This examines vehicle headway in Riyadh (the Capital of · 2014. 5. 14. · headway distributions, such as shifted exponential, Pearson Type II, and log-normal distributions. In the

Modeling vehicle headways for low traffic flows

on urban freeways and arterial roadways

All S. Al-GhamdiAssociate Professor, King Saud University, College of Engineering,Riyadh, Saudi Arabia,EMail: [email protected]

Abstract

One of the important microscopic flow characteristics that affects safety, level ofservice, driver behavior, and capacity of the roadway is vehicle headway. Thetime between arrivals of successive vehicles passing a point on roadway is calledvehicle headway. This study aims at analyzing the vehicle headway for urbanfreeway and arterial roadway sections in Riyadh (capital of Saudi Arabia) andcomparing them with results from international research. Only low volumetraffic conditions is of the concern in this research. The paper describes themathematical distribution for vehicle headway data on two types of roadways(freeways and arterials). The study found that shifted exponential distributionand gamma distribution appear to reasonably fit data from freeways and atrerials,respectively.

1 Introduction

The time between arrivals of successive vehicles passing a point on

a roadway is known as vehicle headway. This headway is one of the

important microscopic flow characteristic that affects safety, level of

service, driver behavior, and capacity of the roadway \ In practice,

the leading edges of two consecutive vehicles are used whether the

measurements are taken automatically, by detectors, or manually, byobservers.

Transactions on the Built Environment vol 41, © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

322 Urban Transport and the Environment for the 21st Century

This examines vehicle headway in Riyadh (the Capital of

Saudi Arabia). Although many studies associated with vehicle

headway have been conducted in the past, in particular in the

United States and some western nations, their results may not be

applied directly to conditions in Saudi Arabia, due to differences in

driving behavior. In addition, there is a clear lack of research in the

area of vehicle headway in Saudi Arabia. Two primary motivations

are behind this kind of research. The first reason is to develop

mathematical distribution models for vehicle headway in Riyadh.

The second reason is to compare these models with corresponding

models from international experience. The study also presents pure

traffic engineering work in this fast developing country where such

work does not now exist. The vehicle headway on both freeway

sections and arterial sections was investigated in this research. This

paper describes the mathematical distribution models that were

developed in this research study for the vehicle headway data

collected from two types of urban roadways (freeways and

arterials) in Riyadh.

It should be made cleared that this study addresses only the

random case which occurs under low-volume traffic conditions. It

would be more insightful to investigate the intermediate case for

moderate flow levels which is currently a hot research subject. This

is true in developed countries but not in developing ones, as shown

from the literature review. However, since this type of research is

the first such in a developing nation, the author preferred to begin

with the simple case in order to learn from it for future research on

the intermediate and high flow levels. Mathematical models were

developed for this purpose. The following sections in this paper

provide the details of the methodology of data collection and

analysis carried out throughout the study.

2 Literature Background

The subject of the time spacing between successive arrivals of

vehicles on roadways (vehicle headways) has been studied by

researchers for a number of years. Several past models have

attempted to describe mathematically the distribution of vehicle

headways. The interest in this modeling was, in particular, for

traffic simulation purposes. Simulation techniques in traffic

applications require a headway prediction model to generate


Urban Transport and the Environment for the 21st Century 323

vehicle headways. The development of vehicle simulation models

began with Adams^ and Schuhl.^ Adams considered the mean time

which elapses before a time gap of prescribed size appears in a

flow of vehicles having a negative exponential distribution. Drew"*

discusses the theoretical concepts and the basis for mathematical

models developed by various researchers to predict vehicle

headways. Because of the poor agreement between the frequencies

of headways observed in practice and the frequencies predicted by

the negative exponential distribution, Drew* suggested advanced

headway distributions, such as shifted exponential, Pearson Type

II, and log-normal distributions. In the TRB monograph, Gerlough

and Huber^ provide an extensive discussion of various headway

models developed by different researchers. Khasnabis and

Heimbach^ developed a headway-distribution model for two-lane

rural highways. They showed that none of the existing models (the

Negative Exponential, Pearson Type III, and Schuhl models

provided satisfactory results for the wide range of traffic volumes

tested. Alternatively, a modified form of the SchuhP model

provided the most reasonable approximation of the arrival patterns

noted in the field. Some researchers supported the claim that when

traffic flows are light (less than about 500 veh/h) headways follow

the negative exponential distribution/ Griffith and Hunt*

mentioned that most investigators seem to agree that no single

distribution will adequately describe the headway distribution evenat the same point on the same road at the same time on successive

days. Collecting time headways in single lanes of traffic at 45

urban sites, typically in busy High Streets, in the U.K., Griffith and

Hunt* found that a simple distribution of Double Displaced

Negative Exponential Distribution (DDNED) provided a good fit to

the observed headways at the vast majority of sites.

May' provided a well-summarized review of the basic

concepts of vehicle headway distributions. He indicated that

vehicle headways can be classified in terms of the level of traffic

flow rate: a random distribution for low flow level, and

intermediate distribution state for moderate flow levels, and a

constant distribution state for high flow levels. He added that the

negative exponential distribution is the mathematical distribution

that represents the distribution of random vehicles such as vehicle



headways (first classification) and that it has some strengths and

weaknesses.

Hoogendoorn and Botma* used a simple analysis to derive

a model equivalent to Branston's Generalized Queuing model for

the description of time-headway distributions describing car-

following behavior. They assumed that the total headway is the

sum of two independent random variables: the empty zone and the

free-flowing headway. The parameters of the model can be utilized

for examining various characteristics of both the road (e.g.,

capacity), and driver-vehicle combinations (e.g., following

behavior). Luttinen^ described the distributional properties of

headways by density estimate, coefficient of variation, skeweness

and kurtosis. He tested the hypothesis of exponential tail and the

independence of consecutive headways using Monte Carlo methods

and autocorrelation analysis, respectively. Luttinen* found that

local conditions, such as road category, speed limit and flow rate,

have a considerable effect on the statistical properties of headways.

From the above review it can be seen that the vehicle

headway is a flow characteristic that can be used in describing

driving behavior. Moreover, no unique distribution exists for

modeling the vehicle headway. A factor that plays a primary role in

this matter is traffic flow. The previous modeling attempts were

more successful with time headway for low to medium levels of

traffic flow in which successive vehicles are almost independent of

each other (random case). However, the modeling process is vague

for higher levels of traffic flow. Thus, this study focuses on road

sections when the random case exists.

3 Data Collection and Reduction

The data collection for this study was conducted using TDC-8

equipment (an electronic count board with inbuilt computer

programs produced by Jamar Technologies^ that can be used in a

variety of data-collection studies in traffic). The TDC-8 counter

was set, for the purpose of this study to observe vehicle headway

data on a 15-minute basis. The observed data was then transferred

to a PC and made ready for the analysis stage, using a spread sheetsoftware.

Twelve urban sites, six three-lane-freeway sections (Table

1) and six two-lane-arterial sections (Table 2) in Riyadh, were



selected for collecting vehicle headway data. The data were

collected during off-peak times and normal traffic and

environmental conditions. The flow rate for the freeway sections

(posted speed is 120 km/h) ranged between 300 and 1300

veh/h/lane (only two sites had flow rates greater than 1,000

veh/h/lane). This level of traffic flow represents random conditions.

The percent of trucks was less than 10. The field data were

observed for each lane (Lane 1= median lane, Lane 2 = central

lane, Lane 3 = shoulder lane). A total of 10,351 timed headways

were observed at all freeway sites.

Table 1. Urban freeway sections for collecting time headway data.

Site

Khurais Road

Khurais Road

Khurais Road

Khurais Road

Khurais Road

Khurais Road

Ring Road-East Part

Ring Road-East Part

Ring Road-East PartRing Road-East Part

Ring Road-East Part

Ring Road-East Part

Ring Road-North Part






Lane Type

Median

Shoulder

Central

Median

Shoulder

Central

Median

Shoulder

CentralMedian

Shoulder

Central

Median

Shoulder

Central

Median

Shoulder

Central

Direction

Westbound

Westbound

Westbound

Eastbound

Eastbound

Eastbound

Southbound

Southbound

Southbound

Northbound

Northbound

Northbound

Eastbound

Eastbound

Eastbound

Westbound

Westbound

Westbound

Flow

(veh/h/lane)

707*

768*

682*

609*

757*

546*

398

599*

532*567*

790*

536*

397

595*

552

301

496

519*Due to equipment setting data in this section were collected over 30 minutes.

For the arterial type of roadway, a total of 2,329 vehicle

headways were collected at six sections on Dhabab road (major

divided arterial roadway in Riyadh with two lanes in each

direction). The posted speed limit is 50 km/h and the range of lanevolumes is between 400 and 1050 vehicle/h/lane (based on 15-



minute flow rate). The volume is almost divided evenly between

the two lanes in each direction (50%:50%). The data were collected

during off-peak times and during normal traffic and environment

conditions. At each of the six sites, vehicle headway data were

observed for the shoulder lane and the median lane (in the same

direction: northbound or southbound), hence, twelve data sets (six

sets for each type of lane) were collected. It should be mentioned

that arterial sites were selected so as to lessen the effect of

upstream traffic signals as much as possible. Table 2 presents a

description for the sites and the size of data collected.

Table 2. The arterial sites and the size of data collected.

Lane No.

1

2

3

4

5

6

7

8

9

10

11

12

Lane Type

Median

Shoulder

Shoulder

Median

Shoulder

Median

Median

Shoulder

Median

Shoulder

Median

Shoulder

Direction

Southbound

Southbound

Northbound

Northbound

Northbound

Northbound

Southbound

Southbound

Northbound

Northbound

Southbound

Southbound

Flow (15-minute

data*)

230

197

263

157

252

212

260

204

157

148

148

101Note: all the sites are located at Dhabab road, a major arterial in Riyadh.*Unlike on freeway sites, data were collected for only 15 minutes at a site.

4 Analysis and Results for Vehicle Headway

The analysis began with testing the homogeneity of data for each

set in order to investigate the possibility of combining data

observed at different sites. The homogeneity test also was

conducted for data collected from different lanes. The other side ofthe analysis deals with fitting data to proper mathematical

distributions. Several mathematical distributions, suggested from

past research (e.g., negative exponential, shifted exponential, log-

normal, and gamma), were tested to find the best fit. The goodness-



of-fit testing procedures was used for this purpose. The following

sections present the analysis of the freeway data set followed by the

arterial data set.

4.1 Freeway Data Set

The vehicle-headway data are analyzed below and summary

statistics are given in Table 3. The mean values for vehicle

headway are 5.751, 5.345, and 4.881 sec for median, central, and

shoulder lane, respectively. It can be seen from the standard

deviations in the fifth column in Table 3 that there is some

variability among lanes. The means among lanes are also different

as shown from ANOVA analysis in Figure 1. Pairwise comparison

was also used to observe which lanes are significantly different.

The result showed that while the means for the median lane and

central lane are not statistically significant, the mean for the

shoulder lane is significantly different from other lane means

(p<0.006). Harriett's test for variance homogeneity showed no

reason for not combining data for the same lane from different

sites. In addition, the homogeneity test showed that the variability

among data, for the three lanes, over all sites, is statistically

significant (p=0.000) as shown in Figure 1. Hence the data for each

lane type was analyzed individually and three data sets were

formed from the freeway data, i.e., a data set for each lane type.

Table 3. Summary statistics for time headway data for freeway

sites.

Lane

Median

Central

Shoulder

Sample

size

2979

3367

4005

Mean

(sec)

5.751

5.345

4.881

Median

(sec)

3.3

2.8

2.5

St. Dev.

(sec)

9.264

9.740

10.072

St.Er.

0.170

0.168

0.159



One-Way Analysis of Variance

Analysis of Variance on Response

Source DF SS MS F p

Factors 2 676.3 338.1 3.95 0.020

Error 10348 885912.1 85.6

Total 10350 886588.4

Individual 95% CIs For Mean

Based on Pooled StDev

Level N Mean StDev + + +-

1 2979 5.458 7.339 ( * )

2 4005 4.881 10.072 ( * )

3 3367 5.345 9.740 ( * )

Pooled StDev = 9.253 4.90 5.25 5.60

Homogeneity of Variance

Response Response

Factors FactorsConfLvl 95.0000

Bonferroni confidence intervals for standard deviations

Lower Sigma Upper n Factor Levels

* 7.3394 * 2979 1

* 10.0715 * 4005 2

* 9.7403 * 3367 3

Bartlett's Test (normal distribution)

Test Statistic: 354.797

p value : 0.000

MTB>

Figure 1. ANOVA and Bartlett's test output.



The shifted exponential distribution gave a very reasonable

fit for the vehicle headway data for each lane as shown in Figures

2, 3, and 4. The amount of shift (77) for each lane type was not the

same. The general probability density function (pdf) for a

continuous random variable, in our case the time headway T, is

given as:̂

/(f) = - - - (̂'-̂ /(m-?) f > 0; m > 0 (1)(m-?7)

Where

m = mean of distribution (l/m is the parameter of the distribution)

(sec).

77 = the amount of shift (sec).

and hence the probability of a headway equal to or greater than t

sec can be expressed as:

and the cumulative density function (cdf) can be obtained by

subtracting the above form from one as:

0 = 1-̂ '"̂ '"̂ (3)

For the median lane the optimal shift parameter is 1 sec and the pdf

with estimated parameters is given below:

(4.751)

For the central lane the optimal shift parameter is 1.2 sec and the

pdf is:

f(t) -(4.145)



Also for the shoulder lane the optimal shift parameter is 1.3 sec and

thepdf is:

(3.581)

Table 4 summarizes the three models for the freeway sections.

Table 4. The p.d.f. and c.d.f. forms for the three lanes.

Lane

Median

Central

Shoulder

Shift*

(?)(sec)

1

1.2

1.3

p.d.f

fff\ * -(,-1) /(4.751)/ \l ) — t(4.751)

f(f\ - 1 -(«-1.2)/(4.145)

(4.145)

fff\- 1 -(/-1.3)/(3.581)

(3.581)

c.d.f

p(h>t) = e-w™»

p(h>t) = e-«-̂ *̂

p(h>t) = e-«-w-*»

^Optimal values.Note: The significance of the goodness-of-fit was tested at 5% level.

ObservedEstimated

4 6 8 10 12 14 16 >16

Vehicle Headway (se

Figure 2. Shifted exponential distribution for vehicle headways for

median lane.



35003000 -I25002000-115001000500 H0

ObservedEstimated

1.2 2 4 6 8 10 12

Vehicle Headway (se

14 16 >16


central lane.

ObservedEstimated

1.3 6 8 10 12

Vehicle Headway (se

14 16 >16


shoulder lane.

4.2 Optimization of Shift Parameter

Fitting a shifted exponential distribution to field data requires

estimating two parameters. One is the mean of measurements from

the origin (m). The other is the shift of data with respect to the



origin (77). The variance can then be estimated easily from these

two parameters: (m-ffY • One difficulty associated with the

shifted exponential distribution is to determine an estimate for the

shift parameter in the model. Gerlough and Huber^ refer to this

parameter as the minimum allowable headway, i.e., a region of the

distribution in which headways are prohibited. They indicate that

some writers have maintained that a deterministic prohibited period

(i.e., a deterministic minimum headway) is philosophically

unacceptable and they would rather have a period during which the

probability of an arrival is very low but not zero. In this study the

calculated value of chi-square test statistic (%*) obtained from the

goodness-of-fit test was used as a criterion for estimating the

optimal shift parameter. We know from the chi-square goodness-

of-fit testing procedures that if the observed frequencies do not

differ much from the expected frequencies, the value of the test

statistic %l is small. As the observed frequencies begin to differ

from the expected frequencies, the values of %l will increase

because the statistic squares these differences, weights them by the

reciprocal of the expected frequencies, and adds the resulting

ratios. Thus a small value of %] supports the null hypothesis that

the random variable conforms to the specified theoretical statistical

distribution. The idea therefore is to alter the value of the parameter

to achieve the best minimum (minimum of minimum) calculated

chi-square value (%l). The value of the shift parameter that gives

this minimum is, hence, the optimal value. The optimal values of

the shift parameter for the three lanes are given in Table 4. Figures

5, 6, and 7 show the graphs for the relationship between the %l

and 77. Each graph has a unique minimum value which is used in

the developed model.



60

40-

30-

-I 20 H

u 10-

0.5 0.6 0.7 &8 0.9 1 11 1.2 1.3 1.4 1.5

Amount of Shift (sec)

Figure 5. Goodness-of-fit measure at different values of shift for

median lane.

180

* 140-2" 120-g loo-'s 80-| 60 -| 40 -^ 20-

0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7



central lane.



120

§ 100 -

# 80-

u 60-"OI 40 -

3 20-

0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7



shoulder lane.

It should be noted that no remarkable difference in the parameter

for the shifted exponential distribution was found between the

minimum value observed and the optimal value obtained using the

chi-squared method for all three-lane types.

4.3 Arterial data:

A total of 2,329 observations for arterial sites were analyzed in this

study. Data were first classified by lane type (median/shoulder) and

direction (northbound (n/b) and southbound (s/b)), as presented in

Table 5. ANOVA and homogeneity test (Bartlett's test for

homogeneity of variance, as illustrated in Figure 8) for the arterial

data showed that type of lane is not a significant factor but

direction (northbound and southbound) is. In other words, the data

for the same direction from all sites regardless of the lane type can

be pooled together in one sample. Thus, two data sets were

prepared for analysis, that is, the northbound data set (1189) and

the southbound data set (1140). While it was expected that the data

for each lane type would be homogenous, this was not the case for

possibly two primary reasons. First, all of the arterial sites werelocated on the same arterial even though the time of collecting the

data was not the same. Second, shoulder parking is prohibited and

land-use activities are very limited along the study sites,

consequently, headways were not influenced by any kind of



interference. Thus, traffic behavior is more likely to be the same or

similar on median and shoulder lanes. A summary of primary

statistics for the pooled data is given in Table 6.

Table 5. Summary statistics for arterial data.

Lane

Shoulder (n/b)

Shoulder (s/b)

Median (n/b)

Median (s/b)

Set

Size*

252

391

212

490

Mean

(sec)

5.075

4.016

4.453

4.281

Median

(sec)

4.2

3.4

3.9

3.2

St.Dev.

(sec)

2.931

3.086

2.786

3.321

S.E.

(sec)

^ 0.185

0.156

0.191

0.15*The size of data is not fully consistent with that in Table 2 due to problems withdumping the data from the TDC-8.


Response Headway

Factors Lane

ConfLvl 95.0000

Bonferroni confidence vehicles for standard deviations


* 3.06793 * 643 1

* 316807 * 702 2

Harriett's Test (normal distribution)Test Statistic: 0.690

p value : 0.406


Response Headway

Factors Direction

ConfLvl 95.0000

Bonferroni confidence vehicles for standard deviations


* 3.21983 * 881 3

* 287935 * 464 4

Bartlett's Test (normal distribution)

Test Statistic: 7.387

p value : 0.007

Figure 8. Output for Bartelett's test for arterial data.



Table 6. Summary statistics for arterial data after pooling.

Lane

Road (n/b)

Road (s/b)

Set

Size

464

881

Mean

(sec)

4.791

4.163

Median

(sec)

4.2

3.2

St.Dev.

(sec)

2.879

3.22

S.E.

(sec)

0.134

0.108

Gamma distribution seems to reasonably fit arterial data.

Gamma distribution is a continuous distribution resulting from its

relationship to a function called the gamma function. The

probability density function (pdf) of the gamma distribution for a

continuous random variable X is given as

ix>0 (6)

where a and p are called the shape and scale parameters,

respectively. The parameters of gamma distribution can be easily

obtained by using the following definitions of the mean and the

variance of this distribution:

mean of data = ju = o,p (by definition), thus a = —

(by definition), thus a =variance of data = cr =cr'

?

(7)

(8)

By equating equations (7) and (8) , the parameter estimates for the

gamma distribution can be obtained.

For the northbound traffic on arterial sites, gamma distribution with

parameters a =1.64 and p = 2.7 was found to fit the data quite

satisfactory , as depicted in Figure 9. The pdf is

/(*;!.64,2.7) =i

2.7"*r(1.64)(9)



1200

E 3003o

3 4 5 6 7Vehicle Headway (sec)

ObservedEstimated

9 10 >10

Figure 9. Observed and estimated data for northbound on arterial

roadway.

For the southbound traffic, as shown from Figure 10, the

consistency between the observed and estimated data is reasonable,

and hence gamma (1.67,2.49) describes the real data. The pdf is

;l.67,2.49) =1

2.49'*T(1.67)x>0 (10)

Table 7 presents the pdf and cdf functions for gamma distributionfor the arterial data.

o 1200

300O

3 4 5 6 7Vehicle Headway (sec)

Observed- Estimated

Figure 10. Observed and estimated data for southbound on arterial

roadway.



Table 7. The p.d.f. and c.d.f forms for the three lanes.

a1.64

1 67

P2.7

249

p.df1 064 i/27

/W.64,_7) 7̂̂ r(1.64)"

•f 1 „•! /:T •*) /1Q\ -067 -x/2492.49""r(1.67)

c.d.ff 1 -064 -x/27 7f(,,1.64,..7) J2_,M^^

F(x'l 67 2 49) - f t~°"e~''™''dt

Notel: The upper model is for the n/b direction and the lower for s/b direction.Note2: The significance of the goodness-of-fit was tested at 5% level.

5 International Comparison

It should be emphasized that even international research does not

agree upon a unique distribution model for the random case. As can

be seen from the literature review, it was difficult to find studies

that match this study in order to conduct subjective comparison.

However, general comparison can be made. Therefore, most of the

mathematical distributions suggested by international researchers

were attempted in this study. Only shifted exponential distribution

and gamma distribution gave a reasonable fit. Unlike gamma

distribution, the shifted exponential distribution has been

mentioned in past research as a proper fit for random case. Thus,

this study introduced gamma distribution as a suitable distribution

for arterial sites.

6 Conclusions

A large set of data, relating to vehicle headways, in single lanes of

traffic, was collected at twelve urban sites (six freeway and six

arterials) in Riyadh. The data were collected during off-peak

periods with normal traffic and weather conditions. A total of

10,351 and 2,329 vehicle headways were collected for freeway and

arterial sites, respectively.For freeway data, the homogeneity test showed that data

from each lane (median, middle, and shoulder lanes) can be

combined from the different study sites. Therefore, analysis was

done for each lane set of data. The mean vehicle headways were

5.75, 5.35, and 4.88 sec. for median, central, and shoulder lanes,

respectively.



On the other hand, the same test on arterial data showed that

the lane is not a significant factor but the direction, namely, the

headway data observed at median and shoulder lanes for the same

direction (northbound or southbound) can be combined. Thus, it

was not possible to combine data from northbound and southbound

even for the same type of lane. The mean vehicle headways were

4.79 and 4.16 sec for northbound and southbound lanes,

respectively.It was found that shifted exponential distribution provided a

decent fit to the observed headways for the freeway data, while the

gamma distribution seems reasonable for arterial data. Models for

both types of distribution were developed for each lane of the

roadway. An optimal value for the shift parameter for the

exponential distribution was obtained by considering the chi-square

value computed from the goodness-of-fit test as the criterion for

optimization. One main advantage of these distributions is the ease

with which it can be used for drawing samples for use in different

studies, especially simulation studies.The following recommendations are reached from this

research.

• For future research in this regard, data should be collected

from one site to cancel out the variability related to different

sites. This study shows some variability in the data from one

site to another which might affect the results. It would be

more accurate to have all the data from the same site, so that

this type of variability will not exist.

• No remarkable difference was found between the minimum

allowable headway and the optimal value obtained for the

parameter of shifted exponential distribution using minimum

chi-squared criterion, therefor, one would use the minimum

allowable headway directly.

• This study focused on random traffic conditions. Further

research should be conducted for congested conditions.

• It seems from the analysis that the effect of direction is not

significant in the mathematical distribution found to fit the

freeway data even though the homogenity test shows that it is

not suitable to combine data for both directions (i.e., gamma

distribution fits data from both directions). Therefore, in



future analyses of this kind, it could be enough to analyze

data for one direction only.

Acknowledgement

The author would like to express his thanks to Mr Al-Dossier and

Mr. Al-Bishi and the transportation laboratory technicians at the

College of Engineering at King Saud University for their valuable

help in collecting the data used in this research.

References

1. May, AD Traffic Flow Fundamentals. Prentice Hall Inc.,

New Jersey, pp. 11-36, 1990.

2. Adams, W.F. Road Traffic Considered as a Random Series.

Journal of Institute of Civil Engineering, 1936.

3. Schuhl, A. The probability Theory Applied to Distribution of

Vehicles on Two-Lane Highways. In Poisson and Traffic,

Eno Foundation, Saugatuck, CT, 1955.

4. Drew, D.R. Traffic Flow Theory and Control. McGraw-Hill,New York, 1966.

5. Gerlough, D.L., and Huber, M.J. Traffic Flow Theory: A

Monograph. Special Report 165, TRB, National ResearchCouncil, Washington, DC, 1975

6. GTD. Annual Statistical Report. General Traffic Directorate,Saudi Arabia, Riyadh, 1997.

7. Khasnabis, S., and Heimbach, C.L. Headway-Distribution

Models for Two-Lane Rural Headways. Transportation

Research Record 772, TRB, National Research Council,

Washington, DC , 1980.



8. Griffiths, J. D , and Hunt, J G Vehicle Headways in Urban

Areas. Journal of Traffic Engineering + Control, pp. 458-462,

October 1991.

9. Hoogendoorn, S. P., and Botma, H. Modeling and Estimation

of Headway Distribution. Transportation Research Board,

76* Annual Meeting, Washington, DC, 1997.

10. Luttinen, R. T Statistical Properties of Vehicle Time

Headways. Transportation Research Board, 76* Annual

Meeting, Washington, DC, 1992.

11. User's Manual. Jamar Technologies, Inc., Horsham, PA,

USA, 1997.

12. Bain, L. J. and Engelhardt, M. Introduction to Probability and

Mathematical Statistics. Duxbury Press, Boston, USA, 1987.


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This examines vehicle headway in Riyadh (the Capital of · 2014. 5. 14. · headway distributions, such as shifted exponential, Pearson Type II, and log-normal distributions. In the