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
Transactions on the Built Environment vol 41, © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
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
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324 Urban Transport and the Environment for the 21st Century
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
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Urban Transport and the Environment for the 21st Century 325
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
Ring Road-North Part
Ring Road-North Part
Ring Road-North Part
Ring Road-North 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-
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326 Urban Transport and the Environment for the 21st Century
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-
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Urban Transport and the Environment for the 21st Century 327
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
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328 Urban Transport and the Environment for the 21st Century
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.
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Urban Transport and the Environment for the 21st Century 329
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)
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330 Urban Transport and the Environment for the 21st Century
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.
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Urban Transport and the Environment for the 21st Century 331
35003000 -I25002000-115001000500 H0
ObservedEstimated
1.2 2 4 6 8 10 12
Vehicle Headway (se
14 16 >16
Figure 3. Shifted exponential distribution for vehicle headways for
central lane.
ObservedEstimated
1.3 6 8 10 12
Vehicle Headway (se
14 16 >16
Figure 4. Shifted exponential distribution for vehicle headways for
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
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332 Urban Transport and the Environment for the 21st Century
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.
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Urban Transport and the Environment for the 21st Century 333
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
Amount of Shift (sec)
Figure 6. Goodness-of-fit measure at different values of shift for
central lane.
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334 Urban Transport and the Environment for the 21st Century
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
Amount of Shift (sec)
Figure 7. Goodness-of-fit measure at different values of shift for
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
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Urban Transport and the Environment for the 21st Century 335
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.
Homogeneity of Variance
Response Headway
Factors Lane
ConfLvl 95.0000
Bonferroni confidence vehicles for standard deviations
Lower Sigma Upper n Factor Levels
* 3.06793 * 643 1
* 316807 * 702 2
Harriett's Test (normal distribution)Test Statistic: 0.690
p value : 0.406
Homogeneity of Variance
Response Headway
Factors Direction
ConfLvl 95.0000
Bonferroni confidence vehicles for standard deviations
Lower Sigma Upper n Factor Levels
* 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.
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336 Urban Transport and the Environment for the 21st Century
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)
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Urban Transport and the Environment for the 21st Century 337
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.
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338 Urban Transport and the Environment for the 21st Century
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.
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Urban Transport and the Environment for the 21st Century 339
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
Transactions on the Built Environment vol 41, © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
340 Urban Transport and the Environment for the 21st Century
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.
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