<第 19 回 ITS シンポジウム 2021>

高速道路における専用車線からの協調型自動運転車合流機会の評価のためのギャップ分布モデリングカラ ジャヤ ヴェルシニ*1
東京大学大学院

鳥海 梓*2

工学系研究科大口 敬*2

社会基盤学専攻*1

東京大学 生産技術研究所*2

高速道路に協調型自動運転車（CAV）専用車線を設ける場合，専用車線の出口では，CAV が一般車線上の手動運転車に合流する必要性が生じる．本研究では，このような合流を想定した専用車線出口の設置位置の検討を行うことを目的として，手動運転車が形成するギャップ分布の分析を行った．阪神高速道路および中央自動車道により取得されたデータを対象に，異なる位置および交通条件下でのギャップ分布を合成ガンマ分
布によりモデル化し，この分布に基づいて，CAV が手動運転車に合流するのにかかる余剰時間を算出した．
その結果，上り坂が続くほど余剰時間は長くなる傾向があり，そのような地点への CAV 専用車線出口の設置は望ましくないことが示唆された．

Modelling Motorway Gap Distribution for Evaluating Merging Opportunityof Connected-and-Automated-Vehicles from Dedicated Lanes.
Jaya Varshini Kala*1

Azusa Toriumi*2

Takashi Oguchi*2

Department of Civil Engineering, The University of Tokyo, Japan*1
Institute of Industrial Science, The University of Tokyo, Japan*2
This study investigates geometric impacts on the gap distributions of Human-Driven-Vehicles (HDVs) to consider
locations of exits for dedicated Connected-and-Automated-Vehicle (CAV) lanes, where CAVs must merge into HDVs.
For that, the gap distributions are modelled by the combined Gamma distributions for different locations and traffic
conditions using Hanshin and Chuo Expressways’ data. Based on the estimated model, the required extra time for CAV’s
merging is hypothetically computed by assuming the random arrival of HDVs on the normal lane at different locations.
The result showed that the waiting time tends to increase in a long stretch uphill, suggesting avoiding such locations tolocate an exit.
Keywords: Gap distribution, Time occupancy, Combined model, CAV dedicated lane

1. Introduction
1-1 Background
Over the last decade, automated driving technology has
been developed rapidly around the world. Connected andAutomated Vehicles (CAVs) are expected to boost
capacity and minimize congestion by avoiding natural
speed reductions induced by road geometry by leveraging
vehicle-to-vehicle (V2V) and vehicle-to-infrastructure(V2I) technology.
At the present stage, CAVs that are partially automatedand can-do conditional automation with the interventionof human drivers are in mass production. However, thetechnology requires further development and time before
a CAV can be fully automated. Current CAV technology is
not feasible in all cases, even on motorways, which
involve motorized traffic only, from the on-ramp or to theoff-ramp and under various traffic conditions.
A possible way to early introduce CAVs is by providing
dedicated CAV lanes. By analyzing heterogeneous flow
models, Ye et al. [1] suggested CAVs can revive their true
potentials earlier in dedicated CAV lanes than inheterogeneous traffic flow. Assuming that dedicated CAV
lanes are installed on the median side of the motorway,
there will be fewer conflicts from vehicles merging fromthe on-ramp or vehicles diverging to the off-ramps;
rightmost lane in case of left-handed traffic in Japan as
illustrated in Figure 1. This study considers the exit for
dedicated CAV lanes. Even though homogeneous traffic
flow can realize smooth and efficient operation inside thededicated CAV lanes, CAVs must merge into HumanDriven-Vehicles (HDVs) on the normal lane at the exit,
influencing both the dedicated CAV and normal lane.
Therefore, such an exit’s design and operation are criticalelements for the dedicated CAV lanes.
1-2 Literature review
Existing studies state that the merging requires a complex
combination of merging plan choice, gap acceptance,
target gap selection, and acceleration decisions [2] and isalso affected by road design and traffic flow [3]. Dülgar etal. [4] have revealed that the merging time threshold,which is the time needed to complete a CAV merge, is a
crucial parameter for automated vehicles. By analyzing
different gaps of 2, 4, 5 and 6 s, Dülgar et al. [4] concluded
that for a safe merging, the minimum critical gap is 4 s.
On the other hand, to understand given conditions for<shoulder>

Normallanes
for HDVs

Dedicated
CAV lane<median>Traffic flow

Exit
Gap

Merging*in the case of left-hand traffic.

Figure 1 Schematic of a dedicated CAV lane

merging CAVs, gap or headway distributions of the HDVs
need to be analyzed. These distributions have long been
researched as one of the traffic flow characteristics. For
example, Zhang et al. [5] considered different headway
characteristics under different traffic flow conditions andconcluded that the headway distribution shapes aredifferent for different traffic flow levels and time periods.
Ha et al. [6] summarized different statistical models,
including combined and mixed models that fit theempirical gap distributions.
1-3 Objective
This study attempts to utilize gap distributions of HDVs
for evaluating the merging opportunity of CAVs from thededicated lanes. Here, parameters that determine CAVs’
merging performance (i.e., critical gap) are assumed to begiven based on the existing research.
The fundamental hypothesis of this study is that gap
distributions of HDVs may be affected by road geometry
and traffic conditions; this makes the merging opportunityof CAVs different along a motorway section. Therefore,
gap distributions are modelled for different locations andtraffic conditions and merging opportunities are evaluatedunder different conditions. This study helps us to discuss
preferable/avoidable motorway sections and traffic
conditions to locate an exit of the dedicated CAV lanes.
2. Methodology
2-1 Gap distribution modelling
In this study, the gap distribution at any location wasmodelled using the combined Gamma distribution:𝑓(𝑡) = 𝜑 ∗ 𝑔(𝑡) + (1 − 𝜑) ∗ ℎ(𝑡)where both 𝑔(𝑡) and ℎ(𝑡) follow Gamma distribution.
It can be imagined that there are two types of gaps, theones of vehicles following their leaders and the others ofvehicles not following. Existing studies have shown thatthe combined Gamma distribution can represent such acharacteristic well [7]. The parameters of the distributions

Cumulative
percentage
(CDF)

100%

0%

Available
percentage

Gap distribution of
HDVs in the adjacentnormal lane

Available gaps
for CAVs

Critical gap(given)Gap[s]

Figure 3 Schematic examples of “required extra time"

Figure 2 Concept of “available percentage”

were initially set by dividing the gaps into two clusters
using k-mean clustering [8] and identified through the
maximum likelihood estimation using MATLAB.
The errors between the estimated and observed “available
percentage” was calculated to evaluate the models. Here,the smallest gap which allows CAVs from the dedicatedlane to merge into HDVs is called a “critical gap”. The“available percentage” is the percentage of available gapsthat are longer than a “critical gap”, as illustrated in Figure
2. In this study, 4 s was assumed for the critical gap of
CAV according to Dülgar et al. [4].
2-2 Occupancy time distribution modelling
Occupancy time in this study is defined as the time whena location on the motorway is occupied by vehicle length.
CAVs cannot merge during such an occupancy time. In
this study, the occupancy time distribution of each location
was modelled by a single Gamma distribution.
2-3 Evaluation of merging opportunity
As mentioned above, the available percentage is one of theindices for evaluating the merging opportunity of a CAV
in case an exit of the dedicated CAV lane is located at eachlocation along the motorway section. In addition, based on
the above-developed models, merging opportunity wasevaluated using the “required extra time”. This index
represents the time duration that a CAV needs, in addition,
to merging from the hypothetical dedicated CAV lane into
the normal lane of HDVs at the specific location due to thelimited arrivals of available gaps ( ≥ the critical gap).
Travelling with acceleration/deceleration of merging
CAVs along the continuous locations and merging CAV’s
queuing is not considered.
The following steps computed the required extra time:1) The occupancy time of an HDV was randomly
extracted from the occupancy time distribution modelusing inverse transform sampling.
2) Similarly, the gap after the HDV of step 1 was
randomly extracted from the gap distribution model.
3) Steps 1 and 2 were repeated for the arrival of 100
HDVs. Thus, an arrival pattern of HDVs was obtained.
4) The required extra time of the CAV can be calculated
as the elapsed time from its arrival at the location to
the beginning of the gap in which the CAV can merge,
namely, greater than the critical gap (4 s in this study).
Examples of the required extra time depending on
CAV’s arrival time are illustrated in Figure 3.
Assuming a CAV may arrive at an arbitrary time, therequired extra times were calculated by shifting thearrival time of the CAV with 0.01 s. The average of allthese cases was taken as the “required extra time”.
5) Steps from 1 to 4 were repeated 50 times, and theaverage of these iterations was taken.
2-4 Study sites and data
Two sites were analyzed for this study. The detailed
geometry of these sites is illustrated in Figure 7(a).
Site 1 is in the inbound direction of Route 11 near
Tsukamoto junction of Hanshin Expressway, an urban
expressway in Osaka. It is a two-lane section with an Sshaped curve, a slight change in gradient, and merging
from the on-ramp. Around the on-ramp, lane change is
prohibited. The data is from image sensing called “ZenTraffic Data [9]”, collected for five different time periodsin September 2018.
Site 2 is in the section between Sagamiko and SagamikoEast interchanges (outbound direction from Tokyo) of
Chuo Expressway, an intercity expressway. It is a two-lane
section without any on-ramp or off-ramp. The gradientchanges along the road section from - 4.5% to 3.5% (sag).
Video recording was conducted on two weekdays, 25thand 26th November 2020, from 8.00 a.m. to 4.00 p.m.

4%available perdentage
Estimated CDF

60%

Estimated PDF (following + non-following)
Estimated PDF (following)
Estimated PDF (non-following)
Observed
Relative Frequency

40%
20%
0%
0

2

critical gap
4gap [s]6810Error [percentage point]

Observed CDF

80%

PDF or CDF

Site1
90-120[veh/5min] 120-150[veh/5min] 150-180[veh/5min]Figure 4 Example of gap distribution

3. Results
3-1 Modelling
An example of the combined Gamma distribution gap
model (at 43.6 kp in Site 2, 90-120 veh/5min) is shown in
Figure 4. Errors in the available percentage, the estimatedminus the observed, were within ±2 percentage points,
as shown in Figure 5. The error is accounted for less than
20% in percentage errors for most cases, except for thecase of 150-180 veh/5min of Site 1 that showed low
accuracy. The model showed better accuracy for Site 2
than for Site 1.
𝜑 value in the equation in 2-1, indicating the ratio of
vehicles following their leader, tended to increase with thevolume, particularly in Site 1, as shown in Figure 7(b).

0%
5.0

4.5

4.0

3.5

-2%
4%
2%

60-90
[veh/5min]30-60[veh/5min]

-2%

43.0

3.0

Site2

90-120[veh/5min]

0%
42.5

43.5

44.0

44.5

kp

Figure 5 Errors in available percentage
20

1

Estimated CDF
Observed CDF

16

0.8

12

PDF

Assuming that the dedicated CAV lanes are usually in therightmost lane (Figure 1), CAV’s merging opportunity isdetermined by the gaps in the median lane. Therefore, ourstudy examined gaps on the median lane. Because gaps
may differ by type of the leading and following vehicles
and there were limited samples of large vehicles, gaps
where the leading and the following vehicles are small
were only analyzed.
It is known that gap distributions are different for different
traffic conditions [5]. Therefore, the average velocity andvolume were calculated every 5 min for both sites. This
paper focuses on the uncongested conditions; the caseswith an average velocity greater than 60 km/h in Site 1 and
80 km/h in Site 2 were only analyzed. Traffic volume wasclassified by every 30 veh/5min, but due to the limited
observations, available traffic-volume classes were notcompletely the same for both sites. The sample size ofeach class is mostly above 200, not smaller than 100.

2%

0.6
Estimated PDF

80.4

Observed
Relative Frequency

4

CDF

100%

0.2

0

0
0.0

0.3
0.6
Occupancy time [s]0.9

Figure 6 Example of occupancy time distribution

Geometric impacts were not obvious, but the increase of
𝜑 from around 4 kp under 90-120 and 120-150 veh/5min
in Site 1 might be affected by the change of the gradientfrom negative to positive. Although the on-ramp at 3.8 kp
might also be the cause, lane-changing from the shoulder
lane to median lane was less because of the lane change
prohibition up to 3.5 kp. These models’ shape and scaleparameters did not show significant relation with thevolume and geometry.
Regarding the time occupancy distribution model, an
example for the same case with Figure 4 is shown in
Figure 6. It was generally distributed between 0.1-0.4 s.
3-2 Evaluation of merging opportunity
Available percentage and required extra time were shownin Figure 7(c) and (d). The required extra times tended to
be longer when the available percentages were small; thisis reasonable because a smaller available percentagemeans fewer opportunities to face a gap greater than thecritical gap.

In general, the available percentage decreased with theincrease in traffic volume while the required extra timeincreased. Thus, with higher traffic volume, the dedicated
CAV lane might no longer be feasible. An exit should be
carefully designed so that CAVs do not fail their merging,for example, by providing an appropriate length of
merging lane, V2I technology that monitors and provides
the arrivals of HDVs to merging CAVs, etc.
In Site 1, the available percentage decreased, and therequired extra time increased under 90-120 veh/5min fromaround 4.0 kp; this may be due to the impact of gradientchange from negative to positive, as also mentioned for 𝜑in 3-1. Another cause might be the illegal lane changing
upstream despite the lane change prohibition from 4.2 kp
to 3.0kp, which suggests that an exit of the dedicated CAV
lane should be avoided in such a section with both change
in gradient and a high possibility of lane changing. Underhigher volume, the required extra time became constantly
long, irrelevant to the gradient.
In Site 2, a decrease in the available percentage and anincrease of the required extra time were found in the uphillsection under 90-120 veh/5min. Similar to Site 1, this may
be an impact of the continuous positive gradient. However,even though the gradient is much steeper in Site 2, thechange in the available percentage and required extra time
were not more significant than in Site 1. Furthermore, thistrend was not seen under lower volume classes, suggestingthat merging opportunity is not much influenced bygeometry under very light traffic.
4. Future Works
The results from the analysis implied that the impacts ofpositive gradient on gap distributions depend on the traffic
volume conditions, but which could not be strongly
confirmed due to the limited observations. For further
validation, an investigation by increasing the number ofsample sites and traffic conditions is necessary. Interactionof gaps between median and shoulder lane, the impact ofon-ramp and off-ramp will also be considered.
Furthermore, because the current study evaluated merging
opportunities based on simplified indices such as theavailable percentage and required extra time, theevaluation methodology will be improved by considering
the relative velocity between CAVs and HDVs and otherparameters.

Acknowledgements
This work is partially supported by the UTokyo –Tsinghua
Collaborative Research Fund. Zen Traffic Data were
provided by the Hanshin Expressway Co. Ltd. Data
collection on Chuo Expressway was supported by Nippon
Expressway Research Institute Co, Ltd., Hachioji
Regional Head Office of Central Nippon Expressway Co,
Ltd., and Highway Planning Inc.
References[1] Ye, L., & Yamamoto, T. (2018). Impact of dedicatedlanes for connected and autonomous vehicle on traffic
flow throughput. Phys. A, 512, 588-59.
[2] Choudhury, C. F., Ramanujam, V., & Ben-Akiva, M.
E. (2009). Modeling acceleration decisions forfreeway merges. Transp. Res. Rec., 2124(1), 45-5.
[3] van Beinum, A., Farah, H., Wegman, F., &Hoogendoorn, S. (2018). Driving behaviour at
motorway ramps and weaving segments based on
empirical trajectory data.Transp. Res. C, 92, 426–441.
[4] Y Dülgar, Y., Rehborn, H., Molzahn, S. E., Koller, M.,
Menth, M., Kerner, B., & Schreckenberg, M. (2018).
A study for merging of automated vehicles. in 27th
Aachen Colloq. Automob. and Engine Technol., 2018.
[5] Zhang, G., Wang, Y., Wei, H., & Chen, Y. (2007).
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freeway loop event data. Transp. Res. Rec., 1999, 141149.
[6] Ha, D. H., Aron, M., & Cohen, S. (2012). Time
headway variable and probabilistic modeling. Transp.
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[7] Konda, H., & Nakamura, H. (2020). Methodology to
Evaluate the Quality of Service of Traffic Flow onIntercity Expressway Sections by using Follower
Percentage. Transp. Res. Rec., 2674, 6, 200-210.
[8] Gabriel Ramos Llordén (2021). Gamma Mixture
Modelestimation
withEMalgorithm (https://www.mathworks.com/matlabcentr
al/fileexchange/53028-gamma-mixture-modelestimation-with-em-algorithm), MATLAB Central
File Exchange. Retrieved June 5, 2021.
[9] Hanshin Expressway Co. Ltd., Zen Traffic Data,
https://zen-traffic-data.net/

Site1
Right

Curvature[%]Gradient[%]

0.5

5.0
-0.5

4.5

Left
flow

4.0

3.5

43.5

44.0

44.5

0
42.5
43.0
Downhill
-5

43.5

44.0

44.5

43.0

43.5

44.0

44.5

43.0

43.5

44.0

44.5

43.0

43.5
kp

44.0

44.5

-0.5

Merging section (onto the shoulder lane)

Uphill

0.0
5.0
-5.0

4.5
Downhill

4.0

3.5

3.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
4.5

4.0

3.5

40%

30%

30%

20%

20%

10%

10%

0%
4.5

4.0

3.5

3.0

0%
42.5

18

18

12

12

6

6

0
5.0

4.5

4.0

Uphill

0.0
42.5

3.0

40%

5.0

Left
flow

5

1.0

5.0

(d)
Required extra time [s]

43.0

3.0

Lane-change prohibited

3.5

3.0

60—90

90—120042.5

kp
Traffic volume:

30—60

Gap measurement points

Right

0
42.5

0.0

5.0

(b)
φ in the combined model(a) Geometry

0.5

(c)
Available percentage

Site2

120—150

150—180 [veh/5min]Figure 7 Results of gap distribution modeling, available percentage, and required extra time