SA. a) Lb DW — Cts

An Observational Study of Freeway Lane-Changing Behaviour.

by

M. Rafik Nemeh

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Civil Engineering (Transportation Division)

APPROVED:

SA. a) Dr. Siamak A. Ardekani (Chairman)

Lb DW — Cts Dr. Richard Walker VV Dr. Toni Trani

December, 1988

Blacksburg, Virginia

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LD 5655 V85S 1988 (433

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An Observational Study of Freeway Lane-Changing Behaviour. | by

M. Rafik Nemeh

Dr. Siamak A. Ardekani (Chairman)

Civil Engineering (Transportation Division)

(ABSTRACT)

Every one who has driven on a freeway has observed the phenomenon of lane-

changing. This phenomenon is, of course, caused by the desire of most of the drivers not

to be in a slow-moving lane. Therefore, the average driver who finds himself in such a

lane moves into a neighboring faster lane, usually after a certain time-lag. This time-lag

depends on the dynamic characteristics of the vehicle, the availability of acceptable gaps,

and the driver risk, which is the value the driver places on the probability of collision

during a maneuver, i.e. the higher the perceived probability of collision, the higher the

time-lag.

Modelling of the lane-changing phenomenon has been the objective of many inves-

tigators in the past. As will be shown later in this study, lane-changing is a very impor-

tant component in highway traffic flow.

In this study, a mathematical model to describe the lane-changing behaviour is

suggested based on the lane-changing hypothesis that whenever there is a lane-changing

maneuver, the average speed of the neighboring lane is faster than the average speed of

the current lane.

A set of data has been collected by a methodology which involves aerial photo-

graphic technique. The collected data are then used to test the validity of the lane-

changing hypothesis, to calibrate and validate an existing lane-changing model, and to

develop a gap acceptance function for freeway lane-changing maneuvers.

Acknowledgements

I would deeply like to thank Dr. S. Ardekani for his limitless valuable suggestions

and comments. I also appreciate Dr. R. Herman’s valuable discussions with Dr. S.

Ardekani; it proved to be helpful.

Sincere appreciation is expressed to Dr. S.D. Johnson for his kind assistance in us-

ing the Mann Mono-Digital Comparator.

I am grateful to the member of my committe, Dr. R. Walker and Dr. T. Trani, for

their support and guidance.

I am also grateful to my parents for making my education all these years possible.

Gratitude is also expressed to my friends who supported me with thier encourage-

ment.

Acknowledgements iii

Table of Contents

1.0 Background And Study Objectives ....... 0.2 ce cece ee ec ee eee eee eee eeee 1

1.1 Introduction 2.0... cece ee eee ee ee ee eee ee eee eee teens ]

1.2 Study Objectives . 0... ccc ec eee eee eee ee eee ete een eens 2

2.0 Aerial Photographic Observations ........... cc cece ec eee e eee e cree re eeeene 4

2.1 Introduction ©. 1... . ce ee ee eee ee eee ee ee tee eens 4

2.2 Specifications 2.0... cee ee eee ee eee ee ee eee eee eens 5

2.3 Reduction 2.0... 0... cece eee ee eee eee e eee n eens 5

2.3.1 Speed Measurement ........ 0... ec ec eee ee eee eee tenet eens 9

2.3.2 Sources of Errors 2.2... ec eee eee tee tee eee e nes 11

3.0 DATA ANALYSIS 2.0... 0. cece ccc ccc eee ee eter eee eet e eee e eens 19

3.1 Review of An Existing Model .... 2.0... ccc ee eee eee eens 19

3.2 Testing the Lane-Changing Hypothesis ........ 0... 0 ccc cece eee eee cece eees 21

3.3 A Gap Acceptance Function For Freeway Lane-Changing Maneuvers ............. 27

3.3.1 The Model 0... cece eee ec ene eee eee teenies 28

Table of Contents iv

4.0 Theory oo. ccc ccc cee ee cere eee eee teehee eee eee eee ee eee eee eens 32

4.1 The proposed model ... 0... cece ee eee ee eee teen nas 32

4.2 Calibration of the proposed model ....... 0... 0... cece ee eee eee ee 34

5.0 Applications 2.0... ccc ccc cc ec eee ee eee eee eee eee ee eee eee eee eens 39

5.1 Simulation problem 2.0... 0.0 eee ee nee e teen eee e eens 39

5.2. Delay problem 2.1... cc eee ee ee rt eee eee eee 40

6.0 Discussions and Recommendations ..........: ccc v creer c cree crease eeseeees 46

Appendix A. Data Obtained from Photographs ......... 0c ces eee e cece cece eceeees 49

Appendix B. Computer Program ...... 0... ccc ce cee eee eee ee eee renee eens 57

REFERENCES 2... cc ee ee eee ee ewes bem e ence eee e eee ee eeees 58

a 61

Table of Contents ¥

List of Illustrations

Figure Jo. ce ec ee eee ee ee ee eee ee nee eee tee

FIg“ure 2. ieee ee ee ee ee ee eee ee eee eee eee eens

Figure 3. cic eee ce eee ee ee ee eee eee eee eee ees

Figure Go cic ee ee ee ee ee ee ee eee eee eee

Figure 5. ee eee ee ee ee ee eee eee ee eee eee

Figure 6. ccc ec ee eee ee ee ee eee ee ee eee ee ee eee eee

Figure Joc cee ee ee eee eee ee ee eee ees

Figure 8. cic ee ee ee eee ee ee ee ee eee eee eee ee

Figure 9. eee ee eee ee eee eee eee eee ees

Figure 10, fc cc ee ee ee eee eee ee ee eens

List of Illustrations

List of Tables

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

1. The angles of rotation between master and conjugate frames. ........ 13

2. The coordinates and the apparent velocities of parked vehicles. ....... 17

3. Qand K data, .. cee ee ee ee ees eee eee 22

4. Analysis of variance. . 2... ee ee eee eee ee eee 24

5. Average Speeds in(mph) 1... cc ccc cee ee eee eee ees 26

6. Chi-square test analysis. 26... cece eee eee eee eee 31

7. Number of cases of lane-changes. 2... 0... cece cee eee te ee ees 36

8. Arbitrary data to illustrate the calibration of the model. ............ 38

9, Current lane (Veh No. Sand 6) 2... ce ee ee ees 49

10. Neighboring Lane (Vehs No. 3 and 4) 2... . cee ce eee eee 51

11. Coordinates of the control points 2.6... .... eee eee eee eens 53

12. Coordinates of vehicle number 2. 0... . eee ee ee eee eee 54

13. Concentration data 2... eee eee eee ens 55

14. Gap acceptance distribution data 2... .. .. eee ee ee ee eens 56

List of Tables Vii

1.0 Background And Study Objectives

1.1) Introduction

Modelling of the lane changing phenomenon between lanes of a unidirectional

traffic has been extensively used in simulation of traffic interactions along a roadway as

well as in understanding the dynamic characteristics of a group of vehicles along a

highway section.

The work by D. Gazis, R. Herman and G. Weiss (Ref 1) has been one of the ear-

liest attempts to model lane-changing phenomenon. The model expresses the rate of

change of lane densities as a function of a sensitivity coefficient and the relative lane

density at time t and at equilibrium. The model addresses questions of stability of flow

along a multilane roadway as a function of products of the sensitivity coefficient and the

time lag involved, much like the analysis of car- following models (Ref 2) . A limitation

of Gazis, et al (Ref 1) is the assumption that oscillations in lane densities take place

about an equilibrium density distribution. Munjal and Pipes (Ref 3) relaxed this as-

sumption by incorporating a conservation equation of flow into the Gazis, et al model.

Background And Study Objectives 1

Makigami, et al (Ref 4) adopted Munjal and Pipes work as well as a stochastic de-

scription of lane-changing as a Markov process (Ref 5) to develop a model for uncon-

gested flow conditions. An embedded assumption of stochastic models (Refs 4,5) is the

existence of steady state conditions since the transition probability matrices used in the

Markov process are not time dependent. Michalopoulos and Beskos (Ref 6) continued

the work of Gazis, et al and Munjal and Pipes to develop a set of three macroscopic

continum models. The first model employs a separate conservation equation for each

lane, while the second mode! employs a single equation for all lanes but considers the

street width as well. Finally, the third model considers the street width as well as a

momentum equation to account for exchange of momenta among lanes.

1.2 Study Objectives

A major handicap in the works reviewed in the introduction is the lack of field ob-

servations to validate and calibrate the mentioned models. This limitation arises mainly

due to the structural complexity of the models. Field observations of variables such as

lane density and its oscillations require repeated observations employing elaborate data

collection techniques such as aerial photography taken frequently and under variety of

traffic conditions. Thus, the need exists for a simple macroscopic lane-changing model

which is easy to calibrate based on field observations.

The main objective of this research is the development of a lane changing model

which expresses the number of lane changes per unit distance from one lane to the next

as a function of the speed differential between each pair of lanes.


The lane-changing model, as will be discussed in more details in chapter four, is

suggested based on the observation that the stimulus of the lane-changing maneuver is

the reduction in the speed rather than the increase of concentration of the current lane.

Although higher concentrations imply lower speeds, but there are cases where the drivers

in the current lane perceive reduction in the speed while the concentration has not ac-

tually changed. A case in point is when passenger car drivers change lane simply because

of the presence of a truck in their lane. So such influences on individual perception can

be taken care of through the error terms introduced in the model.

In this work, analysis of aerial phtographs in pursue of the above-mentioned ob-

jective is introduced in chapter two. Review of an existing lane-changing model, statis-

tical analysis of the data collected , and the development of a gap acceptance function

for freeway lane-changing maneuvers are discussed in details in chapter three. The pro-

posed model as well as the calibration procedure is discussed in chapter four. Applica-

tions of the results are given in chapter five. Finally, discussions and recommendations

are presented in chapter six.


2.0 Aerial Photographic Observations

2.1 Introduction

Aerial photographs, although cumbersome and time consuming to reduce, are

means of collecting a very large amount of traffic data in a short period of time. Time-

lapse aerial photography has been used over the years in various traffic studies.

In freeway studies aerial photography has been employed, for example, to determine

the effect of bottlenecks on freeway traffic (Refs 7,8,and 9), to make estimates of travel

time delay and accident experience due to freeway congestion (Ref 10), to study merging

freeway operations (Ref 11) and freeway interchange operations (Ref 12), to determine

the traffic flow characteristics of a facility (Refs 13,14,15 and 16), and to study headway

and speed distributions and their correlation in freeway traffic (Ref 17).

In the arena of non-freeway traffic, aerial photographs have been used, for example,

to measure the vehicular concentration in a network of streets (Ref 18), to conduct

origin-destination surveys (Ref 19), to perform parking studies (Ref 20), and to measure

the effectiveness of traffic control systems in a network (Ref 21).

Aerial Photographic Observations 4

In the present work data have been collected from aerial photographs to calibrate

and validate an existing lane-changing model, to test the validity of the hypothesis that

whenever there is a lane-changing maneuver the average speed of the neighboring lane

is faster than the average speed of the current lane, and to develop a gap acceptance

function for freeway lane-changing maneuvers.

2.2 Specifications

The aerial photographs were taken in 1982 by the Texas State Department of

Highways and Public Transportation along the Interstate Highway 35 through down-

town Austin as well as along the Interstate Highway 30 through downtown Dallas.

A Cessna 206 turbo-engine aircraft, a 153.28 mm RC10 wild Lens cone camera, and

a 9” by 9” diapositive color film were used. The aircraft was flying at 120 mph at an al-

titude of 3000 feet above street level.

2.3 Reduction

The first step in the reduction of the aerial photographs was to locate vehicles in-

volved in a lane-changing maneuver. A light table was used for this purpose. Care was

taken to identify and exclude those lane-changes for the purpose of exiting the freeway.

Vehicles changing-lane were located by visual inspection. Each pair of consecutive

frames were inspected for lane-changing maneuvers at the same time. It should be

Aeria] Photographic Observations 5

pointed out that, at low speeds, vehicles changing-lane could be detected easily due to

the fact that the vehicle changing-lane might be located at the center of the two lanes.

At high speeds, however, extra effort was required to locate a vehicle just having com-

pleted a lane-change since the vehicle would have moved from one normal position on

the first frame to another normal position on the next frame during the photo inter-

exposure time.

The second step in the reduction of the aerial photographs was to assign a scale to

each frame. It must be noted that aerial photographs are perspective projections so that

a location with higher elevation is closer to the camera lens and thus its image has a

greater scale (Ref 22). However, for relatively flat topography one may assume a con-

stant average scale for each photographic frame. As will be discussed later in the section

on sources of errors, this is not an unreasonable assumption for the study areas in

Austin and Dallas.

The scale determination for each frame was made through measuring photo dis-

tances between fixed monuments on the ground (control points) as well as the ground

distances between those points. The control points were selected after the photographs

were taken. The ground measurements were made by Herman et al (Ref 5,10, and 23)

using a Keuffel & Esser Electronics Distance Measurement apparatus. The altitude of

the aircraft has varied with respect to a different set of photographs and that caused the

range of scale to vary from 1” = 471.2’ to 1” = 537.3’. As will be shown later, 1” = 500°

is a reasonable scale to be used in the calculation presented later in chapter 3.

A Mann Mono-Digital comparator was used to determine the cartesian coordinates

of the location of each vehicle shown in Figure 1 as well as to determine the coordinates

of the control points. The comparator was interfaced with an IBM computer to save the

coordinates of vehicles in a data file. The control points were fixed objects shown in both

the master and the conjugate frames. The position of each vehicle was represented by


eee —| em Oee aeeaea Se

Figure 1. : Illustration of vehicles positions

Aerial Photographic Observations

the x,y coordinates of its left front corner. Table 9 through 14 in appendix A provide the

data obtained from the photographs. Table 9 shows the coordinates of the two front

vehicles in the current lane, namely, vehicle numbers 5 and 6. Table 10 shows the coor-

dinates of the two front vehicles in the neighboring lane; vehicle numbers 3 and 4. The

average speeds in the current lane and the neighboring lane are computed using the

values in Table 9 and 10, respectively. Table 11 shows the coordinates of the control

points. Table 12 shows the coordinates of vehicle number 2. The duration of gap ac-

ceptance is computed from the speed of vehicle number 2 which is shown in Table 14.

To determine the concentration in the current lane and the neighboring lane, the

group of vehicles infront of the lane-changing vehicle is counted and the distance occu-

pied by that group of vehicles is also measured. The data are shown in Table 13. The

distance between vehicles number 2 and 3 has been measured for the gap acceptance

distribution analysis and is recorded in Table 14.

The determination of the elapsed time between two consecutive photographs, 6dr,

was made by reading the image of a clock on each frame. The close divisions were to

the nearest second and were interpolated to the nearest tenth of a second. It must be

noted that an error of 0.1 seconds in 6f, would only result in a 3 to 5 percent error in the

value of mean speed for the corresponding pair of frames. As can be seen later, since

speed differentials between two neighboring lanes are of primary interest, such small

systematic errors are not critical.


2.3.1 Speed Measurement

To determine vehicle speed, the photo displacement of each vehicle, A/ , was

measured as:

Al = [(X,-X,)°+(¥, - ¥,) 1"? (2.1)

where (X,,Y;) and (X,,Y,) are the coordinates of the vehicle on the master and the

conjugate frames , respectively. The above computation assumes that vehicles travel on

a straight line during the elapsed time between two consecutive exposures.

Given the short elapsed time, dt, between successive frames, the above assumption

does not introduce a significant error in the measurements. For example (Ref 23), for a

6t of 2.5 seconds and a scale of 1” = 500’, a vehicle moving at 30 mph would travel only

110 feet corresponding to a displacement of 5588 microns on the photographs. Let us

now assume that the above vehicle has been actually travelling at 30 mph along the

zigzag path ABCDE, rather than the straight path ACE, as shown in Figure 2 (Ref 23)

which schematically depicts a hypothetical case of an extreme lane-changing maneuver.

Then the actual distance travelled in 2.5 seconds in AB+ BC+ CD+ DE= 110’ while the

photographic estimate of the travelled distance is ACE= 108.72’, a discrepancy of only

1.3 percent.

Once a Al is determined in microns using the above procedure, it is converted to the

distance travelled on the ground, AL, through the relation AL = Al x (scale in feet per

microns), However, in measuring, AL , it is, of course, necessary that the coordinates

of a vehicle on the master and the conjugate frames be measured with reference to a

common coordinate system. This was achieved through the transferring of the cartesian

coordinate system of the conjugate frame to that of the master frame, using the fixed


SIN (160°) a ———— ‘ « ‘ 27.8' x Soy 1 94-36

—< 2 x $4.36' « 108.72' |

The schematic path of a vehicle conducting an extreme lane- changing maneuver during the 2.5 seconds elapse time between two successive photographs. The hypothetical vehicle has travelled the zigzag path ABCDE (a distance of 110 feet at running speed of 30 mph) while the photo reduction procedure

Figure 2. : has measured the length ACE = 108.72 feet (Ref 23).


control points that appeared on both frames. The transfer of coordinates allowed for not

only a shift in X and Y but also a rotation of the conjugate coordinate system relative

to the master coordinate system, as shown schematically in Figure 3 (Ref 23).

Once the three transformation parameters X, ,¥Y,,8 were determined the vehicle co-

ordinates on the conjugate frame, (X’,Y’), were expressed relative to the master frame

coordinate system, namely,

X = X’cos (0) — Y’sin (8) + Xp (2.2)

Y = X'sin (6) + Y’cos (0) + Yo (2.3)

The angles, as shown in Table 1, were computed and found to be very small so the

effect of rotation was not significant in the analysis.

2.3.2 Sources of Errors

The vehicle speeds obtained in the manner decribed in the previous section are

subject to errors from a number of different sources (Ref 24). The more significant of

these sources are the non-level topography of the test area, parallax, relief displacement,

tip and tilt of the airplane, and the operator.

The non-flat topography of the area is a major source of error. Unlike a map which

is an orthographic projection and has a uniform scale, an aerial photograph is a per-

spective view and its scale varies from point to point due to variation in terrain elevation

in addition to parallax, etc. For example, the areas on the photograph with higher

ground elevations are closer to the lens of the camera and thus have a larger scale, since

scale = (focal length of the camera)/(aircraft altitude - ground elevation). As a result

Aerial Photographic Observations 1!

\ _ x!

\ a

\ 0’ — — —

_—— x Xo.Yo)

ae [. \ _ \ — 0 \ xX

\ \

Schematic diagram showing the relative positions of cartesian coordinate systems of a pair of successive aerial photographs. In transferring the coordinates of one frame to the other a shift in origin of (.X,, Y,) as well as a rotation 6 of the conjugate coordinate system relative to the master coordinate

Figure 3. ; System was assumed (Ref 23).


Table 1. The angles of rotation between master and conjugate frames.

Case No. The angles

(Degree)

l 0.02044 2 -0.01769 3 -0.00655 4 -0.02332 5 -0.01741 6 -0.00659 7 0.014007 8 0.021146 9 -0.00838 10 -0.00225 11 0.087017 12 0.080403 13 0.020352 14 0.023851 15 0.014019 16 -0.05018 17 0.045280 18 0.063834 19 0.060679 20 0.021083 21 0.008597 22 0.032594 23 0.010979 24 0.000929 25 0.071337 26 -0.01279 27 0.030762 28 0.019379


of variation in terrain elevation, using a constant average scale for an aerial photograph

is bound to produce errors so that a vehicle travelling on top of a hill at speed V appears

to have moved a longer distance during time t than a vehicle travelling at the same speed

V at the bottom of that hill, 1.e. V(bottom) < V < V(top) as shown in Figure 4 (Ref 23).

Parallax is another systematic source of error in photographic reduction. Parallax

is defined as the apparent displacement of the position of an object with respect to a

frame of reference due to a shift in the point of observation (Ref 22). Using the aerial

photographic plane as a reference frame, parallax exists for all images appearing on

successive photographs and is larger the greater the elevation of the point. This apparent

movement between successive exposures takes place parallel to the direction of flight.

The parallax phenomenon affects the transformation of coordinate systems since the

fixed control points used in the transformations are assumed to have the same elevation.

Thus, the control points used in these transformations were chosen to not vary sub-

stantially in elevation and at the same time be widely scattered in the network area.

The relief displacement, while not as major an error source as the parallax, does

generate problems such as the masking due to highrise buildings. The relief displacement

is defined as the shift in position of an image caused by the relief or the height of the

object (Ref 22). In vertical photographs, i.e. those taken when the focal plane of the

camera is parallel to the ground, the relief displacement occurs along radial lines through

the point in the photograph located directly below the camera lens (the principal point).

The relief displacement is greater, the farther the object is from the principal point

and the greater the height of the object. Consequently, the determination of the posi-

tions of vehicles which have greater heights or are further away from the principal point

is subject to greater magnitude of error. However, since the vehicle heights are negligibly

small compared to the flight altitude, the errors due to relief displacement are not con-

siderable.


|

\ Comera Focal | y Length

EIT TTT MNT TTT

L

Me 7V7T777T7 TTT

L

Schematic diagram showing the effect of variations in terrain elevation on the scale of an aerial photographs. Note that while ground distance AB and CD are equal, due to variations in terrain elevation their images on the photographic plate abcd are not of equal length, hence (ab/AB) is not equal to (cd/CD). A vehicle travelling the length AB=L during At would appear to have moved a shorter distance on a pair of time-lapse photographs

Figure 4. : than a vehicle travelling the same length CD=L (Ref 23).


An assumption in the reduction of the aerial photographs is that they are vertical

photographs. Such is not the case if the aircraft is tipped or tilted with respect to the

plane of its altitude at the time of exposure. Since tips and tilts are usually no greater

than three degrees, the errors due to tip and tilt can be considered neglgible (Ref 25).

The non-systematic operator errors are present in all steps of the photo reduction

process but the two most error-prone steps have been the proper placement of the

comparator reticle on the desired points and the reading of the clock image on each

frame.

To correct for each of the above stated systematic errors individually requires the

lay out of many control points prior to photography as well as a tremendous increase

in the level of effort required for reduction of photographs. Moreover, for the purposes

of these studies, where one often deals exclusively with averages, such tedious efforts to

secure high levels of accuracy are not warranted. The question that arises is whether

or not the directions and magnitudes of errors from these sources are random enough

to yield meaningful averages (Ref 23).

To investigate the above question a study of the apparent speed of parked vehicles

is undertaken. Table 2 shows the coordinates of ten parked vehicles scattered in a pair

of frames. The average velocity of these ten vehicles is 1.28 miles per hour which is rel-

atively small enough i.e. the effect of the mentioned systematic errors can be neglected.

A study by Herman et al (Ref 23) was also undertaken to investigate the above

question. Figure 5 (Ref 23) shows the vector fields of velocities for a three pairs of

frames. Also shown in Figure 5 the speed and drift angles histograms corresponding to

the velocity vectors. As can be seen from the histogram of drift angles in Figure 5, the

angles are rather uniformly distributed i.e. the errors in magnitudes of these velocities

are essentially random and the resulting estimate of the average running speed can be

assur.ied unbiased.


Table 2. The coordinates and the apparent velocities of parked vehicles.

Case No. X1l{mm)

1.819 5.263 15.984 20.587 25.743 30.955 39.939 51.098 80.588

0 111.057

mm OOO ION On BW

th


X2(mm)

1.828 5.245 16.012 20.556 25.898 30.769 39.950 51.126 80.871 111.110

Y1(mm)

2.044 2.249 1.951 2.120 2.079 3.005 2.496 2.060 5.024 4.820

Y2(mm)

2.245 2.227 2.054 2.357 2.082 2.876 2.568 2.234 4.982 4.695

V(mph)

1.58 0.22 0.84 1.88 1.22 1.78 0.58 1.39 2.25 1.07

17

370 f r “Ty

os a

- TY

7

WO.

OF GAC HvATIONS

a —y

340+

iJ TY

310 + \ OF a et cles RT te | {fs

280+ \ JN a by ~

g ME Tb 25 » 250 \ am ~ 2 N —t A . WM \

WW \ aa /

= 220/ SL / \ \ z NN ; g | fee Lf I] . 190 - eG . ? /

/ HE 2 ro

160 | \ t so - ~ oN Ae

. ~ = ~ & “ON 130+ <

— . \ ~e —

0° 100 130 160 190 220. 250 280 310 7 X- COORDINATE (i0> MICRONS)

Figure 5. : The vector fields of velocities of parked vehicle (Ref 23).


3.0 DATA ANALYSIS

3.1 Review of An Existing Model

Gazis et al (Ref 1) presented the lane-changing model, equation (3.1), without field

verfication and it was presented solely based on the personal observations from the

driver’s seat. A solution to that model was obtained for a system of differential difference

equations with a time-lag corresponding to the interaction of two lanes. Then, the sol-

ution was generalized to any number of lanes.

In this work, using the data collected from aerial photographs, regression analysis

has been performed to find the coefficients of the Gazis model and to determine its va-

lidity.

The lane-changing model by Gazis et al (Ref 1) on a two-lane highway is presented

as:

Q, = — Q, = a{ K,(t— T) — K,(t — T) — (Kyo — Kj ,)} (3.1)

DATA ANALYSIS 19

where

Q, is The rate of exchange of vehicles (i= 1,2), a is a sensitivity coefficient describing

the intensity of interaction between lanes. In the simplest case, a can be assumed con-

stant. K, and K, are the concentrations of lane I and 2, respectively. T is an interaction

time lag; a value of zero could be assumed for simplicity i.e. once the driver makes a

decision to proceed with his intended maneuver, he crosses immediately from the current

lane ito the neighboring lane i+1 regardless of any other factors that might influence

his maneuver such as the effects of weather, availability of acceptable gap, and a variety

of other judgemental factors. Finally, K,, and K,, denote the equilibrium densities of the

two lanes.

Extension of the above model to more than two lanes is presented in details in

Gazis et al (Ref 1). The above model was suggested based on the following simplifying

assumptions :

1. The effect of exits and entrances has not been taken into account, i.e. there is con-

servation of the number of vehicles, and lane-changing does not take place due to

exit and/or enterance ramps.

2. There is a set of traffic densities for a set of homodirectional lanes which, if ob-

tained, is ideally acceptable to the drivers. Density oscillations occur about this

equilibrium density distribution.

3. The lane concentrations are assumed independent of a longitudinal position coordi-

nate taken along the highway.

Let K, be the difference of the concentrations between two neighboring lanes 1.e.

DATA ANALYSIS 20

K, = K,(t— T) — K\(t— 7)

To find whether the dependent variable Q, and the independet variable K;, are correlated

or not, a plot of Q, versus K, is shown in Figure 6. The data for Q, and K, values are

shown in Table 3. Figure 6 shows clearly that the data points form almost a vertical

straight line. Furthermore, The analysis of variance, as shown in Table 4, reveals that

the value of R? is 0.0001.

Therefore, it appears that there is absolutely no relation between the rate of ex-

change of vehicles between two neighboring lanes and the difference of the deviations

of their concentrations from their equilibrium values.

3.2 Testing the Lane-Changing Hypothesis

It has been hypothesized that whenever there is a lane-changing maneuver, the av-

erage speed of the neighboring lane is faster than the average speed of the cuurent lane.

To test the validity of this hypothesis, speed data which were collected from aerial pho-

tographs can be employed. Twenty eight data points (Table 5) are used in the analysis.

The points are assumed to be independently and identically distributed. The two sample

t-test has been employed. let yu, be the expected value of the speed of vehicles in the

current lane and p, be the expected value of the speed of vehicles in the neighboring lane.

The null and alternative hypothesis can then be expressed as follows :

Hy: Uy - by, = 0

Hy: wy - Wy, < 9

where the t-statistics is computed as:

DATA ANALYSIS 21

Table 3. Q and K data.

Q : The rate of exchange of vehicles K : The difference of concentrations between two neighboring lanes

Case No. Q(Veh./hour-lane) K(Veh./mile-lane)

] 823 -10 2 2595 -12 3 2367 -68 4 443 -116 5 1244 -7 6 2281 -10 7 1489 21 8 4376 -31 9 4663 -42 10 3864 -28 11 1838 -8 12 2158 -17 13 1571 -5 14 1159 -10 15 1928 -12 16 526 -5 17 412 0 18 609 0 19 S77 -5 20 2056 0 21 2703 0 22 2821 -28 z3 1030 -7 24 627 -2 25 184 -66 26 95 -63 27 1127 -21 28 1791 -21

DATA ANALYSIS

5000 r 4

r a

t 4000 + a ;

# 3000f ., £ , 4 A

& L a“ < 2000 We 4a

1000 f a4 a 4

0 4a. — - -200 -100 0 100

K (Veh./Mile-lane)

Figure 6. : A plot of Q versus K

DATA ANALYSIS 23

Table 4. Analysis of variance.

SAS

DEP VARIABLE: Q ANALYSIS OF VARIANCE

SUM OF MEAN SOURCE DF SQUARES SQUARE F VALUE PROBOF

MODEL ] 5780.12444 5780.12444 0.004 0.9516 ERROR 26 4 =40079711.98 1541527. 38 C TOTAL 27 =6400854992.11

ROOT MSE 1241 .583 R-SQUARE 0.0001 DEP MEAN 1691.321 ADJ R-SQ -0.0383 c.V. 73.40903

PARAMETER ESTIMATES

PARAMETER STANDARD T FOR HO: VARIABLE ODF ESTIMATE ERROR PARAMETER=0 PROB > ITI

INTERCEP 1 1701.95372 291.89584 5.831 0.0001 K l 0.51955341 8 .48471435 0.061 0.9516

DATA ANALYSIS 24

feos = (3.2)

with a degree of freedom of 2(n-1); n=n, = 7,

From Table 5, the following values can be computed : The average speed of the current

lane V, = 43.696, the average speed of the neighboring lane V, = 50.15, the standard

deviation of the current lane S$, = 20.007, the standard deviation of the neighboring lane

S, = 15.504, t,,, = -1.349, degree of freedom = 54, and the total number of observa-

tions n = 28.

For t,, = - 1.349 and a degree of freedom of 54, the null hypothesis is rejected at a

0.0935 level of signifigance.

Thus, the hypothesis that the average speed of the neighboring lane is faster than

the average speed of the current lane is accepted with a high level of confidence. More-

over, the result of this statistical test is significant in the sense that the lane-changing

model can be characterized by a model which considers speed differentials between two

neighboring lanes.

DATA ANALYSIS 25

Table 5. Average Speeds in (mph)

Case No.

Woo

TN

iA BWN—

DATA ANALYSIS

Current lane Vil

Neighboring lane V2

7.9 15.9 32.9 38.3 54.9 48.6 58.5 61.9 61.1 61.3 59.7 52.4 60.5 63.2 64.2 Sl 51.2 39 59.6 68.5 68.6 53.5 40.9 54.7 29.4 24.7 49.8 52

26

3.3 A Gap Acceptance Function For Freeway

Lane-Changing Maneuvers

Many studies have been conducted to develop gap acceptance distributions for all

possible maneuvers such as gap acceptance distributions at stop controlled intersections,

on entry ramps to multi-lane divided highways (Ref 26), etc. However, not much atten-

tion has been paid to model gap acceptance distributions due to lane-changing maneu-

vers. As will be shown later in chapter five, The significance of gap acceptance

distributions due to lane-changing maneuvers is that the gap acceptance function can

be employed to determine the expected delay to a single vehicle waiting for a sufficiently

large gap to change lanes.

To model gap acceptance, it is assumed that each driver has a “critical gap”. A driver

would accept a gap (i.e. proceed with his intended maneuver) in the traffic stream if the

duration of the presenetd gap is longer than his critical gap. The critical gap is modelled

as a random variable since it varies both across and within drivers. The decision of ac-

cepting a gap is affected by the number of gaps rejected by the driver before a sufficient

one is spotted.

Given a distribution of critical gaps in the population, one can define gap accept-

ance functions. Such functions relate the probability that a randomly chosen driver

would accept a certain gap to the characteristics of this gap. The most important char-

acteristic of the gap is its duration in seconds.

Several probability density functions have been used to describe the distribution of

the critical gap. Drew et al (Ref 27), Cohen et al (Ref 28) and Solberg et al (Ref 29) have

used the lognormal distribution; Miller (Ref 30) and Daganzo (Ref 31) have suggested

DATA ANALYSIS 27

the normal distribution; Blunden et al (Ref 32) have used the gamma distribution; and

Herman and Weiss (Ref 33) have utilized the exponential distribution. In the following

section, the gap acceptance function that best describes the data collected is discussed.

3.3.1 The Model

The data includes 25 observations collected from aerial photographs. Three data

points are deleted from the analysis because vehicles number 2 (Figure 1) are not shown

on the master frames to compute the gap lengths. Figure 7 shows a histogram of the

data. Note that the class interval formula has been used in plotting the histogram. The

formula is defined by :

range

V=T43.3 logN (3.3)

where W is the width of interval, range is the maximum value minus the minimum value

of the observations considered in the analysis (Table 14), and N is the number of ob-

servations.

The skewed-shape histogram of Figure 7 suggests that gap lenghts might be well

described by a function with a long tail distribution such as the Weibull, The Gamma ,

or the Lognormal distribution. To determine the function that best describes the dis-

tribution, chi-square tests have been performed. From Table 14, the mean and the vari-

ance for the gap lengths were found to be 5.621 and 13.075, respectively. Since the

integrals of the Gamma and Lognormal density functions are difficult to determine, a

simple computer program is developed to estimate numerically the area under the curves

of the mentioned density functions. The program is included in Appendix B. The com-

DATA ANALYSIS 28

15 ¢

lor

Frequency

YY WY/NINZNUIMZAZ 125 375 625 875 11.25 13.75 > 15.00

Gap length (sec)

Figure 7. : Gap Lengths Histogram

DATA ANALYSIS 29

puted values of X? as well as the parameter values of the above three functions are shown

in Table 6. For a=0.05 and degree of freedom of four, the critical value of X? is 9.48.

Table 6 shows that the Lognormal function has the lowest value of X? . Hence, the dis-

tribution is well described by the Lognormal! function.

I may remark the fact of having a few number of observations in some classes of the

histogram. However, for the purpose of our study, an effort is centered arround theore-

tical discussions, i.e. this fact can conservatively be accepted.

As will be shown in chapter five, it is worthy to note that one of the main uses of

gap acceptance functions is in developing expressions for delay problem.

DATA ANALYSIS 30

Table 6. Chi-square test analysis.

Density functions Parameters x

Lognormal function

_ =(Inx ay pw = 1.5533 f(x) = xV2xo “xP 20? fx > 0 2.9228

o= 0.5885 0 otherwise

Gamma function

Brest eb ; =

fix) = Tap ifx>0 a 2.417 5.9191 B = 2.326

0 otherwise

Weibull function

a8? x07! 7 iP fx>0 =

Ax) = 0 otherwise * 1.6 7.1398

= 6.2693 cD |

DATA ANALYSIS

4.0 Theory

4.1 The proposed model

Due to the structural complexity of the existirg lane-changing models (Refs 1,6), a

model that is easy to calibrate and validate based on field observations is needed. The

result of the hypothesis test earlier is significant, since it suggests that the lane-changing

model can be built to give the probability of lane-changing as a function of speed dif-

ferentials between two neighboring lanes.

The model of lane-changing phenomenon is suggested based on the following as-

sumptions:

1. The effect of exit and entrance ramps has not been taken into account.

2. Drivers make their decision to change lanes according to their perception of the

speeds in their current lane and the neighboring lane. Then upon the availability of

Theory 32

a gap of proper length, they proceed with the lane-changing maneuver to the lane

the speed in which is preceived to be considerably higher.

3. The speeds in the current lane and the neighboring lane are assumed to be inde-

pendently and identically distributed.

4. The number of lane changes per unit distance is a power function of the speed dif-

ferentials between each pair of lanes i.e. the number of lane changes (N) from the

current lane i to the neighboring lane i+ 1 can be written as:

N=alV,— Viy1)" (4.1)

5. Let V, be the measured speeds in lane i and U, be the perceived speeds in lane i so

that U, = V, + e, where e, is the driver perception error terms. It is assumed _ that

the error terms are independently and identically distributed Gumbel variates, i.e.

the fraction of the total lane changes (P,,,,) that take place from lane i to lanei+1

can be given by a binomial logit function as:

eV. é

i

P...,=—-— (4.2) WAL es gg dMaat

The proposed model can then be expressed in the general form as :

b cy, a(V;~— Visi) e [

ai, Nie = (4.3) cv. e ‘+e

where

a,b,c, and d are the model parameters.

Theory 33

Equation (4.3) expresses the number of vehicles leaving lane i and entering the

neighboring lane i+ 1 as a function of the speed differentials between each pair of lanes.

4.2 Calibration of the proposed model

The first step in the calibration of the model is to form table 7. This table contains

the total number of cases of lane-changes for all different combinations of the averages

of speeds in the current lane V, and the neighboring lane V;. The values, in table 7 and

8, are arbitrarily chosen for illustration purposes. Building such a table requires a great

deal of observations. The class interval W, for illustration, has been taken as five miles

per hour for both V, and V,. It should actually be calculated using the class interval

formula (Equation 3.3). Based on Table 7, a three dimensional graph (Figure 8), can be

plotted to study the shape of the histogram of the two neighboring lane speeds.

Once table 7 is formed, it is a matter of using an appropriate computer software,

such as SAS, to find the values of the parameters a,b,c, and d.

In table 8, the first column has the average speeds of the current lane; the lane from

which the lane changing maneuver was initiated. Column two has the average speeds

of the neighboring lane. Column three presents the number of vehicles changing lane

per unit distance. Finally, column four shows the values of the fractions of the total lane

changes that take place from the current lane to the neighboring lane.

For the purpose of performing regression analysis, the logarithm of both sides of

equation 4.1 can be taken, i.e. equation 4.1 can then be expressed as:

Y=A+ BX (4.4)

Theory 34

where

Y = log, N, A = log, a, B = b, and X = V,- V,,,.

With similar mathematical manipulation, equation 4.2 can also be expressed as:

Y=AX, + BX. (4.5) 1 2

where

Y=ViyA= X= B= 4, and X, = log, ( 5 - 1).

The second step in the calibration is to perform regression analysis between column

1, 2, and 3 to find the values of the parameters a and b and between column 1, 2, and

4 to find the values of the parameters c and d.

An attempt is initiated to complete Table 7 using the collected data. A few cells are

only filled out, i.e. calibration of the model is not possible at this stage; more data points

are needed.

Applications, using the binomial logit lane-changing model as well as using the de-

termined gap acceptance function, are described in details in the next chapter which in-

cludes a simulation and a delay computation problem.

Theory 35

Table 7. Number of cases of lane-changes.

Soe

se Lessssssssesdessessssansfosssisiiasditieescieseeccccosteee aE

992188 EZ.

vslessessesessteceuaeees selisccevsvecedeavscacsacadacssvssacestsssavsssseatacsevavseestevecsevscs deseeee B97 10S

ll Sh oe bb

sccececsscicaseacasatedecesesessestsesese OST LSP

fh,

bl EEE Eb.

ss scecsevsclensescecceetacsevsvsestaccacsecscedessvesavarteceavevscestecsees SOL OP

BOL

EE

Jliseecsevsctaceesesatelassesssvssedeseessesssstsssee, OU7 USE

beetles

beth Otol

cscccteesecebeeceee. seslesssceecscelsseescsevselacseacessatacssssnssvatecsens det, OF

ele 76. Pt

ERE SE

ccccctecbisccscsedecscssssvscdessvsvsssscdessssestecdsereecessesteecs O27 1'S2

ieee Beccles

teh csc MecetePiccsssetealccceteaesestebeccedessssbeteeteecccccetc.,,

827102 veeceeesees E

EE

EE Eb

EE

z= st

ecesesevaetesseesssesetossesseeees SR bbe,

bh ceeclevscessesesdeesess

ob 1/01 seestseetececertecsess

ee

«scssssseestecsvevecscedeasesesvavsteScasecvslecsasacenes 1

cdeecescssrsederseee QhTHS sesecssaestesessetseuetscsersessestesesnessvsvtuesavarsatsteasssetasaete

Guseerectescsecesetebesesete Preceded

coccccfeececcccct

870 09-

Se'6g-

'08:06- Sb isb-

Ob Ob

I'sese- Vosoc- \'SZSz- -'0Z02- Srsii

ror: Ol-t'S:

S-0: ZANLA

=INImile]ilolr|olalOla/ 8 /'

sto]

zi [

st J]

of |

6 [

8 [

2 2 |

9 |

s |

+» J

¢ [

zf]esf

Theory

Figure 8.

Theory

SAS

: CURRENT LANE SPEED IN MPH. : NEIGHBGAING LANE SPEED IN MPH. : NO. OF CASES.

: Histogram of the two neighboring lane speeds.

$7.50

37

Table 8. Arbitrary data to illustrate the calibration of the model.

Theory

V1 v2 N P

30 35 5 5/K 4I 48 13 13/K 50 52 15 15/K 39 45 8 8/K 45 58 13 13/K

3 13 4 4/K 60 65 16 16/K 38 43 8 8/K 15 25 2 2/K 28 40 8 8/K

K : Total number of cases. N : Number of vehicles that change lane for a specific set of speeds.

38

5.0 Applications

5.1 Stmulation problem

The suggested binomial logit lane-changing model (Equation 4.2) can be combined

with the distribution of the successive gaps in the traffic stream to evaluate the condi-

tional probability that a driver would change lane for specific speeds in the current lane

and the neighboring lane given that a certain gap of length t is available. Both distrib-

utions are assumed independent, i.e. the conditional probability can be written as:

eV; oo

Pin =a | (08 (5.1) e ite ter

where

V, and V, are the speeds of the current lane and the neighboring lane respectively.

@ (t) is the probability density of the successive gaps in the traffic stream.

Applications 39

A numerical example cannot be illustrated at this stage since the binomial logit

model has not been calibrated yet. However, once the parameters c and d are estimated,

it is straight forward to determine the conditional probability.

The significance of the conditional probability is that it can be used as an input data

in traffic simulation packages. For example, suppose that the conditional probability of

lane changing is 0.40, i.e. for given speeds in the current lane and the neighboring lane,

40 percent of the vehicles in the slow-lane of the highway would change lanes, given that

a suitable gap is available.

5.2 Delay problem

The problem that is considered here concerns the delay to a single vehicle waiting

for a sufficiently large gap to change lanes. The systematic applications of renewal the-

ory techniques (Ref 34) offers a method of solving this kind of problem.

The probability of changing lane when confronted with a gap of duration t is given

by the gap acceptance probability a(t). From the data collected, it has been shown that

the form of « (t) can be approximated by the Lognormal ditribution probability, namely:

(In tn? t+dt elt 262 J

a(t) = | mm $1 (5.2) t ty/ 2no”

where

pw = 1.5533 and o= 0.5885.

Applications 40

It is assumed that successive gaps in the neighboring lane are independent random

variables with an exponential probability density @ (t), namely,

$(t) = > etl (5.3)

where

A is the mean of gap length.

@ (t) dt is the probability that a certain gap is between t and t+ dt seconds long.

The probability density for the first gap (Ref 35) is given by @, (t) where

Mo oo(t) =—>——_ (5.4)

| th(r)dt 0

The probability density for the delay time is denoted by Q (t). Q(t) dt is the prob-

ability that the waiting driver will be delayed for a time T such that < T<1+ 61. Let

Q*(s) denote the Laplace transform of Q (t). Also, let ‘¥,(t) and ‘¥(t) be defined by:

Fo(t) = do(AC1 — a(2)] (5.5)

¥() = OCI — a(9] (5.6)

‘V3 (s) and ‘’*{s) are their respective Laplace transforms. &, and @ are defined as the mean

values of a(t) averaged with respect to ¢, (t) and @(t) such that:

Gp = |“ aliddbo(e)5r (5.7) 0

Applications 41

i= | *altb(t)ot (5.8) 0

Maradudin et al (Ref 35) showed that Q*(s) can be written as:

Q*(s) = «5 (s)/[1 — ¥"(s)] + & (5.9)

From the above equation, the mean delay time (Ref 35) is given by:

p= [overt 0 | Y(t (5.10)

In Figure 9, several curves of the mean delay time (r) versus the mean of the gap

lengths (A) are plotted for different values of o, where o is the shape parameter of the

gap acceptance function. The scale parameter » has been kept constant when plotting

the curves, where n= 1.5533. As can be seen from the graph, the critical value of the

mean gap length J , using the determined parameters of n= 1.5533 and o =0.5885, is

about 7.5 seconds. Moreover, the mean delay time (f) is very sensitive for a gap length

value less than 7.5 seconds, i.e. a small change in J would result in a considerable change

of the mean delay time (7).

The probability that a single vehicle in the current lane will experience no delay in

merging to the neighboring lane is given by:

Po = & (5.11)

‘Transparency’ (Ref 35), which is denoted by (J), is another parameter that can be

used to characterize the properties of a highway. It is defined as the percentage of time

during which a waiting driver would say that it is safe to change lanes. This parameter

Applications 42

MEAN DELAY

TIME (SEC)

1000

100 F

sigma = 0.588 w= 1.55335

10 F sigma = 0.2

, sigma = 0.4

| 7 sigma = 0.8

A T F —~T ? l

0 9 10 1S

MEAN OF GAP LENGTH (SEC)

Figure 9. : A plot of the mean delay time versus the mean of the gap lengths

Applications 43

is a function of the gap acceptance probability a(t) as well as the distribution of the

successive gaps in the neighboring lane ¢ (t) and is defined as:

| EY) + a()O() 5 J=(1++—~ 7 (5.12)

| a(t)D(r)dz 0

where

O(1) = | * O(0)6t

In Figure 10, a plot of the transparency (J) versus the mean of the gap lengths (/)

is shown. The figure shows clearly that the transparency vlaues vary considerably for gap

length values less than 10 seconds. Moreover, the Transparency parameter can be em-

ployed to introduce a new method for measuring the level of service as a function of the

gap length. For example, level of service A can defined with a Transparency range of 80

to 100 percent. The level of service is known to indicate the level of performance under

a set of traffic conditions.

In general, the ‘Transparency’ parameter is important because it characterizes the

overall ability of the traffic to delay a driver from changing lanes.

Discussions and recommendations are presented in details in the next chapter.

Applications — 44

100

; 80

~ 60

40 o =

x 20

0

-

L y= 1.5533

. sigma = 0.5885

0 20 30 44 50

MEAN OF GAP LENGTHS (SEC)

Figure 10. : A plot of the transparency versus the mean of the gap lengths.

Applications 45

6.0 Discussions and Recommendations

The important discovery in this study, which is based on the lane-changing hy-

pothesis, is that the lane-changing model can be characterized by a mathematical model

which considers speed differentials between two neighboring lanes.

As shown in chapter two, the collected data are acquired by a methodology which

involves aerial photographic technique. Aerial photography is recognized as a potential

tool to be used in solving traffic operations problems.

The data, which have been collected from aerial photographs, are unfortunately not

sufficient to investigate the reasonableness of the assumptions as well as to determine

the values of the four parameters of the proposed model, namely a,b,c and d. We could

not obtain more than twenty eight data points because the reduction of the aerial pho-

tographs is involved and time consuming. Thus, it should be recognized that this work

is not extensive in the sense that the proposed model has not been calibrated and vali-

dated yet. So further data reduction regarding the determination of the parameters val-

ues are recommended. However, the method that should be followed in calibration of

the proposed model has been presented and explained in details in chapter four.

Discussions and Recommendations 46

To reduce the errors related to the reduction procedure of aerial photographs as well

as to improve the results obtained from the analysis of the collected data, the following

observations are recommended:

The speeds in the current lane and the neighboring lane were found by determining

the speeds of the two vehicles ahead of the vehicle changing lane. Finding the aver-

age speeds in both lanes using more than two vehicles would improve the outcome

from the collected data by reducing the errors involved in the calculation of speeds.

One shoud pay attention to the fact that the gaps between vehicles ahead of the

vehicle changing lane should be relatively small, since large gaps may create a con-

siderable variation in the calculation of the average speeds of these vehicles.

No more than one observations in one pair oi frames should be made; that would

eliminate any doubt about the independency of the data.

The null hypothesis of the statistical test was rejected at an acceptable level of sig-

nificance; the computed observed t value is very close to the acceptable region.

However, more data points could improve the result of the test.

Due to relief displacement problems, the control points should be chosen not to

have a high altitude. Painted marks in a parking lot would be a good example of

contro] points.

This study has presented information about gap acceptance behaviour for freeway

lane-chagning maneuvers and it shows that the Lognormal ditribution can be utilized to

model gap acceptance behaviour. Moreover, it is shown that the gap acceptance func-

tion can be used to determine the average time a vehicle is trapped in a slow lane before


it can change lane. The determination of the delay is significant for highway operational

efficiency.

In this study, using the data points collected, regression analysis has been performed

to dertermine the validity of Gazis et al model (Ref 1). The analysis performed reveals

that Gazis model appears not to be adequately describing the data obtained. However,

caution must be used in accepting the regression analysis results as final since the real

proof would only come with a more data-intensive effort.


Appendix A. Data Obtained from Photographs

Table 9. Current Jane (Veh No. 5 and 6)

Case No. X1(mm)

]

2

3

-0.517 -2.004 -126.684 27.571 8.082 7.017 4.417 4.395 22.463 11.530 8.338 6.455 -18.566 -23.279 -17.347 -18.851 -19.112 -21.615 -21.914 -25.256 -29.783 -32.288 -44.317 -50.410 0.103 -11.586 -21.103 -37.577 20.978 26.939

X2(mm)

-1.966 -3.376 -124,934 -125.904 6.926 5.893 4.621 4.862 15.635 2.454 0.249 -0.520 -25.620 -29.766 -26.049 -27.497 -26.987 -28.794 -28.68 | -32.49] -36.586 -39.327 -51.580 -59.278 -9.115 -22.779 -30.493 -46.935 31.818 37.773


Y 1(mm)

18.434 17,926 -66.043 -67.032 12.905 16.931 32.384 33.922 -4.588 5.807 7.712 9.671 33.418 38.216 31.849 33.664 33.563 36.218 35.957 39.439 43.397 46.099 60.819 67.343 7.101 6.878 6.324 6.271 6.375 6.244

Y2(mm)

17.764 (Veh. no. 5) 17.302 (Veh. no. 6) -62.088 -62.466 16.801 20.829 32.622 34.064 2.107 14.301 14.718 16.064 39.184 43.638 38.829 40.362 40.695 43.171 43.372 46.991 50.179 53.123 65.522 72.818 7.322 7.542 7.147 7.506 6.315 6.400

49

Continue - Table 9

Case No. X1(mm)

16

17

18

19

20

21

22

23

24

25

26

27

28

-5.572 -39.632 3.780 -31.617 -29.163 -44.094 8.032 12.589 15.279 19.024 22.265 36.691 12.350 16.026 16.810 27.816 9.205 38.671 9.089 8.850 9.031 9.074 6.308 15.005 9.057 10.785

X2(mm)

-16.588 -49.419 -3.056 -41.082 -39.610 -54.327 24.717 28.574 25.676 29.868 32.799 46.955 24.267 27.267 26.044 36.491 22.545 51.429 9.171 8.987 9.296 9.118 12.555 22.473 15.616 17.435


Y1(mm)

10.304 10.252 10.462 10.139 7.776 7.860 6.138 6.225 7.225 7.252 7.376 7.381 7.561 7.422 8.112 8.067 1.980 2.181 7.204 8.803 7.686 9.112 10.785 20.799 14.446 16.270

Y2(mm)

10.824 10.996 10.465 10.632 7.865 7.973 6.405 6.264 6.243 6.367 6.149 5.596 6.924 6.692 9.021 9.271 0.166 5.342 7.726 8.996 7.666 9.197 16.873 27.069 21.762 23.480

50

Table 10. Neighboring Lane (Vehs No. 3 and 4)

Case No. Xl{mm) X2(mm) Yi({mm) Y2(mm)

l -0.050 -1.806 19.177 18.400 (Veh. no. 3) -1.564 -3.221 18.857 18.035 (Veh. no. 4)

2 -126.338 -125.409 -66.632 -62.688 -127.107 = -126.425 -67.471 -62.972

3 8.275 6.836 14.386 19.912 6.653 6.163 21.376 26.956

4 5.377 6.788 38.508 44.690 7.936 9.819 - 53.292 59.837

5 8.471 0.301 7.439 14.559 6.572 -0.486 9.491 16.080

6 1.342 -5.684 13.742 20.248 -5.071 -11.382 19.855 25.651

7 -17.804 -26.354 31.651 38.604 -19.449 -27.831 33.224 40.095

8 -20.312 -29.078 34.395 42.017 -26.045 -34.674 40.451 48.049

9 -20.416 -28.602 34.147 42.162 -26.273 -34.172 40.253 48.412

10 -20.703 -28.412 33.960 42.313 -26.460 -34.206 40.053 48.602

11 -27.008 -35.150 39.759 47.762 -29.641 -37.261 42.256 50.025

12 -40.572 -48.584 56.258 61.791 -45.568 -53.700 61.418 66.927

13 5.585 -4.605 6.417 6.554 -16.969 -27.561 6.296 6.685

14 -33.657 -44.912 5.564 6.729 -49.853 -60.108 5.334 6.961

15 39.786 50.887 7,284 7.541 56.947 69.797 8.138 8.071

16 -33.074 -44.189 9.875 10.385 -58.239 -68.396 9.694 10.551

17 0.773 -10.601 9.816 10.100 -33.286 -43.329 9.570 9.957

18 -5.892 - 16.857 7.960 8.339 -26.842 -37.014 8.352 8.475

19 39.113 56.911 7.053 6.421 71.237 89.008 5.890 4.690

20 22.815 33.882 7.850 6.493 28.227 39.479 7.816 6.227


Continue - Table 10

Case No. X1(mm)

21

22

23

24

25

26

27

28

22.741 28.132 12.409 44.887 20.964 24.246 47.899 61.765 9.902 9.623 9.889 9.586 24.412 24.412 12.382 22.027

X2(mm)

33.829 39.426 25.941 56.101 31.340 33.900 60.417 71.480 9.603 9.634 9.713 9.853 32.618 32.618 19.915 30.865


Y1(mm)

8.003 7.909 8.237 6.236 8.762 8.790 4.197 10.627 6.030 9.104 7.859 11.177 30.227 30.227 17,220 27.599

Y2(mm)

6.680 6.469 7.498 5.498 9.862 9.862 8.804 16.125 12.200 15.652 12.600 16.375 37.210 37.210 25.282 37.168

52

Table {1. Coordinates of the control points

Case No. X1(mm)

| -129.020 2 -129.020 3 -2.661 4 -2.609 5 2.437 6 2.362 7 2.223 8 2.314 9 2.310 10 2.209 1} 2.106 12 2.107 13 3.340 14 3.371 15 12.395 16 4.567 17 10.823 18 0.899 19 7.037 20 2.127 21 2.047 22 4.378 23 6.556 24 3.043 25 15.524 26 15.561 27 4.318 23 4.625

X2(mm)

-127.771 -127.771 -2.690 -2.657 2.288 2.213 2.181 2.213 2.389 2.403 2.082 2.120 3.913 3.819 12.792 4.54] 10.670 0.795 7.927 2.312 2.195 4.426 6.842 3.092 15.785 15.600 4.648 4.752


Y1(mm)

-61.631 -61.631 5.324 5.088 6.296 6.503 6.439 6.333 6.433 6.346 6.365 6.509 3.996 4.155 5.550 1.832 1.422 0.990 4.450 3.126 3.170 5.293 6.504 0.766 6.976 6.934 2.562 2.337

Y2(mm)

-57.858 -57.858 5.152 5.099 6.347 6.440 6.455 6.501 6.371 6.460 6.474 6.600 3.930 4.002 5.418 1.697 1.250 0.969 4.524 2.971 2.985 3.127 6.672 0.672 6.886 6.837 2.327 2.478

33

Table 12. Coordinates of vehicle number 2.

Case No. Xl(mm) X2(mm) Y l(mm) Y2(mm)

3.588 1.251 20.157 19.250 2 -124.777 = -123.943 -64.603 -61.170 3 16.130 12.346 -1.060 6.738 4 8.061 6.876 14.505 19.909 3 27,555 20.129 -10.579 -2.646 6 17.368 8.457 -1.720 6.284 7 -14.781 -24.238 28.841 36.592 8 -20.974 -29.413 33.71) 41.780 9 -31.899 -39.878 46.440 52.795 10 38.026 23.235 6.069 5.578 11 5.064 -6.035 5.920 6.018 12 15.086 25.365 7.146 6.831 13 20.741 9.785 9.788 9.534 14 10.139 0.601 9.888 9.270 15 11.143 4.230 7.708 7.892 16 3.645 19.793 6.670 6.996 17 3.051 16.013 7.966 7.242 18 5.125 16.033 8.070 7.354 19 7.132 19.190 8.387 7.548 20 -2.264 9.324 8.653 8.866 21 3.489 15.919 2.363 0.274 22 12.014 10.432 -3.132 3.278 23 11.691 10.403 -2.066 3.273 24 -17.615 -10.572 -18.231 -8.986 25 6.020 13.834 9.897 18.700


Table 13. Concentration data

Case No. Road Length (photo inches)

] 3.00 2 2.50 3 2.00 4 1.00 5 1.50 6 1.00 7 1.00 8 1.00 9 0.75 10 0.75 11 1.25 12 1.25 13 2.00 14 1.00 15 3.50 16 2.00 17 2.00 18 2.00 19 2.00 20 2.00 21 1.25 22 0.375 23 1.50 24 4.00 25 1.75 26 1.50 27 2.00 28 1.00


Number of Vehicles Current

lane

Neighboring lane

33 29 6

ROR Um

UA

Ss NS ON

Te RW

BWI

55

Table 14. Gap acceptance distribution data

Case No. Distance between Vehs no. 2 and 3

(photo inches)

] 0.125 2 0.120 3 0.750 4 1.000 5 0.750 6 0.875 7 0.375 8 0.625 9 0.500 10 1.310 1] 1.500 12 1.000 13 2.100 14 0.370 I5 0.560 16 1.120 17 0.687 18 0.687 19 0.1875 20 0.813 21 1.750 22 0.375 23 0.438 24 2.500 25 0.375


Speed of Vehicle number 2.

(mph)

10.5 14.8 50.5 32.2 58.2 64.2 65.5 62.5 54.6 86.2 64.6 55.1 52.4 45.7 38.6 54.1 66.9 66.5 52.2 47.0 56.3 30.5 27.2 53.7 56.8

Gap Length

(secs)

4.06 2.77 3.07 10.58 4.39 4.65 1.95 3.41 3.12 5.18 7.91 6.19 13.66 2.76 4.95 7.06 3.50 3.52 1.22 5.89 10.60 4.19 5.48 15.88 2.51

56

Appendix B. Computer Program

SrxBSSFSPsSTesBtFSSASIECFESESFERSrTBSAATCSAKCKASPSASESSTSRSETSFSBSBAKCKAARDSEBETFTEFSSCSAIEE: 'se2e2

20 ‘=== eens:

30 '=="2 BY: M. Rafik Nemeh sess: &0 'sezze2 =zece:

50 ‘== This program computes the integral of a function f(x) eazs: 60 '=2== between the limit X=a and X=b using Trapezoidal rule. esas: JQ ‘s25= secs:

80 terrae scet srs See terse sess ee stresses eter tresses tstt sere essere sete ere etree ss tseaesresss

90 ''

100 br nr rrr rere tre tene Define function- ------ rrr tr r rte rr errr 110 '' 320 DEF FNFX(X)=(.103474)*(X72.617)%(2.7182381188 > (-X/2.326))} 230 ' 140 CLS 150 ' 160 '---- rrr rrr tren eecn INPUT -------77-7->- Tt te tt et en ne 170 ' 180 PRINT TAB(20); “THIS PROGRAM COMPUTES TH= INTEGRAL OF A FUNCTION"; 190 PRINT TAB(20); "USING TRAPEZOIDAL RULE." 200 PRINT: PRINT 210 220 PRINT TAB(10); "ENTER LOWER LIMIT OF INTEGRAL"; : INPUT A 230 PRINT TAB(10); "ENTER UPPER LIMIT OF INTEGRAL";: INPUT B 240 PRINT TAB(10); “ENTER n SUBINTERVALS";: INPUT N 250 260 ' 270 br nm mrt t rrr ett t tree Computations 280 '' 290 H=(B-A)/N 300 FA=FNFX(A) 310 FB=FNFX(B) 320 SUM = 0 330 FOR I = 1 TO N-1 340 X*A+I*H 350 SUM = SUM + 2 * FNFX(X) 360 NEXT I 370 INTEGRAL * (H/2)* \FA+tFE+SUM) 380 '

ww, ew ewer wm een were w wee twee wen! Ce we ew ewe ew eee ee

ween www were mee nwnenwrewe en ewenenen ew new ew eee wwe aw oo

410 PRINT: PRINT &20 PRINT TAB(10);"THE COMPUTED VALUE OF THE INTEGRAL IS: "; INTEGRAL 430 PRINT 440 PRINT TAB(10);"STRIKE 1 TO USE THIS PROGRAM AGAIN" 450 PRINT TAB(10);"OR STRIKE 2 TO QUIT"; &60 INPUT Z 470 IF 2 = 1 GOTO 90 ELSE 480 &80 END &90 '

Appendix B. Computer Program 87

REFERENCES

10.

11.

12.

13.

Gazis, D.C. , R. Herman, and G.H. Weiss (1962), “Density Oscillations Be- tween Lanes of A Multilane Highway,” Ops. Res. 10, pp. 658-667.

Chandler, R.E., R. Herman, and E. W. Montroll (1958), “traffic dynamics: Studies in Car Following,” Ops. Res. 6, pp. 165-184.

Munyjal, P.K., L.A. Pipes (1971), “Propagation of on-ramp density perturba- tions on unidirectional two and three lane freeways,” Transp. Res. 5, pp. 241-255.

Makigami, Y., T. Makanishi, M. Toyama, and R. Mizote (1981), “On a similation model for the traffic stream on freeway merging area,” In Proc. of 8th Intntl. Symp. On Transp. and Traf. Theory, pp.63-72.

Roberch, J., (1976). “Multilane traffic flow processes: Evaluation of queueing and lane-changing patterns,” Transp. Res. Rec., 596, pp 22-29.

Michalopoulos, P.G., D.E. Beskos (1984) “Improved Continuum Models of Freeway Flow,” Ninth International Symposium on Transportation and Traffic Theory VNU Science press, pp 89-111.

Marino, R. “Freeway inventory Geometric Bottleneck Congestion,” First In- terim Report, california Transportation Agency, Department of Public Works, Division of Highways - District 7, February 1970.

Munjal, P.K., Y.S. Hsu, R.L. Lawrence, 1971, “Analysis and Validation of Lane-Drop Effects on Multi-lane Freeways,” Transportation Research , Volume 5, pp. 257-266.

Goodwin, B.C., R.L. Lawrence, 1972 “Investigation of Lane Drops,” Highway Research Record 388, pp. 45-61.

Biggs, R.G., M.J. Misleh, 1971 “ I.P.E. 408 U.S. 59 (South West Fwy) in Houston, Control 27-13, Study Results Freeway Surveillance and Control,” Texas Highway Department, Houston, Texas.

Buhr, J.H., D.R. Drew, J.A. Wattleworth, and T.G.Williams, 1967 “A Nation- wide Study of Freeway Merging Operations,” Highway Research Record 202, pp. 76-122.

Johnson, R.T., L. Newman, 1968 “East Los Angles Interchange Operation Study,” Highway Research Record 244, pp. 27-46.

Taylor, J.I., 1965 “Photogrammetric Determinations of Traffic Flow Parame- ters,” Ph.D. dissertation, Ohio State University

REFERENCES 58

14,

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16,

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25.

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27.

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29,

Treiterer, J., J.1. Taylor, 1966 “Traffic Flow Investigations By Photogrammetric Techniques,” Highway Research Record 142 pp. 1-12.

Yu, J.C., J. Lee, 1973 “Internal Energy of Traffic Flows,” Highway Research Record 456 pp. 40-49.

Munjal P.K., Y.S. Hsu, 1973 “Experimental Validation of Lane-changing Hy- potheses from Aerial Data,” Highway Research Record 456, pp. 8-19

Breiman, L., R. Lawrence, D. Goodwin, and B. Bailey,1977 “The Statistical Properties of Freeway Traffic,” Transportation Research, Vol. 11, No. 4, pp. 221-228.

Godfrey, J.W., 1969 “The Mechanism of a Road Network,”, Traffic Engineer- ing and Control, , Vol. 11, No. 7, pp. 323-327.

Desforges, O., 1976 “Une Methode D’Enquete Origine-Destination Par Pho- tographies Aeriennes,” Monograph, Institute of Transport Research, France.

Ruhm, K., 1971 “Traffic Data Collection and Analysis by Photogrammetric Method,” Traffic Engineering and Control, Vol. 13, No. 8, pp. 337-341.

Holroyd, J.. D. Owens, 1971 “Measuring The Effectiveness of Area Traffic Control Systems,” TRRL Report LR420, Transport and Road Research Lab- oratory, Crowthorne, England.

Brinker, R.C., P.R. Wolf, 1977 Elementary Surveying, Chapter 25, Sixth Edition, New York, T.Y. Crowell Company.

Herman, R., S. Ardekani, 1984, “Characterizing The Quality of Traffic Service In Urban Street Networks,”, Center For Transportation Research, The Uni- versity of Texas at Austin, Project 3-8-80-304.

Peleg, M., L. Stoch, and U. Etrog, 1973 “Urban Traffic Studies from Aerial Photographs,” Transportation, Vol. 2, No. 4, pp. 373-390.

Davis, R.E., F.S. Foote, J.M. Anderson, and E.M. Mikhail, 1981 Surveying Theory and Practice, Chapter 16, Sixth Edition, New York, McGraw-Hill Inc..

Radwan, A.E., K.C. Sinha, 1980 “Gap Acceptance and Delay at Stop Con- trolled Intersections On Multi-Lane Divided Highways,”, ITE Journal, March.

Drew, D.R., L.R. LaMotte, J.H. Buhr, and J.A. Wattleworth, 1967 “Gap Ac- ceptance in The Freeway Merging Process,” Report 430-2, Texas Transporta- tion Institute.

Cohen, E., J. Dearnaley, and C.E. Hansel, 1955 “The risk taken in crossing a road,” Oper. Res. 6, pp. 120-128.

Solberg, P., J.D. Oppenlander, 1966 “Lag and gap acceptance at a stop- controlled intersection” Highway Research Record 118, pp. 48-67.

REFERENCES 59

30.

31.

32.

33,

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35.

Miller, A.J., 1972 “Nine Estimators of Gap Acceptance Parameters,” Sth In- tern. Symp. on The Theory of Traffic Flow and Transp., pp. 215-235. New York.

Daganzo, C.F., 1980 “Estimation of Gap Acceptance Functions and Their Distribution Across the Population from Gap acceptance Data,” Transporta- tion Research.

Blunden, W.K., C.M. Clissold, and R.B. Fisher, 1962 “Distribution of Accept- ance Gaps for Crossing and Turning Manuevers,” Proc. Aust. Rd. Res. Board Il, pp. 188-205.

Herman, R., G.H. Weiss, 1961 “Comments on the Highway Crossing Problem,” Oper. Res. 9, pp. 828-840.

Doob, J., 1948 “Renewal Theory from The point Of View of The Theory of Probability,” Transp. Am. Math. Soc. 63. pp. 422.

Weiss, G.H., A.A. Maradudin, 1962 “Some problems in Traffic Delay” Opr. Res. 10, pp. 74-104,

REFERENCES 60

Vita

Name

Nationality

Date of Birth

Education

Honors

and

Activities

Vita

M. Rafik Nemeh

Syria

November 7,1964

Master of Science in Civil Engineering, (Transportation Division),

Virginia Polytechnic Institute and State University, Blacksburg,

Virginia, December 1988, (GPA: 3.784/4.0)

B.S., Civil Engineering, August 1987, Virginia Tech.

Overall GPA: 3.25/4.0

Minor: Engineering Science And Mechanics, GPA: 3.61/4.0

¢ Member, American Society of Civil Engineers.

¢ Member, Institute of Transportation Engineers.

¢ Member, International Club at Virginia Tech.

¢ Member, Golden Key National Honor Society.

61

SA. a) Lb DW — Cts

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