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Evaluating the Time Headway Distributions in
Congested Highways
Sara Moridpour School of Civil, Environmental and Chemical Engineering, RMIT University, Melbourne, 3001, Australia
Email: [email protected]
Abstract—Time headway is a significant traffic flow
parameter that affects the capacity and safety of highways
and freeways. Time headways are broadly used in different
areas of traffic and transport engineering such as capacity
analysis, safety studies, car following and lane changing
behavior modeling, and level of service evaluation. In this
paper, the time headway distribution is investigated for an
urban highway at different traffic flow rates during
congestion. To analyze the headway characteristics, the time
headways to the preceding and following vehicles are
analyzed for heavy vehicles and passenger cars. The
trajectory data used in this study was provided for a highway
section in California: Berkeley Highway, I-80. Appropriate
models of headway distribution are selected for heavy vehicles
and passenger cars using Chi-Square test. Using the selected
models, headway distributions are predicted for each vehicle
type at different traffic flow rates. The results confirm
existence of different time headway distributions in vicinity of
heavy vehicles and passenger cars which is due to the
difference in the behavior of drivers in vicinity of heavy
vehicles and passenger cars under congestion.
Index Terms—headway distribution, highways, heavy
vehicles, passenger cars
I. INTRODUCTION
Time headway or headway is defined as the time
between two consecutive vehicles (in seconds) when they
pass a single point on a roadway [1]. Headway is measured
as the time between the same common features of two
consecutive vehicles (e.g. front bumper). Time headway is
one of the important microscopic traffic flow parameters
which is extensively applied in planning, analysis, design
and operation of roadway systems [2]-[7]. Therefore, it is
essential to accurately evaluate this parameter based on
real behavior of drivers [8]-[10]. Understanding drivers’
behavior in selecting their desired headway is important in
order to have better traffic planning and policy making in
different traffic conditions. This is due to the fact that time
headways and their distributions would influence different
traffic flow parameters such as capacity, level of service
and safety [11], [5]. Precise modeling and analysis of
vehicle headway distributions is required to maximize
roadway capacity and minimize the delays that vehicles
experience [12], [5]. Furthermore, headway analysis is
used in understanding the reasons of accidents as well as
evaluating policies to enhance road safety. In general,
Manuscript received January 5, 2014; revised April 9, 2014.
existing studies mainly ignore the safe headway
requirements in capacity analysis and safety studies. This
may cause inaccurate estimation of traffic flow
characteristics specifically on roadways with traffic flows
less than the perceived capacity [13].
II. LITERATURE REVIEW
Many factors influence the headway distribution of
vehicles including traffic volume, proportion of heavy
vehicles, lane position, road structure, time of the day and
weather condition [14]. Mei and Bullen [15] investigated
different statistical distributions for time headways
measured on a four-lane highway during the morning
peak traffic. According to their results, lognormal
distribution with a shift of 0.3 or 0.4 seconds was the best
fit for the time headways in high traffic volumes.
Sadeghhosseini [16] analyzed time headways at flow
rates varying from 140 to 1704 vehicles per hour per lane
on interstate highways of Illinois, U.S. In his study, using
a lognormal distribution with a shift of 0.36 seconds was
recommended to generate the time headways. In another
study by Arasan in 2003, the headway distribution was
investigated for a four-lane divided urban arterial in
Chennai City in India [11]. In this study, negative
exponential distribution was found to be suitable for
modeling headways at different lanes and over the entire
range of traffic flows. In 2006, Bham and Ancha
analyzed the time headway of drivers in a basic freeway
section as well as a ramp merge, a lane drop and a ramp
weaving section [17]. According to their study, shifted
lognormal distribution provided an accurate fit for all
studied areas. Zwahlen et al. [18] evaluated the
cumulative headway distributions at different traffic
flows and traffic lanes in Ohio freeways in the U.S. Their
results showed that the headway distributions at different
lanes are almost the same for similar hourly traffic flows.
Previous studies are mainly based on the entire time
headway data collected from a highway/freeway section
regardless of considering the vehicle types. Furthermore,
previous studies were mainly undertaken under light to
medium traffic flow conditions. In this paper, headway
distributions are analyzed and compared for heavy vehicles
and passenger cars under heavy traffic conditions. To
evaluate the headway characteristic, the time headways to
the preceding (front) and following (rear) vehicles are
separately evaluated for each vehicle type. To better analyze
the headway distribution in the vicinity of heavy vehicles
Journal of Traffic and Logistics Engineering Vol. 2, No. 3, September 2014
©2014 Engineering and Technology Publishing 224doi: 10.12720/jtle.2.3.224-229
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and passenger cars, the headways are evaluated at different
traffic flows. Then, simple mathematical models are
suggested to estimate the parameters of the front and rear
headway distributions at different traffic flow rates. This
paper is structured as follows. The following section
explains the dataset used in this study. The methodology as
well as the appropriate models of headway distributions
which are selected for each vehicle type (heavy vehicles and
passenger cars) is explained after. The relationship between
the parameters of the front and rear headway distributions
for heavy vehicles/passenger cars and traffic flows is also
analyzed. The final section summarizes the results of this
paper and provides directions for future research.
III. DATASET
Trajectory data used in this study was provided for a
highway section in California: Berkeley Highway (I-80).
The schematic illustration of this highway section is shown
in Fig. 1. The section of I-80 is 503 meters long and
comprises five main lanes with one auxiliary lane. There is
one on-ramp in this section and one exit off-ramp
downstream of the section [19]. There are no lane
restrictions for heavy vehicles in this section. The data
were collected from 4:00 to 4:15 PM and 5:00 to 5:30 PM
using a video capture rate of 10 frames per second. The
data was collected using seven video cameras mounted on
a 30-story building. The dataset was provided in clear
weather, good visibility, and dry pavement conditions. The
dataset has classified vehicles as automobiles, heavy
vehicles and motorcycles. Table I shows the traffic flow
parameters for the section of I-80. For the time period that
the data was collected, the proportion of heavy vehicles is
4.7%, 3.8% and 2.7% of the total traffic at 4:00 to 4:15 PM,
5:00 to 5:15 PM and 5:15 to 5:30 PM, respectively.
Figure 1. Schematic illustration of lane configuration for section of I-80.
The trajectory dataset used in this study makes it
possible to determine the time and space headways
between the heavy vehicles or passenger cars and their
surrounding vehicles at discrete time points. The vehicles
(front and rear vehicles) which are considered for
headway distribution analysis are presented in Fig. 2. In
this figure, the subject vehicle can be either a heavy
vehicle or a passenger car. In this study, the headways are
measured as the time between the front bumper of the
subject vehicle (heavy vehicle or passenger car) and the
front bumper of the front/rear vehicles which are called
front/rear time headways. Due to the noise in the NGSIM
dataset, the dataset was aggregated at each 0.5 second
time interval. Then, the aggregated trajectory data at each
0.5 second time interval (2 observations per second) was
used in this study.
Rear Vehicle Subject Vehicle Front vehicle
Figure 2. The subject vehicle and the preceding and following vehicles.
To analyze the time headway in the vicinity of each
vehicle type, a sample size of 50 heavy vehicles and 50
passenger cars were randomly selected. The main
statistical characteristics of time headways for the selected
heavy vehicles and passenger cars are presented in Table II.
TABLE I. TRAFFIC FLOW PARAMETERS IN THE SECTION OF I-80.
Time Interval Traffic Flow
(veh/hr)
Speed
(km/hr)
Density
(veh/km)*
Level Of
Service (LOS)
4:00 to 4:05 PM 8436 32.3 261 E
4:05 to 4:10 PM 7968 28.7 278 E
4:10 to 4:15 PM 8028 25.2 319 E
5:00 to 5:05 PM 8124 27.5 295 E
5:05 to 5:10 PM 7752 23.2 334 E
5:10 to 5:15 PM 5988 15.1 397 E
5:15 to 5:20 PM 7836 21.9 358 E
5:20 to 5:25 PM 7284 21.2 344 E
5:25 to 5:30 PM 6024 15.9 279 E
Total 7493 23.1 324 E
* Density is calculated as the number of vehicles per kilometer length of
all lanes.
TABLE II. STATISTICAL CHARACTERISTICS OF HEADWAYS FOR
SELECTED HEAVY VEHICLES AND PASSENGER CARS.
Time Headway
Characteristics
(sec)
Heavy Vehicles Passenger Cars
Front Headway
Rear Headway
Front Headway
Rear Headway
Mean 3.23 2.96 1.48 1.05
Median 2.91 2.42 1.17 0.87
Minimum 0.25 0.11 0.09 0.07
Maximum 12.78 9.36 4.91 4.58
IV. METHODOLOGY
To identify the appropriate model for headway
distribution, the statistical models should be applied to fit
the data. To be consistent with the results from majority
of the previous studies, shifted lognormal distribution is
Journal of Traffic and Logistics Engineering Vol. 2, No. 3, September 2014
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applied in this paper to present the time headways.
Lognormal is a well-known distribution model which is
frequently used to represent time headways in many
studies. Lognormal distribution is also proposed to model
time headways under car-following situations [20]. The
mathematical equation of the shifted lognormal
distribution is as follows:
t;
2
))(ln(exp
2)(
1) , ,|(
2
2t
ttf (1)
where, t is the time headway, is the shift value in
seconds, and μ and σ are parameters of lognormal
distribution known as location and scale parameters,
respectively. The two parameters are estimated from the
observed data (sample size = n) using Equations 2 and 3.
n
tn
i
i
1
)ln(
(2)
2
1
1
2
1
))(ln(
n
tn
i
i
(3)
To identify the shift value of the front and rear
headway distributions for each vehicle type, the
lognormal distribution model with shifts ranging from 0.0
to 1.0 seconds (with steps of 0.02 second) are examined.
The goodness of fit of the models is checked using Chi-
Square test with 95% confidence level. The null
hypothesis for each test is presented as follows:
The compatibility hypothesis of time headway
distribution with fitted model is rejected (h = 1) or not
rejected (h = 0). (4)
In this study, the most appropriate front and rear
headway distributions for each vehicle type are
determined using two steps. At the first step, the
goodness of fit on the distribution of front and rear
headways is examined for each vehicle type (heavy
vehicles and passenger cars). For that, the p-value
parameter is used. In a Chi-Square test with 95%
confidence level, larger p-values (p-values should be
larger than 0.05) represent a more compatible model. At
the second step, the headway distributions are obtained
for the selected front and rear headway distribution
models for heavy vehicles and passenger cars at different
levels of traffic flow. For each vehicle type, the goodness
of fit of headway distributions is examined at traffic
flows at each 5 minute time interval (Table I).
V. RESULTS AND DISCUSSIONS
As explained in the previous section, the goodness of
fit models on the headway distributions is examined to
model the time headway distributions for each vehicle
type. At the first step, 204 Chi-Square tests are conducted
for models with different shifts using SPSS software. For
each test, the parameters of the model are estimated from
the time headway data. The results of each step are
presented in Table III which shows the values of ‘h’ for
Chi-Square tests on all headways collected for each
vehicle type. The values of h equal to 1 represent
rejection of hypothesis test and its values with zero
represents approval of the hypothesis test.
TABLE III. RESULTS OF CHAI-SQUARE TEST FOR FRONT/REAR
HEADWAY DISTRIBUTIONS FOR EACH VEHICLE TYPE.
Shift
Heavy Vehicles Passenger Cars
Front
Headway
Rear
Headway
Front
Headway
Rear
Headway
0.00 1 1 1 1
0.02 1 1 1 1
0.04 1 1 1 1
0.06 1 1 1 0
0.08 1 1 0 0
0.10 1 1 0 0
0.12 1 0 0 1
0.14 1 0 0 1
0.16 1 0 0 1
0.18 1 0 1 1
0.20 1 0 1 1
0.22 1 0 1 1
0.24 1 0 1 1
0.26 0 1 1 1
0.28 0 1 1 1
0.30 0 1 1 1
0.32 0 1 1 1
0.34 0 1 1 1
0.36 0 1 1 1
0.38 0 1 1 1
0.40 0 1 1 1
0.42 1 1 1 1
0.44 1 1 1 1
0.46 1 1 1 1
0.48 1 1 1 1
0.50 1 1 1 1
0.52 1 1 1 1
0.54 1 1 1 1
0.56 1 1 1 1
0.58 1 1 1 1
0.60 1 1 1 1
0.62 1 1 1 1
0.64 1 1 1 1
0.66 1 1 1 1
0.68 1 1 1 1
0.70 1 1 1 1
0.72 1 1 1 1
0.74 1 1 1 1
0.76 1 1 1 1
0.78 1 1 1 1
0.80 1 1 1 1
0.82 1 1 1 1
0.84 1 1 1 1
0.86 1 1 1 1
0.88 1 1 1 1
0.90 1 1 1 1
0.92 1 1 1 1
0.94 1 1 1 1
0.96 1 1 1 1
0.98 1 1 1 1
1.00 1 1 1 1
Journal of Traffic and Logistics Engineering Vol. 2, No. 3, September 2014
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a) Front time headway distribution
b) Rear time headway distribution
Figure 3. Selected models fitted on distribution of front and rear headways of heavy vehicles.
a) Front time headway distribution
b) Rear time headway distribution
Figure 4. Selected models fitted on distribution of front and rear
headways of passenger cars.
As it is shown in Table III, the lognormal distribution
models are generally well-fitted to headways. For heavy
vehicles, the lognormal distribution models with shifts
ranging from 0.26 to 0.40 seconds are well-fitted to front
headways. Meanwhile, lognormal distribution models with
shifts ranging from 0.12 to 0.24 are fitted to rear headways
of heavy vehicles. For passenger cars, the lognormal
distributions with shifts ranging from 0.08 to 0.16 and
shifts from 0.06 and 0.10 are well-fitted to front and rear
time headways, respectively. This shows the larger front
and rear time headways in the vicinity of heavy vehicles
compared to the corresponding values in passenger cars.
The larger values of the front time headways in heavy
vehicles may be due to the operational limitations
(acceleration, deceleration, maneuverability) of heavy
vehicles compared to passenger cars. The larger values of
the rear time headways for heavy vehicles may be due to
the safety concerns of the drivers following a large heavy
vehicle. Selected models fitted on distribution of front and
rear headways of heavy vehicles and passenger cars are
presented in Fig. 3 and Fig. 4, respectively.
TABLE IV. ESTIMATION RESULTS OF FRONT/REAR HEADWAY DISTRIBUTIONS OF EACH VEHICLE TYPE.
Traffic Flow
(veh/hr) Heavy Vehicles Passenger Cars
Front Headway Rear Headway Front Headway Rear Headway
μ σ μ σ μ σ μ σ
5988 4.26 1.13 3.50 1.04 1.97 0.90 1.47 0.89
6024 4.23 1.14 3.46 1.07 1.94 0.94 1.43 0.91
7284 3.54 1.09 3.21 1.03 1.69 0.96 1.21 0.87
7752 3.28 1.12 3.08 0.97 1.55 0.87 1.02 0.89
7836 3.17 1.03 2.96 1.00 1.47 0.92 0.94 0.86
7968 3.11 1.01 2.89 0.89 1.39 0.76 0.86 0.77
8028 3.02 1.07 2.73 0.90 1.34 0.81 0.81 0.81
8124 2.84 1.04 2.68 0.86 1.26 0.79 0.75 0.83
8436 2.76 0.96 2.43 0.82 1.15 0.75 0.66 0.79
Journal of Traffic and Logistics Engineering Vol. 2, No. 3, September 2014
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To further analyze the headway distributions in the
vicinity of heavy vehicles and passenger cars, the
headways are evaluated at different traffic flows. Therefore,
at the second step, the goodness of fit of headway
distribution models is examined at traffic flows at each 5
minute time intervals (Table I). The shifted lognormal
models are fitted on front and rear headway data for heavy
vehicles and passenger cars from traffic flows at each 5
minute time intervals. Estimation results of the parameters
of headway distributions are presented in Table IV. In
general, lognormal distribution is identified by two
parameters including location (μ) and scale (σ). The results
of Table IV show the influence of changes in traffic flows
on the parameters of the lognormal distribution. In other
words, the influence of changes on traffic flows on the
time headway patterns are presented in this table.
To better understand the relationship between traffic
flows and time headway patterns, the parameters of the
headway distributions (μ, σ) can be calculated as a
function of traffic flows. Therefore, linear regression
models are used to simply estimate the parameters of the
front and rear headway distributions for each vehicle type
(Equations 5 and 6).
qba
(5)
qdc
(6)
where, μ is the location of the distribution, σ is the scale
of the distribution, q is the traffic flow, and a, b, c and d
are parameters. The estimation results from the regression
modeling are presented in Table V. By having the
parameters of each headway distribution model, the front
and rear headway distributions can be obtained for heavy
vehicles and passenger cars at different traffic flow rates.
According to the results from Table V, the location of
the front and rear headway distribution models (μ) which
represents the mean value of the headways can be
accurately estimated (R2 values of more than 0.87) for
each vehicle type. However, the scale parameter (σ)
which shows the standard deviation of the time headways
can be estimated with lower accuracy (R2 values of more
than 0.56).
TABLE V. ESTIMATION RESULTS OF HEADWAY DISTRIBUTIONS’ LOCATION (Μ) AND SCALE (Σ) PARAMETERS.
Vehicle
Type
Headway
distribution
μ σ
a b R2 c d R2
Heavy
Vehicles
Front
Headway 7.946 -0.006 0.988 1.160 -0.019 0.724
Rear
Headway 5.783 -0.004 0.872 1.580 -0.008 0.749
Passenger
Cars
Front
Headway 3.864 -0.003 0.943 1.429 -0.009 0.689
Rear
Headway 3.392 -0.003 0.946 1.151 -0.000 0.561
VI. CONCLUSIONS
In this paper, headway distributions were analysed for
heavy vehicles and passenger cars under heavy traffic
conditions. To comprehensively evaluate the headway
characteristic, the time headways to the preceding (front)
and following (rear) vehicles were separately evaluated
for each vehicle type. To better analyse the headway
distribution in the vicinity of heavy vehicles and
passenger cars, the time headways were evaluated at
different traffic flow rates. Then, simple mathematical
models were suggested to estimate the parameters of the
front and rear headway distributions at different traffic
flow rates for heavy vehicles and passenger cars.
According to the results from this study, lognormal
distribution models were generally well-fitted to time
headways. For heavy vehicles, the lognormal distribution
models with shifts ranging from 0.26 to 0.40 seconds were
well-fitted to front headways and lognormal distribution
models with shifts ranging from 0.12 to 0.24 were fitted to
rear headways of heavy vehicles. Meanwhile, the
lognormal distributions with shifts ranging from 0.08 to
0.16 and shifts from 0.06 and 0.10 were well-fitted to the
front and rear time headways of passenger cars,
respectively. This shows the larger front and rear time
headways in the vicinity of heavy vehicles compared to the
corresponding values in passenger cars. The larger values
of the front time headways in heavy vehicles may be due to
the operational limitations (acceleration, deceleration,
manoeuvrability) of heavy vehicles compared to passenger
cars. The larger values of the rear time headways for heavy
vehicles may be due to the safety concerns of the drivers
following a large heavy vehicle. The results from this
paper show the existence of difference in the behaviour of
drivers in the vicinity of heavy vehicles and passenger cars
under heavy traffic conditions.
To better understand the relationship between traffic
flows and time headway patterns, the parameters of the
headway distributions (μ, σ) were calculated as a function
of traffic flows. Therefore, linear regression models were
used to simply estimate the parameters of the front and
rear headway distributions for each vehicle type and
different traffic flow rates under heavy traffic conditions.
By having the parameters of each headway distribution
model, the front and rear headway distributions can be
obtained for heavy vehicles and passenger cars at
different traffic flow rates.
In this study, the headway distributions were evaluated
based on entire headway data collected from a highway
section without separating data of each lane. However,
each lane has different traffic flow characteristics which
Journal of Traffic and Logistics Engineering Vol. 2, No. 3, September 2014
©2014 Engineering and Technology Publishing 228
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may influence the time headway distributions. Exclusive
analysis of the headway distributions for each lane can be
a direction for future research. This may assist in more
accurate traffic planning and policy making at different
traffic conditions.
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Sara Moridpour holds a Bachelor of Civil Engineering and Masters degree in
Transportation Planning and Engineering from Sharif University of Technology, Iran. She also
received her PhD degree from Monash
University. She has 9 years of work and research experience in the field of traffic and
transport. Her main research interests include on driving
behavior modeling and analysis, micro
simulation, transport network modeling and optimization. She has been lecturer in the School of Civil,
Environmental and Chemical Engineering, RMIT University, from 2010.
Journal of Traffic and Logistics Engineering Vol. 2, No. 3, September 2014
©2014 Engineering and Technology Publishing 229