 633 459 veh h ( ) = If alulated using the urrent HCM proedure, the apaity would be 633 veh/h and the delay would be 2.5 s/veh. For this eastbound through oveent, the apaity would be overestiated by 38% and the delay underestiated by 37%. It would be the opposite for the other oveents, in whih apaity would be underestiated and the delay would be overestiated at this intersetion. This exad = 0. 368 55. + 0. 368 2. 5 33. 9 sveh 00 FIGURE 4 0 0 20 30 40 50 60 70 80 Pedestrian Volue, ped/hr Effet of pedestrians on eastbound apaity. ple deonstrates that the urrent HCM proedure ould produe signifiant errors in apaity and delay estiations. PEDESTRIAN EFFECT ON CAPACITY On the basis of the proposed analytial odel above, the effets of pedestrian rossings on intersetion apaity and delay an be exained. For the intersetion illustrated in Figure, a wide range of pedestrian volues and different yle lengths are exained for eah oordinated seiatuated signal operation. The effets of pedestrians on the inor street through oveent (i.e., eastbound through) are illustrated in Figure 4. Figure 5 presents the pedestrian effets on the ajor street through oveent (i.e., northbound through), and Figure 6 illustrates the pedestrian effets on the intersetion apaity. For the inor street, Figure 4 reveals soe interesting findings. At first glane, the results ay see to be ounterintuitive beause higher apaities are ahieved with lower yle lengths for the eastbound Capaity (NB), vph 3800 3400 3000 2600 2200 800 400 000 FIGURE 5 C=40 C=20 C=00 C=80 C=60 0 0 20 30 40 50 60 70 80 Pedestrian Volue, ped/hr Effet of pedestrians on northbound apaity.
30 Transportation Researh Reord 207 Total Capaity, vph 8000 7600 7200 6800 6400 6000 5600 5200 4800 4400 4000 FIGURE 6 approah. This an be well explained by a loser exaination. For the ase of alulating the apaity with pedestrians, 2,, the eastbound phase always has the Walk plus FDW pedestrian tie. With the inrease of yle length, the g/c ratio dereases, resulting in less apaity. Although the probability of having a pedestrian all inreases with the inrease of yle length, the final weighted apaity still dereases. For the ajor street through oveent, the pedestrian effet on the apaity is just the opposite of the inor street, as indiated in Figure 5. The apaity dereases with the inrease of pedestrian volue, and a shorter yle length results in a larger derease in apaity. This is aused by the derease in the ajor street green tie due to the inrease of inor street green tie needed to aoodate inor street pedestrian splits. Figure 6 shows the overall intersetion apaity dereases with the inrease of the pedestrian volues. This is logial beause soe effetive green ties are used to aoodate the needs of inor street pedestrian alls. The ajor street, whih has higher traffi deand, has less apaity and ore delay. Therefore, the overall intersetion has less apaity when pedestrian volue inreases. Also on the basis of the odel results in Figure 6, pedestrians have less effet on longer yle lengths than on shorter yle lengths. This is reasonable beause, for the sae green tie redution on the ajor street, the resulting g/c ratio would ause a larger apaity redution on a shorter yle length than on a longer yle length. Overall, results fro the proposed odel see to be reasonable. The pedestrian effets on apaity during a wide variety of pedestrian and yle length senarios are well represented by the proposed analytial odel. SUMMARY AND CONCLUSIONS C=40 C=20 C=00 C=80 C=60 0 0 20 30 40 50 60 70 80 Pedestrian Volue, ped/hr Effet of pedestrians on intersetion apaity. A proposed analytial odel and its ipleentation ethod for analyzing atuated signalized intersetions with onsideration of the stohasti effet of pedestrian rossings were presented in this paper. Model results were opared against and validated by traffi irosiulation odel. The proposed odel overoes the shortoings of the urrent HCM proedure in whih pedestrian arrivals are treated as stati. The odel an help researhers to better understand the pedestrian ipat on various signal operation senarios. The ipleentation flowhart an also assist pratitioners in realistially oputing the apaity and delay of intersetions with atuated signal ontrols. Based on the ontents of this study, the following onlusions an be drawn: On the basis of the odel validation results using irosiulation, the proposed odel produed delay results that were onsistent with results fro SiTraffi. However, the urrent HCM proedure had apparent deviations fro the siulation results. The proposed odel iproved the urrent analytial proedure in dealing with pedestrian effets on apaity and delay estiation. On the basis of the saple alulation in this study, the urrent HCM analytial proedure ould produe signifiant errors, espeially for a inor street with low pedestrian volues. The error in the partiular saple alulations was ore than 30%. An ipleentation fraework was proposed to inorporate the odel into the urrent HCM proedures and to allow analysts to ondut apaity and delay analysis onsidering stohasti pedestrian all events. For oordinated seiatuated signal ontrols, one interesting finding fro this study was that higher pedestrian volues resulted in higher apaity and lower delay for the pedestrian-onurrent oveent. The reason is that the inor street oveent phase would be likely to get the pedestrian split, Walk plus FDW, whih was generally greater than that based on the vehile deand. Another interesting finding was that a longer yle length atually resulted in a lower apaity for the onurrent oveent. However, longer yle lengths produe an overall higher apaity for the intersetion. This paper proposed an analytial odel for dealing with pedestrian ipats on apaity and delay. It deonstrated the appliation of the odel using a seiatuated signal ontrol senario. Further study would be to expand the odel to handle fully atuated signalized intersetions. REFERENCES. Highway Capaity Manual. TRB, National Researh Counil, Washington, D.C., 2000. 2. Tian, Z., T. Urbanik, K. Kair, M. Vandehey, and H. Long. 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