Exercise 2 - FLOW AND HEADWAY MEASUREMENT Aditya Nugroho (HT083276E)

EXERCISE 2

FLOW AND HEADWAY MEASUREMENT AND ANALYSIS

CE 5203 TRAFFIC FLOW AND CONTROL

ADITYA NUGROHO

HT083276E

DEPARTMENT OF CIVIL ENGINEERINGNATIONAL UNIVERSITY OF SINGAPORE

2010

2

Department of Civil Engineering

CE 5203 Traffic Flow and Control

1.0 INTRODUCTION

The headway between vehicles in a traffic stream is of fundamental importance in trafficengineering applications. In this Exercise 2 the objectives are:

To understand how traffic flow is measured microscopically using statisticaldistributions

Describe the traffic conditions (quantitatively) on ClementiAvenue 6 at this roadsection

Measure and analyze traffic with a headway distribution

2.0 HEADWAY METHOD

2.1 Time headway measurementTime headways have been measured using video camera and and the data has been extractedusing appropriate computing software called MultiTimer. In this Exercise 2, several pointsshould be considered before do the measuring of time headway.

A reference line- a certain line should be set as a reference to count vehicles. Headway record- Headway were recorded when the vehicles pass a reference point on

a lane measured from front bumper to front bumper. Headway record error measurement in observation- A MultiTimer software was

installed to record the headways, by clicking the computer keyboard, during theplayback of the video. It was noted that it was difficult to extract the headway byclicking the software when front bumper of vehicles pass a reference point duringheavy traffic flow, because of the quick arrival of vehicles in succession (on severaloccasions with zero headway).

Headway record error measurement suggestions- It is suggest that to solve theproblem on headway record by using video, the video should played in a computerusing full motion video software, which would allowed the playback of the video at0.25 times the actual speed with clear visual. This method helped in easy extraction ofheadway data during heavy traffic-flow conditions and to avoid errors in measured ofheadway data.

Vehicles type headway- Although it is important to classify the headway by vehiclescomposition where headway distributions will vary depending on the traffic mix in thetraffic streams. However, to simplify our exercise, we do not consider vehicle type. Itis assumed that the traffic in this road section are homogenous where all vehicles areidentical following perfect lane discipline.

Therefore headway can be measured by following equations:

( ) ( ) ( )1h x T x T xi ii (1)

where, 1( ) and ( )i iT x T x is the elapsed time between successive vehicles respectively in trafficstreams.

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2.2 Statistical distribution methods, parameter estimation and hypothesisIn order to get generally applicable results about the properties of headways, it is necessary tomeasured headway into a statistical distribution techniques by following this steps.

Selecting statistical distribution techniques Model parameters Goodness of fit test (Nonparametric test)

Four theoretical simple distributions were selected for a more detailed analysis:a) Negative exponential distribution - interarrival time distribution of a totally random

arrival process.b) Shifted exponential distribution - avoids the problem of extremely short headways by

setting a threshold for short headways.c) Gamma distribution - generalization of the exponential distributions.d) Lognormal distribution - has a theoretical connection to the car-following models.

Table 1 Properties and parameter of statistical distribution

Properties NE SE Gamma Lognormal

PDF 1 ,0,

heF(h)=

( )1 ,0,

heF(h)=

[ , ( )]

( )

0,

,h

F(h)=

ln( )

0,

,h

F(h)=

pdf ,0,

hef(h)=

( ),0,

hef(h)=

1

( )( )( )

0,

,hh ef(h)=

1 ln( )

0,

,hhf(h)=

Mean1

(h)=

1(h)= +

(h)= +

1 22(h)= +e

Median 21

LnMd(h)=

2LnMd(h)= +

1(0,5)Md(h)=F

Md(h)= e

Variance 2

2

1(h)=

2

2

1(h)=

2

2(h)=

22 2 2

1e(h)=e

Skewness 3 2(h)= 3 2(h)= 3

2(h)=

3 1( 2)(h)=

Kurtosis4 9(h)= 4 9(h)= 4

63(h)= 4 3 2

4 +2 +3 3ω ω ω(h)=

*NE=Negative exponential, SE=Shifted exponential

In order to test goodness of fit (GOF), the GOF can be measured by non parametric testnamely Kolomogorov-Smirnov test. The Kolmogorov-Smirnov (K-S) statistic D is the largestabsolute vertical difference between Fo(x) and Fe( x ):

| ( ) ( )|D=max F x F xo e (2)

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Where Fo(x) and Fe( x ) are the observed and expected distributions.

The null hypothesis (H0) that the distributions are equal, is tested against the alternativehypothesis (HA) that the distributions are not equal. The null hypothesis should be rejected ifthe distributions differ.

3.0 Descriptive statistic of headway data

The headway data was collected from a four-lane divided arterial road in Clementi Avenue 6nearby tunnel nosing. A video camera was mounted on the roadside, opposite to a referencepoint (tunnel nosing), to continuously capture the moving traffic across this reference point.The collection of data was done during 30 minutes in morning peak hours. A summary ofempirical headway data regarding to mean, maximum, minimum headways observed atvarious number of sampling at four corresponding lanes is indicated on Table 1.

Table 1 Descriptive statistic vehicles headway data

Properties Lane 1 Lane 2 Lane 3 Lane 4 CombinedNo of vehicles (N) 417 307 198 272 1194Mean 4.354079 5.954151 9.080269 6.662511 6.074226Median 2.7965 4.39 5.703 4.336 3.992Mode 1.5 2 2.922 1.422 1.375Standard Deviation 4.490037 5.431602 9.563156 6.28141 6.423677Kurtosis 8.342999 7.027718 5.942939 5.761006 10.70902Skewness 2.495149 2.328692 2.165551 2.066255 2.697092Minimum 0.531 0.375 0.563 0.781 0.375Maximum 31.703 35.172 61.203 38.734 61.203

0%

5%

10%

15%

20%

25%

30%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Freq

uenc

y

Vehicles headway (s)

Lane 1

Lane 2

Lane 3

Lane 4

Figure 1. Frequency Distribution of Observed Headway

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The average headway for all lanes is calculated to be 6.08 s, the headway ranges fromminimum of 0.37 s to maximum of 61.20 s and the standard deviation is 6.42 s. With the largegap between minimum and maximum headway, therefore choice of appropriate class intervalfor grouping the data is a vital step in the analysis of headway data. In order to simplify ourexercise, the class intervals of 1s were considered by ignoring the headway more than 30s(Appendix 1).

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Freq

uenc

y

Vehicles headway (s)

Lane 1

Lane 2

Lane 3

Lane 4

Figure 2. Cumulative Distribution of Observed Headway

The frequency of vehicles headway of given class interval at each lane were plotted togetherin Figure 1 and 2. The figure reveals that all lanes, approximately ranging from 16%-22% ofvehicles were accepted class headway of 1 s to 2 s.

4.0 Simple distribution of combined lanes headways and goodness of fit

The evaluation of the headway distribution models is based on three considerations: Reasonability. The parameters of simple distribution models can give additional

information on the properties of traffic flow. Applicability. Parameter estimation should not be too complicated. Validity. The model should give a good approximation of the empirical headway

distributions. This is tested by the goodness of fit tests.

Following figures represent of four theoretical simple distributions of empirical headwaysdata.

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0

0,2

0,4

0,6

0,8

1

1,2

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0,18

0 5 10 15 20 25 30

Di s

trib

ut i

on f

unc

t ion

Pr o

b.

den

si t

yNegative Exponent Distribution

Prob. density Distribution function

Figure 3 Combined lanes Negative exponential distribution model

0

0,2

0,4

0,6

0,8

1

1,2

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0,18

0,2

0 5 10 15 20 25 30

Di s

trib

ut i

on f

unc

t ion

Pr o

b.

den

si t

y

Shifted Exponent Distribution


Figure 4 Combined lanes Negative exponential distribution model

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0

0,2

0,4

0,6

0,8

1

1,2

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0 5 10 15 20 25 30

Di s

trib

ut i

on

fu

nc

t io

n

Pr o

b.

den

si t

yGamma distribution

Pro b. d en sity Distributio n fun ction

Figure 5 Combined lanes Gamma distribution model

0

0,2

0,4

0,6

0,8

1

1,2

0

0,05

0,1

0,15

0,2

0,25

0,3

0 5 10 15 20 25 30

Di s

trib

ut i

on f

unc

t ion

Pr o

b.

den

si t

y

Log-normal Distribution


Figure 6 Combined lanes Log-normal distribution model

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From the above figures we can conclude that the probability density function displays theshape of the distribution better than the probability distribution function. Concerning theparameter estimation, the headway distributions are strongly skewed to the right. Table 2 and3 summarized all distribution properties and parameter estimation.

Table 2 Summary of simple distribution properties

Negative Exponential Shifted Exponential Gamma Distribution Log-normalMu 6.074 Mu 1.817 Alpha 6.793 Alpha 1.217Lambda 0.165 Lambda 0.175 Beta 1.118 Beta 0.301Var(X) 36.9 Var(X) 5.69 Var(X) 8.50 Var(X) 34.51Std. Dev. - Std. Dev. - Std. Dev. 2.91 Std. Dev. 5.87Mode 0 Mode 0.375 Mode 6.48 Mode 1.76Med(X) 0.693 Med(X) 4.335 Med(X) 7.23 Med(X) 2.72Skewness 2 Skewness 2 Skewness 0.767 Skewness 6.18Kurtosis 9 Kurtosis 9 Kurtosis 0.883 Kurtosis 110.94

Table 3 Estimation parameters

Distribution Estimation ValueNegative exponential h 0,165

Shifted exponential1

, min( )hh

0,175

Gamma2 2

,h ss h

6,793 =1,118

Lognormal mean(ln ), =stdev(ln )h h 1,217 =0,301

The goodness of fit was tested using the nonparametric Kolmogorov-Smirnov (K-S) test. Theresults of nonparametric K-S tests including null hypothesis (H0) and alternative hypothesis(HA) are presented in the table below.

Table 4 Summary of goodness of fit test

Distribution K-S Test H0 HANegative exponential 0.216 Accepted RejectedShifted exponential 0.381 Accepted Rejected

Gamma 0.478* Rejected AcceptedLognormal 0.189 Accepted Rejected

*The critical D value is 0.41 for a level of significance of 0.05

From the above table we can conclude that the lognormal distribution goodness-of-fit resultsare better than other simple distributions.

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5.0 Conclusion

The data in this study are from Clementi Avenue 6 four lane roads divided by tunnel roadtowards PIE. Such factors as speed limit, lane and shoulder width, shape of tunnel nosing,proportion of heavy vehicles, may have an effect on the headway distribution. Our knowledgeof these relationships is, however, very limited. The headway distribution model should bevery adaptive, but not too flexible. It should have all the right parameters, but none too many.

In this exercise, the proposed procedures give a scientific foundation to identify and estimatestatistical models for vehicle headways, and to test the goodness of fit. It has been shown thatthe statistical methods in the analysis of vehicle headways should be further extend to analysemore accurately of the results. Such the method to estimate of scale, location and shapeparameter distribution is important in identification of process data properties.

6.0 References

Highway Capacity Manual. 2000. Special Report 209, 4th Ed., TRB, National ResearchCouncil, Washington, D.C.,

Luttinen, R. Tapio, Statistical Analysis of Vehicle Time Headways. Helsinki University ofTechnology, Transportation Engineering, Publication 87. Otaniemi, 1996.

May, A. D. (1990), Traffic Flow Fundamentals, Prentice-Hall, Inc., Englewood Cliffs, N.J.

Traffic Flow Theory A State-of-the-Art Report (2001). Committee on Traffic Flow Theoryand Characteristics (AHB45). TRB, National Research Council, Washington, D.C.,

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Appendix 1 Headway class interval

Class Lane 1 Lane 2 Lane 3 Lane 4 Combined1 47 12 7 4 702 119 50 30 57 2563 55 41 25 39 1604 38 39 11 29 1175 38 36 15 19 1086 28 26 17 19 907 14 21 13 20 688 17 17 4 9 479 12 10 8 9 39

10 9 9 6 7 3111 15 9 6 11 4112 4 5 5 7 2113 3 11 6 6 2614 2 2 4 6 1415 2 4 7 6 1916 5 4 5 3 1717 - - 2 7 918 2 4 2 4 1219 1 - 1 - 220 2 1 - 2 521 - 1 4 3 822 - - 1 3 423 2 - 1 1 424 2 - - 1 325 - - 3 - 326 1 1 1 - 327 - 1 2 - 328 - 3 2 - 529 1 1 - - 230 - - - - -

Total 419 308 188 272 1,187

Exercise 2 - FLOW AND HEADWAY MEASUREMENT Aditya Nugroho (HT083276E)

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