A STUDY OF THE PATTERN OF HEADWAYS

ON AN URBAN FREEWAY

I

CANQ VO

I 224 la Voirie

i Québec Jean-Luc Simard, B.A,Ing.P, B.Sc.A,M.Sc.A.

da Ingénieur en circulation

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:1 V;o111'. Cédr

,

„ett.th

A STUDY- OF THE PATTERN OF HEADWAYS

ON AN URBAN IeREEWAY

by

J. L. SIMARD Reçu CEflt 4Ett'OCWEfirfAT/ON

DEC 4 1980

TRAOSPOTS QUÉBEC

A thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Applied Science in Civil Engineering

in the Faculty of Civil Engineering,

at the University of Toronto, Ontario

August 1962

ACKNOWLEDGYŒNTS

The author of this thesis is very much indebted to

the Canadian GOod Roads Association whose sponsorship enabled

him to carry out this study und to the Quebec Department of

Roads, especially their Chief Engineer, Mi'. Arthur Branchand,

and their Chief Traffic Engineer, Mi'. Henri Perron, for

granting him a full year leave of absence in order that the

work might be completed.

Special acknowledgments are due to: Professor M. M.

Davis who as director of this project provided mudà valuable

information and assistance and also reviewed the thesis; Mi'..

W. Q. Macnee, the chief Traffic Engineer of the Department

of Highways of Ontario, for supplying the photographie

equipment and the films from which the data involved in this

thesis have been extracted; Mi'. J. L. Vardon, Planning

Engineer in the same Department, for his comments and

assistance; Mr. R. Wormleighton, professor in the Faculty

of Mathematics at the University of Toronto, for his advice

in statistical nattera; •the authorts wife, Suzanne, for her

invaluable assistance in extracting the data from the films.

TABW OF CONTENTS

Chapter Page

ACKNOWLEDGMENTS ii

LIST OF TABLES

LIST OF FIGURES vi

I. INTRODUCTION 1

H. EXPERIMENTAL,PROCEDURE 3 The field equipment 3 The projection -equipment 5 The study locations .. 8 Extraction of data 14

III. EXPERIMENTAL RESULTS 18 The concept of headway 18 The Poisson formula 20 Samples . . 28 Sample means and variances 29 Hourly volumes 30 Qbserved Headways 30 The Erlang Distribution 38 Nature of the Erlang Distribution 40 Test of Goodness of fit . 44 Simulation by parallel arrangement 49 Relation of parameter K to volume 53 Relation of parameter K to type of lane . 58

IV. " CONCLUSIONS . . '69:

APPENDIX "A" - Tables of observed and theoretical distributions of Headways at ail study locations 73

APPENDIX "B" - Curves of observed and theoretical distributions of Headways at ail study locations

APPENDIX "C" - Curves (plotted on semi-log paper) of observed and: theoretical distributions of Headways at ail study locations 107

90

Table of Contents continued

APPENbIX "D" - Tables of chi-square tests when fitting observed data to Poisson and Erlang (K=2) distributions 124

REFERENCES 157

LIST OF TABLES

Table No.

Pace

Summary of

Summary of variances,

the projected Hourly volumes 15

calculated sample means, sample hourly volumes and Of ............ 31

Summary of chi-square test results

L. Classifiction of K values in relation to types of lane and study locations ... . 59

General analysis of variance table. Two-way classification with interaction ......... 60

Analysis of variance table. Ah l three types of lane included 61

,Classification of K values in relation to study locations and two types of lane only .. 63

8. Analysis of variance table. Passing lane • excluded 64

A-1 to A-16. Theoretical and Observed Distribution of Headways at ail individual study locations ...... 73-89

D-1 to D-32. Chi-Square Tests for ail individual study locations when fittïng data to Poisson and Erlang (K=2) .. 124156

I

I

ILIST OF FIGURES

I Figure Nod Page

I 1. Arriflex 16 m.m. camera used to obtain

the data • • 4

I2. Tower from which the films were taken .....

3. Pro jector used to take the data from the films .. ... 7

I 4.

... Sketch showing the general location of the reference line •• • • • • • ••• •• • • 10

I 5. Sketch showing the study locations 11

I

6. Typical traffic volumes obtained from automatic traffic recordera .. 13

7. A typical headway distribution curve 22

.0bserved distribution of headways in a driving lane; Dufferin Street . 23

Observed distribution of headways in a passing lane; Islington Avenue .. 24

I 10. Observed distribution of headways in a deceleration lane; Dixon Road . 25

I 11. Observed distribution of headways in

through lanes combined; Avenue Road . . 26

I 12. Erlang arrivai or service-time distribution.

Probability that the next arrivai will occur after time interval t 39

I 13. Family of arrivai distributions correspond- ing to a parallel arrangement of exponential channels ... .. • • •• • . 51

I 14. Influence of lane volume on parameter K 56

I 15. Influence of total volume of combined

through lanes on parameter K . . 57

I vi

List of Figures continued

B-1 to B-16 Theoretical and Observed Distribution of Headways at ail individual study locations 90-106

C-1 to C-16 Theoretical and Observed Distribution of Headways (plotted on semi-log paper) at ail individual locations 107-123

vii

I. INTRODUCTION

The purpose of this thesis was to study the

longitudinal distribution of vehicles in the traffic stream.

In respect of roads with small volumes of traffic, we knew

that a good approach to the problem may be dotained by

assuming that the traffic is fortuitously distributed and

follows closely enough a quite- simple probability function

named Poisson distribution (3).

However, because of the nature of the Poisson

formula ald the way it has been derived, we had serious

doubts that it would not work as well for average or

relatively high traffic volumes. Since ail the traffic

theory regarding the longitudinal pattern of the traffic

flow is based on this function, we thought that it was very

worthwhile to check its accuracy at different volumes and

possibly determine the conditions for which it applies.

At high volumes, the traffic is not as fortuitously

dis tributed as in the case of low volumes sLnce there are

so many factors influencing the driver operation that we no

longer have this complete randomness upon which the Poisson

function is based. In these conditions, it was felt that

another mathematical tool was needed which was not based on

this "complete randomness". This tool should possibly work

-1-

2

for either the opposite extreme from randomness which is

reguIarity and can be nearly approached during congestion,

or for the whole range of possibilities between randomness

and regularity.

A. K. Erlang (5) has developed such a distribution

and used it to describe telephone traffic. The Erlang

distribution is "less random" than the Poisson distribution

in the sense that it predicts a more regular and determined

flow than the latter. Since -We expected. to observe a

relative regularity of the traffic flow at high volumes,

the Erlang function might possibly work. Furthermore,

there might be such an analogy between road traffic and:

telephone.. traffic that we could possibly apply to road

traffic the theory as developed in telephone by A. K.

Erlang.

II. EXPERIEENTAL PROCEDURE

THE FIELD EQUIPMENT

The data upon which we based our calculations

have been extracted from some films tàken by the Dept. of

Highways of Ontario in previous studies.

The camera used to tale the films was an rArriflex"

16. m.m. professional cine camera, see Fig. No. 1. Although

it was equipped with wide-angle, telescopic and normal lenses,

-0/ily the normal ions was used.. It was also powered by a

storage battery and the frame speed could be set at any

given speed by a quick adjustment. A small amount of

experimenting showed that a speed of eight frames per

second would give the best results as far as the number of

vehicles observed per'length of film and the movement of a

particular vehicle in any one frme are.concerned.

The film spools contained approximately 400 feet

of film which produced approximetely thirty minutes of film

without changing spools. Although it would have been

desirable to have no interruptions in filming during this

period, it was occasionally necessery to stop the camera

to meke speed adjustments. Filking was aLso stopped if

traffio came to a stop either on the through lanes or on

the deceleration lame. This stop condition contributed

-3-

» Arriflex 16 mm. 'camera used to obtain the data

FIGURE 1

nothing to the study and in ail cases the Cause was beyond

the camera range and therefore of no interest in this

study.

To obtain the required field of view, the camera

was located on a touer constructed of portable tubular steel

scaffolding (Fig. No. 2). The touer was approximately fifty

feet high and guys were used to minimize swaying. The

distance between the touer and the nearest lane of the

highway was in the vicinity of 100 feet. The touer was

plainly visible to the motorists, particularly when it was

manned. In order to minimize any effect on the habits of

the drivers, it was in place approximately one week before

any camera work was carried out. Personnel also moved

around at the top of the tower during this familiarization

period. As far as could be determined, the touer had no

effect on the operating characteristics of the vehicles

on the through lanes or on the ramp.

THE PROJECTION EQUIPMENT

The projector used for this phase of the study

was a Bell end Howell 16 m.m. suent time and motion study

type (àee Fig. No. 3). The film could be run forward,

reversed or stopped at any time. The projector was

specially fitted for manual frame advancement for detailed

examination of each individual frame of the film. A counter

Tower from which the films were taken

6

FIGURE 2

JUN • 62

i Abe • alifflimiuren.

Projector used to extract data from the films

7

FIGURE 3

was attached to the projector for recording the number of

frames which had been viewed between the passage of successive

vehicles. The projector was equipped with a normal

projection lens with a focal length of 2 inches and required

a projector distance of about 20 feet to obtain a picture

size of 4 X 3 ft.

THE STUDY LOCATIONS

Although the Dept. of Highways of Ontario had a

great number of films available which had been taken in

previous studies, maay of them could not be used for the

purAse of this work for some reason or another. One of

the major reasons preventing their use was in most cases

due to the fact that it was often extremely difficult to

determine with a reasonable accuracy the precise instant

when the oncoming vehicles reached a reference une drawn

on the screen for the purpose of calculating the headways.

Most of the films having indeed been taken at intersections

for specific ramp studies, our reference une was often

located so far in the field of vision as to make accurate

determinat ion of the instant that a given part of any

particular vehicle crossed the une practically impossible.

Another reason which prevented their use was that on many

films the field of vision was so small that it did not

show a point approximately halfway between the beginning of

the full width of the deceleration lane and the island nose,

which was the location where we referenced the vehicles.

Fig. No. 4 is a sketch showing the general location of the

reference limé. Ideally, we would have liked to reference

the vehicles before they reach the taper section of the

deceleration lane, but no film provided us with this

opportunity. In order that ours data reflect the existing

conditbns of the traffic flow, we had to locate our

reference line before the nose of each intersection so

that we take an account of the vehicles using the

deceleration lane; we had to reference them also at a point

far aaough from the beginning of the deceleration lane so

that the through traffic previously influenced by the

vehicles using the deceleration lane would be given enough

time to readjust their operation as if they had not been

influenced by the former.

After examination of ail the films that were

available, four films were finally selected for longitudinal

movement analysis. Ah l films show major intersections with

the Highway #401, an urban freeway in the northern part of

Metropolitan Toronto. Figure No. 5 is a sketch showing

the study locations. To simplify the naming of the sites

throughout the rest of the thesis they will be called only

by the nome of the street which intersects Highway en.

REFERENCE UNE

ISLAND NOSE DECELERAT10N LANE -

_LDRIVING LANE

IPASSING LANE

MINI MI Mill MINI RIB BIBI MM MI MIR IIIIIII 111111 111111 Mil MI MM Mill MINI

SKETCH SHOWING THE GENERAL LOCATION OF THE REFERENCE LINE

FIGURE

ISLINGTON AVE

0

0

0 >c

k I PL, !NIG AVE

Sketch Showing The Study Locations

FIGURE

12

While each intersection has a different layout, this was not

taken into consideration in oui' study since the longitudinal

movement of vehicles which we study is not influenced by

the intersection layout. Each intersection is in flat

terrain and the geometric features of both the inter-

changes and the highway at each location have the same

standard level of design so that neither the geometric

design of the highway nor the physical nature of the

highway have any effect on the spacing of vehicles at any

particular location.

_The Annual Average Daily Traffic volume at these

locations as determined by the Ontario Department of

Highways was reported as being between 36,000 and 67,000

vehicles per day for both directions of travel. Some

typical traffic volumes taken at each location with

portable automatic traffic recorders are shown in Fig. No.

6; these volumes have been recorded during one week just

prior to the filming. The highway was originally designed

as a rural freeway, but due to its proximity to Toronto and

the large volumes of traffic, it can be classed more

accurately as an urban freeway.

Ah l the films used in this study were taken

during that period of Ume when there was the greatest

combination of through and diverging traffic. This

happened from 7:30 a.m. to 8:30 a.m. for westbound traffic

c0 O' zj-

e <4 •=4 a a a GO C>

N- cc as

pi

If\

il••••••

HDecelerationLane 2 Through

14.00

300 200

100

O

4.11.

NO.«

-h Pi P-4 Pi

111 N-- c0

Lanes

1600 0.4.• 1200

L. 800 -

400

0

<4 2-4 Pi

o• _I- Ir\ • .o

600_____

500 400

300L

200,

100'

0

z ‹

z Z. zP-4 P-I P-I <4 <4

Z Pi

N- c0 Cl` 4- Ir\ N- 00 0% _..-

ISLINGTON AVENUE tr) 700 600

500

11- 00

300

200

100

DUFFERIN STREET

DIXON ROAD

Tif

2400

2000 1600

1200

800

400

o

2800 2400

2000

1600

1200

800

400

0

2800 2400

2000

1600

1200

800

14. 0 0

O

EQ

UIV

AL

EN

T H

OU

R LY

VO

LUM

ES

1

P-4 Pi

c0

o

700 600

50Ô

1400

300

200

100

o

AVENUE ROAD

FIGUE 6: TYPICAL TRAFWIC VOLUMES OBTAIN.b1) PROM AUTOMATIC TRAFFIC RECORDERS

on Highway #401 and from 4:30 p.m.to 5:30 p.m. for eastbound

traffic. Although it might have been preferable that the

films be taken at different periods of the day in order to

deal with the greatest range of traffic volumes, we feel

that the volumes obtained for each individual lane at each

intersection provided us with a range of volumes good anough

for the purpose of this thesis. Table No. 1 shows these

hourly volumes projected in each case from our population

samples. In other words we think that the volumes of

traffic which were analysed show anough variations so as

to serve our purpose of studying the longitudinal flow

pattern of traffic at different volumes.

EXTRACTION: OF DATA.

Our work consisted of observing at each location

the headways in individual lanes and in the two combined

through lanes. Although the method used was very tedious

and required a considerable period of time, it was very

simple. It consisted of turning the projectOr manually

until the right front tire of a vehicle waà approximately

even with our screen reference line and then of recarding

the corresponding Crame number. Knowing the camera speed,

which in our case was always 8 Crames per second the

headways cou1d be simply obtained by subtracting the

consecutive dounter readings and then dividing by the.

MI Mal »I 11•11 MEI 111111 111•11 MI MI MI 11111111 MIR MM Mill IBM

TABLE NO. 1

SUMMARY OF THE PROJECTED HOURLY VOLUMES (IN VPH)

LOCATION DriIVING LANE PASSIN2T LANE DEC ,7T,ERATION LANE TOTAL ALL LAUES

DIXON RD. 460 658 846 1964

ISLINGTON AVE. 562 1096 674 2332

DUFFERIN ST. 950 1614 328 2892

AVENUE RD. 852 1788 622 3262

16

Camera speed, which was 8 frames per second— We fellowed the

saine method for the four study locations. At each location,

we ran the 'film three times, first recarding simultaneouely

the headways of both the driving and passing , through lm es,

secondly recording the deceleration lane headways and

thirdly recording the headways for the two combined through

lanes.

At first, we recorded the headways for periods

of fifteen minutes, which was approximately half of the film

spools, and when headways for each lane had been recorded,

we automatically reren the film in its entire length, which

corresponded approximately to a period of time of thirty

(30) minutes; this turne counting the vehicles in each lane.

The next step was to project these volumes to hourly volumes

by assuming a linear relation which we felt would be

accurate enough since the filming having been done at peak

hours, the volumes of traffic observed were so heavy and

the traffic flow so regular as to assume the saine regularity

for the second half hour. This has proved to be true at a

later stage of this thesis when we decided to reduce out"

s amples to 200 vehic les to simplify the calculations and we

found negligible differences between the volumes projected

from the time corresponding to the first 200 headways and

those projected from the volumes observed during the entire

length of the spools. The headway sam.-oie means calculated

17.

from both sets of data also showed negligible differences.

Therefore, we have gocd reasons to affirm that the estimated

hourly volumes projected from our sample means closely

represent the existing hourly volumes at the tue of filming.

In any future similar type of study, we would recommend

that traffic counts be taken at the tue of filming so as

to Sive an opportunity ta check the estimated figures.

III. EXPERIMNTAL RESULTS

THE CONCEPT OF HEADWAY

The longitudinal pattern of traffic flow can best

be analysed by the measurement and examination of the gaps

between vehicles. But this can be done in several ways

One way is to measure the distance in feet between vehicles.

Another way is to masure the distance from the beginning

of one vehicle to the beginning of the next vehicle. In

either case, individuel or average spacings between

vehicles can be determined. However, from observation, it

has been found that the minimum or ildesirable" spacing

(i.e. the distance to the vehicle ahead that the motorist

accepts as a safe distance so that he will have enough

time to react in case of emergency) varies as some function

of the velocity at which the vehicles are travelling.

Thus while a distance of 50 feet to the preceding vehicle

may be acceptable at a speed of twenty Cive miles per hour,

the motorist would no doubt prefer a greater distance if his

speed is sixty miles per hour. Therefore we can see that

it is not suf fiole nt to compare the spacings of vehicles in

terms of linear distances without cons idering al so the

vehicle speeds.

-18-

19

A more convenient method to measure these gaps is

in ternis of units of tue rather than distance, the second

being the usuel unit used. The elapsed tue from the passing

> of the front (or any other part) of a vehicle past a fixed

point until the sanie part of the next vehicle passes. the

sanie point is termed a "head.way". The examination of

head.ways in a traffic stream does not have to be correlated

with speed.s since a desirable headway does not vary much

with speed. The variation of- d.esirable head.way as related

to variation of speed is very small end can be neglected.

Another reason why the "headway" concept hes to be preferred

to- a linéar spacing concept in stream analysis is that while

the latter is a direct measure of traffic d.ensity, the

head.way is a direct measure of the rate of traffic flow,

which is .easier to determine and. at a lower cost than the

traffic density. Therefore, this concept has been used

through.out the length of this thesis.

A study conducted by O. K. Norman, of which thé

main results have been summarized in Figure 7-6 of TraffiC

Engineering by Matson, Smith and Hurd (7) s.uggests that a

head.way greater than fine (9) seconds is c.cnsidered to mean

that the following vehicle is operating in a free flow

condition and is not influenced by the vehicle ahead.. For

head.ways stnaller than nine (9) seconds, the a.bsolute speed

of the rear vehicle begins to fall rapidly to approach the

20

speed of the vehicle ahead, and there a marked drop in the

relative sbeed of the first and trailing vehicles, thus

indicating that the trailing vehicles adjust their speed

to the leading vehicle, and therefore that the former is

operating under restricted flow conditions. The preceding

vehicle is considered to exert some influence over the

behaviour of the following vehicle.

Although we expected to observe in our study a

great number of headways smaIler than nine (9) seconds,

thus losing that "complete randomness" upon which ail the

theory of probability and statis tics is based, we decided

that a statistical approach was justified since we felt

that there still was left in our traffic stream a certain

degree of undetermination and randomness. Furthermore, this

was our main objective to find out where a statistical

approach could be applied or at what point sudà an approach

ceased to be realistic.

THE POISSON FORMULA

Previous observations have shown that the distri-

bution of headways in a traffic stream does not display a

central tendency about the mean, i.e. is net a "normal"

distribution in à statistical sense. The experience bas

shown that most of the drivers tend to travel at headways

less than average while only a féw drivers exceed the

21

average value, usually by a much larger amount. Fig. No. 7

is a typical headway distribution curve. This suggests that

the headway distribution might follow the function derived

from probability laws known as trie Poisson distribution. If

we examine our own data for one of each type of lanes

studied (see Fig. 8, 9, 10, 11), it becanes readily apparent

that the resulting curves have approximately the same shape

as the previous typical headway distribution. It seemed

therefore reasonable to analyse in more detail our experi-

mental observations, even if it was known that theoretical

and observed results would not coincide exactly.

The Poisson distribution is given by:

-M x M P(x) - xl (1)

where P(X) = thejprobability of x events occurring. x = 1,2,.....n.

m = the expected number of events occurring on any given observation, i.e. the mean of x,

(x) or 7 = m.

e = the base of Naperian logarithms = 2.71828.

Applied to traffic, the definition of terms

becomes:

P(X) = probability of the arrivai of x vehicles at a point during a given length of turne.

m = mean number of vehicles arriving in the given length of turne = tV .

3.65-6

t = given time length of gap (sec.)

MI Mal MM 11•11 MI lila Mil MINI MIR MI IBM IIMI MI Mill MIR MI

PER CENT OF TOTAL VEHICLE S TRAVELLING AT

IMIMP

Mar

1_i Ii IlinniniTiT1 I I rrni t t 10 15 20 25 30 35 14.0 45 rs.)

H EADWAY - SECONDS

FIGURE 7: A TYP ICAL HEADWAY DISTRIBUTION CURVE

MI MI MI 11•1 IBM WIN Mill 111•1 MI IBM Mil IBM »I MIR MM Mill MM AT

GI

V4N

1 11.:;11 f

eiA

Y

10 11 12 13 14 15 HEADWAY SECONDS

FIGURE 8: OBSERVED DISTRIBUTION OF HEADWAYS IN A'DRIVING LANE (DUItFERIN STREET) \--)

MB MB MI 111111 Mn Mil MM 11111 MIN 11•11 11311 Inn MI 111111 an Inn MB

CENT OF TOTAL VERICLES TRAVELLIN G AT UVEN HEADWAY

50

30 _

20 _

10

0 1 2 3 14 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 HEADWAY - SECONDS

FIGURE 9: OBSERVED DISTRIBUTION OF HEADWAYS IN A PASSING LANE (ISLINGTON AVENUE)

Mil MI ail MI MI IIIMI IIIIIII MIN MM MI MI 1111111 OIS Mlle MI Mil MI IBM

9 10 11 12 13 114. 15 16 17 18 HEA.DWAY - SECONDS

FIGURE 10: OBSERVED DISTRIBUTION OF HEADWAYS IN À DECELERATION LANE (DIXON ROAD)

lm am am me oui mi mi mi mi lm lm am am am mur lm sa lm mi

1-1 L. 5 6 7 HEADWAY --.SECONDS

9 10 •

FIGURE 11: OBSERVED DISTRIBUTION OF HEADWAYS IN THROUGH LA:ES COl'IBINED (AVENUE ROAD)

27

V = hourly volume in VPH.

If the hourly volume is not known, m =

where x = the'average of observed headways. Since a

headway corresponds to a period of time during Which no

vehicle arrives, substituting zero (0) to x in the above.

formula (1) gives:

P(0) = e-mm° = e-m (2) 01

which may be interpreted as the probability of occurrence

of headways equal to or greater than a selected time t.

Following What was mentioned above, formula (2) can be

written in two different ways, as,follows:

e-vt/3600 (3)

where P(t) = the probability of occurrence of a headway greater than t seconds

and V and t are defined as above;

or P(t) = e-t/7 (4)

where P(t), t,, ead 7 are defined as previously.

Since the hourly volumes were not known at the

Ume of computations, we used formula (4) rather than

formula (3) but we would have obtained the sanie results

in using the latter since the hourly volumes have been

based upon the headway sample means and since (as previously

stated) the proportion between their Ume of observation

and the hourly volume was a linear one.

SAMPLES

As previously mentioned, when we began extr.acting

the data with the projector pur samples were drawn from a

film running turne of approximately ha1f a film spool or 15

minutes. The result was that our sample varied from lane

to lane and from location to location, since we dealt with

different volumes during the same elapsed period of turne.

At a later stage however we realized that using a constant

sample would greatly simplify the calculations and make them

less tedious. Therefore, instead of drawing our sample

from a constant time of observation, we observed a constant

number of cars, no matter what was the elapsed turne of

observation. We felt thata sample of 200 cars would be

sufficient to obtain a statistically stable sample and

therefore decided to analyse this sample at each location.

This sample stability was well damonstrated since we

obtatned approximately the saine hourly volumes and the saine

sample means when we calculated them using in ell cases.the

two sets of samples.

28

SAMTLE MEANS AND VARIANCES

There were two ways to calculate the sample mean.

One was to observe ail headways of our saoules, sum them up

and then divide by the total number of observed headways,

which in our case was 200. Since this method required that

we knew the exact value of each headway, it was rejected

since it involved a cons iderable amount of calculations

If we consider that we had to study a total of. sixteen (16)

samples, each one involving two hundred (200) figures.

Another one was to divide the total time of observation by

the number of vehicles observed. This method, much simpler

than the latter, was used. The total time of observation

could be easily calculated by recording the frame nuMber

corresponding to the two hundredth (200TH) vehicle and then

dividing by.the camera speed (8 Crames per second). If we

let Y = the frame number corresponding to the passage of

the 200th car, the sample mean is given by:

TOTAL TIME OF OBSERVATION = y = y

7 - TOTAL NUMBER OF OBSERVED VEHICLES 8 x 200 1600 e

The sample mean 7 being knawn for each set of

data, the sample variance S2 could easily be obtaîned by

using the well known formula:

s2 = r (Xi-)2 fi 1-1 n-1

where S2 = the sample variance

29

30

= the sample mean

K = the number of cells

xi = the cell midpoints

fi = the cell frequencies

n = the total number of observations = I. fi = 200 1=1

HOURLY VOLUMES

The hourly. Volumes pro jected from our samples

have been obtained as follows:

Hourly vOlume = No.of observed headways x camera speed x 3600 4. Prame number corresponding to the 200thvehicle

= 200 x 8 x 3600 1, Y

As we noted earlier, we have good reasons to belleve

that the'theoretical hourly volumes calculated by the above

formula coincide clos ely with the actual volumes which would

have been recorded, had we counted the traffic volumes during

one hour at the time of the filming. Table No. 2 gives 'a

summary of the calculated sampie meals sample variances

and hourly volumes for each individual lane and through

lanes combined,

OBSERVED HEADWAYS

Having a mathematical tool to work with, it is now

possible to examine the data obtained in the present thesis

MI MM MOI MI Mal NUI MM MB Mil IBM MW Mil MB MI 11111 MB 'MI

TABLF, NO. 2.

SUMMARY OF CALCULATED SANPLE Prq1 ANS (7), SAMPLE VARIANCES (S2 ), HOURLY VOLUMES AND 1/5E.

LOCATION TYPE OF LANE

SAMPLE

i'r ee)

1 SAMPLE VARIANCE(S2 )

HOURLY VOLUME(VHP) x

DRIVI7G 7.85 0.127 30.16 460 DIXON ROAD PASSING 5.48 0.182 32.18 658 DECELERATION 4.26 0.235 16.72 8116

DRIVIE'l 6.42 0.156 17.96 562 ISLINGTON AVE. PASSING 3.28 0.305 9.78 1096 DECELERATION 5 .3 5 0.187 , 28 . 53 674

DRIVING 3.80 0.263 6.45 950 DUFPERIN ST. PASSING 2.23 0.)1118 1.64 DECELERATION 10.97 0.091 102.32 Il

%

DRIVING 4.23 0.236 8.97 852 AVENUE ROAD PASSIN 2.01 0.498 1.88 1788 DECELERATIŒT 5.80 0.172 28.43 622

DIXON ROAD 2 THROUGH L. 3.22 0.311 7.à5 1118

ISLINGTON AVE. 2 TILROUGH L. 2.17 0.461 3.79 1658

DUFFERIN ST. 2 THROUGH L. 1.40 0.712 1.23 2564

AVENUE ROAD 2 THROUGH L. 1.36 0.733 1.29 2640

32

and see how close they fit with the theoretical Poisson

distribution. In order to do this, we first calculated the

theoretical Poisson curve for each set of data, using the

formula P(t) = in which X" h.ad already been calculated

(see Table No. 2). This was easily d.one with the help of

Exponentiel Tables (9) . The next step was to classify oui'

observed headways in celle of one (1) second intervals and

to tabulate the observed cumulative cell frequencies. Since

the theoretical distribution-is a 100%-cumulative frequency

distribution (i.e. it gives the % of headways greater than

a given value), we also tabulated. the 100%-cumulative cell

frequencies of the observed headways in order to plot and

compare the theoretical curves with the one obtained from

our data. Tables A-1 to A-16 (in Appendix A) and Fig. B-1

to B-16 .(in Appendix B) show the observed resulte and their

corresponding curves together with the theoretical results

and their corresponding curves; also included are the

theoretical results and curves obtained from the Erlang•

distribution, used at a later stage of this thesis.

When we observe the Tables and Figures mentioned

above, we notice great discrepancies between the experimental

and theoretical results. bre certainly cannot conclude by a

more observation of the curves a very close fit between the

observed headways and the theoretical probability curves.

This wasto be expected because or the nature of the Poisson

33

distribution (3). Being derived as a limiting fOrm of the

binomial distribution it applies only when the number of

possible events is large (theoretically, infinite) and the

probability of occurrence of any individual event in the

time interval considered is small. An example of the kind

of data which we would expect to find dis tributed in the

Poisson form would be the number of accidents happening at

a very busy intersection; or the frequency of headways of

a given length on a road where the traffic volume is

extremely small. This is obviously not the case at our

study locations. The traffic volumes observed at the time

of filming are so heavy that the probability of occurrence

of a headway of any small time length is very large and the

number of possible large headways is very small. The

reason for studying the headways on the deceleration lanes

was precisely that when we began our study, we thought that

these lanes would carry the small volumes for which a

Poisson distribution might apply. Unfortunately, after

projection of our data to hourly traffic volumes, we

discovered that only one out of the four (4) deceleration

lanes studied had a reletively small volume. This happened

in the deceleration lane at Dufferin St., where the projected

hourly volume was 328 vehicles per hour. Of all the Figures,

Fig. B-9 is probably the one where we can observe the

closest fit between the experimental curve and the theoretical

34

Poisson curve. Although the two curves do not fit perfectly,

they are very close to each other for headways greater than

four (4) seconds. A statistical test performed at a later

stage of this thesis to check the goodness of fit of the

experimental and theoretical curves will show that even if

the two curves above mentioned donot fit statistically,

they fit much better than any other emilar set of two

curves at each study location. Although the statement of

definite conclusions would have to be supplemented by

other studies, we feel that this is an indication that the

Poisson distribution may be the best model to describe the

headway,distribution at small volume of traffic. This also

might be an indication of the volume which should be

defined as a "small volume" in Œrder tat the Poisson

function gives a fit.

From the observation of the Figures, we notice

that ail curves follow approximately the saee general

pattern. But there are so many variations among them that

we can hardly find a relation between the traffic rate of

flow and the goodness of fit of the observed headway

distributions with the theoretical Poisson distribution,

If any trend is to be observed, it might be that in most

cases (if not in ail) the dis crepancy between the

theoretical and experimental curves tends to decrease as

the headway time length "t" increases. This suggests that

35

the observed data might be represented better by two

distributions, one for small spacings and one for lerge

spacings. This idea of a "double distribution" was first

suggested by Greenshields (2) who once obtained a good fit

with a distribution for headways less than 4 seconds and o

another for headways of more than 4 secon3s. It would

therefore be interesting to check whether our data follow

this kind of pattern.

As already maltioned, the Poisson function

applied to the distribution of headways is of the general

form y = ex.(see eg. 2) which may be written:

logey = X.

Thus the equation plotted on semilog-paper becomes a straight

lins with a negative slope since x = -m. When me plot our

data on senilog-paper, we get the curves shown on the Figures

C-1 to C-16 in Appendix C. It anpears that the greatest

discrepancies occur for large headways but this is not the

case and simply due to the semi-log scale. In only 3 cases

out of 16 the data were more closely fItted by to straight

unes. And of these three, it will be seen later that one

fits the Poisson distribution at the 2.5% level and therefore

should nOt give a better fit with two straight unes. The

fact that the combined through lanes of Dixon Road (sec

Figure C-13) appear to be best represented by two unes when

36

in fact it fits the Poisson formula 1s here again due to the

semi-log scale. Indeed, we had, to break the une for only a

few points iihich are found in the larger headway range where

we find the minimum deviations between the experimental and

theoretical results but at the sanie time the maximum

discrepancies due to the semi-log scale. In the Islington

driving lane (Fig. C-4) and deceleration lane (Fig. C-6),

the only cases loft where we have the two straight linos

pattern, the change of direction on the Unes occurs at

approximately 13 and 9 seconda respectively. Althaugh the

intersection of the two straight unes is very sensitive

with respect to position of the two Unes and these unes

are only-band fitted within a specified. range we do not

believe that this is enough to explain the discrepancy in

the "location of the breaks" of the curves which in

Greenshields' study occurred. at 4 seconds. From ail this,

it appears that the theory developed by Greenshields con

hardly be generalized but is rather determined. and influenced

in each case by different location factors. Other studios

previously done on the sanie subject by J. L. Vardon (11)

and J. W. Wise (13) also showed great discrepancles with

Greenshielda' curves and therefare prove that we cannot

generalize this concept of a multiple distribution, each

random in its limited case, as one which would apply in ail

cases.

37

Greenshields (2) has also shown that for each

plotted point there is a car responding range of expected

error or natural uncertainty caused by the fact that nunless

the sample is very large there is always a difference

between the sample values and those of the universen. This

natural uncertainty, based on the standard deviation of the

sample, is obtained from the formula:

(5)

where n = the total number of happenings recorded

fo = the acCumulated frequency,

Since in our atudy n was always equal to 200, the factor

n is sovery close to unity that it can be negIected and n-1 the equation becomes:

Z =Vfo (1 - fo)

These figures have been calculated for each point and plotted

in Figures C-1 to C-16. With a few exceptions for points in

the (0-2) second interval, the unes drawn through the

experimental points stay within the natural uncertainty

range, from which we can conclude that the data can be

represented by a straight une. Even in the combined

through lanes of Dixon Road and the driving and deceleration

lanes of Islington Avenue where the data were fitted by

two straight lines, a single line would stay very closely

38

within the expected range of error.

Although the best fitting curve appeared to be in

most cases a single straight linejit is easily observable

from the Figures C-1 to C-16 that (with the exception of

Fig. C-9) the distribution is not a Poisson one. Everywhere,

esnecially in the small headway range, we observe great

variations between the two distributions and the slope of

the experimental and theoretical curves are different. The

fact hawever that the experimental data can be closely

fitted on a semi-log paper by a straight une means that

the model best representing them still has some exponential

form, more elaborate than the Poisson model and with a

different slope.

THE ERLANG DISTRIBUTION

From Tables A-1 to A-16 and Figures B-1 to B-16,

we observe many discrepancies between the experimental

distribution and the Poisson distribution. But a general

pattern common to ail cases is that in the smaller headway

range the experimental curve lies somewhere well above the

Poisson curve. Thus the model that we need to renresent

our data is one which would give greater probability values

in this range of small headways. We felt that the best

distribution serving Ulis nurnose is the A. K. Erlang

distribution. Figure 12 shows the Erlang distribution

39

ERLANG ARRIVAL or SERVICE-TIME DISTRIBUTION. Probability that the next arrivai will occur

after time interval t.

FIGURE 12

for several values of the parameter K; where K=1, the

distribution is the simple exponentiel or Poisson case. The

Erleng distribution bas been extensively used in telephone

traffic for different purposes such as to calculate the lino

holding or waiting time and we felt that similarities

between telephone and highway traffic in crowded situntions

are such that it might well be applicable to highway traffic.

NATURE OF TUE ERLANG DISTRIBUTION

The Erlang distribution can be thought upon as if

every oncoming vehicle had to pass throtj a series of

exponential,channels, called phases, be held in the

exponentiel channel for a variable time and thaireleaSed

from the channel before the following vehicle can be

accommodated. Each phase is of the exponentiel or Poisson

type but the resulting distribution is not exponentiel.

There is a possibility of forming any kind of distribution

pattern by simply adding more and more phases. These

distributions, called Erlang distributions, provide a family

of service-time distributions which range from the completely

random exponential type to the completely regular service-

time situation. This kind of service-time distribution can

be easily interpreted as appl:ying to a freeway toll booth

operation. In such a case, it is perfectly imaginable that

a vehicle has to go through any number of phases that we went

Li

before being accommodated by the toll booth facility and the

following vehicle being allowed in the sanie booth. This is

not so easily interpretable when applied to the headway

distribution. It is indeed hardly imaginable that the

traffic flow actually goes through such phases. Or if it

does so, it is still more difficult to physically separate

its operation into distinct phases. The only physical

interpretation that we cari think of would be to assume a

traffic stream composed of two types of drivers, each type

representing a different phase. One phase would include

ail those drivers who are always in a hurry and therefore

always keep the distance between their vehicle and the

leading vehicle (the headway) to a minimum. As soon as

they can find in the parallel traffic lane a minimum

acceptable gap, they shift lanes and overtake the leading

vehicle. The other phase would be composed of ail the

drivers who, not being in a hurry, don't mind to have any

gap langth between their vehicle and the leading vehicle.

These drivers are assumed to operate their vehicles as if

they were not influenced at ail by the operation of the

other vehic les in the traffic stream. Although this inter-

pretation may not be totally in conformity with the nature

of the Erlang service-timn distribution types, it is the

best that we could think of. On the other hand, it was not

so important to us to visualize a physical interpretation

42

to a mathematical model as to find a mathematical model which

would best fit our experimental data. The Erlang distribution

being "less random" than the Poisson distribution since it

predicts fewer very short or very long intervals between

arrivais, it appeared to be a mathematical model that might

give a better fit to our observed headway distribution than

Poisson.

The Erlang distribution is given by the formula:

K=1 A0(t) = e

_KAt (KAt)n (6)

n=0

where:

Ad(t) = the'probability that no arrival'occurs4n time t after the previous one.,

À = the mean rate of arrivai.

,K = the number of phases in the system.

n = the number of states in tha system.

e = the base ofNaperian Logarithms = 2.71828.

Applied to traffic, the equation .(6). becomes:

-K)tt K=1 e P(t) = (KAt)n/n1 (7)

n=o

where,

P(t) = the probability of occurrence of a headway greater than t seconds.

A = 15, 7 being the average of observed headways.

L13

Substituting K=1 in (7) gives:

P(t) = which is the familiar exponential or

Poisson distribution (see eq. 4).

After a careful examination of the family of

Erlang curves as shown in Figure 12, we felt that the

Erlang distribution with two phases (or K=2) had the best

chances to fit our experimental data. In order to check

this, we tried to fit the experimental data of the Avenue

Road driving lane with a three phase Erlang distribution

(K=3); a test of goodness of fit proved that an Erlang

curve with K=2 gives a botter or closer fit. Then we

decided to analyse the fit obtained from the latter at ail

study locations. Substituting K=2 in eq.(7) gives:

P(t) = (1 2 At) e-2At (8)

As we had previously done for the Poisson distribution,

by substituting different values to "t" in .eq. (8), we

obtained for ail sets of data the theoretical Erlang

distribution with K=2. These results and curves have been

summarized in Tables A-1 to A-16 and Figures B-1 to B-16.

In ail cases, as we expected, this distribution

gives a much closer fit than the Poisson distribution in

the smaller headway range. Then follows a transition

range of headways in which we observe great discrepancies

between the experimental distribution and either one of

Poisson or Erlang. In this range, we don't observe either

any regularity: in some cases, Poisson gives a better fit,

in other cases it is Erlang. In the larger headway range,

the discrepancies between the experimental data and the

theoretical Poisson or Erlang curves decrease appreciably

but here again we donet observe any regularity. As for the

intermediate range, the experimental distribution sometimes

lies between the Poisson and Erlang distributions, sometîmes

getting closer to either one-of them but remaining inside;

sometimes it goes outside either one of them. In general

however, the observed distribution tends to be closer to

the Erlang distribution.

From observation of the results and curves, it

does mot appear that our observed data can be represented

by either one of the theoretical distributions. Further-

more, if one of them is to give a fit, we can hardly

determine which one it would be since, although the Erlang

distribution generally lies closer to the observed data,

the determination of a goodness of fit based on a simple

examination of curves is usually very misleading.

TEST OF GOODNESS OF FIT

Several statistical tests of significance exist

which allow us to determine with more certainty if the

experimental and theoretical data coincide. The chi-square

(X2) test is the most appropriate.to the present application.

By definition Y X2 = 2: ( f'-f .:02. (9).

i=1 ft

where,

fo = the observed frequency for any class interval.

ft = the computed or theoretical frequency for the same class interval.

K = the number of .class intervals.

If n = the total number of observations, then ft = n pi,

pi being the probabilities associated with the class

intervals. When we compare the statistics obtained from

eq. (9) with the X2 distribution with degrees of freedom

V=K-Y, •whère Vis the number of linear restrictions imposed

on the difference (fo-ft), we can test if the experimental

data can be represented by any given theoretical distribution.

The X2 tabulated values are the maximum values which the

sample statistic can assume and yet stiil be considered as

representing that the experimental curve does flot differ

significantly from the theoretical curve. In other words,

if the semple statistic obtained from eq. (9) is smaller

than the tabulated value at a given significance level are

there is no evidence to indicate that the experimental

distribution differs from the theoretical distribution,

the other hand, if the semple statistic exceeds the

tabulated value at the sanie significance level q,, we have

(1-cteo of chances of being right when we affirm that the

experimental distribution differs from the theoretical data.

We could also say that the probability is less than ar % that '

the experimental and theoretical curves are the sanie.

Common significance levels are 0.01, 0.025 and 0.05.

We can now use this test to check which one of

either Poisson or Erlang distribution represents more closely

our experimental data. For this purpose, we compared the

experimental observed frequencies (f o ) with the theoretical

expected class frequencies (ft) derived from both distributions.

As previously stated, the expected class frequencies are

obtained by multiplying the number of observed frequencies

(n) by the probability (pi) associated with each class

interval. Although these probabilities could have be.en

obtained in tables in the case of the Poisson distribution,

the fact that there were not such tables for the Erlang

distribution lead us to decide not to use the Poisson tables

but rather to use the saine procedure for both distributions

and calculate them directly from our results summarized in

Tables A-1 to A-16. These tables give the probability of

occurrence of a headway greater than a given time t, both

for Poisson and Erlang distributions. The class interval

probabilities (pi) which were needed for the chi-square

tests could simply be obtained by subtracting successively

these tabulated values. Chi-square tests of significance

have been performed for ah l study locations, each turne

47

comparing the observed data with both Poisson and Erlang

distributions. These are shown in Tables D-1 to D-32 of

Appendix D. The fact that the class interval probabilities

(pi) indicated in the above Tables have four (4) figures

while the probability values of Tables A-1 to A-16 from

which they have been derived have only three (3) figures

is simply due to rounding of the latter figures. The chi-

square test results have been summarized in Table 3, from

which we can make a few observations. Firstly, only three

(3) out of sixteen (16) sets of experimental data gave a

statistical fit. At Dufferin deceleration lane and Dixon

,driving lane, we get a fit with Erlang distribution at the

1% level and a fit with Poisson at the 2.5% level in the

combined thrbugh lanes of Dixon Road. The t/;affic volumes

at these locations were respectively 328, 460 and 1118 VPH.

This is totally opposite to our expectations, since we

always felt that the Poisson distribution had best chances

to give a fit for small traffic volumes and Erlang for heavy

volumes. Secondly, out of the thirteen (13) study locations

remaining, most of them (11) gave a closer fit with the

Erlang distribution, with only two (2) being fitted more

closely by the Poisson distribution. Thirdly, the combined

through lanes, with the exception of Dixon Road which gave

a fit with Poisson, ail fit very nearly with Erlang.

From ail this we can hardly suspect the existence

MI MI MI MM MI MI Mal MI 1311 11•1 IBM 1111111 Mil MM MI MI MI IM

TABTP1 NO. 3

SUMMARY OF CRI-SQUARE TEST RESULTS

LOCATION TYPE OF LANE

VOLUM STATISTICS (POISSON)

CHI-SQUARE VALUE ( rg)

DIFFER- ENCE

STATISTICS (ERLANG)

CHI-SQUARE VALUE (32.-1,)

DIFFER- EICE

CL0S2, FIT TC

DUFFERIN DECEL. 328 30.59 27.69 2.90 23.30 27.69 4.39

t d t t

: bd b d bd pl pl bl

ITJ t i1 PI w

1 71 1

71 t1

r

1-1 ri

CQ

! 7:›1

C0 U5

Ç)

Q CD Li) Q (.1 L

I)

ç) QQ o p ci)

Ç4 Q

DIXON DRIVING 460 )01 .77 27.69 17.08 25.69 29.14 3.45 ISLINGTON DRIVING 562 65.39 24.73 40.66 26.74 214.73 2.01 AVENUE RD. DECEL, 622 76.50 26.22 50.23 80.51 26.22 54.29 DIXON PASSING 658 47.13 26.22 20.91 87.03 26.22 60.81 ISLINGTON DECEL. 674 71.61 24.73 46.88 57.141 24.73 32.68 DIXON DECRTi. 846 76.60 23.21 53.39 60.814: 21.67 39.17 AVENUE RD. DRIVING 852 69.21 21.67 47.54 29.30 21.67 8.13 DUFFERIN DRIVIR7T 950 127.39 20.09 107.30 60.16 20.09 40.07 ISLINGTON PASSING 1096 71.81 20.09 51.72 46.06 18.48 27.58

DUFFERIN PASSING 1614 115.69 16.81 98.88 36.95 15.09 21.86

AVENUE RD. PASSIM 1788 163.70 15.09 148.61 71.32 13.28 58.04-r DIXON 2 LARES 1118 14.95 16.01 1.06 53.52 18.48 35.014

ISLINGTON 2 LARES 1658 31.12 16.81 14.31 20.06 18.48 1.58

DUFFERIN 2 LANES 112640 37.25 13.28 23.97 11.04 9.21 1.83

AVENUE RD. 2 LARES m2564 59.24 13.23 45.96 11.17 9.21 1.96

NOTE: x - Fits at the 2.5% level. * - Fits at the 1% level.

14-9

of any relationship betWeen the traffic volumes and the

application of either Poisson or Erlang. It appears however

that the Erlang distribution generally represents the headway

distribution better than the Poisson distribution, although

neither one gives an accurate representation of the actwaL

observations.

SIMULATION BY PARALLEL ARRANGEMENT'

The next step was.to try another probability

distribution which would fit the experimental data closer

than the previous distributions. The Erlang distribution

was defined for integer values of the parameter K and greater

than unity. Me feltthat it would be interesting to ti'y a

distribution where the value of K is smaller than unity.

Such a value would correspond to a situation "beyond"

Poisson, which might •be called hyper-random. We felt fully

justified in trying this new approach since the authors of

a recently published research project (4) on the saine

subject declared: "there seems to be some evidence that a

multi-lane freeway with very high traffic volumes might

be hyper-random".

A model which gives such hyper-exponential

curves is one, as in the case of the Erlang distribution,

that assumes exponential channels. But unlike Erlang,

where these channels were arranged in series, this one

would assume 'a parallel arrangement of channels (8). It

is represented by the density function':

.de-2ent (1-01 e -2(1-01nt So(t) (10)

When applied to traffic, eq. (10) becomes:

-24t P(t) = '75F (1.4 () e-2(1—e) '

where P(t) and have been defined in other sections of this

thesis and o4nti. Our procedure was first to determine the parameter 4- ,

which was given by the equation of the second degree:

r + 2 sr (1-41] =

j (12)

wheré j eqUals the expression in the brackets. (jeol) and

s2 = the-sample variance. Then, having determined 0- , we

could calculate P(t) from eq. (11), for, different values of

t. When 1ooking at the curve patterns of eq.(10) which are

shown in "ligure 13, we had good reasons to believe that

some Of our set of data, especially when the observed dis-

tribution lies for a great part below the Poisson curve,

would assume a similar type àf distribution with different.

rand j values.

When we solved eq.(12) by substituting theits ànd

s2 is values of ah l sets of data, we obtained imaginary roots

for d' in ail cases but one, ald it was in the passing lane

set)

or

Aste

-1

FAMILY OF ARRIVAL DISTRIBUTIONS CORRESPONDING TO A PARALLEL ARRANGEMENT OF EXPONENTIAL CHANNELS.

FIGURE 13

of Dixon Road. The resulting equation was

4.07e2 _ /-1--07tr + 1 = 0, whose solution gives (r= 0.462..

Since s2 = j 2, 32.18 = (5.48)2 j, from which j = 1.07.

Thus the theoretical distribution.at the passing lane. of

Dixon Road would be an hyper-exponential one but located very

close to the exponential or Poisson curve. This agrees

with the results obtained when performing the chi-square

tests. In this case we had found out a closer relation

with Poisson than with Erlane but not a statistical fit.

The only other case where Poisson gave a closer fit than

Erlang was in the deceleration lane of Avenue Road.

Logically we would have expected here a fit with an hyper-

exponential- curvb, although less than at Dixon Road since

the discrepancies in the small headway range are larger.

The fact that we did not obtain aly real equation for d-

in this case would indicate that the experimental distri-

bution is better represented by a theoretical distribution

located somewhere between Poisson and. Erlang. We feel that

the same conclusion could. be applied to ail cases where

we did not obtain a fit with either Poisson or Erlang but

where Erlang gave a closer fit. Although we did not compare

our data with an Erlang distribution with K=3, we believe

that the best fitting distribution is of the Erlang type

with a K value between 1 and. 2.

53

RELATION OF PARAMETER K TO VOLUME

Since the Erlang type of distribution appeared to

us as a model susceptible to represent our experimental data,

the next logical step was to find out if there is a linear

correlation between the K values and the traffic volumes.

If we could find such a correlation, we possibly could try

to fit our data with a Gamma function which is an Erlang

type distribution but has the advantage to work for any

value of K. The coefficientS of variation of K was 1/(77

and was equal to the sample standard deviation divided by

the sample mean. Thua

1 = s , from which we get:

(13)

The K values were first calculated for ail individual lanes

and plotted against the traffic volumes. Immediately it

was obvious that the large amount of scatter ruled out a

perfect relationship. The data was then analysed statistically

to see if there was any linear relationship at ail. The

square of the correlation coefficient for pairs of measure-

ments was calculated since this value would be a measure of

the linear dependence of one set of values on •the other.

The traffic volumes were designated as xi while corresponding

values of K were designated as Yi. 12 pairs of measurements

54

were suitable for malysis.

Then the square of the correlation coefficient

(rx72) is given by:

2 En - I xi yi] 2

rxy = En I (xi2)- (!xi)2][n (yi2) Œ

yi )23

In the present case,

n = 12; Exi = 10,450;Z yi = 20.23; Ixiyi = 19,323.78

i(xi)2 = 11,256,428; 5:(yi)2 = 39.28; Œx)2 = 109,202,500;

CEn)2 = 409.25.

Substituting these values in eq.(14) gave

r2 y = 0.261, therefore the K factor is only 26.1% linearly

dependent upon the volume of traffic.

The above test was then repeated using values for

the combined through lanes. In this case,

n = 4;1Exi = 7980;ZYi = 5.65; 2:xiyi = 11461.90;

E(x1)2 = 17,542,584; 1(Yi)2 = 8.04; (Exi)2 = 63,680,40o;

GEyi)2 = 31.92; r2xy = 0.371.

Then, in the case of the combined through lanes, the K

factor is only 37.1% linearly dependent upon the volume.

The only conclusion to be drawn from these figures would be

that the K factor assumes a slightly more linear dependence

upon the volume when we consider the traffic stream as a

whole rather than when we consider each individual lane.

Although the values of 26.1% and 37.1% are quite

low, it was felt that a "best" straight lime should be

(14)

determined in any case. This une is calculated by the

method of least squares and has the form:

=

where 7 and 7 are the means of the x's and the yts,

and,

y.x (rxi)

n I(xi)2 - (E x)2

In the case of individual lanes,

= 10,450 = 870.83; 7 = 20.23-= 1.69 12 • 12

and for the:combined through lanes,

7 = 790 = 1995; 7 = = 1.41 4

Substituting these values of 7 and 7Y- and the above values

for n,rxi,Zyi,IExiyi, Y(xi)2, (x1)2 in eq.(15), we

respectively obtained:

0.00079; and by.x = 0.00012.

Thus, the "best" straight unes for each individual lane

and the combined through lanes are respectiyely:

y - 1.69 = 0.00079 (x - 870.83) (une A)

y - 1.41 = 0.00012 (x - 1995) (une B)

These unes have been plotted in Figures 14 and 15.

55

MI NB MI MI 1•11 MB MI MB MB MB BU 111111

5.0_

4.0

3.0

A 2.0

1.0

PA

RA

ME

TE

R

0 C I iobo 5 JO

272 0 00

LANE TRAFFIC VOLUME (VPH)

FIGURE 14: INFLUENCE OF LANE VOLUME ON PARAMETER K

4. 0

3.0

P A

RA

ME

T E

R

2.0

1.0

0.0

am mu am am am am ma mi ma mu am ma am ma ma ma ma

13 g

g

tIM

1000

1 1 1 I 1 1 I 1 I 1 1 I 1 1 1 1 I '

1500 2000 2500 3000

TWO COMBINED THROUGH LANES TRAFFIC VOLUME (VPH)

FIGURE 15: INFLUENCE OF TOTAL VOLUME OF COMBINED THROUGH LANES ON PARAMETER K

58

It should be emphasized that these two straight

unes are not necessarily the best curves to fit the data

but are merely the best "fitting" straight unes. We

know from the values of the correlation coefficients that

there is a very small linear relationship between the

parameter K and the traffic volume. The true relationship

is certainly a more elaborate one and would be represented

by an equation of a higher degree. It was not felt however

to be worthwhile to find out this true mathematical relation-

ship since this would lead to a theory too elaborate for

practical application. For ail practical purposes, especially

if accuracy is not required, we believe that these unes

are reliable enough to show which value of K corresponding

to a given volume should be used when working with a

Gamma function.

RELATION OF PARAMETER K TO TYPE OF LANE

From what we have seen, we know that the relation-

ship between the K values and the volumes of traffic is

not a simple one. Letts now reclassify these K values, but

this time in relation to the type of lane and the study

location. Table 4 'shows such a classification.

59

TABLE L.

CLASSIFICATION OF K VALUES IN RELATION TO TYPES OF LANE AND STUDY LOCATIONS

TYPE OF STUDY LOCATTOU . TOTAL LANE Dufferin Dixon Rd. Avenue Rd. Islington

Decelera- tion 1.18 1.09 1.18 1.00 4.45

Driving 2.24 2.04 1.99 2.30 8.57

Passing 3.03 0..93 2.15 1.10 7.21

TOTAL 6.45 4.o6 5,32 , 4.40 20.23

From observation of the above table there appears

to exist a relationship between the parameter K and the type

of lane. We notice thatall values of K are close to unity

for the deceleration lanes, and close to 2 for the driving

lane; we observe no regularity in the case of the passing

lanes. However, it is not certain that the type ar lane

is the only influencing factor. It is also possible that

these values could have been influenced by the study

location itself. The best way to determine the interaction

of these factors was to carry out a well known statistical

procedure, the analysis of variance.

Since the method of performing an analysis of

variance can be found in every textbook of Statistics, it

was not felt necessary to explain it here. A summary of

this table is included, however, as well as the computations

60

leading to our results. Table 5 shows the table used to

analyse our data and the resultà are s'ummarized in Table 6.

TABLE 5

GENERAL ANALYSIS OF VARIANCaTABLE. TWO-WAY CLASSIFICATION WITH INTERACTION.

Source of Estimate

Sum of Squares S. of. S.

Degrees of F eedom(D.F.)‘

Mean Square M.S.

MEAN

BETWEEN ROWS

BETWEEN- COLUMNS

INTERACTION

TOTAL

T..2 1

ni - 1

nj - 1

(ni-l)(nj-1)

='ninj

/

S.of S./ni-1

S.of S./nj-1

S.of S./(ni-l)(nj-1)

N

571c2._1 - T..2

4- nj N

.T..2 _ T ..2 L-ni N

2:Xb1:1.2 - nj

4.- ni N

• Z X2 ij N ij

where Xij = the observations classified in ni rows and ni

columns; N = nini; T.j = 2:Xij; Ti. =IXii; T.. = ni ij

In our case (see Table 52), ni = 3; nj = '4; N = 12;

Z_X2ij = (1.18)2 + + '(1.10)2 = 39.27; ij

T..2 = (20.2 3)2 = 34.10; ZT.2 J = ()t5)2 -F. (8.57)2 +(7.21)2 N. 12 nj 4 4

= 3 .3I; T• = (6.)5)2 t (4,06)2 + (5.32)2 + (4.40) 2 ni 3 3 3 3 .

= 35.24.

ni

61

Thus the variation due to the type of lane = 36.31 - 34.10

= 2.21; the variation due to the study location = 35.24

314.10 = 1.14; the interaction = 39.27 - (34.10 + 2.21 + 1.14)

= 1.82.

Ide can now set up the following analysis of variance table:

TABLE 6

ANALYSIS OF VARIANCE TABLE. ALL THREE TYPES OF LANE INCLUDED.

SOURCE OF, ESTIMATE

SUM.OF SQUARE S. of S.

D.F. MEAN SQUARE

Mean

Between Types of iJele

Between StUdy Locations

Interaction

Total

34. 0

2.21

1.14

1.82

, 39.27

2

3

6.

12

2.21

1.14

1.82

4

4

4

2

3

6

=

=

=

1.05

0.38

0.303

The above resuIts allow us to carry out a few tests of

significance:

1. Le-Us check the hypothesis that the study location

does not affect 'the factor K at the 5% level.

From our table, 0.38 = 1.26 - 0.303

From the F-distribution F(36, 0.05) = 4.76

Since 1.2644.76, this means that the test is not

1

1

62'

significant and therefore the stated hypothesis is true.

Thus the study location does not affect the K factor at

the 5% level. This conclusion is very satisfactorily

accepted since if the opposite had been true, it would

have indicated that our results are simply due to chance

and not conclusive in any way.

2. Let's now check the hypothesis that the type of

lane does not affect the factor K at the 5% level.

I . From our table, 1.105 =-3.64

0.303

From the F-distribution, F(2e6,0.05) = 5.14

Since 3.64 45.14, this means that the test is not

significant and therefore the type of lane does not affect

the K factor. In other words, the obtained results give no

evidence of a relationship between the type of lane and the

value of K. It need not be emphasized that this is a very

disappointing conclusion since such a relationship had

appeared strongly possible from observation of the results.

We believe that the fact that the -lest did not turn signi-

ficant can be explained by either one of the following

causes: the number of observations was insufficient, thus

providing no solid ground for a statistical analysis; or,

what is more probable, the discrepancies between the K

values of the passing lanes are so great that they may

cause the test not to be significant. Although the first

cause was quite possible and certainly cannot be eliminated,

1

63

we felt that it was beyond the scope of this thesis to

investigate it. The second cause could be simply checked

by analysing the variance of the data for te driving and

deceleration lanes, thus excluding the passing lanes where

the greatest variations in the value of K were found. An

analysis of variance was then carried out for the data of

Table 7:

TABLE 7

CLASSIFICATION OF K VALUES IN RELATION TO STUDY LOCATIONS AND TUO TYPES OF LANE ONLY.

TYPE OF LANE

S _ TOTAL Dufferin Dixon Rd, Avenue Rd.. Islinaton

Decelerà- tion ' 1.18 1.09 1.18 1.00 • 4.45

Driving 2.24 2.04 1.99 2.30 8.57

Total 3.42 3.13 3.17 3.30 13.02

In this case, ni = 2; ni . 4; N . 8;

E2X1i = (1.18)2 + .... + (2.30)2 = 23.40; T..2 = (13.02)2 ij N

= 21.19; 2:T.2j = (441.5)2 + (8.57)2 = 23.31; nj 4 4

I y Ti.2 = (3.42)2 + (3.13)2 + (3.17)2 4- (3.30)2 = 21.22; '"" ni 2 2 2 2

The variation due to type of lane = 23.31 - 21.19 = 2.12;

the variation due to study location = 21.22 - 21.19 = 0.03;

From this, the following analysis of variance table (Table

8) was set up:

TABLE 8

ANALYSIS OF VARIANCE TABLE. PASSING LANE EXCLUDED.

SOURCE OF ES IMATE

SUM OF SQUARE •

D.F. MEAN SQUARE u

Mean 21.19 1

Between Types of Lane 2.12 1 2.12+1 = 2.12

BetWeen Study Locations 0.03 3 0.0343 = 0.01

Interaction 0.06 3 0.06+i = 0.02

Total 23.40

Ne can now verify the same hypotheses as previously:

Hypothesis: the stndy location does not affect the

parameter K at the 5% level.

From our table, 0.01 = O. 0.02

From the F-distribution, F(3,3,0.05) = 9.28

Since 0.54.9.28, the test is not significant. There-

fore there s definitely no evidence to indicate that the

study location affects the value of K.

Hypothesis: the type of lane does not affect the

parameter K at the 5% level.

From our table, 2.12 = 106 0.02

6)4

65

From the F-distribution, F(1,3,0.05) = 19.13

Since 106».10.13, the type of lane is revealed as a

highly significant effect. This is a very interesting

conclusion that we had expected when we first observed

Table 4 and suspected a trend. between the type Of lane and

the factor K.

According to this, the headway distribution of an

urban freeway would be theoretically represented by different

distributions when we cons ider each Iane separately. The

deceleration lane headways would. be best represented by an

Erlang distribution with K=1, which is the Exponential or

Poisson distribution; in the driving lane, a good theoretical

model would be an Erlang distribution with K=2; finally, the

passing lane assumes so many Variations that a specific

Erlang curve does not apply in ail cases but any one or

the family couId apply in. individual cases. These

conclusions 'coincide well enough with what had already

been obtained when performing •the chi-square tests to

Compare the goodness of fit of Erlang (K=2) and Poisson

distributions to our experimental data. These tests had

first indicated a better fit with Erlang (K=2) than with

Poisson in ail driving lanes. In one occasion (Dixon Road),

there was a statistical fit and one was very close

(Islingtor). However, there is partial disagreement when

we corne to the deceleration lanes. Indeed, only one out

66

of four deceleration lanes (it was at Avenue Rd.) had given

a closer fit with Poisson when checking with the chi-square

test. Two had showed a closer fit and one had given a

statistical fit with Erlang (K=2), although in this last

case a fit with Poisson was very close, the difference

between the chi-square statistics and the tabulated value

being as small as 2.90. These discrepancies are not easily

explained but we think that the conclusions drawn from the

analysis of variance table are much more reliable than those

suggested by the chi-square tests. While the analysis of

variance gives a positive evidence when it is significant,

a significant chi-square test only provided us with a

negative and indirect evidence if we consider the criteria

that we used to decide which one of Erlang (K=2) or Poisson

distribution best represented our experimental data. This

criteria was based upon the minimum difference between the

statistic obtained when fitting our data to either one of

Poisson or Erlang and some tabulated values. This will be

best illustrated by the following example: In the Islington

deceleration lane, the inference that Erlang (K=2) gives a

better fit than Poisson was based upon the fact that the

difference between the statistic derived from Erlang (57.41)

and the chi-square tabulated value (2)1.73) is smaller than

the corresponding values (71.61 and 24.73 respectively)

obtained when fitting our data to Poisson. We believe that

67

such an inference still provides evidence but rather negative

as compared with the evidence provided by an analysis of

variance table. Since the closer fit to Erlang has also

been decided according to the same criteria in the case of

the Dixon Road deceleration lane, there is really only one

case where the condusionsdrawn from the analysis of variance

table and the chi-square method basically disagree. This

happens in the Dufferin deceleration lane for which a chi-

square test provided signific-ant forPoisson and not

significant for Erlang (K=2). Here the previous attemptS

to explain the apparent contradiction between the conclusions

inferred by both statistical methode do not hold since,

being not significant, the chi-square test provides us with

a positive evidence that we cannot attenuate. Although the

difference between the statistic obtained when fitting our

data'to Poisson and the tabulated value is very small (2.90),

the test with Poisson is not significant and therefore, even

if this gives ground to infer that Poisson nearly gives a

fit, the conclusion suggested by the non-significant test

is much more positive and muet be the one accepted.

Thus, with the exception of this last case, the

apparent dis crepancies between the trends suggested by the

analysis of variance and chi-square methods are not serious

and can be partly explained. Therefore, we have good

reasons to affirm that there is a relationship between the

68

value of the parameter K and the individual lanes of an

urban freeway, as demonstrated by the previous analysis of

variance. It has also been suggested to apply Poisson in

the case of a deceleration lane and Erlang (K=2) in the case

of a driving lane. It should be emphasized however that

they are not necessarily the distributions which will give

the best fit in ail individual cases. The best fitting

distribution would be one with a value of K calculated in

each case from the experimenthl data. For ail practical

purposes however, we believe that the distributions with

K=1 and K=2 should give the best overall fit when applied

to deceleration and driving lanes respectively.

IV. CONCLUSIONS

This study has attempted to investigate the

distribution of headways on Highway 401, an urban freeway.

The statistical approach was an attempt tomake the analysis

general enough so that the results could have application

to other locations. It is hoped that the observed headway

distributions on Highway 401-and their statistical analysis

can be extended and applied ta the longitudinal flow pattern

of any urban freeway. The inferences drawn from our study

should certainly be supplemented by more data, but never-

theless they do establish a foundation for further

investigation.

The analysis performed in this thesis suggested

the following conclusions:

The photographie method, although it is accurate

and perhaps the most useful tool in a comprehensive study

of 'the traffic flow, is very tedious whenextraeting the data

from the films. Another method might be used more

advantageously when studying only the longitudinal traffic

flow characteristics.

The headway distribution of an urban freeway

shows many discrepancies with the Poisson and Erlang (K=2)

probability distributions. However, the Erlang distribution

-69-

generally gives a closer statistical fit to,observed data

than Poisson and should therefore be preferred to the latter

to describe the pattern of arrivais on a 4-1ane urban

freeway. For ail practical purposes, calculations based on

an Erlang (K=2) distribution are mostly in the 'safe side

and certainly more accurate than when based on a Poisson

distribution. Our present traffic theory actually based on

Poisson would undoubtedly gain in accuracy if it was modified

and based on Erlang (K=2).

The Erlang distribution fits the observed

distribution of headways most closely when such a distri-

bution is considered for the combined through lanes rather

than for each lane taken individually.

The probability of a linear relationship between

the traffic volume and the parameter K of the Erlang

distribution is very small. This probability appears to

increase when the traffic lanes are considered in combination

instead of individually. The assumption of a linear relation-

ship however is very useful and could be advantageously

used when great accuracy is not required. A Gamma function

should then be used, the value of the parameter K being

taken from either one of the Fig. là_ or 15.

An interesting result is that there appears to

exist a correlation between the type .of lane and the

parameter K. Accordingly, an Erlang distribution with R=2

727 would generally apply to the headway distribution in the

driving lane and an Erlang distribution with K=1 (or

Poisson) would apply to the deceleration lane. This

was ex'pected in the lattei case since the vehicles diverging

in the deceleration lane are likely drawn at random from

the main through traffic stream, thus suggesting the use of

the Poisson distribution. The fact that Poisson did not

give a fit in this case could be explained by the fact

that we don't have a "complete randomness" since the

diverging vehicles were influenced by the through traffic

stream before performing their diverging manoeuver.

The "two straidat unes theory" as suggested

by Greenshields (2) did not apply to Highway 401. It would

therefore appear that his theory cannot be generalized to

ail conditions, or should certainly be modified.

A model of hyper-exponential distribution has

been tried but proved unsuccessfu1 to fit the observed

data. It is felt however that further investigation with

a different model could turn out more successful and should

therefore be carried out.

A statistical model, by nature, isa mathematical

formula which intends to describe some observed data with

the greatest accuracy possible. It is therefore impossible

that it fit3perfectly the data, otherwise it becomes a

physical law based on certainty and can no longer be called

TABLE A-9

OBSERVED AND EXPECTED DISTRIBUTIONS - OF HEA..DWAYS

DECELER.A,TION LANE - DUFFERIN

CLASS INTERVAL

(SEC)

FREQUENCY (fo)

CUMULATIVE FREQUENCY

CUMULATIVE FREQUENCY

% OF HEADWAYS GREATER THAN THE CLASS INTER-VAL UPPER LIMIT

SHOWN 100% Theo-

retical Poisson

Theo-retical Erlang- (K=2)

0.0-0.9 1 1 99.4 91.3 98.5

1.0-1.9 13 14 91.9 83.4 94.8

2.0-2.9 15 29 83.2 76.1 89.6

3.0-3.9 21 50 71.1 69.5 83.5

4.0-4,9 8 58 66.5 63.4 76.9

5.0-5.9 13 71 59.0 57.9 76.2

6.6-6.9 lo 81 53.2 52.9 63.6

7.0-7.9 6 87 49.7 48.3 57.3

8.0-8.9 12 99 42.8 44.1 51.3

9.0-9.9 6 105 39.3 40.3 45.7

10.0-10.9 7 112 35.3 36.8 40.6

11.0-11.9 8 120 30.6 33.6 35.9

12.0-12.9 4 124 28.3 30.6 31.6

13.0-13.9 5 129 25.4 28.0 27.8

14.0-14.9 7 136 21.4 25.5 24.3

;---15. o 37 173 o.o - -

N.173

•

83

TABLE A-10

OBSERVED AND EXPECTED DISTRIBUTIONS OF HEADUAYS.

DRIVING LAUE - AVENUE ROAD

CLASS INTERVAL

(SEC)

•

•

OBSEMVED FREQUENCY

(fo)

CUMULATIVE FREQUENCY

CUMULATIVE FREQUDECY

% OF H_MDWAYS GREATER THAN THE CLASS INTER-VAL UPPER LIMIT

SHOWN 100% Theo-

retical Poisson

Theo-retical Erlang (K=2) -

0.0-0.9 3 3 98.5 79.0 91.2

1.0-1.9 49 52 74.0 62.4 75.6

2.0-2.9 43 95 52.5 49.3 58.6

3.0-3.9 28 123 38.5 38.9 43.7

4.0-4.9 21 144 28.0 30.7 31.7

5.0-5.9 11 155 • 22.5 24.3 22.6

6.0-6.9 • 15 170 15.0 19.2 15.8

7.0-7.9 12 182 9.0 15.1 10.9

8.0-8.9 5 187 6.5 12.0 7.5

9.0-9.9 4 191 4.5 9.4 5.1

10.0-10.9 1 192 h..0 7.5 3.5

11.0-11.9 2 194 3.0 5.9 2.3

12.0-12.9 o 194 3.0 4.6 1.6

13.0-13.9 2 196 2.0 3.7 1.0

14.0-14.9 2 198 1.0 2.9 •

0.7

15.0-15.9 1 199 0.5 2.3 0.4

16.0-16.9 1 200 0.0 1.8 0.3

N=200

II II II II

TABLE A-11

OBSERVED AND EXPECTED DISTRIBUTIONS OF HEADWAYS.

PASSING LANE - AVENUE ROAD

CLASS INTERVAL

(SEC)

OBSERVED FREQUINCY

(fo)

CUMULATIVE FREQUENCY

CUMULATIVE FREQUENCY

% OF HEADWAYS GREA TER THAN THE CLASS INTER-VAL UPPER LIMTT

SHOUN no% Theo-

retical Poisson

Theo- retical Erlang (K=2)

0.0-0.9 14 III 93.0 60.8 73.7

1.0-1.9 116 130 35.0 36.9 40.8

2.0-2.9 43 173 13.5 22.5 20.1

3.0-3.9 14 187 6.5 13.6 9.3

4.0-4.9 6 193 3.5 8.3 4.1

5.0-5.9 2 195 2.5 5.0 1.7

6.o-6.9 2 197 1.5 3.1 0 . 7

7.0-7.9 2 199 0.5 1.9 0.3

8.0-8.9 o 199 0.5 1.1 0.1

9.0-9.9 0 199 0.5 0.7 0.05

10.0-10.9 0 199 0 . 5 0.4 0.02

11.0-11.9 o 199 0.5 0.3 0.009

12.0-12.9 1 200 0.0 0.2 0.003

N=200

85

TABLE A-12

OBSERVED AND EXPECTED DISTRIBUTIONS OF HEADWAYS.

DEC-4TRRATI0N LAME -• AVENUE, ROAD

CLASS INTERVAL

(SEC)

•

OBSERVED FREQULNCY

(fo)

CUMULATIVE FREQUSTCY

CUMULATIVE FREQUdICY

% OF HEADWAYS GREATER THAN THE CLASS INTER-VAL UPPER LIMIT

SHCUN 100% Theo-

retioal Poisson

Theo-retical Erlang CK=2)

0.0-0.9 1 1 99.5 84.2 95.3

1.0-1.9 50 51 74-5 70.9 84.8

2.0-2.9 35 86 57.0 59.7 72.4

3.0-3.9 28 114 43.o 50.3 60.0

4.0-4.9 11 125 37.5 42.3 48.7

5.0-5.9 20 145 27.5 35.6 3g.9

6.0-6.9 7 152 24.0 30.0 30.7

7.0-7.9 11 163 19.5 25.3 23.9

8.0-8.9 5 168 16.0 21.3 13.5

9.0-9.9 3 171 14.5 17.9 14.3

10.0-10.9 4 175 12.5 15.1 10.9

11.0-11.9 2 177 11.5 12.7 8.3

12.0-12.9 4 181 9.5 • 10.7 6.2

13.0-13.9 2 183 8.5 9.0 4.7

14.0-14.9 3 186 7.0 7.6 3.5

1.s 15 14 200 0.0 - - N=200

TABTR A-13

'OBSERVED AND EXPECTED DISTRIBUTIONS OF HEADWAYS.

COMBINED THROUGH LAMES - DIXON ROAD

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

CUMULATIVE FREQUENCY

CUMULATIVE FREQUENCY

% OF HEADWAYS GREATER 'l'HAN THE GLUS INTER-VAL UPPER - LIMIT

SHOWN '100% Theo7

retical Poisson

Theo- - retical Erlang (K=2)

M-0.9 47 47 76.5 73.3 87,1

1.0-1.9 59 106 47.0 53.7 64.7

2.0-2.9. 29 135 32.5 39.3 44-3

3.0-3.9 15 150. 25.0 23.8 29.0

13 163 18.5 21.1 18.3

5.0-5.9 lo 173 13.5 15.5 _11.4

6.0-6.9 5 178 11.0 11.3 6.9

7.0-7.9 5 183 8-.5.. 8.3 4.1

8.0-8.9 9 192 4.0. 6.1 2.4

9.0-9.9 3 195 25

10.0-10.9 2 197 1.5. 3.3 0.9

11.0-11.9 2 199 O. 2.4 • 0.5.

.12.0-12.9 0 199• 0.5 1.8 0.3

13.0-13.9 1 200 0.0- 1.3 • 0.2

N=200

86

87

TABLE A-14

OBSERVED AND EXPECTED DISTRIBUTIONS OF HEADWAYS.

COMBINED THROUGH LAESS - ISLINGTON AVE.

•

CLASS INTER VAL

(SEC)

OBSERVED FREQULN C Y

(fo)

CUMULATIVE ii REQUEPICY

CUMULATIVE FREQUEN CY

100%

% OF HE A . DkA YS GREATER THAN THE CLASS INTER - VAL UPPER LIMIT

SHOWN Theo- reti cal Poisson

Theo - retical Erlang , (K=2)

0.0-0.9 44 44 78.0 63.1 76.4

1.0-1.9 75 119 40.5 39.8 45.0

2.0-2.9 33 .152 , 24.0 25.1 23.7

3.0-3.9 18 170 15.0 15.8 11.7

4.0-4.9 9 179 10.5 10.0 5.6

5.0-5.9 9 188 6.0 6.3 2.9

6.0-6.9 6 194 3. 0 •

4.0 1.2

7.0-7.9 1 195 2.5 2.5 0.5

8.0-8.9 2 197 1.5 1.6 0.2

9.0-9.9 2 199 0.5 1.0 0.1

10.0-10.9 0 199 0.5 0.6 0.04

11.0-11.9 1 200 0.0 0.4 0.02

N=200

TABLE A-15

'OBSERVE]) AND EXPECTED DISTRIBUTIONS OF HEADWAYS.

COMBINE]) THROUGH LAWES - DUFFERIN STREET

CLASS INTERVAL

(SEC)

OBSEKVED FREQUE\ICY

(fo)

CUMULATIVE FREQUENCY

CUMULATIVE FREQUFDCY

100%

% OF HEADWAYS , GREATER THAN THE CLASS INTER-VAL UPPER LIMII

SHMN Theo- retical Poisson

Theo-retical Erlang

K=2)

0.0-0.9 0.0-0.9 61 61 69.5 49.1 553.4

1.0-1.9 87 148 26.0 24.1 22.3

2.0-2.9 33 181 9.5 11.8 7.4

3.0-3.9 14_ 195 2.5 5.8 2.2

4.0-4.9 3 198 1.0 2.8 0.7

5.0-5.9 1 199 0.5 1.4 0.2

6.0-6.9 0 199 0.5 0.7 0.05

7.0-7.9 0 199 0.5 0.3 0.01

8.0-8.9 1 200 0.0 0.2 0.004

N=200

88

•

TABLE A-16

OBSERVED AND EXPECTEDDISTRIBUTIONS OF HEADWAYS.

COMBINED THROUGH LANES - AVENUE.ROAD

CLASS INTERVAL

(SEC)

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0.0-0.9 62 62 69.0. 48.1 56.9

1.0-1.9 95 157 21.5 23.1 21.0 .

2.0-2.9 26 183 8.5 11.1 6.6

3.0-3.9 9 192 4.0 5.3 1.9

4.0-4.9 4 196 2.0 2.6 o.6

5.0-5.9 2 198 1.0 1.2 02

6.0-6.9 1 199 0.5 0.6 0.05

7.0-7.9 1 • 200 0.0 0.3 0.01

N=200

89

APPENDIX "B"

Curves of observed and theoretical distributions of Headways at ail study locations.

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,APPENDIX 'D"

Tables of Chi-Square Tests when fitting observed data to Poisson and Erlang (1.2) distributions.

MS 111111 IBN OBI 11111 MIN MI 1•111 MB IBN NB IIIMI MM

TABLE D-1

CHI-SQUARE TEST TO CBECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

DRIVING LANE - DIXON ROM)

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

THEORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT

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Cel

c'r\ 0 s.0

C\I 111

s.0 C\I C\1

H cp

529 22.04 1.0-1.9 24 0.1050 9 0.43 2.0-2.9 19 0.0925 0 0.00 3.0-3.9 22 0.0815 36 2.25 4.0-4.9 18 0.0718 16 1.14 5.0-5.9 8 0.0632 25 1.92 6.0-6.9 13 0.0555 4 0.36 7.0-7.9 15 0.0491 25 2.50 8.0-8.9 16 0.0431 49 5.44 9.0-9.9 8 0.0381 0 0.00

10.0-10.9 13 0.0335 36 5.14 11.0-11.9 4 0.0295 4 0.67 12.0-12.9 7 0.0259 4 0.80 13.0-13.9 6 0.0229 1 0.20 14..p-14.9 7 i 0.0202 1.88

D.e 15 26 19 0.1488 64 N=200 Z 41 )1.77

=27.69 x2 (13,0.01)

44.77 >27.69 Poisson does not fit at the 1% significance level.

MI MI MM IBM Mill 1•11 MIN Mil MI MM MI MM

TABLE D-2

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTTON TO OBSERVED DATA:

DRIVING LANE - DIXON ROAD

CLASS INTMVAL

(SEC)

OBSERWED FREQUENCY

(fo)

PROBABILITY (ERLANG) (K=2)

THEORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 1 0.0273 5

-d-H (NI ‘Y-1 0

Ocr l H el

(-(1-..- 0

O N

(NI H

H

16 3.20 1.0-1.9 24 0.0653 13 121 9.31 2.0-2.9 19 0.0851 17 4 0.24 3.0-3.9 22 0.0925 19 9 0.47 4.0-4.9 18 0.0924 18 o 0.00 5.0-5.9 8 0.0877 18 100 5.56 6.0-6.9 13 0.0802 16 9 0.56 7.0-7.9 15 0.0720 14 1 0.07 8.0-8.9 16 0.0633 13 9 0.69 9.0-9.9 8 0.0549 11 9 0.82 10.0-10.9 13 0.0471 9 16 1.78 11.0-11.9 4 0.01 0 8 16 2.00 12.0-12.9 7 0.0336 7 o 0.00 13.0-13.9 6 0.0285 6 o 0.00 14.0-14.9 7 0.0235 5 4 0.80

5;15 19 0.1063 21 4 0.19 N=200 I:=25.69

X2(14,0.01) = 29°14

25.69 429.14 Erlang fits at the 1% significance level.

0",

Rio WB ma lm immi au lm mi mu au lm mi dm mi lm lm mu

TABLE D-.3

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSEEVED DATA:

PASSING LANE - DIXON ROAD

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBUILITY (POISSON)

THEORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 15 0.1664 33

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f\)

H H U

3 N

HN

O C

o-P-

CO C

D 32à 9.82 1.0-1.9 56 0.1387 28 784 28.00 2.0-2.9 27 0.1156 23 16 0.70 3.0-3.9 11 0.0964 19 64 3.37 4.0-4.9 5.0-5.9

16 11

o.08011 0.0670

16 13

o 4

0.00 0.31

6.0-6.9 12 0.0558 11 1 0.09 7.0-7.9 7 0.01165 9

à 2.00 8.0-8.9 4 0.0388 8 9.0-9.9 lo 0.0324 7 9 1.29

10.0-10.9 4 0.0269 5 1 0.20 11.0-11.9 4 0.0225 5 1 0.20 12.0-12.9 13.0-13.9 1à.0-14.9

Ili 3)

8 0.0187 0.0157 0.0130

4 3 10 3

4 0.40

ei5 15 0.0652 13 4 0.31 N=200 Z =47.13

X2 (12,0,01)

47.13 >26.22 Poisson does 'not fit at the 1% significance level.

MI MM Mill Mil MM MI »I MI Mill ail MI Mal MOI

TABLE D-4

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF LhLANG (K=2) DISTRIBUTION TO OBSERVE]) DATA:

pAssING.LANE - DIXON RoAD

CLASS INTERVAL

(SEC)

OBSERVE]) FREQUENCY (fo)

PROBABILITY (ERLANG)

(K=2)

THEORETICAL FREQUENCY

(fT) (fo-fT) ( -fT)2 (fo-rie

fT

0.0-0.9 15 0.0522 10 5 25 2.50 1.0-1.9 56 0.1133 23 33 1089 47.35 2.0-2.9 27 0.1326 27 o o 0.00 3.0-3.9 11 0.1292 26 15 225 8.65 4.0-4.9 16 0.1159 23 7 4-9 2.13 5.0-5.9 11 0.0983 20 9 dl 4.05 6.0-6.9 12 0.0810 16 16 1.00 7.0-7.9 7 0.0647 13 6 36 2.77 8.0-8.9 4 0.0490 lo 6 36 3.60 9.0-9.9 10 0.0418 8 2 4 0.50

10.0-10.9 4 0.0309 6 2 4 0.67 11.0-11.9 4 0.0229 5 1 1 0.20 12.0-12.9 0.0178 4 13.0-13.9 1

4 V 0.0132 3 9 1 1 0.11

14.0-14.9 e15

3 15

0.0094 0.0278

2 6 9 81 13.50

N=200 Z=87.03

2 X (12,0.01) = 26.22

87.03 >26.22 .> Erlang does not fit at the 1% significance level.

MI MM Mil MI MI MI OBI MM MN IBM MI 1111M MI" Mil Un MM

TABLE D-5

CHI-SQUARE 'I.ST TO CHECK THE GOODNESS OF 'FIT OF POISSON DISTI-ŒBUTION TO OBSERVED DATA:

DECELERATION.LANE - DIXON ROAD

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

THEMETICAL FREQUENCY

(fT) (fo-fT) 2 (fo-fT) (fo-fT)2

fT

0.0-0.9 5 0.2094 42 37 1369 32.60 1.0-1.9 63 0.1656 33 30 900 27.27

2.0-2.9 42 0.1309 26 16 256 9.85 3.0-3.9 4.0-4.9

25 12

0.1035 0.0818

21 16

L. 4

16 16

0.76 1.00

5.0-5.9 6.0-6.9

9 10

0.0647 0.0511

13 10

4 o

16 0

1.23 0.00

7.0-7.9 5 0.0404 8 3 9 1.13 8.0-8.9 5 0.0320 7 2 4 0.57

9.0-9.9 5 0.0252 5 o o 0.00 10.0-10.9 4 0.0200 4 11.0-11.9 3 0.0158 3 12.0-12.9 3 16 0.0120 2 13 3 9 0.69 13.0-13.9 5 0.0103 2 14.0-1)1.9 1 0.0079 2

e 15 3 . o2 94 6 3 9 1.50 N=200 2E=76.60

X2 (10,0•01) = 23.21

76.60 >23.21 Poisson does.not fit at the 1% significance level.

MW Mil MIN IBM MI Mal OMM IIIIIII URI MI 1111111 MI MIS Nal Mil 111111 MM

TABIE D-6

CHI-SQUARE TEST 5D CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

DECELERATION LANE - DIXON ROAD

CLASS INTERVAL (SEC)

OBSERVED FREQUENCY (fo)

PROBABILITY (LhLANG) (K=2)

THEORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT) 2 (fo-fT) 2

fT 0.0-0.9 5 0.0812

\A-) o

o -p

-

H

H

11 121 7.56 1.0-1.9 63 0.1610 31 961 30.03 2.0-2.9 42 0.1695 8 64 1.88 3.0-3.9 25 0.1488 5 25 0.84 4.0-4.9 5.0-5.9

12 9

0.1199 0.0919

12 9

6.00 4.50

141

6.0-6.9 10 0.0677 4 16 1.14 7.0-7.9 5 0.0491 5 25 2.50 8.0-8.9 5 0.0345 2 4 0.57 9.0-9.9 5 0.0245 o o 0.00

10.0-10.9 4 0.0167 11.0-11.9 3 0.0113 12.0-12.9 3 19 0.0083 8 64 5.82 13.0-13.9 5 0.0050 14.0-14.9 1 0.0034

3;1.5 3 0.0072 N=200 X=60.84

X2(9,0.01) = 21.67

60.84 >21.67 Erletx;does not fit at the 1% significance level.

MIN UN 1111111 MM MIS MM Mi Mil UNI Mill MM MM 11•11 Mi

TABLE D-7

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

DRIVING LANE - ISLINGTON AVE.

CLASS INTERVAL (SEC)

OBSERVED FREQUENCY (fa)

PROBABILITY (POISSON)

THEORETICAL FREQUENCY

(f T) (fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 1 0.14/01

-

I cy\ Ir\ c‘i co

,0 c-n

H O

c0 t‘•-•.0 \

N

NN

H

NN

NH

HH

HH

CO 0 CO

CO N

H

H H

crl,.0 N

H H

784 27.03 1.0-1.9 15 0.123S 100 4.00 2.0-2.9 30 0.1057 64 2.91 3.0-3.9 4.0-4.9

36 18

0.0905 0.0774

324 4

18.00 0.25

5.0-5.9 20 0.0662 49 3.77 6.0-6.9 15 0.0567 16 0.36 7.0-7.9 9 0.0484 1 0.10 8.0-8.9 9 0.0415 1 0.13 9.0-9.9 6 0.0355 1 0.14 10.0-10.9 9 0.0303 9 1.50 11.0-11.9 11 0.0260 36 7.20 12.0-12.9 3 0.0222 13.0-13.9 7 0.0190 14.0-14.9 2 0.0163 15.0-15.9 1 21 0.0139 0 0.00 16.0-16.9 2 0.0119 17.0-17.9 1 0.0102 18,0-13.9 3 0.0087 19.0-19.9 2 0.0074

N=200 r=65.39

,2 = 24.73

65.39 >24.73 Poisson does not fit at the % significance level,

1111/ IIIIIII 1111.11 MN Ili MM UNI I» Mil lila

TABLE D-8

CHI-SQUARE TEST TO CHECK TEE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBS-à),VED DA,TA:

DRIVING LANE - ISLINGTON AVE.

CLASS INTERVAL

(SEC)

OBSERVED FREQUEIICY

(fo)

PROBABILITY (ERLANG)

(K=2 )

THEORETICAL FREQUENCY

(fT) (f0-fT)

2 (fo-fT) (fo-fT)2 fT

0.0-0.9 1 0.0397 8 7 à9 6.13 1.0-1.9 15 0.0902 18 3 9 0.50 2.0-2.9 30 0.1108 22 8 64 2.91 3.0-3.9 36 0.1139 23 13 169 7.35 4.04 .9 18 0.1075 22 4 16 0.73 .0-5.9 20 0.0962 19 1 1 0.05

0.0-6.9 15 0.0332 17 2 4 0.2h 7.0-7.9 9 0.0704 14 5 25 1.79 8.0-8.9 9 0.0585 11 2 4 0.36 9.0-9.9 6 0.0475 9 3 9 1.00

10.0-10.9 9 0.0385 7 2 4 0.57 11.0-11.9 11 0.0316 6 5 25 4.17 12.0-12.9 3 0.0245 4 13.0-13.9 7 0.0193 3 14.0-14.9 2 0.0154 3 17 4 16 0.94 15.0-15.9 1 21 0.0121 2 16.0-16.9 2 0.0092 2 17.0-17.9 1 0.0077 1 18.0-18.9 3 0.0051 1 19.0-19.9 2 0.0049 1

N =200 1=26.74

2 , " (11,0.01) = 24.73

26.74 > 24.73 Erlang does not fit at the 1% significance level.

OUI SM MIS Mi MM UN MINI Mill MI IIIIIII MI OMM MM OMO OUI 11111 il! MI

TABIE D-9

CRI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TOOBSERVED DATA:

PASSING LANE - ISLINGTON AVE.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

fo

PROBABILITY (POISSON)

THEORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT 0.0-0.9 1.0-109 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9 8.0-8.9 9.0-9,9

10.0-10.9 11.0-11.9 12.0-12.9 13.0-1309 14.0-14.9 15.0-15.9 16.0-16.9 17.0-17.9 18.0-18.9 19.0-19.9

4

12 77 35 19 16 10

8 8 3 3 1 3 1 0 1 1 1 0 o

1

12

0.2629 0.1937 0.1429 0.1053 0.0776 - 0.0572 0.0422 0.0310 0.0230 0.0168 0.0127 0.0090 0.0068 0.0049 0.0037 -0.0027 0.0020 0.0015 0.,0011 0.0008

53 39 29 21 15

-11 8 6 5 3 3 2 1 1 1 1 0 o o o

12

r---- .0 C\I r-1

1-1 N N

O

1681 1111111

31.72 37.03 1.24

, 0.19 0.07 0.09 0000 0.67 0.80

0.00

36 4 1 1 o 4 4

0

N=200 Z=71.81

X2

(8,0.01) = 20.09

71.81 > 20.09 Poisson does not fit at the 1% significance level.

1-à Lx) U.)

Mi MM Mil Mil IIMII MIS MI IIIIIII MM MM Mil MI IVO Mill Mlle MOI

TABLE D-10

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

PASSING LANE - ISLINGTON AVE.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (ERLANG)

(K=gi_

THEORETICAL FREQUENCY

fT (fo-fT) (fo-fT)2 (fo-fT)2

fT 0.0-0.9 12 0.1251 25 13 169 6.76 1.0-1.9 77 0.2196 4)i 33 1089 2)4.75 2.0-2.9 35 0.2013 4o 5 25 0.63 3.0-3.9 19 0.1540 31 12 144 4.65

25 16

4.0-4.9 5.0-5.9

16 lo

0.1080 0.0722

21 14

5 L ,4 1.19 1.14

6.0-6.9 8 0.0460 9 1 1 0.14 7.0-7.9 8 0.0291 6 2 4 0.67 8.0-8.9 3 0.0181 4 9.0-9.9 -3 0.0110 2

10.0-10.9 1 0.0063 1 11.0-11.9 3 0.0045 1 12.0-12.9 1 0.0022 0 13.0-13.9 0 15 0.0017 8 7 Lj..9 6.13 14.0-14.9 1 0.0009 o 15.0-15.9 1 0.00°4 o 16.0-16.9 1 0.0003 o 17.0-17.9 o 0.0001 o 18.0-18.9 o 0.0001 o 19.0-19.9 1 0.00003 o

N=200. . 1E=46 .06

,2 = 18.48 (7,0.01) 1-1

46.06 >18.48 Erlang does not fit at the 1% significance levai.

MINI IIIIIII MI MI MINI MOI MI Mal Mil Mal 111111 1111•11 RIB- -MM Mill Mill Ni

TABLE D-11

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT CF POISSON DISTRIBUTION TO OBSERVED DATA:

DECELERATION LANE - ISLINGTON AVE.

CLASS INTERVAL

SEC

oBSorTED FREQUENGY

fo

"OreBILITY (POISSON)

THEOnETICAL FREQUENCY

(fT) (fo-fT) (fo-fT) (fo-fT)2

fT

0.0-0.9 1 0.1706 34 33 1089 32.03 1.0-1.9 43 0.1414 28 15 225 8.04 2.0-2.9 47 0.1174 24 23 529 22.04 3.0-3.9 25 0.0973 19 6 36 1.89 4.0-4.9 5.0-5.9

18 15

0.0807 0.0670

16 13

2 2

4 4

0.25 0.31

6.0-6.9 12 0.0555 11 1 1 0.09 7.0-7.9 10 0.0461 9 1 1 0.11 8.0-8.9 7 0.0382 8 1 1 0.13 9.0-9.9 2 0.0317 6 4 16 2.67

10.0-10.9 2 0.0263 5 3 9 1.80 11.0-11.9 3 0.0218 4 12.0-12.9 2 0.0181 4 \ 13 5 25 1.92 13.0-13.9 2 \ 8 0.0150 3 14.0-14.9 1 0.0124 2

e 15 10 0.0605 12 2 4 0.33 N=200 =71.61]E

X2

(11,0.01) = 24.73 71.61 >24.73 Poisson does not fit at the 1% significance leve1.

nal MB 11111 MM MB MI MI MB IBM WIN NB MB MB 111111 1111111

TABLE D-12

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

DECELERATION LANE - ISLINGTON AVE.

CLASS INTERVAL

(SEC)

OBSERVED FREQUINCY

(fo)

PROBABILITY (ELLANG)

(K=2)

TERORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT) 2 (fo-fT) 2

fT

0.0-0.9 1 0.0547

(\I

N-1

) crl

\O GO

•0 d

ce

('J U

\ H

C\J C

\J (N.1 C‘.I

(-1 H

r-I

10 100 9.09 1.0-1.9 43 0.1180 19 361 15.04 2.0-2.9 47 0.1364 20 400 14.81 3.0-3.9 25 0.1313 1 1 0.04 4.044.9 18 0.1168 5 25 1.09 5.0-5.9 15 0.0984 5 25 1.25 6.0-6.9 12 0.0801 4 16 1.00 7.0-7.9 10 0.0634 3 9 0.69 8.0-8.9 7 0.0498 3 9 0.90 9.0-9.9 2 0.0378 6 36 4.50

10.0-10.9 2 0.0294 4 16 2.67 11.0-11.9 3 0.0219 12.0-12.9 2 0.0158 13.0-13.9 2/ 8 0.0126 4 16 1.33 14.0-14.9 1 0.0086

e:15 10 0.0245 5 25 5.00 N=200 2 =57.41

,2 X (11,0.01) 24.73

57.41 >24.73 -Erlang does not,fit at the 1% significance level.

MI lin Mil MI 1•11 1311 1•11 8Z11 1111111 1•11 MM MB MI 11•1

TABLE D-13

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

DRIVING LANE - DUFFERIN ST.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

Tliki;ORETICAL FREQUENCY

(fT) f -fT) ( -fT) (fo-fT)2

fT

0.0-0.9 3 0.2313 46 43 1849 40.20 1.0-1.9 51 0.1777 36 15 225 6.25 2.0-2.9 72 0.1367 27 45 2o25 75.00 3.0-3.9 20 0.1051 21' 1 1 0.05 4.0-4.9 16 0.0807 16 o o 0.00 5.0-5.9 7 0.0621 12 5 25 2.08 6.0-6.9 11 0.0477 10 1 1 0.10 7.0-7.9 6 0.0367 7 1 1 0.14 8.0-8.9 5 0.0282 6 1 1 0.17 9.0-9.9 3 0.0217 4

10.0-10.9 2 0.0167 3 11.0-11.9 12.0-12.9

1 1 9

0.0128 0.0099

3 ].5 2 6 36 2 .40

13.0-13.9 0 0.0075 2 14.0-14.9 2 0.0059 1

N=200 E=127.39

2 X (8,0.01) = 20.09 127.39 >20.09 Poisson does not fit at the 1% significance level.

UNI MM MINI 1MM 1M1 Mal MB IBM 1M11 MB Inn MI MI Mn

TABLE D-14

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

DRIVING LANE - DUFFERIN ST.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (ERLA.HG)

(K=2)

TH1,11 0RETICAL FREQUENCY

(fT) (fo-f ) (fo-fT)2 (fo-fT) 2

fT

0.0-0.9 1.0-1.9

3 51

0.0931 o.1853

20 37

17 14

289 196

14.45 5.30

2.0-2.9 72 0.1845 39 33 1089 27.92 3.0-3.9 20 0.1534 31 11 121 3.90 4.0-4.9 16 0.1170 23 7 49 2.13 5.0-5.9 7 0.0847 17 10 100 5.88 6.0-6.9 11 0.0590 12 1 1 0.08 7.0-7.9 6 0.0409 8 2 4 0.50 8.0-8.9 5 0.0266 5 o o 0.00 9.0-9.9 3 0.0179 4

10.0-10.9 2 0.0122 2 11.0-11.9 1 9 0.0072 1 9 0 0 0.00 12.0-12.9 1

1 0.0046 1

13.0-13.9 0 0.0036 1 14.0-14.9 2 0.0014 0

N=200 1=60.16

x2= 20.09

60;16 >20.09 Erlang does not fit at the 1% significance leve1.

Mill MINI 11111 MI MIN Mil MI 11•11 IMIM RIB

TABLE D-15

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERTM DATA:

PASSING LANE - DUFFERIN ST.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

THEORETICAL FREQUENCY

(fT) f -fT) (f0-fT)2 (fo-fT)2

fT

0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9 8.0-8.9

15 87 55 26 10 4 0

il 3

0.3611 0.2307 0.1474 0.0942 0.0601 o.o385 0.0245 0.0158 0.0100

C‘I O

'N as C

O If \ cr ‘N

C

r-i

57 41 26 7 2 4 5 2

3249 1681 676 49

à 25

4

45.13 36.54 23.31 2.58 0.33 2.00 5.00 0.8o

N=200 1:=115.69

2 v — (6,0.01) = 16.81

115.69 > 16.81 Poisson does not fit at the 1% significance level.

ma ma am am am ma am am mi am am am mic am ma am ami

TABLE l>16.

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

PASSING LANE - DUFFERIN ST.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (ERLAUG) (K=2)

THEORETICAL FREQUENCY

(fT) (fo-fT) (fo-fT)

2 (fo-fT) 2

fT

0.0-0.9 15 0.2261 45 30 900 20.00 1.0-1.9 87 0.3088 62 25 625 lo.o8 2.0-2.9 55 0.2143 43 12 lhh 3.35 3.0-3.9 26 0.1238 25 1 1 o.04 L0-4.9 10 0.0651 13 3 9 0.69 5.0-5.9 4 0.0326 7 3 9 1.29 6.0-6.9 0 0.0155 3 7.0-7.9 2)3 0.0073 2 i 6 3 9 1.50 3.0-8.9 i3 0.0056 1

N=200 1=36.95

2 v " (5,0.01) 7 15.09

36.95 > 15.09 Erlang does not fit at the 1% significance

Mil MINI 1•111 11111111 1•11 111111 Mil MM Mill OMM

TABLE D-17

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISIUIBUTION TO OBSERVED DATA:

DECELERAT ION LANE - DUFFERIN ST.

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABTF,ITY (POISSON)

THEORETICAL FREQUENCY

(fT) fo-fT) ( -fT)2 (fo-fT) 2

fm 0.0-0.9 1 0.0870 15 14 196 13.07 1.0-1.9 13 0.0794 1)1 1 1 0.07 2.0-2.9 15 0.0725 13 2 4 0.31 3.0-3.9 21 0.0662 11 10 100 9.09

8 0.0605 10 2 4 0.40 5.0-5.9 13 0.0551 9 4 16 1.78 6.0-6.9 10 0.05014 9 1 1 0.11 7.0-7.9 6 0.0460 8 2 4 0.50 8.0-8.9 12 0.0420 7 5 25 3.57 9.0-9.9 6 0.0384 7 1 1 0.14

10.0-10.9 7 0.0350 6 1 1 0.17 11.0-11.9 8 0.0320 6 2 Li 0.67 12.0-12.9 4 0.0291 5 1 1 0.20 13.0-13.9 5 0.0267 5 o o 0.00 14.0-1)1 .9

e 15 71

371 44 0.0243 0.2554

4 1 45 I 49

5 25 0.51

N.173 2:=30.59

X2(13,0.01) = 27.69

30.59 >27.69 Poisson does not fit at the 1% significance level. -P-

11111 111M1 ION 11111 MI 111111 MI an BIR Bal MB an MB NUI MB lin BU

TABLE D-18'

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

DECELERATION LAUE - DUtiVERIN ST.

GLUS I DITERVAL I (SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (ERLANG)

(K=2)

THEORETICAL FREQUEUCY (fT)

(fo-fT) (fo-fT)2 (fo-fT)2 fT

0.0-0.9 1.0-1.9

1 t 14 13 0.0147 0.0375

C.- - O• H

H R..1

H H

0 0

0• CO

(\J

H H

H r-I H

H H

1

\71

K) l,-)O

FV-P-

R.)\

-r11-

1 1--'W

O 0"

16 1.60

2.0-2.9 15 0.0522 36 4.00 3.0-3.9 21 0.0611 100 9.09 4.0-4.9 8 0.0657 9 0.82 5.0-5.9 13 0.0669 1 0.36 6.0-6.9 10 0.0659 1 0.09 7.0-7.9 6 0.0633 25 2.27 8,0-8.9 12 0.0599 4 0.40 9.0-9.9 6 0.0560 16 1.60

10.0-10.9 7 0.0512 4 0.44 11.0-11.9 8 0.0h71 o 0.00 12.0-12.9 h 0.0424 9 1.29 13.0-13.9 5 0.0386 4 0.57 14.0-14.9 7 0.0343 1 0.17

37 0.21i32 , 25 0.60 N.173 2:=23.30

2- 27 ° 69

23.30 427.69 Erlang fits at'the I% significance level.

Mn 111111 MB lila 111111 MI MB an MI 11111 1111111 1M11 11•11 nal MI

- TABLE D-19

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

DRIVING LANE - AVENUE ROAD

CLASS INTERVAL .

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILIT/. (POISSON)

THEORETICAL FREQUENC/

(f T) (fo-2T) (fo-fT)2 (fo-fT)2

fT 0.0-0.9

n

cnco

H H U

1N

11\--

- N

g-1 r-I

0.2212 44 41_ 1681 38.20 1.0-1.9 0.1660= 33 16 256 7.- 76 2.0-2.9 0.1312 26 17 289 11.12 3.0-3.9 0.1035 21 7 49 2.33 4.0-4.9 0.0818 16 5 25 1.56 5.0-5.9 0.0646 13 2 ' 4 ç 0.31 6.0-6.9 0.0510- 10 5 25 2.50 7.0-7.9 0.0403 8 4 16 200 8.0-8.9 0.0318 6 1 1 0.17 9.0-9.9 0.0252 5 1 1 0.20

10.0-10.9 0.0198 4 11.0-119 0.0157 3 . 12.0-12.9 0.0125 3 13.0-13.9 0.0097 2 16 7 49 3.06 14.0-14.9 '0.0077 2 15.0-15.9 0.0061 1 16.0-16.9 0.0048 1

]:=69.21

X2(9,0.01) = 21.67 69.21 > 21.67 *)* Poisson does not fit at the 1% significance level.

Mill Mill MM MM IM 1•111 111111 MI MI MI MM Mal MI Mal IBM Bill

TABLE D-20

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

DRIVIM LANE - AVENUE ROAD

CLASS OBSERVED PROBABILITY THEORETICAL INTERVAL FREQUENCY (ERLANG) FREQUENCY (fo-fT) -fT)2 (fo-fT)2 (SEC) (fo) K=2 (fT) fT 0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 LL.0-4.9 5.0-5.9 6.0-6.9

3 49 43 28 21 11 15

o.0880 0.1556 0.1700 0.1492 0.1200 0.0915 0.0677

18 31 34 30 24 18 14

15 18 9 2 3 7 1

225 324 81 4 9

/4-9 1

12.50 10.45 2.38 0.13 0.38 2.72 0.07 7.0-7.9

8.0-8.9 9.0-9.9

12 5 4

0.0486 0.03wp 0.0241

10 7 5

2 2 1

4 4 1

0.40 0.57 0.20 10.0-10.9 1 0.0162 3'

11.0-11.9 2 0.0120 2 12.0-12.9 0 0.0070 1 13.0-13.9 14.0-14.9

2 9. 0.0058 0.0034

1 1

9 o 0. 0.00

15.0-15.9 21 1 0.0022 1

16.0-16.9 1 0.0016 0 N=200 1:=29.80 -

X2'(9,0.01) = 21.67

29.30 >21.67 Erlang does flot fit at the 1% significance level.

111111 Inn MB IMBI MIN MM

TABLE D-21

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

PASSING LANE - AVENUE ROAD

CLASS INTERVAL

(SEC)

OBSERVhD FREQUENCY

(fo)

PROBABILITY (POISSON)

THEORETICAL FREQUE£Y

(fT) fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9 8.0-8.9 9.0-9.9

10.0-10.9 11.0-11.9 12.0-12.9

a 43 14

6 2 2 2 0 0 o o 1

5

0.3923 0.2383 0.1)01 9 0.0881 0.0535 0.0325 0.0197 0.0121 0.0073

0.00)1 4 0.0028 0.0016 0.0010

78 48 30 18 11

7 4 2 2 1 1 o 0

•

10

64 68 13 4 5

, 5

5

4096 4624 169 16 25 25

25

52.51 96.33 5.63 0.89 2.27 3.57

2.50

N=200 1=163.70

15.09

163.70 )1.15.09 Poisson does not fit at the 1% significance level.

MB Bal Mail IMMI IBN Inn BU Mal Bal MI MB MI MB Bal MM Mill Mail

TABLE D-22

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

PASSING LANE - AVENUE ROAD

CLASS OBSERV1D PROBABILITY THEORETICAT, 2 INTERVAL YREQUENCY (ERLANG) FREQUaTCY (fo-fT) (fo-fT)2 (fo-fT)

fT (SEC) (fo) (K=2) (fT)

0.0-0.9 0.2627 53 39 1521 28.70 1.0-1.9 a 0.3292 66 50 2500 37.88 2.0-2.9 43 0.2071 41 2 4. 0.10 3.0-3.9 14 0.1083= 22 8 64 2.91

1.60 4.0-4.9 6 0.051)1 10 4 16 5.0-5.9 2 0.0239 5 6.0-6.9 2 0.0102 2 7.0-7.9 2 0.0045 1 8.0-8.9 0 7 0.0017 0 8 1 1 0.13 9.0-9.9 0 0.0005 0 10.0-10.9 0 0.0003 0 11.0-11.9 0 0.0001 0 12.0-12.9 1 0.00006 0

N=200 • 1:=71.32

X2(4,0.01) = 13.28

71.32 >13.28 Erlang doeà not fit àt the 1% significance level.

MM Mi Mill Mil lila Mil MI Mil Mill MI IBM IBM 11111 Mil MM

TABLE D-23

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

DEC7TŒRATI0N LANE - AVENwl: ROAD

CLASS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

THEORElICAL FREQU7N- CY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 1 o.1680

o

••••••••"..".%

G• •.0 C

r\ H 0 C

O £

'-"O

Crl C

C\I r-I r-I f--1

r-I r-I

33 1089 32.03 1.0-1.9 50 0.1331 23 529 19.59 2.0-2.9 35 0.1120 13 169 7.68 3.0-3.9 23 0.0943 9 81 4.26 4.0-4.9 11 0.0794 5 25 1.56 5.0-5.9 20 0.0669 7 49 3.77 6.0-6.9 7 o.o563 4 16 1.45 7.o-7.9 11 0.0474 1 1 0.10 8.0-8.9 9.0-9.9

5 3

0.0399 0.0336

3 4

9 16

1.13 2.29

10.0-10.9 11.0-11.9

4 2

0.0283 0.0239

2 3

4 9

0.67 1.8o

12.0-12.9 4 0.0200 13.0-13.9 2 i9 0.0169 1 1 0.10 14.0-14.9 els 3

14 0.0142 0.0758 1 1 0.07

N=200 1.=76.50

X2 = 26.22 (12,0.01) 76.50 >26.22 Poisson does flot fit at the 1% significance level.

11111111 Mil MM MI MIR IMIII MM MM MI MIR MI Mill MI MM MI

TABLE D-24

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED DATA:

DECELERATION LANE - AVENUE ROAD

CLASS INnevAL

(SEC)

OBSEttVED FREQUENCY

(fo)

PROBABILTTY (ERLANG)

(K=2)

THEORETICAL YREQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT 0.0-0.9 1 0.0472

--N1

[MW

,4)

o• 0

VIA

- Ft 1 -

1 •0 8 64 7.11

1.0-1.9 50 0.10101 29 841 40.05 2.0-2.9 35 O.12L4 10 100 4.00 3.0-3.9 28 0.1238 3 9 0.36 4.0-4.9 11 0.1130 12 /l4. 6.26 5.o-5.9 20 0.0984 0 0 0.00 6.0-6.9 7 0.0821 9 81 5.06 7.0-7.9 11 0.0673 2 4 0.31 8.0-8.9 5 0.0543 6 36 3.27 9.0-9.9 3 0.o426 6 36 4.00

10.0-10.9 4 0.0339 3 9 1.29 11.0-11.9 2 0.0260 3 9 1.80 12.0-12.9 4 0.0202 13.0-13.9 9 0.0153 o o 0.00 14.0-14.9 3 0.0120

*15 14 0.0351 7 49 7.00 N=200 1=80.51

0.01) = 26.22

80.51 > 26.22 Erlang does not fit at the 1% significance level.

11111 11111 111111 111111 MI MI MM MIN MB MI 11111 111111 Bal MM

TABLE D-25

CHI-SQUARE TEST T0 CHECK Tib GOODNESS OF FIT OF POISSON DISTRIBUTION .TO.OBSERVED DATA:

COMBINED THROUGH LANES - DIXON ROAD

CLASS INIRVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

THEORTICAL FREQUENCY

(CT) f -fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 47 0.2673

HH

UT

t f N

.)

0 0

C\J H

cel

r-i (Y•N

C\J

36 0.68 1.0-1.9 59 0.1958 400 10.26 2.0-2.9 29 0.1435 0 0.00 3.0-3.9 15 0.1052 36 1.71 4.0-4.9 5.0-5.9 6.0-6.9

13 10

5_

0.0770 0.0565 0.0413

4 1 9

0.27 0.09 1.13

7.0-7.9 5' 0.0303 1 0.17 8.0-8.9 9 0.0222 9-.0-9.9 3 0.0163

10.0-10.9 2 17 0.0119 9 0.64 11.0-11.9 2 0.0087 12.0-12.9 0 0.0065 13.0-13.9 1 0.0046

N=200 Z=14.95

X2 = 16.01 (7,0.025)

14.95 Z.16.01 Poisson fits at the 2.5% significance level.

Mil MI 11•11 AMI Rial Ili MI MIN IMIII OMM MI MM MI Mil MI

OHI-SQUARE TO ,OH CKTHEG0NESS OF FIT OF EALA:Nèz;'-(e=2) - DIeTRIBLIZION,- Tbs..0§..mvi) DATA:

OblviBINED-_TIIRO:UGH LANE-à- - -DIXOÉ IIOAD . .

CLASS INTERVAL

(SEC)

OBSER-Vh?, 0 • FR 0ENCY E0 -. •

(fo)

PROBABILITY

(K=2)

`IUEORET:ILICAL FREcraZ CY

(fil.)1 fo-fT) (f -fT)2 (fo-fT)2

fT

0.0-0.9 47 0.1291 26 21 ) 011 16.96 1.0-1.9 59 -Ô.2214.2 45 114. 196 4.36 2.0-2.9 29 0.2033 41 12 1)14 3.51 3.0-3.9 15 0.1535 31 16 256 8.26 4.0-4.9 13 0.1°66 - 21 8 6LL 3.05 5.0-5.9 10 00697 14 4 ' 16 1.14 6.0-6.9 5 -0.0hh5 9 4 16 1.78 7.0-7.9 5 o.o279 6 1 1 0.17 8.0-3.9 9 - 0.0163 • 9.0-9.9 3 :0.0100

31 2

10.0-10.9 2 17 .3 -.0053 1 lo 100 14.29 11.0-11.9 , 2 00035 1 12.0-12.9 .. o O.0024 o 13.0-13.9 1 ' 0.0008 0

N=200 - r=53.52

.)ç,2 - (7 -0.01)

:ï3.52 >i8.48 Y Er1ang doe.s ribt fit at the,10/„signi_ficance level.

Bal MI MB MB MI MB MI MIN MI MI MB MB MB IMM ,—

TABLE D-27

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

COMBINED THROUGH LAIES - ISLINGTON AVE.

CLASS INTERVAL (SEC)

OBSERVED FREQUENGY

(fo)

PROBABILITY POISSON)

THEORETICAL FREQUENCY (f T)

- (fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 YI 0.3693 74- 30 900 12.16 1.0-1.9 75 0.2330 47 28 784 16.68 2.0-2.9 33 0.1469 29 4 16 o.55 3.0-3.9 18 0.0926 19 1 1 0.06

9 0.0582 12 3 9 0.75 5.0-5.9 9 0.0371 7 2 4 0.58 6.0-6.9 6 0.0233 5 1 1 0.20 7.0-7.9 1 0.0146 3 8.0-8.9 2 0.0092 2 9.0-9.9 2 6 0.0059 11 7 1 1 0.14

10.0-10.9 o 9.0036 1 11.0-11.9 1 0.0023 o

N=200 E =31.12

X2(6,0.01) = 16.81

31.12 >16.81 poisson does not fit at the 1% significance level.

IMIll MI MI 111311 MM MI MI Mil Mil Mal Mil IBM Mil

TABLE D-28

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVED -DATA:

COMBINED THROUGH LANES - ISLINGTON AVE.

CLASS INTERVAL

(SEC)

OBSERVED FREQUEMCY

(fo)

PROBABILITY (ERLANG)

(K=2)

THEORETICAL FREQUENCY

(fT) (fo-fT) f -fT)2 (fo-fT)2

fT

0.0-0.9 44 0.2356 47 3 9 0.19 1.0-1.9 75 0.3145 63 12 1244 2.29 2.0-2.9 33 0.2130 43 10 loo 2.33 3.0-3.9 18 0.1197 24 6 36 1.50 4.0-4.9 5.0-5.9

9 9

0.0617 0.0268

12 5

3 4

9 16

0.75 3.20

6.0-6.9 6 0.0168 3 7.0-7.9 1 0.0069 1 8.0-8.9 2

2 1 12 0.0031 1 5 7 9.80

9.0-9.9 0.0009 o 10.0-10.9 o 0.0006

. o

11.0-11.9 1 '0.0002 0 y=20.06 N=200

y2 " (5,0.01) = 18'48

20.06 >18.48 Erlang does not fit at the 1% significance level.

MM MI Mill IBM MI MM IBM MM Mil MI Mil Mil

TABLE D-29'

CHI-SQUARE TEST TO.CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:-

COMBINED THROUGH LANES - DUFFERIN ST.

CLASS INTERVAL

(SEC)

OBSERVED FREQUIMCY

(fo)

PROBABILITY (POISSON)

THEORETI CAL FREQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT

0.0-0.9 61 0.5093 102 41 1681 16.48 1.0-1.9 87 0.2500 50 27 729 14.58 2.0-2.9 33 0.1226 25 8 64 2.56 3.0-3.9 14 0.0601 12 2 4 0.33 4.0-4.9 3 0.0296 6 3 9 1.50 5.0-5.9 1 0.01/01 3 6.0-6.9 2 0.0072 11 5 3 9 1.80 7.0-7.9 o 0.0035 1 8.0-8.9 1 0.0017 o

N=200 E =37.25

(4,0.01) = 13.28

37.25 713.28 Poisson does not fit at the 1% significance level.

UMM Mill IIIMI MI Mi MI Ilall MM MI MI MI Mil IBIS MM MM URI Mil

TABLE -D-30

CHI-SQUARE TEST TO CHECK THE GOGDNESS OF FIT OF _a(LANG (K=2) DISTRIBUTION TO OBSERVED DATA:

COMBINED THROUGH LANES - DUFYERIN ST.

CLASS INTERVAL (SEC)

OBSERVED FREQUENCY (fo)

PROBABILITY (ERLANG) (K=2)

THEORETICAL 10.a-EQUENCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fT 0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.o-5.9 6.0-6.9 7.0-7.9 8.0-8.9

=

61 87 33 14 1 3 1 0 0 1

19

0.4165 0.3603 0.3494 0.0517 0.o156 o.0046 0.0014 0.0004 0.00006

83 72 30 10 3 1 0 114 0 0

22 15 3

5

484 225 9

25

5.83 3.13 0.30

1.78

N=200 z=11.04

X2(2,0,01) = 9.21

11.04 >9.21 Erlang does not fit at the 1% significance level.

Bal Mil MM IIIIIII MIN MI MM Mal MI MM IIIIIII MI MM MM MINI

TABLE .D-31

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF POISSON DISTRIBUTION TO OBSERVED DATA:

COkBINED THROUGH UNES - AVENUE ROAD

GLUS INTERVAL

(SEC)

OBSERVED FREQUENCY

(fo)

PROBABILITY (POISSON)

THEORETICAL FREQUINCY

(fT) (fo-fT) (fo-fT)2 (fo-fT)2

fg'

0.0-0.9 62 0.5195 104 42 1764 16.96 1.0-1.9 2.0-2.9

95 26

0.2497 0.1199

50 24

LL5 2

2025 4

40.50 0.16

3.0-3.9 9 0.0576 12 3 9 0.75 4.0-à.9 4 0.0277 6 2 4 0.67 5.0-5.9 2 0.0133 3 6.0-6.9 1)4 0.0064 1 5 1 1 0.20

7.0-7.9 1 0.0031 1 E=59.24 11=200

"A" ,2,0.01) 13.28 (4 59.24 > 13.28 Poisson does not fit t t e 1% significance level.

_

lia MN MM Mil MM IIIMI MI IIIIIII Mil Mil MI MM Mi MIN Mal INN

TABLE D-32

CHI-SQUARE TEST TO CHECK THE GOODNESS OF FIT OF ERLANG (K=2) DISTRIBUTION TO OBSERVE]) DATA:

COMBINED THROUGH LANES - AVENUE ROAD

CLASS INTERVAL

(SEC)

OBSLRVED FREQUI1NCY

(fo)

PROBABILITY (ERLANG)

(K=2)

THEORETICAL FREQUEN-CY

(fT) (fo-fT) (fo-fT)2 (f0-fT)2

fT 0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9

62 95 26

9 4 2 1 17 1 1

0.4308 0.3596 0.1432 0.0472 0.0134 0.0038 0.0015 0.0004

86 72 29

9 3 1 13 0 o

14 23

3

4

196 529

9

16

2.28 7.35 0.31

1.23

N=200 1=11.17

X(2,001) = 9.21

11.17 > 9.21 Erlang does not fit at:the 1% Significancé

157

REPERENCES

Bennett, C.A., ad Franklin, N.L., Statistical Analysis in Chemistry and the Chemical Industry; John 'Wiley and Sons, 1954.

Greenshields, B.D., and Weida, F.M.; Statistics with applications to Highway Traffic Analysis; the Eno Foundation for Highway. Traffic - Control, 1952,

Gerlough, D.L.; "The use of the Poisson Distribution in Highway Traffic", in Gerlough, D.L., and. Schuhl,. A., "Pesson and Traffic"; the Eno Foundation for Highway Traffic Gontrol, 1955,

Haight, F.A., Whisler, B.F., and Mosher, W.W.Jr.; A new statistical method for describing the distribution of cars on a road; Institute of Transportation and Traffic Engineering, University of California, Los Angeles.

. 5. Jensen, A.; "Traffic theory as am aid in the planning and operation of the Roaa Grid". Ingenitren No.2, vol. 1, 1957. (Danish Engineering Periodical).

Kendall, D.G.; Some problems in the Theory of Queues; Royal Statistical Journal, Series 13, Vol. 13, No.2, 1951.

Matson, T.M., Smith, U.S., and Hurd, F.W.; Traffic Engineering; McGraw-Hill, 1955.

Morse, P.M.; Queues, Inventories and Maintenance; Operation Research Society d' America, publication No.1; John Wiley and Sons, 1952.

. Pearson, K.; Tables for Statisticians and Biometricians; Biometric Laboratory, University College, London, part I, third edition, 1951. Smith, J.G., and Duncan, A.J.; Sampling Statistics and applications; McGraw-Hill, 1943.

Vardon, J.L.; "Some factors affecting mergIng traffic on the outer Ramp of Highway Inter changes"; A Master's degree thesis, Queen's University, July 1959.

Williams, K.M.; "Vehicle operating characteristics on outer loop deceleration lanes of interchanges"; A Master s degree thesis, University of Toronto, October 1960.

Wyse, J.M., "A study of Flow on Highway 401, an Technical Report of the 1960.

some characteristies of Traffic Urban Freeway"; Unpublished Ontario Department of Highways,

olmiliirjElsil[q