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Modeling and Predicting Stochastic Merging Behaviors at Freeway On-Ramp Bottlenecks

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Merging behavior is inevitable at on-ramp bottlenecks and is a significant factor in triggering traffic breakdown. In modeling merging behaviors, the gap acceptance theory is generally used. Gap acceptance theory holds that when a gap is larger than the critical gap, the vehicle will merge into the mainline. In this study, however, analyses not only focus on the accepted gaps, but also take the rejected gaps into account, and the impact on merging behavior with multi-rejected (more than once rejecting behavior) gaps was investigated; it shows that the multi-rejected gaps have a great influence on the estimation of critical gap and merging prediction. Two empirical trajectory data sets were collected and analyzed: one at Yan’an Expressway in Shanghai, China, and the other at Highway 101 in Los Angeles, USA. The study made three main contributions. First, it gives the quantitative measurement of the rejected gap which is also a detailed description of non-merging event and investigated the characteristics of the multi-rejected gaps; second, taking the multi-rejected gaps into consideration, it further expanded the concept of the “critical gap” which can be a statistic one and the distribution function of merging probability with respect to such gaps was analyzed by means of survival analysis. This way could make the full use of multi-rejected gaps and accepted gaps and reduce the sample bias, thus estimating the critical gap accurately; finally, considering multi-rejected gaps, it created logistic regression models to predict merging behavior. These models were tested using field data, and satisfactory performances were obtained.
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Research Article
Modeling and Predicting Stochastic Merging Behaviors at
Freeway On-Ramp Bottlenecks
Jian Sun ,1Kang Zuo,2Shun Jiang,1and Zuduo Zheng3
1Department of Trac Engineering & Key Laboratory of Road and Trac Engineering, Ministry of Education,
To ng j i Un i ve r s it y, C h in a
2WuhanPlanning&DesignInstitute,1250JinghanRoad,Wuhan,China
3School of Civil Engineering, e University of Queensland, St. Lucia 4072, Brisbane, Australia
Correspondence should be addressed to Jian Sun; sunjian@tongji.edu.cn
Received 12 December 2017; Revised 26 March 2018; Accepted 15 April 2018; Published 16 May 2018
Academic Editor: Emanuele Crisostomi
Copyright ©  Jian Sun et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Merging behavior is inevitable at on-ramp bottlenecks and is a signicant factor in triggering trac breakdown. In modeling
merging behaviors, the gap acceptance theory is generally used. Gap acceptance theory holds that when a gap is larger than the
critical gap, the vehicle will merge into the mainline. In this study, however, analyses not only focus on the accepted gaps, but also
take the rejected gaps into account, and the impact on merging behavior with multi-rejected (more than once rejecting behavior)
gaps was investigated; it shows that the multi-rejected gaps have a great inuence on the estimation of critical gap and merging
prediction. Two empirical trajectory data sets were collected and analyzed: one at Yan’an Expressway in Shanghai, China, and the
other at Highway  in Los Angeles, USA. e study made three main contributions. First, it gives the quantitative measurement of
the rejected gap which is also a detailed description of non-merging event and investigated the characteristics of the multi-rejected
gaps; second, taking the multi-rejected gaps into consideration, it further expanded the concept of the “critical gap” which can be
a statistic one and the distribution function of merging probability with respect to such gaps was analyzed by means of survival
analysis. is way could make the full use of multi-rejected gaps and accepted gaps and reduce the sample bias, thus estimating the
critical gap accurately; nally, considering multi-rejected gaps, it created logistic regression models to predict merging behavior.
ese models were tested using eld data, and satisfactory performances were obtained.
1. Introduction
Numerous studies have shown that merging from the accel-
eration lane has a signicant impact on trac operations
at freeway on-ramp bottlenecks [, ] and can also trigger
trac breakdown [–]. Merging behavior is a complex
task, because, typically, the driver has to focus on three or
more vehicles in their current and target lanes in a limited
timeframe []. is contrasts with the driver’s task in the car-
following process where she/he generally only needs to focus
on the immediately preceding vehicle in their current lane
[]. erefore, the study of merging behaviors is a challenging
one.
In recent decades, merging behavior has been extensively
studied, and many models have been developed. Lots of these
models are based on the gap acceptance theory; that is, when
meetingagap,adriverwillcompareittothecriticalgap.Ifthe
gap is larger than the critical gap, the driver will accept it and
merge; if not, he/she will reject it and move on to nd another
gap[,].Intuitively,therearetwotypesofgap:theaccepted
gap and the rejected gap. Many previous studies are related
to the former, while few concern the latter. In addition, to
thebestknowledgeoftheauthors,nostudyhasinvestigated
multi-rejected gaps and their impact on merging behavior.
isstudyaimedtollthisgapintheliteratureby
focusing on the relationship between multi-rejected gaps
and merging behavior. In so doing, it made three main
contributions: rst, it redenes the rejected gap and gives
the quantitative measurement of it and investigated the
characteristics of the multi-rejected gaps in Los Angeles
and Shanghai, respectively; second, it extended the concept
of the “critical gap” that could be a stochastic one when
Hindawi
Journal of Advanced Transportation
Volume 2018, Article ID 9308580, 15 pages
https://doi.org/10.1155/2018/9308580
Journal of Advanced Transportation
taking into account the multi-rejected gaps and used survival
analysis to estimate the merging probability function with
respect to such gaps; nally, considering multi-rejected gaps,
logistic regression models are created to predict merging
behaviors. ese models have been tested using eld data,
and satisfactory performances have been obtained. With
the proposed model, the microscopic simulation is able
toreproducetracoperationmoreaccurately.Infuture,
the autonomous vehicles will benet from the model when
predicting surrounding vehicles near on-ramps and merging
reasonably.
e remainder of this paper is structured as follows:
Section  presents a review of the literature related to
merging behavior; Section  describes the study sites and
data; Section  discusses characteristics of the multi-rejected
gaps; Section  expands the concept of the critical gap which
can be a stochastic one and its estimation method; Section 
describes the logistic models to predict merging behavior
with the consideration of multi-rejected gaps; and Section 
concludes the paper by summarizing its main ndings and
recommending topics for future study.
2. Literature Review
As demonstrated above, this study aimed to analyze the
relationship between multi-rejected gaps and merging behav-
ior. is section discusses the existing literature on the
three specic aspects of this relationship which need to be
addressed.
e rst task is to gain an understanding of the concept
of a “rejected gap.” Initially, it is used in the estimation of
critical gaps at unsignalized intersections []. However, little
attention has been paid to the rejected gap in a merging
area, and most scholars simply regard rejected gaps as non-
merging events. Hou et al. [] note that when a vehicle’s
lateral coordinates remain the same lane or with some
oscillations, it is a non-merging event. Similarly, Meng and
Weng [] dene a rejected gap as the situation in which the
driver fails to merge into the current adjacent lane. Daamen
et al. [] note that drivers prefer to choose an optimum
gap that might reject several acceptable gaps before merging.
Marczak et al. [] rst dene the rejected gap as the gap a
mergercouldhavechosen(butchoosesinsteadtodriveahead
and merge into a gap downstream) and analyze the impact
of one rejected gap. Based on the evidence of these studies,
therefore, we conclude that previous studies either lack a clear
description of rejected gaps or simply consider one rejected
gap. Furthermore, the selection and calculation of rejected
samplesinthesestudiesarealsonotclearlydened.
e second aspect of the relationship between multi-
rejected gaps and merging behavior requiring analysis relates
to gap acceptance theory [–]. is theory is extensively
used in the modeling of merging behaviors, and the critical
gap is its key. e theory is mostly based on the assumption
that rejected gap critical gap accepted gap []. From this
equation, we can easily see that the accepted gap is no less
than the rejected gaps. However, according to our research,
this is not oen the case: the rejected gap can be larger than
theacceptedgap(seeSection.fordetails).
Gap acceptance theory is also widely used in microscopic
trac simulation models. Current simulation models, such as
MITSIM[],CORSIM[],VISSIM[],andTransModeler
[], use critical gaps in dierent ways. For example, risk
factors are used to dene the critical gaps in CORSIM; and
a psychophysical model is used in VISSIM to obtain a critical
gap. e TransModeler denes the linear and non-linear
critical gaps according to a combination of the speed of the
subject vehicle, the lead gap, and the lag gap in the target
lane. However, these simulation models fail to realize the
stochastic nature of the merging behavior. In fact, even on
the basis of gap acceptance theory, merging is a stochastic
rather than deterministic event. In addition, the critical
gap is also stochastic rather than constant. A lognormal
distribution of critical gap is assumed in MITSIM. Although
it notices the stochastic nature of critical gap, it neglects
theinuenceoftherejectedgaponthecriticalgap.Hence,
the assumed distribution of the critical gap is unsuitable
[].
Finally, an accurate prediction of merging behavior or
decisions is the basis of trac simulation and driver assis-
tance systems. is is not only because merging behavior
mightleadtotracbreakdown,butalsobecausealastminute
merging decision will aect subsequent driving behavior and
the behaviors of the following drivers []. Hidas [, ]
developed a merging model including cooperative and forced
merge movement components. Most of existing merging
models considered the eects of the instant speed, the relative
speed, and the gaps of merging vehicle with its assumptive
leadandlagvehicle[,].Withfurtherstudy,moresur-
rounding trac characteristics were taken into consideration
for modeling merge behaviors. Hidas [, ] considered the
eect of the reaction time, the maximum waiting time, and
time distance on merging behaviors. Sun and Eleeriadou
[] modeled this behavior considering driver characteristics.
Recently, Marczak et al. [] take the once rejected gap
before merging into consideration to model the merging
behavior.
As for the approach to model the merging behavior, it
can be either parametric or non-parametric. Discrete choice
models such as the binary logit model, the multinomial logit
model, and the nested logit model are proposed to model
lane-changing behavior []. e models mentioned above
are parametric. In recent years, however, non-parametric
machine learning methods such as decision tree [], fuzzy
genetic algorithm [], and Bayes classication [] have also
been introduced to model lane-changing behavior. However,
in their application of these models, all the above-mentioned
studies neglected the inuence of multi-rejected gaps on the
merging prediction.
In summary, studies reported in the review of the lit-
erature () do not clearly dene and calculate the rejected
gap; () ignore the critical gap which can be a stochastic one
(causedbyrejectedgaps)andtherealrelationshipbetween
theacceptedandrejectedgap;and()failtoconsidermulti-
rejected gaps in most gap acceptance model and merging
prediction models. ese deciencies are addressed in this
study.
Journal of Advanced Transportation
lane 1
lane 2
lane 3
lane 4
Flow Direction
Point: A
Acceleration lane
133 m
(a) Hongxu on-ramp bottleneck in Shanghai (GPS position of point A, .,
.)
lane 1
lane 2
lane 3
lane 4
lane 5
lane 6
Flow Direction
Flow Direction
Acceleration lane
212 m
Point: A
(b) US Highway  bottleneck in Los Angeles (GPS position of point A, .,
.)
F : e schematics of two study sites.
Flow Direction
Rejected gap1 Rejected gap2Accepted gap
ID: 1618
Lag Gap Lead Gap
Pre Gap
V
,'
V
,#
V
,$
ABLG LD
LC PE
F : Descriptionofthevariablesinamergingprocess, where LC is the merging vehicle, LD is the leading vehicle on the target lane, LG
is the following vehicle on the target lane, and PE is the leading vehicle on the initial lane.
3. Study Sites and Data
Aimed to better understand the merging behavior aected
by multi-rejected gaps, two isolated bottlenecks were picked,
and US- NGSIM data set and Hongxu on-ramp data set
were selected to ensure sucient rejected gaps and a relatively
higher data resolution.
3.1. Study Sites. Figure  shows the schematic diagrams of
the two on-ramp bottleneck sites. Figure (a) illustrates the
Hongxu on-ramp bottleneck, on an eastbound section of the
Yan’an Expressway in Shanghai (China), hereaer referred to
as “SH.” ere are three mainline lanes, and the acceleration
lane has a length of  m. Figure (b) illustrates a southbound
segment of the US- in Los Angeles (USA), hereaer
referred to as “LA.” In this case, there are ve mainline lanes,
and the acceleration lane has a length of  m. During the
observation time, both sites are in congested condition, and
thespeedisaboutkm/htokm/hinSHandkm/hto
 km/h LA, respectively. It should be pointed out that there
are also diverging vehicles in LA; however, it is quite less than
the merging vehicles and we remove the merging events that
are obviously aected by the diverging vehicles.
3.2. Data Extraction. To collect data from SH, video cameras
were installed in buildings  m or more tall. e cam-
eras covered  m m upstream and approximately  m
downstream of the bottleneck, where merging behaviors
are prevalent. Trajectories were extracted with the help of
advancedtrackprocessingsoware-George(alsoseein[]).
is soware can manually identify and mark the initial posi-
tion of each vehicle in the video and then track its trajectory.
It can record vehicle position, velocity, acceleration, and other
parameters at . s intervals []. e data for LA are from the
NGSIM dataset, with a time resolution of . s. ese LA data
collection details can be found in Zheng et al. [].
e study focuses on merging vehicles and the vehicles
surrounding them.  and  merging samples were
collected from SH and LA, respectively. Similar to Sun et
al. [] and Marczak et al. [], variables considered in the
analysis are depicted in Figure  (detailed trajectory is show in
Figure)andsummarizedinTable.Forclarity,themeanings
Journal of Advanced Transportation
Position (m)
250
200
150
100
50
0
Time (frame)
4620 4660 4700 4740 4780 4820 4860
Vehicle on shoulder lane
Vehicle on acceleration lane
Rejected Gap 2
Accepted Gap
1610
1620
1613 1630
1618
Rejected Gap 1
(a) e trajectories of merging Vehicle  and its surrounding vehicles
(the red line represents the vehicle on the acceleration lane, while the black
line represents the vehicle on the shoulder lane)
TA;J (s)
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
Time (frame)
4680 4700 4720 4740 4760 4780 4800
85% Point
A: Rejected Gap 1
85% Point
Decision-making
Point
B: Rejected Gap 2
C: Accepted Gap
(b) e calculation of Vehicle ’s rejected gaps and accepted gap
F : An example of a merging process at LA.
T : Potential variables in the merging process.
Indicator Notation Unit
Remaining distance on acceleration lane m
Speed of merging vehicle m/s
Time gap
LD and LG gap
s
LD and LC lead
LC and LG lag
PE and LC pre
Space gap
LD and LG gap
m
LD and LC lead
LC and LG lag
PE and LC pre
Speed dierence
LD and LC lead m/s
LC and LG lag
roughout the article, “gap” indicates time gap unless explicit noted.
of variables are described in Figure  taking Vehicle  in LA
as an example.
lead =
LD −
LC
lag =
LC −
LG
()
where LD is the speed of leading vehicle at the shoulder
lane; LC is the speed of merging vehicle; LG is the speed of
following vehicle at the shoulder lane.
3.3. Dening and Measuring the Rejected Gaps. Previous
studies give the various descriptions of the rejected gap. In
this study, we used the concept of rejected gap as Marczak et
al. [] mentioned that the rejected gap is the gap a merger
couldhavechosen.However,theselectionandcalculation
of rejected samples and their gaps are not clearly explained
in all the previous studies. is study addresses this short-
coming, and the detailed process is shown below. e vehicle
trajectories, the rejected gaps, and the accepted gap for the
mergingprocessofVehicleinLAdatasetareshown
in Figure . e merging vehicle  is faster than other
vehicles. Vehicle  experienced two rejected gaps while
driving on the acceleration lane (the red trajectory). en it
accepted the gap between Vehicle  and Vehicle  and
merged into the mainline (the black trajectory of ). Points
A and B denote two rejected gaps, while point C represents
the accepted gap (see Figure (b)).
In Figure (a), the intersection of the red and black lines
means that the merging vehicle overtakes the vehicle in the
mainline, rather than merging. at is, before this point,
the merging vehicle rejected a gap. e point where a line
changes from red to black is the merging point (when the
midpoint of a vehicle’s front bumper crosses the lane marking,
the corresponding time point is the point at which that
vehicle changes lane). In this instance, Vehicle  entered
the observation area of the mainline at the  timeframe.
Before it merged into the mainline at the  timeframe
(accepting the gap between Vehicles  and ), it has
rejected two gaps: one between Vehicles  and , and
the other between Vehicles  and .
Similarly,foreachpairofleadandlagvehiclesatthe
shoulder lane, we can calculate the gap at each time step (e.g.,
. s for LA data). us, a merging vehicle experiences a series
Journal of Advanced Transportation
Mean speed (m/s)
20
18
16
14
12
10
8
6
4
2
0
Time (min)
01234567891011121314
(a) e mean speed variations in SH
Mean speed (m/s)
Time (min)
15
14
13
12
11
10
9
8
7
6
50 5 10 15 20 25 30 35 40
(b) e mean speed variations in LA
F : e mean speed variations in two datasets.
T : Statistics of trac ow states.
Dataset Duration time (min) Range of Mean speed (m/s) Range of Mean Flow (veh/h/ln)
SH  .. –
LA  .–. –
of gaps before the merge is eventually executed. e accepted
gapinthisstudyisselectedandcalculatedat.sbeforethe
merging point, but which gap should be considered as the
rejected gap for this merging vehicle? ere are two steps to
calculateandselecttherejectedgaps.
Step 1. Calculate the th percentile as the rejected gap (gap ).
It is for the following two reasons: () the th percentile is
oen used in trac engineering; for example, the speed limit
is usually set as the th percentile []; () the th percentile
ofthegapsislargeenoughfordriverstoperceiveandtomake
merging decision accordingly. Meanwhile, we also tested the
th and th percentiles of the rejected gaps and found no
signicantdierenceamongthem.Wehavemadethemean
test and distribution test for the each other of those three
percentiles. It shows that all the valueishigherthan.;
therefore there are no signicant dierences between two test
objectnomatterformeanordistribution.
Step 2. Exclude the unreasonable gaps. ere are several
standardstodothedatacleaning:()themean()and
standard deviation ()ofmergingsamplesgap on both sites
arecalculated,respectively,andtherejectionsampleswhose
gap is smaller than ( − ) are removed; () the gaps that
drivers do not get prepared to merge in are eliminated; and
() the samples that the speed of merging vehicles is too
slow (e.g.,  km/h) to execute a lane change process are also
excluded from the rejected gaps. We believe that these two
steps ensure that the selected gaps are reasonable.
4. Statistical Analysis of Rejected
Gaps during the Merging Process
4.1. Statistics of Trac Flow States. e states of trac ow in
SH and LA datasets are shown in Figure  and Table . e
speed, respectively, range from . m/s to . m/s in SH and
. m/s to .m/s in LA, and during most of the observation
time, the speed is a relative low one. erefore, the proposed
model is suitable for the merging behavior prediction in
middle-and-high-density ow. e key property in this study
is that the merging speed is larger than the mainline speed.
In this condition, merging vehicles are able to overtake the
preceding mainline vehicles and select a suitable gap from
several ones. erefore, the rejected gaps have signicant
impact on merging decision.
4.2. Distribution of Rejection Number. e rejection number
isthenumberofrejectedgapsbeforeasuccessfulmerge.Its
distribution in SH and LA is shown in Figure .
As shown in Figure , as the rejection number increases,
the number of vehicles decreases. Indeed, more than .%
of vehicles in SH and .% of vehicles in LA rejected a gap at
least once. e mean rejection number is . in SH and . in
LA.
4.3. Relationship between Rejected Gaps and Accepted Gaps.
As previously discussed, there might be several rejected gaps
during a vehicle’s merging process. e following analysis
now determines whether the rejected gaps can be larger than
the accepted gap during one merging process.
Journal of Advanced Transportation
Sample Frequence
200
180
160
140
120
100
80
60
40
20
0
Rejection Number
012345678>=9
SH
LA
84
182
64
108
49
89
32
84
31
58
2730
910 1219 482021
F : e distribution of the rejection number.
T : Percentage of merging event that at least one rejected gap
larger than accepted gap for samples with dierent rejection number.
Rejection number
SH .% .% .% .% .%
LA .% .% .% .% .%
First, the distributions of gap and gap for both accepted
gaps and rejected gaps in SH and LA are shown in Figure .
is gure clearly reveals a large overlap for either gap or
gap at both sites, thus indicating that the rejected gaps can be
larger than the accepted gap. Moreover, using -test, rejected
and accepted gaps are statistically compared. Results show
that the mean of accepted gap is signicantly larger than the
mean of rejected gap bothinSHandLA,whilethedierence
is.minLAand.minSH,respectively.Whenitcomesto
gap , the mean of accepted gap is not signicantly dierent to
the rejected gap in SH and LA. And value in SH ( = 0.25)
is larger than that in LA ( = 0.19). One possible explanation
is that drivers in SH are more irrational and thus engage in
more aggressive merging behaviors (e.g., forced merging), as
reported in Sun et al. [, ].
To further verify the ndings above, the percentage of
the merging events with at least one rejected gap larger
than the accepted gap for samples with dierent rejection
number (i.e., rejection number >)arelistedinTable.
Similar results (rejected gaps can be larger than the accepted
gap) are obtained. In particular, Table  shows that the
percentage is relatively large (reach .%) for SH. is
nding indicates that SH drivers are more unpredictable (an
indirect indication of aggressiveness), while LA drivers are
more rational and observant (Figure ).
4.4. Merging Features for Dierent Rejection Numbers. is
section focuses on the merging features related to dierent
rejection numbers.
As for the merging behavior of individual vehicle, we
compared the speed, speed dierences, space gaps, and time
gaps of rejected gaps and accepted gaps of each vehicle.
However, because of the driver heterogeneity, there is no
obviousconclusionobtained.isaspectisthekeypointin
our further research.
As for the aggregate merging behavior of all vehicles, in
accordance with the studies of Marczak et al. [] and Sun et
al. [], variables that have a signicant impact on merging
behavior are analyzed, including gap,,,lag,andgap.
Samples with the same rejection number are put together and
theacceptedvalueisanalyzed.ebox-plotsofthesemerging
features in relation to dierent rejection numbers are shown
in Figure .
Figure  shows that, in both SH and LA, the critical gap
(means of accepted gap) is relatively stable (i.e., mostly s).
is nding not only emphasizes the key critical gap, but also
suggests the appropriate time gap for a merging decision.
As for ,wecanseethat,withtheincreaseinrejection
number, the merging location is closer to the end of the
acceleration lane and the distribution is more concentrate
for both SH and LA. is implies that the more the gaps are
rejected, the less the space can be chosen and the more urgent
the merging event becomes. is phenomenon may cause a
higher probability of executing a forced lane changing and
donates great damage to the bottleneck, which accords with
what Sun et al. [] nds.
in relation to dierent rejection numbers dier from SH
to LA. In SH, the merging speed decreases with the increase
of rejection number, which is opposite to LA. It because that
the speed in the acceleration lane is faster than that in the
shoulder lane in SH, and drivers reject more gaps to reduce
the time cost, while LA drivers reject several gaps to seek a
better lane change condition.
Figure  shows that with the increase of rejection numbers
thespeeddierencebetweenLCandLGinSHisalways
uctuating around . e small speed dierence and time
gap demonstrate the irrational merging behavior, which may
cause more cooperative and forced lane changings. In this
way, the vehicle in the target lane must slow down actively
or passively. According to research nds of Sun et al. [],
there are more forced lane changings in SH. is implies
that merging decisions in SH are more aggressive and selsh.
However, the speed dierence between LC and LG increases
with the increase of rejection numbers in LA (over  m/s),
which implies that merging decisions in LA are more rational.
Figure  show a diversity merging behavior under dif-
ferent rejection number. It means that rejected gaps have
impact on merging behaviors. ese gures also show a
dierent merging preference between drivers in SH and LA.
Specically, drivers in SH are more risk-taking and self-
focused while drivers in LA are more rational and altruistic.
is results in more forced and cooperative lane changings in
SH[]andmorefreelanechangingsinLA.
5. Estimation of Critical Gap and the
Distribution Function
As discussed above, during the merging process, a vehicle
might reject several gaps, and the rejected gaps might be
larger than the accepted one. erefore, it is reasonable
to treat the gap acceptance process of a merging event as
stochastic, rather than deterministic. In other words, whether
drivers choose to merge or not under a given gap can be
characterized as a probabilistic event. Correspondingly, the
Journal of Advanced Transportation
Frequence
0.3
0.25
0.2
0.15
0.1
0.05
0
SA;J (m)
−5 10 25 40 55 70 85 100
SH-A
SH-R
(a) Distribution of 𝑆gap-SH
Frequence
0.25
0.2
0.15
0.1
0.05
0
SA;J (m)
−5 10 25 40 55 70 85 100 115
LA-A
LA-R
(b) Distribution of 𝑆gap-LA
SH-A
SH-R
Frequence
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
TA;J (s)
20246810
(c) Distribution of 𝑇gap -SH
LA-A
LA-R
Frequence
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
TA;J (s)
0246810
(d) Distribution of 𝑇gap -LA
F : e distribution of 𝑔𝑎𝑝 and 𝑔𝑎𝑝 in SH and LA (note that SH-A and LA-A represent the accepted gaps in SH and LA, and SH-R
and LA-R represent the rejected gaps.).
critical gap should also be treated as stochastic. is section
focused on extending the concept and tting the merging
probability distribution function with respect to such critical
gap when considering multi-rejected gaps.
5.1. Estimation of the Critical Gap. Survival analysis [] is
a branch of statistics for analyzing the expected duration
of time until one or more events happen. It is widely used
in medicine, biology, economics, and so on. Like stochastic
capacity [–], the critical gap can also be estimated by
usingsurvivalanalysis.Previous[]researchshowsthatthe
time gap is more eective in predicting the merging behavior
than the space gap. erefore, in this study, gap is seen
as the survival time (analogous to lifetime in lifetime data
analysis); the merging behavior corresponds to the failure
event (analogous to death in lifetime data analysis); the
rejected gap (not merge at a certain gap) is considered as
censored data (analogous to the data that lifetime is longer
than the duration of the experiment in lifetime data analysis)
while the accepted gap (merge at a certain gap) is considered
as uncensored data.
Product Limit Method (PLM), developed by Kaplan and
Meier [], is a non-parametric method of survival analysis
that also includes the semi-parametric method and para-
metric method []. e non-parametric method is usually
used to determine the survival probability under a certain
survival time; the semi-parametric method is oen chosen
when analyzing the inuence of variables on the survival
Journal of Advanced Transportation
SH LA
Time gap between LD and LG - TA;J (s)
10
9
8
7
6
5
4
3
2
1
0
Time gap between LD and LG - TA;J (s)
10
9
8
7
6
5
4
3
2
1
0
Rejection number
012345
Rejection number
012345
Rejection number
012345
Rejection number
012345
Rejection number
01234 5
Rejection number
01234 5
Rejection number
012345
Rejection number
012345
D(m)
Distance to end of acceleration lane -
D(m)
Distance to end of acceleration lane -
120
100
80
60
40
20
0
Merging speed - V(m/s)
Merging speed - V(m/s)
20
18
16
14
12
10
8
6
4
2
0
Speed dierence between LC and LG -
ΔV
F;A (m/s)
Speed dierence between LC and LG -
ΔV
F;A (m/s)
5
4
3
2
1
0
−1
−2
−3
−4
−5
200
180
160
140
120
100
80
60
40
20
0
22
20
18
16
14
12
10
8
12
10
8
6
4
2
0
F : Merging features for various rejection numbers (the upper, middle, and lower lines of the box represent the th, th, and th
percentiles of the data, resp.).
Journal of Advanced Transportation
Survive probability
1
0.8
0.6
0.4
0.2
0
TA;J (s)
0246810
(5, 0.5)
Without
One
All
(a) Results of 𝑇gap -SH
TA;J (s)
0246810
Survive probability
1
0.8
0.6
0.4
0.2
0
Without
One
All
(b) Results of 𝑇gap -LA
F : Results of PLM.
time;andtheparametricmethodisbasedonthespecic
distribution of survival time and aims to build a parametric
model of survival function. In this study, we rstly use
the non-parametric PLM approach to get a distribution of
survival probability. en parametric PLM model is adopted
aer determining the distribution of survival probability.
PLM applies the multiplicative theorem of probability to
calculate the survival probability. is calculation procedure
is outlined below.
Step 1. Sortthesurvivaltime(i.e.,gap) data in ascending
order. Rank =1,2,3,...,.Ifthevalueofcensoreddata(i.e.,
rejected gaps) is equal to non-censored data (i.e., accepted
gaps), the non-censored data is placed ahead.
Step 2. Calculate the survival probability of one rank starting
from =1.
(𝑡) =
𝑖:𝑡𝑖<𝑡
𝑖−𝑖
𝑖,()
where is survival time, that is, a certain gap;𝑡is the
probability that survival time is longer than ,inotherwords,
theprobabilitythatthevehicledidnotmergeatthetimepoint
;𝑖is the number of vehicles with a longer survival time
than 𝑖,thatis,thenumberofvehiclesthatdidnotmergeat
acertaingap ;𝑖is the number of death events at the time
point 𝑖,thatis,thenumberofvehiclesthatmergedintothe
mainline at a certain gap
Figure  shows the survival curve. It demonstrates that,
under the same survival probability, the larger the rejection
number considered, the larger the time gap. Meanwhile,
as for the same time gap, the larger the rejection number
considered, the higher the survival probability. For clarity
and conciseness, Without represents the samples (gaps) with
only accepted gaps (uncensored data); One indicates the
samples with uncensored data and censored data of one
closest rejected gap before merging on time; All includes
uncensored data and all censored data (rejected gaps). In
Figure (a), for example, for the survival probability of .,
the time gap of Without, One,andAll is . s, . s, and . s,
respectively.Forthetimegapof.s,thesurvivalprobability
of Without, One and All is ., ., and ., respectively.
ese trends are the same for both the SH and LA sites.
ese analyses demonstrate that the rejecting behavior (reject
a gap) has an impact on the accepted gaps, and comparing the
results of considering just one rejected gap and multi-rejected
gaps, the survival probability of the time gap is quite dierent.
erefore, multi-rejected gaps should be taken into account
when estimating the critical gap rather than without thinking
about it or just considering one largest rejected gap [].
Furthermore, the slope of curve All in SH is bigger than
that in LA, which shows a faster change of critical gap in
SH. ere are two likely reasons for this: () more vehicles
in SH than in LA rejected at least one gap before merging
(as demonstrated in Section .); () there were a higher
proportion of drivers in SH who rejected gaps that were
bigger than the accepted gaps (as shown in Section .).
5.2. Merging Probability Function with respect to Critical Gap.
Let
()=1−(),()
where ()is the probability of merging under a survival time
. In other words, it is the merging probability function with
respect to critical gaps.
In order to use a parametric PLM, it is essential to deter-
mine the distribution of merging probability with respect to
the critical gap.
 Journal of Advanced Transportation
Distribution function
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
TA;J (s)
012345678910
Parameter free
LA
Weibull distributed
(a)
Distribution function
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
TA;J (s)
0123456 7 8 9 10
Parameter free
Weibull distributed
SH
(b)
F : Weibull distribution function (the data used here are the samples of All).
T : Results of parametric PLM.
Position alpha beta −2
LA . . .
SH . . .
Similar to stochastic capacity [], based on the PLM
result, various function types such as Normal, Lognormal,
Weibull, exponential, and Log-logistic distribution curve are
tested.WendthattheWeibulldistributioncurvebest
performs the function of merging probability with respect to
critical gap, whose function is expressed as follows:
()=1−
−(𝑥/𝛽)𝛼for ≥0, ()
where is shape parameter; is scale parameter.
When applying a parametric PLM, Maximum Likelihood
Estimation (MLE) helps us to get an answer. e Maximum
Likelihood Estimation function for a Weibull distribution is
= 𝑛
𝑖=1 −𝑎 ⋅𝑎−1
𝑖⋅−(𝑞𝑖/𝑏)𝑎𝛿𝑖⋅−(𝑞𝑖/𝑏)𝑎1−𝛿𝑖.()
en by the means of genetic algorithm, we have a relative
betterresultanditislistedinTable.
From Figure , we can see that the Weibull distribution
well ts the merging probability distribution of the critical
gapinbothLAandSH.isresultcanbeappliedtomicro-
scopic trac simulation, where the Weibull distribution
couldbemoreappropriatethanthelognormaldistribution
[]. Furthermore, the critical gap which is stochastic can help
toexplainfree,forced,andcooperativelanechanging[,].
According to gap acceptance theory, we always accept the gap
when it is larger than the traditional critical gap, and this leads
toafreeratherthanaforcedlanechange.However,oncewe
considerthestochasticnatureofcriticalgap,theforcedlane
change could occur because the rejected gap is larger than the
accepted gap. ese forced lane changes might also lead to
better simulation of early onset breakdown phenomena [].
6. The Merging Behavior Prediction Model
In order to quantitatively measure how variables aect merg-
ing behavior, the utility-based logistic regression model is
utilized to predict the merging behavior by considering the
multi-rejected gaps.
6.1. Logistic Regression Model. In the logistic regression
model, the merging utility is modeled as a function of
explanatory variables aecting drivers’ merging behavior.
Maximum likelihood estimation is used to estimate param-
eters. e basic expression in the logistic regression model is
shown in []
logit =ln
1−=+ 𝑘
𝑖=1𝑖𝑖,()
where logit() is the logarithm of the odds of experiencing
an event (the linear relationship of the variables) and is the
probability of an event.
Because not every vehicle has a preceding vehicle, the
explanatory variables pre and pre are excluded from the
model, and the remaining variables (as shown in Table ) are
all taken into consideration. In our study, the condence level
is pre-set as .. e nal models (considering the multi-
rejected gaps) for LA and SH are shown as follows:
Journal of Advanced Transportation 
LA: =1− 1
1+exp 3.566 +0.159lead 0.023 0.241lag +0.12lead ()
SH: =1− 1
1+exp −0.481+ 0.51gap 0.313lead +0.121 0.028+0.345lead 0.036lag ,()
where is the probability of merging into the mainline, with
respecttotheandsamples(includingbothaccepted
andalltherejectedgaps)inSHandLA,respectively.
e value for parameters in () and () is lower than
.. And the Nagelkerke -square is . for the logistic
regression model in SH and . for that of LA. ese
two models well explain the merging behavior while the
Nagelkerke -squareishigherthan.[].
In (), lead,,lag,andlead are the signicant
variables for describing the merging process in LA. e
coecients of lead and lead are positive, which means that
the larger lead or lead is, the higher the merging probability
becomes. e negative variables and lag is an indicator
for the urgency of the merging. e smaller the remaining
distance to the end of the merging point and bigger speed
dierence between merging behavior and putative follower,
the higher the probability of merging into the mainline.
ere are more variables included in () than in (). is
is mainly because of the more complex driving behavior and
trac conditions in SH than LA. ere are three common
variables in the SH and LA models: ,lead ,andlead .
e sign of coecients of and lead for SH is the
same as that for LA; thus, the explanations are the same.
e positive coecient of the variable gap and implies
that there is a higher probability of a vehicle merging into
the mainline when it meets a larger gap or owns a higher
speed.Furthermore,thenegativevariableslag and lead could
indicateapreferenceinSHforchoosingamergingposition
near the leading vehicle on the target lane in time and near the
followingvehicleonthetargetlaneinspace.ismeansthat
the merging vehicles in SH prefer to overtake the following
vehicle on the target lane and run aer the leading vehicle on
the target lane.
e prediction result for dierent rejection numbers is
shown in Table , and is followed by a prediction accuracy
comparison. A probability larger or equal to . is considered
as a merging event, while a probability less than . is taken
as a non-merging event.
In the following table, One represents the samples which
includealltheacceptedsamplesandoneclosestrejectedgap
before merging; Two stands for the samples which contain
all the accepted samples and two closest rejected gaps before
merging, as well as ree,Four,andAll (consisted of all the
accepted and rejected samples).
e variables with in Table  are signicant in the
logistic regression model. Table  demonstrates that variables
have signicant inuences on merging behavior are dierent
when considering various quantities of rejected gaps. As to
SH, the One hasfoursignicantvariablesonlycoveringthe
merging vehicle and the lag vehicle at the target lane. As
to the All, it owns six signicant variables covering all the
vehicles in a merging event, including time gap, space gap,
and speed dierence. Comparing the two models, the latter
one concentrates on more factors and has a higher prediction
accuracy. It implies that the behavior of rejecting a gap has
impact on the subsequent decision, and the signicant factor
changes with the increasing of rejected gaps. us, multi-
rejected gaps matter in the merging behavior. What is more,
as to LA, One has ve signicant variables while All only has
four. However, prediction accuracy of All is higher than One.
It seems that considering multi-rejected gap contributes to
the recognition of signicant factors.
AsshowninTable,themoretherejectionnumber
considered, the higher the prediction accuracy for both sites.
is emphasizes the importance of multi-rejected gaps when
predicting a merging behavior. What is more, relative to the
prediction accuracy in SH, it is much higher in LA for the
One,Two ,ree,andFour. However, when considering all the
rejected gaps, the prediction accuracy for both sites reaches
., and the gap of prediction accuracy between two sites is
lled. Since we have found in Section . that more vehicles
in SH than in LA rejected at least one gap before merging,
models of One,Two ,ree,andFour neglect more rejected
gaps in SH than in LA, which leads to a lower prediction
accuracyinSH.Whenalltherejectedgapsareconsidered
(i.e., no rejected gaps are neglected), the prediction accuracy
for both sites achieves improvement. To some extent, it
further proves the necessity to consider multi-rejected gaps
when analyzing and modeling merging behaviors.
6.2. Discussion. Aer introducing the multi-rejected gaps
intothemodel,thereisanimprovementinaccuracyforboth
sites. However, three related issues are worthy of discussion.
First, local impact factors such as road geometry inuence
themergingbehavior.epreviouswork(Marczaketal.
[]) veried the conclusion. Actually, in our model, the
conclusion is the same. In Table , the distance to the end
of the acceleration lane and the speed dierence between
themergingvehicleandthemainlinevehicleslead or lag
signicantly inuence the prediction of the merging behavior.
e prediction accuracy is still not very high compared
with that found in other studies (.% in Hou et al. [],
and more than % in Marczak et al. []). Although NGSIM
datasetisalsousedinHouetal.[],thedenitionof
“non-merging events” shows big dierence in that study.
Furthermore, there is a large overlap between accepted and
rejected gaps (both the gap and gap)inourdatasets;this
makesitamorecomplextasktodistinguishthemandleadsto
 Journal of Advanced Transportation
T : Signicant variables of logistic regression model.
Variabl e s SH LA
One Two ree Four All One Two ree Four All
∗
∗ ∗
gap ∗∗
lead ∗∗ ∗
lag ∗∗
gap ∗∗
lead ∗∗
lag ∗∗ ∗ ∗ ∗
lead ∗∗∗ ∗
lag ∗∗ ∗ ∗∗ ∗ ∗
T : Prediction accuracy of logistic regression.
Rejection number One Two ree Four All
LA . . . . .
SH . . . . .
True positive rate
1.0
0.8
0.6
0.4
0.2
0.0
True positive rate
1.0
0.8
0.6
0.4
0.2
0.0
False positive rate
0.0 0.2 0.4 0.6 0.8 1.0
False positive rate
0.0 0.2 0.4 0.6 0.8 1.0
LA SH
F : ROC curves of two sites (true positive rate [sensitivity] represents the probability of predicting an accepted state as an accepted
state, while false positive rate [ specicity] is the probability of predicting a rejected state as an accepted state).
arelativelylowaccuracy.Meanwhile,Marczaketal.[]used
a dierent data set. We applied their method to our datasets,
and the accuracy we obtained is only .%.
e eectiveness of the prediction model can be veried
by using the Receiver Operating Characteristic (ROC) curve
[].AsshowninFigure,underthesamefalsepositiverate,
thetruepositiverateinLAishigherthanthatinSH.is
indicates the more eective prediction of a merging event
in LA and the lower probability of making the mistake of
regarding accepted state as a rejected state. Furthermore, the
pointnearthetop-lecornerisrelativelymoresensitiveand
specic; thus, we can nd the most appropriate threshold
value to classify the merge and non-merge behavior. In this
study, Youden’s Index (sensitivity + specicity ) [] is
used to nd the most appropriate threshold value. e largest
Youden’s Index is the threshold, we need to determine the
most appropriate classication threshold, and they are .
and . in LA and SH, respectively. Moreover, the area under
theROCcurve(i.e.,AUC)reectsthepredictioneectiveness
[]. Generally, the AUC between . and . indicates a poor
result; . to . indicate a moderate result; and the model
performs well when the area is larger than .. In this study,
Journal of Advanced Transportation 
the AUC for LA and SH is . and ., respectively. From
this, we can conclude that the model is eective.
7. Conclusions and Further Work
Basedonthetwotrajectorydatasets,oneatYananExpress-
way in Shanghai (China) and the other at Highway  in
Los Angeles (USA), this study analyzed the merging behavior
at the acceleration lane, especially the relationship between
multi-rejected gaps and accepted gaps. From this analysis, we
have drawn the following conclusions.
() e mean rejection number is . in SH and . in
LA. By using -test, both in SH and LA, the mean of accepted
gap is signicantly larger than the mean of rejected gap, while
themeanofacceptedgap is not signicantly dierent to the
rejected gap . Meanwhile, the result shows that the rejected
gap can be larger than the accepted gap.
()erejectedgapshaveimpactonmergingbehaviors,
and the merging preference is dierent between drivers in SH
and LA. With the increase of rejection number, the merging
condition becomes worse in SH while it is improved in LA.
is result illustrates that drivers in SH are more risk-taking
andself-focusedwhiledriversinLAaremorerationaland
altruistic.
() ere is a signicant dierence in the critical gaps
under dierent rejection numbers: the more the gaps are
rejected, the larger the critical gap becomes. For example, for
a merging probability of ., with no rejected gaps considered,
the critical gap is . s in LA and .s in SH; on the other
hand,withalltherejectedgapsconsidered,thecriticalgap
is . s in SH and . s in LA. Meanwhile, survival analysis
was undertaken to better understand the characteristics of the
merging process, and the Weibull distribution function best
ts the merging probability function of critical gap.
() Logistic regression models were developed to predict
merging events. By taking into account multi-rejected gaps,
the signicant variables are more reasonable and ecient,
and the prediction accuracy of merging behavior improves
greatly. Comparing the model with only one closest rejected
gap and all rejected gaps, there is a .% and .%
improvement of the latter one in LA and SH, respectively. On
the other hand, the local factors such as the road geometry
inuence the merging behavior. In the prediction model, the
distancetotheendoftheaccelerationlaneandthespeed
dierence between the merging vehicle and the mainline
vehicles signicantly inuence the prediction of the merging
behavior. Besides the randomness, the merging behavior may
be dynamic because drivers may adjust the crucial gap aer
rejecting some ones, which is to be investigated in the future
work.
is study focuses on the merging behavior in high-
density ow, which has fundamental dierence with the
classical gap acceptance studies. In our study sites, speed of
merging vehicles is larger and drivers are able to select a gap
from several ones at the same time, which is a multinomial
choice. For travel eciency, the driver may prefer the gap
downstream. For urgency, the driver may force in small gaps.
Other factors overweigh the impact of gap. At a stop sign, the
merging or crossing driver selects the oncoming gaps one by
one.Merginginlow-densityowisinthesamecondition,
because the speed of the merging vehicle is smaller than
that of the mainline vehicles. erefore, the classical gap
acceptance theory for these scenarios is a binary choice. For
whatever reasons such as eciency and urgency, the best
choice is to select the gap which is larger than the crucial one.
erefore, this study models a dierent merging behavior but
does not imply that the gap acceptance behavior in other
situations has the same properties.
Drivers in SH are more risk-taking and self-focused;
this leads to more forced lane changings in SH and makes
the modeling of merging behaviors for that site more chal-
lenging. is observation implies the necessity to adequately
accommodate human factors in the microscopic modeling of
driving behaviors (primarily, their lane-changing maneuvers
and car-following behavior), as advocated in the recent liter-
ature (e.g., [, , ]). However, the explicit incorporation
of risk perceptions in the modeling of the merging process
wasbeyondthescopeofthisstudyandisatopicforfuture
research.
Finally, drivers can experience several gaps during the
merging process, and the gaps they nally accept are not
necessarily the optimal ones. With the development of
autonomous and connected vehicles, another area of future
research is the optimization of the merging behavior decision
modeltoensuretheoperationaleciencyofthemergingarea
and the optimal and synchronized merging of all vehicles.
Conflicts of Interest
e authors declare that they have no conicts of interest.
Acknowledgments
e authors would like to thank the National Natural Science
Foundation of China (U and ) and the
Science and Technolog y Commission, Shanghai Municipality
(DZ), for supporting this research.
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