a methodology for modeling pedestrian platoon ...

A METHODOLOGY FOR MODELING PEDESTRIAN PLATOON DISCHARGE AND CROSSING TIMES AT SIGNALIZED CROSSWALKS

Authors Wael K.M. ALHAJYASEEN M.Sc., Department of Civil Engineering Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8603, JAPAN Tel: +81 (52) 789-5175 FAX: +81 (52) 789-3837 Email: [email protected]

Hideki NAKAMURA Professor, Dr. Eng, Department of Civil Engineering Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8603, JAPAN Tel: +81 (52) 789-2771 FAX: +81 (52) 789-3837 Email: [email protected]

Number of Words = 4666 Number of Figures =11 Total Words Count= 4666+11*250=7416 Submission Date: 28 of October, 2008

Alhajyaseen and Nakamura

1

ABSTRACT Optimizing crosswalk configurations including width, position and angle is an important concern to improve the overall performance of signalized intersections. Quantifying the effects of bidirectional flow and crosswalk width on pedestrian walking speed and crossing time is a prerequisite for improving the geometric design and configuration of signalized crosswalks. The time used by a pedestrian platoon to cross a signalized crosswalk is divided into discharge time and crossing time. Discharge time is a function of pedestrian demand and crosswalk width while pedestrian crossing time is basically a function of crosswalk length and walking speed. However when pedestrian demand increases at both sides of the crosswalk, crossing time increases due to interactions between conflicting pedestrian flows. A variety of methods have been developed for determining appropriate pedestrian crossing times. Although many of these methods have useful applications, most of them have shortcomings when considering the effects of bi-directional flow on crossing time. No consideration is given to deceleration or reduction in walking speed that results from the interaction between conflicting flows. Furthermore existing models tend to underestimate the discharge time necessary for a pedestrian platoon waiting at the edge of the crosswalk. This study proposes a new methodology for modeling pedestrian discharge and crossing times. Discharge time is modeled by using shockwave theory while crossing time is modeled by applying aerodynamic drag theory. The developed methodology provides a rational quantification for the effects of bi-directional pedestrian flow and crosswalk geometry on walking speed and crossing time.


2

INTRODUCTION A crosswalk is defined as a portion of roadway designated for pedestrians to use for crossing the street. Crosswalk geometry and configuration including width, position and angle at signalized intersections directly affect the safety, cycle length and resulting delays for all users. However existing manuals do not provide clear specification or guidance for designing crosswalks at signalized intersections. Quantifying the effects of bi-directional pedestrian flow and crosswalk width on pedestrian walking speed and crossing time is a prerequisite for optimizing the geometry and configuration of signalized crosswalks. The objective of this study is to propose a new methodology for modeling the total time necessary for a platoon of pedestrians to cross a signalized crosswalk considering the effect of bi-directional pedestrian flow on walking speed and crossing time. The developed methodology will be utilized as a basis to define the required crosswalk width for different demand volumes and split ratios. In this study, the total crossing time is divided into discharge time and crossing time. Discharge time is a function of pedestrian demand and crosswalk width, and it is modeled by using shockwave analysis. Crossing time is a function of pedestrian demand at both sides of the crosswalk and crosswalk geometry and it is modeled by applying drag theory. Sensitivity analysis is presented to show the sensitivity of developed models to bi-directional flow effects and crosswalk geometry. Finally the proposed models are validated and compared with existing models. LITERATURE REVIEW Few studies addressed the issue of bi-directional pedestrian flow and its impact on crossing time at signalized crosswalks. Most of the existing works in this respect attempted to investigate the impact of bi-directional flow at other pedestrian facilities such as walkways and sidewalks. However the characteristics of the environment as well as the pedestrian arrival pattern at crosswalks is different from other pedestrian facilities. Most crossing time estimations have been based on assumptions providing for start-up delay and a particular walking speed. The Pedestrian chapter of the Highway Capacity Manual (1) and Pignataro (2) have formulations similar to Equation (1).

T I

N L  ( x ped ) Sp w

(1)

Where T is total time required for all the crossing process, I is initial start-up lost time, L is crosswalk length (m), Sp is walking speed (m/s), x is average headway (s/ped./m), Nped is number of pedestrians crossing during an interval p from one side of the crosswalk, and w is crosswalk width (m). Equation (1) shows that the time spent on the crosswalk itself (L/Sp) is independent from the pedestrian demand, bi-directional effect and crosswalk width. The Manual on Uniform Traffic Control Devices (3) provides a procedure to estimate pedestrian crossing time (clearance interval) depending on average walking speed (4 ft/sec) and crosswalk length which is similar to L/Sp in Equation (1). However this procedure does not consider the effect of bi-directional pedestrian flow which is determined by opposite pedestrian flow and crosswalk width.


3

The Japanese Manual on Traffic Signal Control (4) proposes a formula similar to Equation (1), but the initial start-up lost time is included in the discharge time. The proposed model does not consider the bi-directional pedestrian flow effects. Lam et al. (5) investigated the effect of bi-directional flow on walking speed and pedestrian flow under various flow conditions at indoor walkways in Hong Kong. They found that the bi-directional flow ratios have significant impacts on both the at-capacity walking speeds and the maximum flow rates of the selected walkways. However they did not investigate the effect of different walkway’s dimensions on the walking speed, and the capacity of the walkway. Virkler, et al. (6) collected data from some relatively low-volume and high-volume signalized crosswalks and recommended an equation for one-directional flow that also considers platoon size. However they did not consider the impacts of bi-directional pedestrian flows. Golani et al. (7) proposed a model for estimating crossing time considering start-up lost time, average walking speed, and pedestrian headways as a function of the dominant platoon and the opposite platoon separately. The proposed model (Equation (2)) is based on HCM (3) model which was calibrated by using empirical data.

T I

L  Sp

T I

L  Sp

N N    2.09 ped1  0.52 ped 2  w w   N N    0.81 ped1  0.52 ped 2  6.4  w w  

If _ N ped1  5 ped./m (2)

If _ N ped1  5 ped./m

Where Nped1 is the size of the subject platoon and Nped2 is the size of the opposite pedestrian platoon. The proposed model relates the impact of bi-directional flow to the headway between pedestrians when they finish crossing. Therefore it is difficult to see how the interaction is happening and what the resulting speed drop or deceleration is. This study aims to develop a rational methodology that can estimate total pedestrian crossing time as a function of crosswalk geometry, pedestrian demand at both sides of the crosswalk and signal timing. METHODOLOGY The total time needed by a platoon of pedestrians to cross a signalized crosswalk Tt from the beginning of the pedestrian green indication until the pedestrian platoon reaches the other side of the crosswalk can be divided into two main parts: discharge time Td and crossing time Tc. Discharge time Td is the necessary time for a pedestrian platoon to move from the waiting area and step inside the crosswalk. While crossing time Tc is the time which is necessary to cross the crosswalk:

Tt  Td  Tc

(3)

The discharge time Td is a function of pedestrian demand and crosswalk width. The definition of discharge time Td is similar to that of queue discharge time of vehicles waiting at the stop line of a signalized intersection which is usually estimated through shock wave theory, therefore shockwave theory is chosen for modeling pedestrian platoon discharge time.


4

Td

C-g

w

I

Stop line of the crosswalk

D

Stopping shock wave Speed ω1

Space

D D

Un

Starting shock wave Speed ω2 ifo r

m

arr iv

al r ate (

A1 )

ped

δmin

est ria n/s eco nd

Time FIGURE 1 Schematic formation of pedestrian rows at high demand. Crossing time Tc is dependent on pedestrian crossing speed which is affected by the size of opposite pedestrian platoon and crosswalk width. This is analogous to a moving body facing a fluid which causes a reduction in its speed depending on its cross sectional area, the density of the fluid and the relative speed between them. This phenomenon is known as drag force theory and its analogy is used for modeling pedestrian platoon crossing time Tc. For the purpose of this study pedestrian demand is defined as the accumulated number of pedestrians at the edge of the crosswalk during the previous pedestrian flash green and red intervals. MODELING DISCHARGE TIME TD Discharge time Td basically depends on pedestrian arrival rate, pedestrian red interval, and crosswalk width. Shockwave analysis is used to estimate queue discharge time which is equivalent to the time necessary for a pedestrian platoon to discharge at the edge of the crosswalk. Model Assumptions The following assumptions are made for modeling discharge time Td: i) High pedestrian demand is assumed for the model development (FIGURE 1). Later the developed model will be modified to consider the case of low pedestrian demand. ii) Pedestrian arrival rate A1 is assumed to be uniform. Therefore the accumulated pedestrian demand P1 at the beginning of the pedestrian green interval is defined through Equation (4). P1  A1 (C  g )

(4)

Where C is the cycle length and g is the pedestrian green interval. iii) Pedestrian arrival unit is assumed as pedestrian row per second Ar1. FIGURE 1 shows how pedestrian rows are forming at high pedestrian demand case. Assuming that the lateral


5

distance which a pedestrian occupy along the crosswalk width is δ, then the maximum number of pedestrians that can fit through one row Mp along crosswalk width w is estimated through Equation (5). Mp 

w

(5)



To estimate pedestrian arrival rate in the unit of pedestrian row per second, arrival rate A1 is divided by the maximum number of pedestrians that can fit in one row along the crosswalk width w. Ar1 

A1 A  1 Mp w

(6)

Where Ar1 is the arrival rate of the subject pedestrian demand in the unit of pedestrian rows per second. The number of accumulated pedestrian rows Rp at the start of pedestrian green interval is:

R p  Ar1 (C  g ) 

A1 (C  g ) w

(7)

iv) The lateral distance that a pedestrian occupy δ is assumed to be a function of pedestrian demand and crosswalk width. However, for simplification, longitudinal distance between waiting pedestrians D which is the same distance between pedestrian rows is assumed to be constant (FIGURE 1). v) Start-up lost time I is considered as a part of discharge time Td, as shown in FIGURE 1. vi) To apply shockwave theory, the speed of the arrival demand at the sidewalk (pedestrian row per second) and the speed of the discharging demand at the crosswalk should be defined. Speed of the arriving pedestrian rows at the sidewalk (upstream) is assumed to be equal to the pedestrian free-flow speed us at sidewalk. while speed of the discharging pedestrian rows at the crosswalk (downstream) is assumed to be equal to the pedestrian free-flow speed uo at crosswalk, as shown in FIGURE 2. Model Development FIGURE 2 shows that two shockwaves are formed in the waiting area of the crosswalk. The first one is stopping shockwave which results from the continuous arrival of pedestrians to the waiting area that forms new rows. The second one is starting shockwave due to the discharge of the waiting pedestrian rows after pedestrian green indicator is displayed. The speed of the stopping shockwave ω1 is:

1 

Q Q2  Q1  Ar1  A1 w    K K 2  K1 K j  Ar1 u s K j  A1 wu s

(8)


6

Direction of pedestrian flow ω1

ω2

Waiting area Sidewalk (upstream) K1

Q1=Ar1 u1=uc

Crosswalk (downstream)

Q2=0 K2

u2=0 K2=Kj

Stopping shock wave

K3

Q3= Qd u3=u0

Starting shock wave

ω1: speed of stopping shockwave, ω2: speed of starting shock wave, Ar1: pedestrian arrival rate in pedestrian row per second, Qd: discharge rate in pedestrian row per second, us: is pedestrian walking speed at sidewalks, uo: is pedestrian free flow walking speed at crosswalks, Kj: jam density in the unit of pedestrian row per meter.

FIGURE 2 Stopping and starting shockwaves. The speed of the starting shockwave ω2 is:

2 

Qd Q Q3  Q2   K K 3  K 2 Qd uo  K j

(9)

The number of arrived pedestrian rows should be equal to the number of discharged pedestrian rows, therefore:

1 ((C  g )  Td )  2Td

(10)

After substituting Equation (8) and (9) in Equation (10), the discharge time Td becomes:

  A1 w   (C  g )  K  A wu  j 1 s   Td      A1 w Qd    Q u  K   K  A wu j  1 s  d o  j

   

(11)

Parameter Estimation The parameters included in Equation (11) are estimated as follows: i) To define a value for the pedestrian free-flow speed at crosswalks, a 1.5-hour video tape for the crosswalk at the east leg of Nishi-Osu intersection in Nagoya City (6m wide × 25.4m long) was analyzed. Nishi-Osu intersection is characterized by small pedestrian demand with a large crosswalk width. 102 samples of pedestrian’s free-flow speeds were measured. All the considered pedestrians were leading pedestrians and they did not face any opposite flow


7

or turning vehicles. The average free-flow speed for all the samples is uo=1.45 m/s. This value is assumed as the free-flow speed of pedestrians at crosswalks uo. ii) Lam et al. (8) studied pedestrian walking speed at different walking facilities and they found that pedestrian’s free-flow walking speed at outdoor walkways is lower than that of signalized crosswalks by 17%. However, for the purpose of this study pedestrian free-flow speed at sidewalks us is assumed to be 20% less than the free-flow speed at crosswalks. iii) In order to define the jam density Kj, two video tapes (2-hour each) for the crosswalks at the east and west legs of Sasashima intersection in Nagoya City (10m wide) were analyzed. The data was collected in the morning peak hours, when long queues were formed due to the high pedestrian demand. The measured jam densities range between 0.9 – 1.4 ped./m2, while the average is 1.1 ped./m2. The average measured value is assumed as pedestrian jam density Kj. It should be noted that the jam density which is used in the proposed model is the jam density in the unit of pedestrian row per meter. Therefore it is assumed that the minimum lateral distance δmin that a pedestrian can occupy along the crosswalk width is 1.0 meter, including the lateral clearance distance between waiting pedestrians. As a result, the jam density Kj in the unit of pedestrian row per meter (FIGURE 1) is defined in Equation (12).

K j  K j ( ped . / m 2 )   min  1.1 *1.0  1.1

ped.row/m

(12)

iv) Maximum discharge flow rate Qd was observed at crosswalks with very high pedestrian demand. Two video tapes (2-hour each) for the crosswalks at the east and west legs of Sasashima intersection in Nagoya City (10m wide) were analyzed. The average observed discharge rate at high pedestrian demand crosswalks is assumed as the maximum discharge flow rate Qd. The average observed discharge rate is 0.45 pedestrian row per second. After estimating the jam density Kj and the discharge rate Qd, Equation (11) can directly be used to estimate the discharge time Td for high pedestrian demand. However at low pedestrian demand, some modifications should be considered, such as adjusting the lateral distance δ depending on pedestrian demand and crosswalk width. Modification for Low Pedestrian Demand Case At low pedestrian demand many factors such as pedestrian origin and destination can affect pedestrian decisions regarding the waiting position which makes pedestrians form different rows in irregular patterns. Pedestrian arrival rate is assumed to be uniform. FIGURE 3 shows how the pedestrian rows are forming when pedestrian demand is low. The number of pedestrians that forms a row decrease as pedestrian demand decreases, which means δ increases as pedestrian demand decreases. Equation (11) is utilized by using the empirical data collected at Nishi-Osu and Sasashima intersections to estimate the average lateral distance that a pedestrian can occupy δ at different demand values. Then δ is modeled as a function of pedestrian demand per meter width of the crosswalk. FIGURE 4 illustrates the relationship between the average occupied width by one pedestrian δ and pedestrian demand per meter width of the crosswalk. A preliminary statistical analysis was performed to determine the best function to represent the relationship between δ and pedestrian demand per meter width of the crosswalk. The power function was found the best to describe this relationship which is shown in Equation (13).


8

Td

C-g

w

I

Stop line of the crosswalk

D

Space

D

Un

ifo r

m

δ

D

Stopping shock wave Speed ω1

δ

Starting shock wave Speed ω2 arr iv

al r ate (

A1 )

ped

est ria n/s eco nd

Time FIGURE 3 Schematic formations of pedestrian rows at low demand. 8

Occupied lateral distance by one pedestrian δ (m)

7.5

7

P w

  2.5323( 1 ) 0.383

6.5 6 5.5 5

R 2  0.7981

4.5

4 3.5 3 2.5 2

1.5 1 0.5 0 0

1

2

3

4

5

6

7

8

9

10

Pedestrian demand per meter width of the crosswalk (ped./m)

FIGURE 4 Relationship between average occupied width by one pedestrian and pedestrian demand per meter width of the crosswalk. P w

  2.53( 1 ) 0.383  2.53(

A1 (C  g ) 0.383 ) w

(13)

FIGURE 4 shows that when pedestrian demand becomes high, δ becomes very close to 1.0 meter which is in accordance with the previous assumption that δmin is equal to 1.0 meter. After substituting Equation (13) in Equation (11), the resulted formula can be used to estimate discharge time Td for any pedestrian demand volume and crosswalk width (FIGURE 11b)). MODELING CROSSING TIME Tc The force on an object that resists its motion through a fluid is called drag. When the fluid is a gas like air (FIGURE 5a)), it is called aerodynamic drag (or air resistance). While if the fluid is a


9

Moving Body

Air Stream

Ap

ρ

a) Drag force Air or Fluid

P2

Moving Body

u2

u1

w

P1

Lo Opposite Pedestrian Flow

Subject Pedestrian Flow

w: crosswalk width (m), Lo: crosswalk length, P2: opposite pedestrian demand, P1: subject pedestrian demand, u1: speed of the subject pedestrian flow, and u2: speed of the opposite pedestrian flow.

b) Pedestrian demand at both sides of the crosswalk FIGURE 5 Applying drag force concept on bi-directional pedestrian flows at crosswalk. liquid like water it is called hydrodynamic drag. Drag is a complicated phenomenon and explaining it from a theory based entirely on fundamental principles is exceptionally difficult. Pugh (9) described the relation of drag D, the relative velocity of the air or the fluid, and a moving body in terms of a dimensionless group, the drag coefficient Cd. The drag coefficient is the ratio of drag D to the dynamic pressure q of a moving air stream and is defined by Equation (14): D  Cd qAp

(14)

Where D is drag force (kg·m/s2), Cd is drag coefficient (dimensionless), q is dynamic pressure (force per unit area), and Ap is the projected area (m2). The dynamic pressure q which is equivalent to the kinetic energy per unit volume of a moving solid body (Pugh (10)) is defined by Equation (15):

1 q  u 2 2

(15)

Where ρ is density of the air in kilogram per cubic meter, and u is the speed of the object relative to the fluid (m/s). By substituting Equation (15) in Equation (14), the final drag force equation is:


10

1 D  Cd ρu 2 Ap 2

(16)

’Drag Force’ Caused by the Opposite Pedestrian Flow To use the drag force concept to model the interactions between pedestrian flows, the following assumptions are made: i) Opposite pedestrian demand is considered as a homogenous flow (FIGURE 5b)) with a density equal to the number of pedestrian waiting in the beginning of the green interval divided by an area equal to the width of the crosswalk multiplied by 1.0 meter.

 

P2 w *1

ped. / m2

(17)

ii) The subject pedestrian flow is considered as one body moving against the opposite pedestrian flow. The interactions occur along the projected area of all pedestrians in the subject flow which is defined as the sum of the widths of all pedestrians in the subject flow:

Ap  n

(18)

Where Ap is the projected area of the subject pedestrian flow (m), β is the average body width of one pedestrian, and n is a dimensionless number equal to the number of pedestrian in the subject pedestrian flow P1, shown in FIGURE 5b). iii) The initial speed of the subject and the opposite pedestrian flow when they start crossing is assumed to be equal to the free-flow speed uo, therefore the relative speed u becomes:

u  (u1  (u2 ))  uo  uo  2uo

(19)

After substituting Equation (17), (18), and (19) in Equation (16), then the drag force equation becomes: D

1 P Cd 2 (2uo ) 2 n 2 w

(20)

Assuming that the average width of one pedestrian body β is 0.6m, the drag force D becomes:

D

P P 2 1 1 * 0.6 * Cd 2 4u02 n  CD adj 2 uo n 2 w 2 w

(21)

Where CDadj is adjusted drag coefficient (dimensionless), and it is defined according to Equation (22).

C D adj  4Cd  4 * 0.6 * Cd

(22)


11

Assumed Interaction Time

u2=uo

CL 0.5Lo

Distance

P2

Real Interaction Time

u1=uo

P1 Lo/2uo

Tc-(Lo/2uo)

Time Lo: crosswalk length, P2: opposite pedestrian demand, P1: subject pedestrian demand, u1: speed of the subject pedestrian flow, u2: speed of the opposite pedestrian flow, uo: free flow speed, and Tc: time needed by subject pedestrian demand to cross the crosswalk.

FIGURE 6 Time-Space diagram of the conflicting pedestrian flows.

Deceleration of the Subject Pedestrian Flow The net force on a particle observed from an inertial reference frame is proportional to the time rate of change of its linear momentum (Momentum is the product of mass and velocity):

F

dM d (mv) dv   m  ma dt dt dt

(23)

Where m is the mass of the moving body and its equivalent to the subject pedestrian demand P1, and a is the average deceleration of the subject pedestrian flow. The final speed of a moving particle in a straight line with constant average deceleration according to the motion equations is:

u 2f  ui2  2aL

(24)

Where ui is initial speed (m/s) which is assumed to be equal to the free-flow speed uo, uf is final speed (m/s), a is average deceleration of the subject pedestrian demand (m/s2), L is the travelled distance (m). FIGURE 6 shows the projection of pedestrian flow trajectory from both sides of a crosswalk. A major assumption of this methodology is that both opposing flows will start walking with the same free-flow speed uo in a straight line until the middle of the crosswalk where they will meet. In order to avoid the complexity in estimating the real interaction time, the time from the moment when the subject pedestrian flow meets the opposite pedestrian flow at the


12

middle of the crosswalk until the subject pedestrian flow reaches the end of the crosswalk is assumed as the interaction time here. The resulting deceleration is averaged along the assumed interaction time. Therefore the final speed is:

uf 

Lc  Lo 2 Tc  Lo 2uo

(25)

Where Lc is average trajectory length of the subject pedestrian demand, Tc is average crossing time of the subject pedestrian demand, Lo is crosswalk length and uo is free-flow speed. By substituting Equation (25) in Equation (24), the average deceleration of the subject pedestrian demand becomes:

 L  Lo 2   uo   c Tc  Lo 2uo  1  a 2 Lc  Lo 2

2

2

(26)

By substituting the mass and the acceleration in Equation (23), the net force (ped.·m/s2) becomes:  L  Lo 2   u o   c Tc  Lo 2u o  1  F  ma  P1 2 Lc  Lo 2

2

2

(27)

Model Development The drag force caused by an opposite pedestrian flow should be equal to the force that causes the deceleration. By equating Equation (27) and Equation (21), and after solving them for the crossing time Tc, the net equation is: ( Lc 

Tc  uo2 

Lo ) 2

CD adj P2uo2 ( Lc 

Lo ) 2



Lo 2uo

(28)

w

Pedestrian demand is defined as the number of accumulated pedestrians during pedestrian red and flash green signal indications, and those who arrive during the discharge time. Therefore opposite pedestrian demand can be presented as: P2  A2 * (C  g  Td )

(29)

Where P2 is opposite pedestrian demand, A2 is arrival rate of the opposite pedestrian demand (ped./s), C is cycle length, g is pedestrian green interval, and Td is discharge time of the opposite pedestrian platoon.


13

0.05

C Dadj  0.04019 * ( r 0.79086 ) R 2  0.42

0.045

0.04 0.035

Sample size is 35 data points

CDadj

0.03 0.025 0.02 0.015 0.01

Measured values Model

0.005 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Split ratio r

FIGURE 7 Modeling drag coefficient as a function of split ratio r. After substituting Equation (29) in Equation (28), the average crossing time and walking speed of the subjected pedestrian flow are given in Equations (30) and (31) respectively. L ( Lc  o ) L 2 Tc   o 2u o (30) L C D adj A2 u o2 ( Lc  o )(C  g  Td ) 2 u o2  w

uf  u  2 o

CDadj A2uo2 ( Lc  Lo 2)(C  g  Td )

(31)

w

Equations (31) and (30) are the final equations which represent how the walking speed and the crossing time vary according to pedestrian demand combinations from both sides of the crosswalk and crosswalk geometry. Estimating the Drag Coefficient CDadj The value of adjusted drag coefficient CDadj according to aerodynamic drag is dependent on the kinematic viscosity of the fluid, projected area and texture of the moving body. In the pedestrian’s case, this value can here be assumed to be dependent on the pedestrian demand at both sides of the crosswalk and their split ratio. To estimate speed drop and crossing time by using Equations (31) and (30), CDadj was first estimated from empirical data. A 2-hour video tape for the crosswalk at the east leg of Imaike Intersection in Nagoya City (9.6m wide × 21.5m long) was analyzed. The pedestrian demand in each cycle at each direction, the average pedestrian trajectory length, and the average pedestrian crossing time in the same cycle were extracted from the video tape. Then by using Equation (30), CDadj was estimated for 35 data point where the total pedestrian demand was ranging from 5 – 30 pedestrians per cycle (one cycle is 160s). In any case when pedestrians counter a turning vehicle that causes a reduction on their speed or a change on their trajectory, the whole cycle is neglected and removed from the data base. Furthermore if pedestrians walk outside the crosswalk, that cycle is also neglected. After analyzing the available data, CDadj was


14

40

P1 : Subject pedestrian demand, P2 : Opposite pedestrian demand, uo : Free flow speed is 1.45 m/s, Lo : Crosswalk length is 20 m.

Crossing time Tc (seconds)

38 36 34 32

P1,P2 =10,10 P1,P2 = 10,15 P1,P2 = 10,20 P1,P2 =10,30

30 28 26 24 22

20 18

16 14 12 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Crosswalk width w (m)

FIGURE 8 Changes in crossing time with changing demand and crosswalk width. modeled in terms of the split ratio r which is the ratio of the subject pedestrian demand to the total pedestrian demand. Equation (32) defines the split ratio r.

r

P1 P1  P2

(32)

FIGURE 7 shows the relationship between split ratio and CDadj. As split ratio increases the drag coefficient also increases. Due to the limited sample size and the existence of many external factors that can affect pedestrian behavior, the coefficient of correlation (R2) is relatively small. SENSITIVITY ANALYSIS After estimating the drag coefficient, Equation (30) can directly be used to calculate the average crossing time Tc for different demand volumes under different crosswalk widths. FIGURE 8 shows how the crossing time varies with crosswalk width. When crosswalk width becomes larger for a specific demand, crossing time decreases until it becomes almost constant (free-flow condition). But when crosswalk width becomes smaller for a specific demand, crossing time increases, as the interactions between the opposing flows increases, until it reaches a point where the opposing flows block each other causing a drastic increase in crossing time. FIGURE 9a) shows the drop in average walking speed due to the effects of bi-directional pedestrian flow. As the crosswalk width decreases for a specific pedestrian demand, the interactions increase causing reduction in the average walking speed. The drop in the walking speed continues with reducing crosswalk width until a point where the speed drops drastically. This tendency is reasonable if we assume that pedestrian cannot walk outside the crosswalk. Therefore it is expected that as the demand increases for a specific crosswalk width, the average walking speed will drop, until it reaches almost zero where every pedestrian cannot walk any more. Split ratio is one of the important factors that affect pedestrian crossing time. FIGURE 9b) shows the reduction in pedestrian walking speed due to the increasing in the opposite pedestrian demand. Therefore as split ratio decreases the average pedestrian speed also decreases.


15 1.6

Average pedestrian speed (m/s)


1.6

1.4

1.2

1

P1 : Subject pedestrian demand, P2 : Opposite pedestrian demand, uo : Free flow speed is 1.45 m/s, Lo : Crosswalk length is 20 m.

0.8

0.6

P1,P2 = 10,10 P1,P2 = 10,15 P1,P2 =10,20 P1,P2 = 10,30

0.4

0.2

Subject pedestrian demand P1 is constant = 10 ped., uo : Free flow speed is 1.45 m/s, Lo : Crosswalk length is 20 m.

1.4

1.2

1

0.8

w=3 m

0.6

w=4 m

0.4

w=5 m 0.2

w=6 m

0

0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

0

15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Split ratio r

Crosswalk width w (m)

a) Effect of pedestrian demand

b) Effect of split ratio


FIGURE 9 Average walking speeds, pedestrian demand and split ratio. 1.50

w = 2.5 m w=3 m w=4 m w=5 m w=6 m

1.40 1.30 1.20

1.10 1.00 0.90

0.80 0.70 0.60 0.50 0.40

0.30 0.20 0.10 0.00 60

70

80

90

100

110

120

130

140

150

160

Cycle length (seconds)

Assumptions: Free-flow speed uo is 1.45m/s, crosswalk length is 20m, pedestrian arrival rate at each side of the crosswalk is 0.2 ped./sec, split ratio is 0.5, and pedestrian green ratio is 0.25.

FIGURE 10 Average walking speed and cycle length. FIGURE 10 shows how the walking speed drops as a result of increasing the cycle length of the signalized intersection. As the cycle length increases the accumulated pedestrian demand at both sides of the crosswalk increases, causing more interactions between conflicting pedestrian flows which lead to a drop in the average walking speed. VALIDATION AND COMPARISION To validate the proposed models, the average crossing time was measured for 38 samples under different demand ratios and compared with the estimated crossing time from the proposed model. FIGURE 11a) illustrates the differences between measured and estimated crossing times. A paired t-test was performed and the result showed that the estimated values were not significantly


16

Estimated crossing time (seconds)

18

Absolute percentage error range (0.05% - 9.48%) Mean absolute percentage error ( 1.77% ) Root mean square error (RMSE) = 0.44

17.5

17

16.5

16

15.5

15

14.5

14 14

14.5

15

15.5

16

16.5

17

17.5

18

Measured crossing time (seconds)

a) Crossing time model validation 24

observed HCM (2000) Manual on Traffic Signal Control (2006) Proposed model

22

Discharge time (seconds)

20 18

Mean absolute percentage error = 25 % Root mean square error (RMSE) = 2.00

16

14


12 10 8 6


4 2 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

Subject pedestrian demand per meter width of the crosswalk (ped./m)

b) Discharge time model validation FIGURE 11 Validations of the proposed models. different from the observed values at the 95% confidence level. The proposed model produces a mean absolute percentage error of 1.8%. The discharge time Td estimated by Equation (11) is compared with the observed data and the estimated discharge time from the existing formulations in HCM (3) and Japanese Manual on Traffic Signal Control (4), as shown in FIGURE 11b). By comparing the mean absolute percentage error and the root mean square error (RMSE), the proposed model which is based on shockwave analysis, produces more accurate and reliable results. Furthermore the existing


17

formulations tend always to undersestimate the necessary discharge time for a pedestrian platoon. The tendency of the proposed discharge time model is more consistent with the observed data. CONCLUSIONS AND FUTURE WORKS A new methodology for modeling crossing time considering bi-directional pedestrian flow and discharge time necessary for a pedestrian platoon at signalized crosswalks is proposed in this paper. The proposed methodology provides two different formulations for the crossing time Tc and the discharge time Td. Shockwave and drag force theories are successfully utilized for modeling the discharge time and the crossing time respectively. The final formulation of crossing time Tc provides a rational quantification for the effects of crosswalk geometry and bi-directional pedestrian flow on walking speed and crossing time. However, the proposed models do not consider the effects of age and trip purpose on crossing and discharge times. Therefore calibrating the developed models for the effects of age and trip purpose will expand their applicability to different cases such as signalized crosswalks at school zones or crosswalks with high aged pedestrian activities. The nature of the drag coefficient CDadj is a key factor in estimating crossing time. Collecting and analyzing more data are necessary to develop more concrete formula for the drag coefficient with higher correlation.. The application of the proposed methodology is to rationally define the required crosswalk width under different pedestrian demand volumes considering the reduction in walking speed due to an opposite pedestrian flow. Moreover for other practical applications including the estimation of pedestrian crossing speed, crossing and discharge times, the developed models can be simplified and represented in a feasible forms.

REFERENCES 1. Highway Capacity Manual. Transportation Research Board, National Research Council, Washington, D.C., 2000. 2. Pignataro, L. J. Traffic Engineering: Theory and Practice. Prentice-Hall, Englewood Cliffs, N.J., 1973. 3. Manual on Uniform Traffic Control Devices for Streets and Highways. FHWA, U.S. Washington, D.C., 2003. 4. Manual of Traffic Signal Control. Japan Society of Traffic Engineers, 2006. 5. Lam, William H.K., Lee, Jodie Y.S., Chan, K.S., Goh, P.K.: A generalized function for modeling bi-directional flow effects on indoor walkways in Hong Kong. In Transportation Research Part A 37; pp. 789-810, 2003. 6. Virkler, M. R., and Guell, D. L.: Pedestrian Crossing Time Requirements at Intersections. In Transportation Research Record 959, TRB, National Research Council, Washington, D.C., pp. 47–51, 1984. 7. Golani, A. and Damti, H.: Model for Estimating Crossing Times at High Occupancy Crosswalks. In Transportation Research Board, TRB, Annual Meeting, 2007. 8. Lam, William H. K., Cheung, C.-y.: Pedestrian Speed/Flow Relationships for Walking Facilities in Hong Kong. In Journal of Transportation Engineering, Vol. 126; Part 4, pp. 343-349, 2000. 9. Pugh, L. G. C. E.: The influence of wind resistance in running and walking and the mechanical efficiency of work against horizontal or vertical forces. In Journal of


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Physiology, 213; pp. 255-276, 1971. 10. Pugh, L. G. C. E.: The relation of oxygen intake and speed in competition cycling and comparative observations on the bicycle ergo meter. In Journal of Physiology, 241; pp. 795-808, 1974.