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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 Trac Engineering & Key Laboratory of Road and Trac 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 signicant factor in triggering trac 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 inuence 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 signicant impact on trac operations

at freeway on-ramp bottlenecks [, ] and can also trigger

trac 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.

isstudyaimedtollthisgapintheliteratureby

focusing on the relationship between multi-rejected gaps

and merging behavior. In so doing, it made three main

contributions: rst, it redenes 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 benet 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 specic 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 [] dene 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 dene 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 oen the case: the rejected gap can be larger than

theacceptedgap(seeSection.fordetails).

Gap acceptance theory is also widely used in microscopic

trac simulation models. Current simulation models, such as

MITSIM[],CORSIM[],VISSIM[],andTransModeler

[], use critical gaps in dierent ways. For example, risk

factors are used to dene the critical gaps in CORSIM; and

a psychophysical model is used in VISSIM to obtain a critical

gap. e TransModeler denes 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 trac simulation and driver assis-

tance systems. is is not only because merging behavior

mightleadtotracbreakdown,butalsobecausealastminute

merging decision will aect 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 eects of the instant speed, the relative

speed, and the gaps of merging vehicle with its assumptive

leadandlagvehicle[,].Withfurtherstudy,moresur-

rounding trac characteristics were taken into consideration

for modeling merge behaviors. Hidas [, ] considered the

eect of the reaction time, the maximum waiting time, and

time distance on merging behaviors. Sun and Eleeriadou

[] 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 classication [] have also

been introduced to model lane-changing behavior. However,

in their application of these models, all the above-mentioned

studies neglected the inuence of multi-rejected gaps on the

merging prediction.

In summary, studies reported in the review of the lit-

erature () do not clearly dene 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 deciencies 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 aected

by multi-rejected gaps, two isolated bottlenecks were picked,

and US- NGSIM data set and Hongxu on-ramp data set

were selected to ensure sucient 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), hereaer 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), hereaer

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

thespeedisaboutkm/htokm/hinSHandkm/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 aected 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 soware 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 dierence

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. Dening 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

mergingprocessofVehicleinLAdatasetareshown

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 trac 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

oen used in trac 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 signicant dierences 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 Trac Flow States. e states of trac 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 signicant

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 dierent 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 signicantly larger than the

mean of rejected gap bothinSHandLA,whilethedierence

is.minLAand.minSH,respectively.Whenitcomesto

gap , the mean of accepted gap is not signicantly dierent 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 dierent 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 Dierent Rejection Numbers. is

section focuses on the merging features related to dierent

rejection numbers.

As for the merging behavior of individual vehicle, we

compared the speed, speed dierences, 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 signicant 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 dierent 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 dierent rejection numbers dier 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 dierence 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 selsh.

However, the speed dierence 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

dierent merging preference between drivers in SH and LA.

Specically, 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 eective 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 oen chosen

when analyzing the inuence 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 dierence between LC and LG -

ΔV

Ｆ；Ａ (m/s)

Speed dierence 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

aer 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 dierent.

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.WendthattheWeibulldistributioncurvebest

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 trac 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 aect 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 aecting 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 condence 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 +0.159lead − 0.023 −0.241lag +0.12lead ()

SH: =1− 1

1+exp −0.481+ 0.51gap −0.313lead +0.121 −0.028+0.345lead − 0.036lag ,()

where is the probability of merging into the mainline, with

respecttotheandsamples(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,andlead are the signicant

variables for describing the merging process in LA. e

coecients 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

dierence 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

trac conditions in SH than LA. ere are three common

variables in the SH and LA models: ,lead ,andlead .

e sign of coecients of and lead for SH is the

same as that for LA; thus, the explanations are the same.

e positive coecient 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 aer the leading vehicle on

the target lane.

e prediction result for dierent 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 signicant in the

logistic regression model. Table demonstrates that variables

have signicant inuences on merging behavior are dierent

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 signicant variables covering all the

vehicles in a merging event, including time gap, space gap,

and speed dierence. 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 signicant 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 signicant 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 signicant 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. Aer introducing the multi-rejected gaps

intothemodel,thereisanimprovementinaccuracyforboth

sites. However, three related issues are worthy of discussion.

First, local impact factors such as road geometry inuence

themergingbehavior.epreviouswork(Marczaketal.

[]) veried 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 dierence between

themergingvehicleandthemainlinevehicleslead or lag

signicantly inuence 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 dierence 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 : Signicant 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 [ −specicity] is the probability of predicting a rejected state as an accepted state).

arelativelylowaccuracy.Meanwhile,Marczaketal.[]used

a dierent data set. We applied their method to our datasets,

and the accuracy we obtained is only .%.

e eectiveness of the prediction model can be veried

by using the Receiver Operating Characteristic (ROC) curve

[].AsshowninFigure,underthesamefalsepositiverate,

thetruepositiverateinLAishigherthanthatinSH.is

indicates the more eective 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-lecornerisrelativelymoresensitiveand

specic; thus, we can nd the most appropriate threshold

value to classify the merge and non-merge behavior. In this

study, Youden’s Index (sensitivity + specicity −) [] is

used to nd the most appropriate threshold value. e largest

Youden’s Index is the threshold, we need to determine the

most appropriate classication 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 eective.

7. Conclusions and Further Work

Basedonthetwotrajectorydatasets,oneatYan’anExpress-

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 signicantly larger than the mean of rejected gap, while

themeanofacceptedgap is not signicantly dierent 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 dierent 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 signicant dierence in the critical gaps

under dierent 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 signicant variables are more reasonable and ecient,

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

inuence the merging behavior. In the prediction model, the

distancetotheendoftheaccelerationlaneandthespeed

dierence between the merging vehicle and the mainline

vehicles signicantly inuence the prediction of the merging

behavior. Besides the randomness, the merging behavior may

be dynamic because drivers may adjust the crucial gap aer

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 dierence 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 eciency, 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-densityowisinthesamecondition,

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 eciency and urgency, the best

choice is to select the gap which is larger than the crucial one.

erefore, this study models a dierent 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 conicts 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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