MODELING AND ANALYSIS OF PEDESTRIAN FLOW AT SIGNALIZED CROSSWALKS *

Transcription

1 MOELING AN ANALYSIS OF PEESTRIAN FLOW AT SIGNALIZE CROSSWALKS * By Wael ALHAJYASEEN ** and Hideki NAKAMURA ***. INTROUCTION The operational efficiency of vehiclar traffic and pedestrian flo are considered as important concern especially at signazed crossalks here both of them have to share the same space. A crossalk is defined as a portion of roaday designated for pedestrians to se for crossing the street. Crossalk geometry and configration at signazed intersections directly affect the safety, cycle length and reslting delays for all sers. Optimizing crossalk configrations inclding idth, position and angle is an important concern to improve the overall performance of signazed intersections. Pedestrian flo at signazed crossalks can be nidirectional or bi-directional depending on pedestrian demand at both sides of the crossalk. Pedestrian crossing time is basically a fnction of crossalk length and alking speed. Hoever hen pedestrian demand increases at both sides of the crossalk, crossing time increases de to the interaction beteen confcting pedestrian flos. Qantifying the effects of bi-directional flo and crossalk idth on pedestrian crossing speed and crossing time is a prereqisite for improving the geometric design and configration of signazed crossalks. A variety of methods have been developed for determining appropriate total pedestrian crossing times at signazed intersections. Althogh many of these methods have sefl appcations, most of them have shortcomings hen considering the effects of bidirectional flo on crossing time. No consideration is given to deceleration or redction in alking speed that reslts from the interaction beteen the confcting flos. The obective of this stdy is to develop a ne methodology for modeng the bi-directional pedestrian flo at signazed crossalks. The strctre of this paper is as follos: After introdction and teratre revie, total crossing time is modeled. Total crossing time is divided into discharge and crossing times. ischarge time is modeled by sing shockave analysis hile crossing time is modeled by applying drag theory. The fndamental diagrams that represent the relationship beteen speed, flo and density of pedestrian flo are presented to sho the sensitivity of pedestrian flo to bi-directional effects and crossalk geometry. Then proposed models are vadated and compared ith existing models. Possible appcations of the developed models sch as the Keyords: Pedestrian crossing time, Signazed crossalk, rag force, Bi-directional flo, Lane-ke segregation pocy Stdent member, M.Sc., ept. of Civil Eng., Nagoya Univ. (Fro-cho, Chiksa-k, Nagoya , Tel , FAX: , ael@genv.nagoya-.ac.p) Member, Professor, r. Eng., ept. of Civil Eng., Nagoya Univ. ( nakamra@genv.nagoya-.ac.p) evalation of lane-ke segregation pocy ) for pedestrian crossing are discssed. Finally, the paper ends p ith smmary of the reslts, conclsions and ftre orks.. LITERATURE REVIEW The alking speed and/or alking time of pedestrians are of prime importance in stdying the operation and design of pedestrian facities. Fe stdies addressed the isse of bi-directional pedestrian flo and its impact on crossing time at signazed crossalks. Most of the existing orks in this respect attempted to investigate the impact of bi-directional flo at other pedestrian facities sch as alkays and sidealks. Hoever the characteristics of the environment as ell as the pedestrian arrival pattern at crossalks is different from other pedestrian facities. Most crossing time estimations have been based on assmptions providing for start-p delay and a particlar alking speed. The pedestrian chapter of the Highay Capacity Manal ) and Pignataro 3) have formlations similar to Eqation (). L N ped T I ( x ) () S p Where T is total time reqired for all the crossing process, I is initial start-p lost time, L is crossalk length (m), S p is alking speed (m/sec), x is average headay (sec/ped/m), N ped is nmber of pedestrians crossing from one side of the crossalk dring an interval p, and is crossalk idth (m). Eqation () shos that the time spent on the crossalk itself (L/S p ) is independent from the pedestrian demand, bi-directional effect and crossalk idth. The Manal on Uniform Traffic Control evices 4) depends on average alking speed (4ft/sec) and crossalk length to estimate crossing time (clearance interval) hich is similar to L/S p in Eqation (). The Japanese Manal on Traffic Signal Control 5) proposes a formla similar to Eqation (), bt the initial start-p lost time is inclded in the discharge time. This procedre does not consider the effect of bi-directional pedestrian flo. Lam et al. 6) investigated the effect of bi-directional flo on alking speed and pedestrian flo nder varios flo conditions at indoor alkays in Hong Kong. They fond that the bi-directional flo ratios have significant impacts on both the at-capacity alking speeds and the maximm flo rates of the selected alkays. Hoever, the effects of different alkay s dimensions on alking speed and capacity of the alkay ere not investigated. Golani et al. 7) proposed a model for estimating crossing time considering start-p lost time, average

2 alking speed, and pedestrian headays as a fnction of the sbect and opposite pedestrian platoons separately. They fond that the size of the opposite pedestrian platoon can case a significant increase in the crossing time of the sbect pedestrian platoon especially at high demands. The proposed model (Eqation ()) is based on HCM ) model hich as cabrated by sing empirical data. If If _ N ped 5 T I _ N ped 5 T I L S p L S p N.9 Where N ped is the size of the sbect platoon and N ped is the size of the opposite pedestrian platoon. The proposed model relates the impact of bi-directional flo to the headay beteen pedestrians hen they finish crossing. Therefore it is difficlt to see ho the interaction is happening and hat the reslting speed drop or deceleration is. Virkler, et al. 8) collected data from some relatively lo-volme and high-volme signazed crossalks and recommended an eqation for one-directional flo that also considers platoon size. Hoever they did not consider the impacts of bi-directional pedestrian flos. Virkler 9) introdced a method to estimate the reqired pedestrian crossing time nder high-volme conditions and ith bi-directional pedestrian flo. According to the data analysis, it as conclded that the opposing pedestrian platoon size does not add any significant effect on estimating the crossing time of the sbect pedestrian platoon hich is contradicting ith the reslts of Golani et al. 7) and Lam et al. 6) Therefore the dispersion of the sbect pedestrian platoon throgh the crossing process as the only considered factor in the proposed methodology. This stdy aims to develop a rational methodology that can estimate total pedestrian crossing time as a fnction of crossalk geometry, pedestrian demand at both sides of the crossalk and signal timing. 3. TOTAL CROSSING TIME T t N.5 ped ped The total time needed by a platoon of pedestrians to cross a signazed crossalk T t from the beginning of the pedestrian green indication ntil the pedestrian platoon reaches the other side of the crossalk can be divided into to main parts: discharge time T d and crossing time T c. ischarge time T d is the necessary time for a pedestrian platoon to move from the aiting area and step inside the crossalk. While crossing time T c is the time hich is necessary to cross the crossalk: T T T t d c (3) The discharge time T d is a fnction of pedestrian demand and crossalk idth. The definition of discharge time T d is similar to that of vehicles aiting at the stop ne of a signazed intersection. Shockave theory is chosen for modeng pedestrian platoon discharge time. N N ped ped () Crossing time T c is dependent on pedestrian crossing speed hich is affected by the size of opposite pedestrian platoon and crossalk idth. This is analogos to a moving body facing a fd hich cases a redction on its speed depending on its cross sectional area, the density of the fd and the relative speed beteen them. This phenomenon is knon as drag force theory and its analogy is sed for modeng pedestrian platoon crossing time T c. For the prpose of this stdy, pedestrian demand is defined as the accmlated nmber of pedestrians at the edge of the crossalk dring the previos pedestrian flash green and red intervals. 4. MOELING ISCHARGE TIME T d ischarge time T d basically depends on pedestrian arrival rate, pedestrian red interval, and crossalk idth. Shockave analysis is sed to estimate qee discharge time hich is eqivalent to the time necessary for a pedestrian platoon to discharge at the edge of the crossalk. () Model Assmptions The folloing assmptions are made for modeng discharge time T d : i) High pedestrian demand is assmed for the model development (Figre a)). Later the developed model ill be modified to consider lo demand case. ii) Pedestrian arrival rate A is assmed to be niform. Therefore the accmlated pedestrian demand P at the beginning of the pedestrian green interval is defined throgh Eqation (4). P A( C g) (4) Where C is the cycle length and g is the pedestrian green interval. iii) Pedestrian arrival nit is assmed as pedestrian ro per second A r. Figre a) shos ho pedestrian ros are forming at high pedestrian demand. Assming that the lateral distance hich a pedestrian occpies along the crossalk idth is δ, then the maximm nmber of pedestrians that can fit throgh one ro M p along crossalk idth is estimated throgh Eqation (5). M p (5) To estimate pedestrian arrival rate in the nit of pedestrian ro per second, arrival rate A is divided by the maximm nmber of pedestrians that can fit in one ro along the crossalk idth. A A Ar (6) M p Where A r is the arrival rate of the sbect pedestrian demand in the nit of pedestrian ros per second. The nmber of accmlated pedestrian ros R p at the start of pedestrian green interval is: A ( C g) R A ( C g) (7) p r iv) The lateral distance that a pedestrian occpy δ is assmed to be a fnction of pedestrian demand and crossalk idth. Hoever, for simpfication,

3 Space Stop ne of the crossalk Stopping shock ave Speed ω C-g Uniform arrival rate (A) pedestrian/second I T d Starting shock ave Speed ω δmin irection of pedestrian flo ω Waiting ω area Sidealk Q = (pstream) K = K c K Q =A r K =K = s Crossalk (donstream) Q c = Q d c = Time Stopping shock ave Starting shock ave a) Pedestrian ros formation b) Stopping and starting shockaves C: cycle length, g: pedestrian green interval, I: pedestrian lost time, T d: pedestrian platoon discharge time, : crossalk idth, : the longitdinal distance beteen aiting pedestrians, ω : speed of stopping shockave, ω : speed of starting shock ave, A r: pedestrian arrival rate in pedestrian ro per second, Q d: discharge rate in pedestrian ro per second, s: is pedestrian alking speed at sidealks, o: is pedestrian free flo alking speed at crossalks, K : am density in the nit of pedestrian ro per meter. Figre : Schematic formation of pedestrian ros at high demand and reslted shockaves longitdinal distance beteen aiting pedestrians hich is the same distance beteen pedestrian ros is assmed to be constant (Figre a)). v) Start-p lost time I is considered as a part of discharge time T d, as shon in Figre. vi) To apply shockave theory, speed of the arriving pedestrian ros at the sidealk (pstream) is assmed to be eqal to the pedestrian free-flo speed s at sidealk. hile speed of the discharging pedestrian ros at the crossalk (donstream) is assmed to be eqal to the pedestrian free-flo speed o at crossalk, as shon in Figre b). () Model evelopment Figre b) shos that to shockaves are formed in the aiting area of the crossalk. The first one is stopping shockave hich reslts from the continos arrival of pedestrians to the aiting area that forms ne ros. The second one is starting shockave de to the discharge of the aiting pedestrian ros after pedestrian green indicator is displayed. The speed of the stopping shockave ω is: Q Q Ar A (8) K K K A K A r The speed of the starting shockave ω is: Qc Q Qd K K Q K c The nmber of arrived pedestrian ros shold be eqal to the nmber of discharged ros, therefore: (( C g) T d ) T d () After sbstitting Eqation (8) and (9) in Eqation (), the discharge time T d becomes: A ( C g) K A s T d () Qd A Qd o K K A s (3) Parameter Estimation The parameters inclded in Eqation () are estimated as follos: d s o s (9) i) To define a vale for the pedestrian free-flo speed at crossalks, a.5-hor video tape for the crossalk at the east leg of Nishi-Os intersection (6m ide 5.4m long) in Nagoya City as analyzed. Nishi-Os intersection is characterized by small pedestrian demand ith a large crossalk idth. samples of pedestrian s free-flo speeds ere measred. All the considered pedestrians ere leading pedestrians and they did not face any opposite flo or trning vehicles. The average freeflo speed for all the samples is.45 m/s. This vale is assmed as the free-flo speed of pedestrians at crossalks o. ii) Lam et al. ) stdied pedestrian alking speed at different alking facities and they fond that pedestrian s free-flo alking speed at otdoor alkays is loer than that of signazed crossalks by 7%. Hoever, for the prpose of this stdy pedestrian free-flo speed at sidealks s is assmed to be % less than the free-flo speed at crossalks. iii) In order to define the am density K, to -hor video tapes for the crossalks at the east and est legs of Sasashima intersection (m ide 9m long) in Nagoya City ere analyzed. The data as collected in the morning peak hors, hen long qees ere formed de to the high pedestrian demand. The measred am densities range beteen.9.4 ped/m, hile the average is. ped/m. The average measred vale is assmed as pedestrian am density K. It shold be noted that the am density hich is sed in the proposed model is in the nit of pedestrian ro per meter. Therefore the minimm lateral distance δ min that a pedestrian can occpy along the crossalk idth is assmed.m, inclding the lateral clearance distance beteen aiting pedestrians. As a reslt, the am density K in the nit of pedestrian ro per meter (Figre a)) is defined by Eqation (). K K ( ped./ m ) min. ped.ro/m () iv) Maximm discharge flo rate Q d as observed at crossalks ith very high pedestrian demand. To -hor video tapes for the crossalks at the east and est legs of Sasashima intersection (m ide 9m

4 Occpied lateral distance by one pedestrian δ (m) Space Stop ne of the crossalk Stopping shock ave Speed ω C-g Uniform arrival rate (A) pedestrian/second I T d Time Starting shock ave Speed ω δ δ a) Formations of pedestrian ros at lo demand b) Modeng occpied lateral distance δ Figre : Lo pedestrian demand case P a ( ) Parameter Estimate t-vale a b R Pedestrian demand per meter idth of the crossalk (ped./m) b long) in Nagoya City ere analyzed. The average observed discharge rate at high pedestrian demand crossalks is assmed as the maximm discharge flo rate Q d. The average observed discharge rate is.45 ped.ro/sec. After estimating the am density K and the discharge rate Q d, Eqation () can directly be sed to estimate the discharge time T d for high pedestrian demand. Hoever at lo pedestrian demand, some modifications shold be considered, sch as adsting the lateral distance δ depending on pedestrian demand and crossalk idth. (4) Modification for Lo Pedestrian emand Case At lo pedestrian demand many factors sch as pedestrian origin and destination can affect pedestrian decisions regarding the aiting position hich makes pedestrians form different ros in irreglar patterns. Assming a niform pedestrian arrival rate, Figre a) shos ho the pedestrian ros are forming hen pedestrian demand is lo. As pedestrian demand decreases, the nmber of pedestrians that forms a ro decrease, hich means that the lateral distance δ increases as pedestrian demand decreases. Eqation () is tized by sing the empirical data collected at Nishi-Os and Sasashima intersections to estimate the average lateral distance that a pedestrian can occpy δ at different demand vales. Then δ is modeled as a fnction of pedestrian demand per meter idth of the crossalk and is illstrated in Figre b). A preminary statistical analysis as performed to determine the best fnction to represent the relationship beteen δ and pedestrian demand per meter idth of the crossalk. The poer fnction as fond the best to describe this relationship (Eqation (3)). P A ( C g) ( ).533( ) (3) Figre b) shos that hen pedestrian demand becomes high, δ becomes very close to. meter. This correspondes to the previos assmption of δ min =. m. After sbstitting Eqation (3) in Eqation (), the reslted formla can be sed to estimate discharge time T d for any pedestrian demand volme and crossalk idth (Figre 8 b)). 5. MOELING CROSSING TIME T c The force on an obect that resists its motion throgh a fd is called drag. When the fd is a gas ke air (Figre 3 a)), it is called aerodynamic drag (or air resistance). While if the fd is a qid ke ater it is called hydrodynamic drag. rag is a compcated phenomenon, and explaining it from a theory based entirely on fndamental principles is exceptionally difficlt. Pgh ) described the relation of drag, the relative velocity of the air or fd and a moving body in terms of a dimensionless grop, the drag coefficient C d. The drag coefficient is the ratio of drag to the dynamic pressre q of a moving air stream and is defined by Eqation (4): C d qa p (4) Where is drag force (kg m/sec ), C d is drag coefficient (dimensionless), q is dynamic pressre (force per nit area), and A p is the proected area (m ). The dynamic pressre q hich is eqivalent to the kinetic energy per nit volme of a moving sod body (Pgh ) ) is defined by Eqation (5): q.5 (5) Where ρ is density of the air in kilogram per cbic meter, and is the speed of the obect relative to the fd (m/sec). By sbstitting Eqation (5) in Eqation (4), the final drag force eqation is: q.5c (6) d A p () rag Force Cased by the Opposite Pedestrian Flo To se the drag force concept to model the interactions beteen pedestrian flos, the folloing assmptions are made: i) Opposite pedestrian demand is considered as a homogenos flo (Figre 3 b)) ith a density eqal to the nmber of pedestrian aiting in the beginning of the green interval divided by an area eqal to the idth of the crossalk mltiped by.m. P ped./ m (7) * ii) The sbect pedestrian flo is considered as one body moving against the opposite pedestrian flo. The interactions occr along the proected area of all pedestrians in the sbect flo hich is defined as

5 Air Stream ρ A p Moving Body Air Stream P Opposite Pedestrian Flo P L o Moving Body Sbect Pedestrian Flo a) rag force b) Pedestrian at both sides of the crossalk : crossalk idth (m), L o: crossalk length (m), P : opposite pedestrian demand, P : sbect pedestrian demand, : speed of the sbect pedestrian flo (m/sec), and : speed of the opposite pedestrian flo (m/sec). Figre 3: Applying drag force concept on bi-directional pedestrian flos at crossalk the sm of the idths of all pedestrians in the sbect flo: A p n (8) Where A p is the proected area of the sbect pedestrian flo (m), β is the average body idth of one pedestrian, and n is the nmber of pedestrian in the sbect pedestrian flo P, shon in Figre 3 b). iii) The initial speed of the sbect and the opposite pedestrian flo hen they start crossing is assmed to be eqal to their free-flo speed and respectively, therefore the relative speed becomes: ( ( )) ( ) (9) The initial speed is assmed to be constant vale regardless of pedestrian platoon s density. Therefore in the case of ni-directional flo, crossing speed is alays eqal to free-flo speed. After sbstitting Eqation (7), (8), and (9) in Eqation (6), then the drag force eqation becomes: P.5* Cd * *( ) * * n () Assming that the average idth of one pedestrian body β is.6m, the drag force becomes: P.5* Cad * *( ) * n () Where C ad is adsted drag coefficient (dimensionless), and it is defined as: C ad * C () () eceleration of the Sbect Pedestrian Flo The net force on a particle observed from an inertial reference frame is proportional to the time rate of change of its near momentm hich is the prodct of mass and velocity: dm d( mv) dv F m ma (3) dt dt dt Where m is the mass of the moving body and its eqivalent to the sbect pedestrian demand P, and a is the average deceleration of the sbect pedestrian flo. The final speed of a moving particle in a straight ne ith constant average deceleration according to the motion eqations is: al f i (4) Where i is initial speed (m/sec) hich is assmed to be eqal to the free-flo speed, f is final speed (m/sec), a is average deceleration (m/sec ), and L is travelled distance (m). Figre 4 shos the proection of pedestrian flo traectory from both sides of a crossalk. d A maor assmption of this methodology is that both opposing flos ill start alking ith their free-flo speed on a straight ne ntil they meet in the crossalk. The meeting point is dependent on the speed of sbect and opposite pedestrian platoons. The time hen the to pedestrian platoons ill meet is compted by Eqation (5). Lo t (5) The interaction distance l i is assmed to be eqal to the physical depth of the opposite pedestrian platoon. The physical depth of the opposite pedestrian platoon can be estimated by tizing the methodology sed for modeng the discharge time T d. Therefore the physical depth l i of the opposite pedestrian platoon is defined by Eqation (6). Rp P * (6) K * K Where R p is the nmber of accmlated ros of the opposite pedestrian demand at the start of pedestrian green interval. Figre 4 shos that reslting deceleration is averaged along the assmed interaction time. Therefore the final speed can be defined as: l i Lo f Tc ( ) (7) ti Where T c is crossing time of the sbect pedestrian platoon and l i is the physical depth of the opposite pedestrian platoon. If the physical depth of the opposite pedestrian platoon is longer than the remaining crossing distance (L o - t), the interaction distance ill be eqal to L o - t. Therefore hen estimating the interaction distance, the physical depth of opposite pedestrian platoon shold be compared ith the remaining crossing distance for the sbect pedestrian platoon and the smaller one shold be considered as the interaction distance. By sbstitting Eqation (7) in Eqation (4), the average deceleration of the sbect pedestrian platoon becomes: L o i a Tc ( ) l (8) The net force (ped.m/sec ) that cases the deceleration of the sbect pedestrian platoon is defined as the average deceleration a (Eqation (8)) mltiped by the mass of the sbect pedestrian platoon M hich is assmed to be eqal to the sbect pedestrian demand P.

6 .35 t Opposite pedestrian flo (P ) Interaction Time C ad C ad.37r R.84 Samplesize is 38 data points (3) Model evelopment The drag force cased by an opposite pedestrian flo shold be eqal to the force that cases the deceleration of the sbect pedestrian flo. By eqating the to forces, and solving them for the crossing time T c, the net eqation becomes: T c istance Lo t C l Sbect pedestrian flo (P ) t t i t- T c Time L o: crossalk length, T c: crossing time, t: time from the beginning of crossing ntil the sbect and the opposite pedestrian platoons meet in the crossalk, t i: interaction time, l i: interaction distance, and : free-flo speed of the sbect and the opposite pedestrian flo respectively. Figre 4: Time-space diagram of the confcting pedestrian flos ( l ) i o i P ( ) ad L (9) Pedestrian demand is defined as the nmber of accmlated pedestrians dring pedestrian red and flash green signal indications, and those ho arrive dring the discharge time. Therefore opposite pedestrian demand can be presented as: P A *( C g Td ) (3) Where P is opposite pedestrian demand, A is arrival rate of the opposite pedestrian demand (ped/sec), and T d is discharge time of the opposite pedestrian platoon. After sbstitting Eqation (3) in Eqation (9), the average crossing time and alking speed of the sbected pedestrian flo are given by Eqations (3) and (3) respectively. Lo Tc ( ) C A l ( ) ( C g T ) ad i d (3) CadAl i ( ) ( C g Td ) (3) f Eqations (3) and (3) are final eqations hich represent ho alking speed and crossing time vary ith pedestrian demand combinations of bi-directional flo and crossalk geometry. (4) Estimating the rag Coefficient C ad The vale of adsted drag coefficient C ad according to aerodynamic drag is dependent on the irectional spt ratio r Figre 5: Modeng drag coefficient as a fnction of directional spt ratio r kinematic viscosity of the fd, proected area and textre of the moving body. In the pedestrian s case, this vale can here be assmed to be dependent on the pedestrian demand at both sides of the crossalk and their spt ratio. To tize Eqations (3) and (3) to estimate speed drop and crossing time, C ad as first estimated from empirical data. A -hor video tape for the crossalk at the east leg of Imaike intersection (9.6m ide.5m long) in Nagoya City as analyzed. The pedestrian demand in each cycle at each direction, the average pedestrian traectory length, and the average pedestrian crossing time in the same cycle ere extracted from the video tape. Then by sing Eqation (3), C ad as estimated for 38 data point here the total pedestrian demand as ranging from 5 3 pedestrians per cycle (one cycle is 6sec). In any case hen pedestrians enconter a trning vehicle that cases a redction on their speed or a change on their traectory, the hole cycle as neglected and removed from the data base. Frthermore if pedestrians alk otside the crossalk, that cycle as also neglected. After analyzing the available data, C ad as modeled in terms of the spt ratio r hich is the ratio of the sbect pedestrian demand to the total pedestrian demand. Eqation (33) defines the directional spt ratio r. P P r P (33) Figre 5 shos the relationship beteen directional spt ratio and C ad. As directional spt ratio increases the drag coefficient also increases de to increasing pedestrian demand. After estimating the drag coefficient, Eqations (3) and (3) can be sed to estimate the average crossing speed and crossing time T c for different demand volmes nder different crossalk dimensions. 6. FUNEMENTAL IAGRAMS Measred vales Model As an important step to vadate ho the proposed methodology describes the behavior of pedestrian flo, the fndamental diagrams of directional and bi directional pedestrian flo are dran. () For Sbect irectional Pedestrian Flo Figres 6 a) and b) represent the speed-flo and speed-density relationships for the sbect pedestrian flo. The density of sbect pedestrian flo k is defined by Eqation (34).

7 Weighted average crossing speed (m/sec) Weighted average crossing speed (m/sec) irectional crossing speed (m/sec) irectional crossing speed (m/sec).6.4 r =.. r =. r =.8 r =.3.8 r =.4 r =.9.6 r =.7.4 r =.6. r = Sbect pedestrian flo rate q (ped/m/sec).6 r =.6.4 r =.9. r =. r =..8 r =.8 r =.3.6 r =.4.4 r =.5. r = ensity of the sbect pedestrian platoon K (ped/m ) a) irectional speed-flo relationship b) irectional speed-density relationship Figre 6: Fndamental diagrams for sbect directional pedestrian flo at signazed crossalks.6.4 r =.9 or..6.4 r =.9 or.. r =.6 or.4 r =.7 or.3. r =.8 or r =.5 r =.8 or r =.7 or.3 r =.5 r =.6 or Bi-directional pedestrian flo rate q (ped/m/sec) Total pedestrian density K t (ped/m ) a) Bi-directional speed-flo relationship b) Bi-directional speed-density relationship Figre 7: Fndamental diagrams for bi-directional pedestrian flo at signazed crossalks l K P * (34) Where K is the density of sbect pedestrian platoon, P is sbect pedestrian demand, l is physical depth of sbect pedestrian platoon (Eqation (6)) and is crossalk idth. Figre 6 a) shos that as directional spt ratio increases the maximm sbect pedestrian flo increases. Figre 6 b) shos the drop in average alking speed de to the effects of bi-directional pedestrian flo. As the density of sbect pedestrian flo increases either de to decreasing crossalk idth or increasing sbect pedestrian demand, the interactions increase casing redction in the average alking speed. The drop in the alking speed contines ntil a point here the speed drops drastically. This tendency is reasonable if e assme that pedestrian cannot alk otside the crossalk. Therefore it is expected that the average alking speed ill drop as the demand increases for a specific crossalk idth, ntil it reaches almost zero here every pedestrian cannot alk any more. Figre 6 b) shos that the crossing speed of the sbect pedestrian flo also increases at a specific density hen directional spt ratio increases. () For Both Bi-directional Pedestrian Flos Figres 7 a) and b) represent the speed-flo and speed-density relationships for both directions of flo at the crossalk. The total pedestrian density K t at the crossalk is defined according to Eqation (35). P l P l K t K K * (35) * Where K and K are the density of sbect and opposite pedestrian platoon respectively, P and P are sbect and opposite pedestrian demand respectively, l and l are physical depth of sbect and opposite pedestrian platoons respectively and is crossalk idth. Figre 7 a) shos that directional spt ratio has significant impact on the maximm flo rate of a crossalk. It shos that the interactions beteen opposing pedestrian flos increase as directional spt ratio reaches.5 here the loest maximm flo rate q occrs. Frthermore at directional spt ratio of.9 or. hich is very close to ni-directional flo, the maximm total pedestrian flo q that can be achieved is.75 ped/m/sec. Hoever Highay Capacity Manal ) (EXHIBT -3) assmes a maximm ni-directional pedestrian flo rate of.66 ped/m/sec for commters

8 ischarge time (sec) irectional crossing speed (m/sec) Absolte percentage error range =.5%-4.8% Mean absolte percentage error = 3.54% Root mean sqare error (RMSE) = r =. r =. r =.3 r =.4 r =.5 a) Crossing speed model vadation Observed ata Highay Capacity Manal HCM () Japanese Manal on Traffic Signal Control (6) Proposed ischarge Time Model Proposed ischarge Time Model Mean absolte percentage error = 5% Root mean sqare error (RMSE) =. r =.8 r =.7 r =.6 r = ensity of the sbect pedestrian platoon K (ped/m ) r=.5~.5 r=.5~.35 r=.35~.45 r=.45~.55 r=.55~.65 r=.65~.75 r=.75~.85 Highay Capacity Manal HCM Mean absolte percentage error = 4% Root mean sqare error (RMSE) = Japanese Manal on Traffic Signal Control Mean absolte percentage error = 76% Root mean sqare error (RMSE) = Sbect pedestrian demand per meter idth of the crossalk (ped/m) b) ischarge time model vadation Figre 8: Vadations of the proposed models hich is beteen the total maximm bi-directional flo rates at directional spt ratios of.9 or. and.8 or. according to the proposed methodology. This can be referred to the loer assmed free-flo speed and the consideration of aged pedestrians by HCM ). Figre 7 b) shos ho the average crossing speed varies ith total pedestrian density K t. When total density K t decreases either de to redcing pedestrian demand or increasing crossalk idth, crossing speed increases ntil it becomes almost constant (free-flo condition). Bt hen total density K t increases, crossing speed decreases, as the interactions beteen the opposing flos increases, ntil it reaches a point here the opposing flos block each other casing a drastic decrease in crossing speed. 7. VALIATION AN COMPARISION To vadate the proposed models, average crossing speed as measred nder different directional demand ratios and compared ith the estimated speed from the proposed model. Figre 8 a) illstrates the differences beteen measred and estimated crossing speeds. A paired t-test as performed and the reslt shoed that the estimated vales ere not significantly different from the observed vales at the 95% confidence level. The proposed model prodces a mean absolte percentage error of 3.54%. According to the proposed methodology estimated speeds are expected to be loer than observed vales. This tendency is logical since the developed model estimates the speed directly after the interaction ith the opposite pedestrian flo hile observed speed is the average speed throgh all the crossing process and it is measred by dividing pedestrian traectory length to crossing time. Hoever Figre 8 a) shos that estimated speeds for some data points are higher than observed vales. These points ere extracted from the video tapes collected at Imaike intersection hen pedestrian demand as lo. If pedestrians alk sloly at lo demand (mited interactions), this can be referred to their dresired speed hich is not considered in the proposed methodology. The discharge time T d estimated by Eqation () is compared ith the observed data and the estimated discharge time from the existing formlations in HCM ) and Japanese Manal on Traffic Signal Control 5), as shon in Figre 8 b). By comparing the mean absolte percentage error and root mean sqare error, it is clear that the proposed model prodces more accrate and reable reslts. Frthermore the existing formlations alays tend to ndersestimate the necessary discharge time for large pedestrian platoons. The tendency of the proposed discharge time model is more consistent ith the observed data.

9 Sbect Flo P Time (sec) Sbect Flo P Mix-Lane Pocy L o Lane-Like Segregation Pocy L o Opposite Flo P Opposite Flo P T d for lane-ke segregation and mix-lane pocies T t for lane-ke segregation pocy T t for mix-lane pocy T c for mix-lane pocy T c for lane-ke segregation pocy Crossalk idth (m) a) Lane-ke segregation and mix-lane pocies b) Crossing time T c and discharging time T d Assmptions: Crossalk length L o is m, free flo speed of sbect and opposite pedestrian flos ( and respectively) is.45m/s, pedestrian speed at sidealk s is.6m/s, pedestrian demand at each side of the crossalk is 5ped/cycle, directional spt ratio r is.5, am density K is.ped.ro/m, and maximm discharge flo rate Q d is.45ped.ro/sec. Figre 9: Comparison beteen lane-ke segregation and mix-lane pocies 8. MOEL APPLICATIONS The developed models can be tized for ide range of appcations sch as the evalation of pedestrian flo at signazed intersections, assessing pedestrian signal timing and improving the geometric design of signazed crossalks. Teknomo ) proposed a lane-ke segregation pocy as an effective tool to improve pedestrian flo at signazed crossalks. It aims to change the bidirectional flo into ni-directional as shon in Figre 9 a). He conclded that lane-ke segregation pocy (nidirectional) is sperior to mix-lane pocy (bi-directional) in terms of average speed, ncomfortabity, average delay and dissipation time. Hoever in his analysis, the necessary discharging time for a pedestrian platoon and the reslted delay ere not considered. In order to evalate the performance of the lane-ke segregation pocy, Figre 9 b) is presented ith assmed vales. It compares the crossing and discharging times of the lane-ke segregation and mix-lane pocies for the same sbect pedestrian demand volme. If e assme that the existing pocy is mix-lane here opposing pedestrian flos share the same space and the existing crossalk idth is 8m, then the total estimated crossing time T t is 5sec (Figre 9 b)). Hoever if laneke segregation pocy is implemented and the available crossalk idth is divided eqally beteen the confcting pedestrian flos (directional spt ratio is assmed as.5) the total estimated crossing time is 8sec (Figre 9 b)). After the implementation of lane-ke segregation pocy, crossing speed increases and crossing time decreases. While discharging time increases becase of dividing the available crossalk idth hich ill compensate the saving in crossing time and leads to higher total crossing times and delays. Therefore providing ider crossalks for each pedestrian flo at both sides of the crossalk is necessary in order to maintain the same total crossing time. Hoever as crossalks become ider, cycle length ill increase becase of all-red time reqirement. Longer cycle lengths ill case longer delays and deteriorates the overall mobity levels of signazed intersections. Moreover lane-ke segregation pocy may create ne confcts at both sides of the crossalk de to the desired destination of pedestrians hen they exit the crossalk. According to the previos conclsions, it is not recommended to apply the lane-ke segregation pocy nless pedestrian level of service at a specific intersection is in a higher priority than other ser s level of service. Existing methodologies for the estimation of minimm reqired pedestrian green interval depend on constant average alking speed and crossalk length to estimate total crossing time. Therefore developed models can be tized for better estimation of the minimm reqired pedestrian green at signazed crossalks ith bi-directional pedestrian flo. Hoever the main desired appcation behind the proposed methodology is to rationally define the reqired crossalk idth nder different pedestrian demands considering the redction in alking speed de to an opposite pedestrian flo. 9. CONCLUSIONS A ne methodology for modeng crossing time considering bi-directional pedestrian flo and discharge time necessary for a pedestrian platoon at signazed crossalks is proposed in this paper. Shockave and drag force theories are sccessflly tized for modeng the discharge and crossing times respectively. The proposed methodology does not consider the speed variation beteen individals inside the platoon. Meanhile, sch phenomenon can be considered by assming a speed distribtion for the pedestrian platoon instead of average constant speed. The final formlation of crossing time T c provides a rational qantification for the effects of crossalk geometry and bi-directional pedestrian flo on alking speed and crossing time. Hoever, the proposed models do not consider the effects of age, trip prpose and different ambient conditions on crossing and discharge times. Therefore cabrating the developed models ill expand their appcabity to different cases sch as signazed crossalks at school zones or crossalks ith aged pedestrian activities.

10 In the modeng methodology, it as assmed that the initial speeds of the sbect and the opposite pedestrian platoons are constant vales regardless of their densities. This assmption shold be modified to consider the redction in the initial speed de to the increasing in pedestrian platoon s density. The proposed crossing time model is vadated ith a mited data hich does not cover many pedestrian demand combinations, therefore more concrete vadation is reqired. The developed models prodce reasonable fndamental parameters of speed, flo and density. Lane-ke segregation pocy eminates the confcts beteen opposing pedestrian flos hich increase crossing speed; hoever discharge time ill significantly increase de to dividing the available crossalk idth. This leads to an increase in the total crossing time and the resltant total delay. Ths the implementation of ider crossalks is reqired to redce the total crossing time and total delay hich cases longer cycle lengths and may negatively affect on the overall mobity levels of signazed intersections. The proposed methodology ill be tized to rationally define the reqired crossalk idth nder different pedestrian demand volmes considering the bidirectional flo effects. REFERENCES ) Teknomo, K.: Appcation of microscopic pedestrian simlation model. In Transportation Research, Part F 9, pp. 5-7, 6. ) Highay Capacity Manal. Transportation Research Board, National Research Concil, Washington,.C.,. 3) Pignataro, L. J. Traffic Engineering: Theory and Practice. Prentice-Hall, Engleood Cffs, N.J., ) Manal on Uniform Traffic Control evices for Streets and Highays. FHWA, U.S. Washington,.C., 3. 5) Manal of Traffic Signal Control. Japan Society of Traffic Engineers, 6. 6) Lam, Wilam H.K., Lee, Jodie Y.S., Chan, K.S., Goh, P.K.: A generazed fnction for modeng bidirectional flo effects on indoor alkays in Hong Kong. In Transportation Research Part A 37, pp , 3. 7) Golani, A. and amti, H.: Model for Estimating Crossing Times at High Occpancy Crossalks. In Transportation Research Board, TRB, Annal Meeting, 7. 8) Virkler, M. R., and Gell,. L.: Pedestrian Crossing Time Reqirements at Intersections. In Transportation Research Record 959, TRB, National Research Concil, Washington,.C., pp. 47 5, ) Virkler, M. R.: Scramble and Crossalk Signal Timing. In Transportation Research Record 636, TRB, National Research Concil, Washington,.C., pp , 998. ) Lam, Wilam H. K., Cheng, C.-y.: Pedestrian Speed/Flo Relationships for Walking Facities in Hong Kong. In Jornal of Transportation Engineering, Vol. 6; Part 4, pp ,. ) Pgh, L. G. C. E.: The inflence of ind resistance in rnning and alking and the mechanical efficiency of ork against horizontal or vertical forces. In Jornal of Physiology, 3; pp , 97. ) Pgh, L. G. C. E.: The relation of oxygen intake and speed in competition cycng and comparative observations on the bicycle ergo meter. In Jornal of Physiology, 4; pp , 974. MOELING AN ANALYSIS OF PEESTRIAN FLOW AT SIGNALIZE CROSSWALKS * By Wael ALHAJYASEEN ** and Hideki NAKAMURA *** Pedestrian s crossing speed and time are of prime importance in stdying the operation and design of signazed crossalks. Existing methodologies do not consider the effects of bi-directional flo or crossalk geometry on crossing speed and the resltant crossing time. In this stdy a ne methodology is proposed for modeng the total time necessary for a platoon of pedestrians to cross a signazed crossalk, considering the effects of bi-directional flo and crossalk geometry. The fndamental relationships beteen speed, flo and density of bi-directional pedestrian flo are presented. An evalation for the lane-ke segregation pocy for pedestrian crossing is inclded. The final formlation of crossing time T c provides a rational qantification for the effects of crossalk geometry and bidirectional pedestrian flo on crossing time and speed. * 信号交差点における横断歩行者交通流のモデル分析 アルハジヤシーンワエル ** *** 中村英樹 歩行者の横断速度および横断時間は, 信号制御された横断歩道の設計運用の検討に重要であるが, 既存手法では双方向歩行者交通流や横断歩道幅員が横断速度および横断時間に与える影響について考慮されていない. 本研究では, これらの影響を考慮した横断歩行者の横断完了に必要な総時間をモデル化する新しい手法を提案した. 双方向歩行者交通流の速度, 交通量, 密度の基本的な関係を示すとともに, 横断歩行者がレーン状に分離して横断する場合についても評価した. モデル化により, 横断時間 T c は横断歩道の幾何構造と双方向の歩行者交通流が, 横断時間および横断速度に与える影響を良好に再現可能であることが示された.