ंઢมߋ‫ن‬ଇͱྲྀྔ‫ٯ‬సͷؔ܎೔ຊ‫ܕ‬ɾυΠπ‫ܕ‬ͷൺֱ໺ా‫ ޺ق‬1 , ೔Ӭాହ‫ ܒ‬2 , ୞໦ਐҰ 11ࠤլେֶ ޻ֶ‫ڀݚܥ‬Պ ஌ೳ৘ใγεςϜֶઐ߈2ࠤլେֶ ૯߹৘ใ‫ج‬൫ηϯλʔ֓ཁೋंઢͷߴ଎ಓ࿏Ͱ͸ɺ௥ӽंઢͷྲྀྔ͕૸ߦंઢͷͦΕΛӽ͑Δʮྲྀྔ‫ٯ‬సʯͱ‫ݺ‬͹ΕΔ‫ݱ‬৅͕Α͘‫ݟ‬ΒΕΔɻຊߘͰ͸ɺNagel-Schreckenberg ϞσϧΛೋंઢʹ֦ு͢Δ͜ͱͰɺྲྀྔ‫ٯ‬సͷ‫ݯى‬Λ୳Δɻ૸ߦंઢଆ͔Βͷ௥͍ӽ͠Λ཈੍͢ΔυΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇͱͦͷΑ͏ͳ཈੍ͷແ͍೔ຊ‫ܕ‬ͷͦΕΛϞσϧԽ͠ɺγϛϡϨʔγϣϯΛߦ͏ɻ೔ຊ‫Ͱܕ‬͸ɺ௥ӽंઢͷ଎౓͕૸ߦंઢͷͦΕΑΓ΋଎͍͜ͱ͕ɺྲྀྔ‫ٯ‬సʹͱͬͯॏཁͰ͋Δ͜ͱ͕Θ͔ΔɻυΠπ‫Ͱܕ‬͸ɺ૸ߦंઢ͔Βͷ௥͍ӽ͠཈੍͕ྲྀྔ‫ٯ‬సΛ༠ಋ͢Δ͜ͱ͕Θ͔Δɻ

Relation between lane-changing rules and the reverse lane usage
A comparison between Japanese and German types
Toshihiro Noda1 , Yasuhiro Hieida2 , and Shin-ichi Tadaki1
1

Department of Information Science, Saga University
2

Computer and Network Center, Saga University
Abstract

A phenomenon called “reverse lane usage”, in which flow in the fast lane exceeds that in theslow lane, is usually observed. In this paper, we study origins of reverse lane usage using atwo-lane extension of the Nagel-Schreckenberg model. We employ two types of lane-changing
rules: German and Japanese types. The German type rule set contains a rule suppressing
motions of overtaking slow cars on the fast lane through the slow lane. The Japanese typerule set, on the contrary, allows overtaking motions through both lanes. In the Japanese type,the phenomenon occurs only when the maximum speed in the fast lane is faster than that inthe slow lane. In the German type, the rule suppressing overtaking through the slow lane isessential for the phenomenon.

1͸͡ΊʹՃʹ൐͍ɺ૯ྲྀྔʹର͢Δ૸ߦंઢͷྲྀྔ͕ 1/2 Λ
ԼճΔɺͭ·Γ௥ӽंઢͷྲྀྔ͕૸ߦंઢͷྲྀྔΛෳ਺ंઢΛ༗͢Δߴ଎ಓ࿏Ͱ͸ɺ௥ӽߦಈʹΑͬͯҰंઢ࿏Ͱ͸‫ݟ‬ΒΕͳ͍‫ڵ‬ຯਂ͍‫ݱ‬৅͕ൃੜ͢Δɻ
ਤ 1 ͸೔ຊͷೋंઢߴ଎ಓ࿏ʹ͓͚Δ૸ߦंઢͷྲ্ྀճΔɻ͜ͷ‫ݱ‬৅Λʮྲྀྔ‫ٯ‬సʯͱ‫͢ʹͱ͜Ϳݺ‬Δɻಉ༷ͷ‫ݱ‬৅͸ɺυΠπͷ࣮ଌσʔλʹ͓͍ͯ΋ใࠂ͞Ε͍ͯΔ [1, 2, 3, 4]ɻྲྀྔ‫ٯ‬స‫ݱ‬৅ΛγϛϡϨʔྔΛɺ૯ྲྀྔɺͭ·Γ૸ߦंઢͱ௥ӽंઢͷྲྀྔͷγϣϯΛ௨ͯ͡࠶‫͢ݱ‬ΔࢼΈ͕ߦΘΕ͍ͯΔɻNagel࿨ʹର͢Δൺͱͯ͠දͨ͠΋ͷͰ͋Δɻ૯ྲྀྔͷ૿Β͸ Nagel-Schreckenberg ϞσϧʹΑΓɺྲྀྔ‫ٯ‬స1‫ݱ‬৅Λ͍ࣔͯ͠Δ [1]ɻTadaki Β͸݁߹ࣸ૾‫࠷ܕ‬ద‫ݮ‬଎ɿ ઌߦंʹিಥ͠ͳ͍Α͏ʹ‫ݮ‬଎͢Δɻ଎౓Ϟσϧʹɺߴ଎ಓ࿏Λ໛ͨ͠ඇରশతͳंઢม

vn = min (vn , gn − 1)ߋ‫ن‬ଇ‫ंߦ૸ͼٴ‬ઢͱ௥ӽंઢͷ࠷ߴ଎౓ͷࠩΛಋೖ͢Δ͜ͱͰɺྲྀྔ‫ٯ‬సΛ࠶‫͍ͯ͠ݱ‬Δ [2, 3, 4]ɻ͠ϥϯμϜԽɿ ֬཰ p Ͱ 1 ‫ݮ‬଎͢Δɻ

͔͠ɺंઢมߋ‫ن‬ଇ͸ଟ͘ͷཁૉ͔ΒͳΓɺͲͷཁ

vn = max (vn − 1, 0)ૉ͕ྲྀྔ‫ٯ‬సͷओཁҼͰ͋Δ͔͸໌Β͔ʹ͞Ε͍ͯͳ͍ɻ3ຊߘͰ͸ɺೋंઢʹ֦ுͨ͠ Nagel-SchreckenbergϞσϧΛ༻͍ͯɺंઢมߋ‫ن‬ଇͱྲྀྔ‫ٯ‬సͷؔ܎Λ(3)ೋंઢͷ NS ϞσϧNS Ϟσϧʹंઢมߋ‫ن‬ଇΛ௥Ճ͢Δ͜ͱͰɺೋ໌Β͔ʹ͢Δɻಛʹɺ૸ߦंઢ͔Βͷ௥͍ӽ͠ʹରंઢ࿏ͷ NS ϞσϧΛߏ੒͢Δɻ֤࣌ࠁʹ͓͍ͯɺ͢Δ࣮࣭తͳ‫੍ن‬ͷ఺ͰҟͳΔɺ೔ຊ‫ͱܕ‬υΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇΛϞσϧԽ͠ɺͦͷޮՌΛ‫ݕ‬౼͢ΔɻൺֱͷͨΊʹɺ೔ຊ‫ͱܕ‬υΠπ‫ܕ‬ͷ‫ڞ‬௨෦෼ͱҟͳ࠷ॳʹ‫ޙ‬ड़͢ΔंઢมߋΛશंʹରͯ͠ߦ͏ɻͦͷ‫ʹޙ‬ɺલड़ͷ଎౓มߋ‫ͼٴ‬Ґஔͷߋ৽Λߦ͏ɻंઢมߋ‫ن‬ଇ͸େ͖͘ೋͭͷཁૉʹ෼͚Δ͜ͱ͕Δ෦෼Λ෼཭͢Δɻ·ͨɺंઢมߋ‫ن‬ଇͷޮՌΛ௚Ͱ͖ΔɻंઢมߋΛߦ͏ඞཁੑͱɺंઢมߋͷ҆શ઀తʹΈΔͨΊʹɺܾఆ࿦తंઢมߋɺͭ·Γ৚݅ੑͰ͋Δɻ‫͍ͯ͠ߦ૸ࡏݱ‬Δंઢ‫ं઀ྡͼٴ‬ઢͷઌΛຬͨ͢৔߹ʹ͸ඞͣंઢมߋΛߦ͏͜ͱͱͨ͠ɻߦंʹରͯ͠ɺҰఆͷ‫ࢹ( ཭ڑ‬ք d) Ҏ಺ʹ͍ۙͮͨࡍʹɺंઢมߋΛߦ͏ඞཁੑΛ൑அ͢Δɻ͔͠͠ɺ

Slow lane (Meishin 468kp (west bd.), Aug, 1996)
1ंઢมߋ͕ඞཁͰ͋ͬͯ΋ɺྡ઀ंઢ΁҆શʹҠಈ

0.9Ͱ͖ͳ͚Ε͹ंઢมߋ͸Ͱ͖ͳ͍ɻ

0.80.7
flow ratio(2)ຊ‫Ͱڀݚ‬͸ɺ૸ߦंઢଆ͔Βͷ௥͍ӽ͠ͷ‫ن‬ଇ͕

0.60.5ҟͳΔ೔ຊͱυΠπͷ৔߹Λൺֱ͢Δɻ೔ຊͷަ௨0.4‫ن‬ଇͰ͸ɺं͸૸ߦंઢΛ૸ߦ͢Δ͜ͱΛ‫ج‬ຊͱ͠

0.30.2͍ͯΔ (ಓ࿏ަ௨๏ೋे৚)[6]ɻ·ͨɺ௥͍ӽ͠Λߦ

0.1
0050

100

150
200
250
total flow (1/5min)300

͏৔߹ʹ͸ɺ௥ӽंઢ͔Βߦ͏͜ͱͱఆΊΒΕ͍ͯ

350

Δ (ಓ࿏ަ௨๏ೋे৚ୈࡾ߲)ɻ͔͠͠ɺ௥ӽंઢΛਤ 1: ໊ਆߴ଎ಓ࿏ɺ1996 ೥ 8 ݄ͷ࣮ଌσʔλɻԣ࣠͸ 5 ෼ؒͷ૯ྲྀྔɺॎ࣠͸૸ߦंઢͷंઢར༻཰Ͱ͋Δɻ૯ྲྀྔ͸૸ߦंઢͱ௥ӽंઢͷ࿨Λද͢ɻ

૸Δ௿଎ंΛɺ૸ߦंઢଆ͔Β௥͍ӽ͢͜ͱΛ཈੍
͢ΔͨΊͷ۩ମత‫ن‬ଇ͸ͳ͍ɻͦͷͨΊɺ௥ӽंઢΛ૸ߦ͢Δେ‫͕Ͳͳंܕ‬ɺ૸ߦंઢଆ͔Β௥͍ӽ͞
ΕΔ͜ͱ΋͠͹͠͹ൃੜ͢ΔɻυΠπͷަ௨‫ن‬ଇͰ͸ɺ೔ຊͱಉ༷ʹं͸૸ߦंઢΛ૸ߦ͠ͳ͚Ε͹ͳ

Nagel-Schreckenberg ϞσϧΒͳ͍ɻҰํɺ௥ӽंઢ্ͷ஗͍ंʹର͢Δ (૸ߦNagel-Schreckenberg(NS) Ϟσϧ͸ɺ࠷΋Α͘རंઢ͔Βͷ) ௥͍ӽ͠Λ཈੍͢ΔͨΊʹɺ௥ӽंઢ༻͞Ε͍ͯΔަ௨ྲྀηϧΦʔτϚτϯϞσϧͷҰͭʹ஗͍ं͕͍Δ৔߹ʹ͸ɺ૸ߦंઢΛ૸ߦ͢Δं͸Ͱ͋Δ [5]ɻಓ࿏͸ηϧʹ෼ׂ͞ΕɺҰͭͷηϧʹ͸௥ӽंઢ΁Ҡಈ͢Δͱ͍͏‫ن‬ଇ͕ઃ͚ΒΕ͍ͯΔɻҰ୆ͷं͔͠ೖΔ͜ͱ͕Ͱ͖ͳ͍ɻ֤ं͸લํͷۭೋͭͷ‫ܕ‬ͷंઢมߋ‫ن‬ଇͷޮՌΛൺֱ͢ΔͨΊʹɺͷηϧͷ਺ʹԠͯ͡ɺ࠷ߴ଎౓ vmax ͷ੍‫ݶ‬ͷԼͰ‫ڞ‬௨తͳ෦෼ͱҟͳΔ෦෼Λ੔ཧ͠ɺंઢมߋ‫ن‬ଇ2Λ‫ه‬ड़͢ΔɻՃ଎‫ݮ‬଎Λߦ͏ɻҎԼʹࣔ͢଎౓มߋ‫ن‬ଇʹैͬͯશंͷ଎౓Λܾఆͨ͠‫ʹޙ‬ɺશंͷҐஔΛߋ৽͢Δɻ೔ຊ‫ͱܕ‬υΠπ‫ͦܕ‬ΕͧΕͷඞཁੑͷ৚݅Λද 1n ൪໨ͷंͷҐஔΛ xn ɺ଎౓Λ vn ɺઌߦ͢Δ n+1ʹ·ͱΊΔɻ૸ߦंઢ͔Β௥ӽंઢ΁ͷҠಈͰ͸ɺ૸ߦंઢΛઌߦ͢Δंͷ௥͍ӽ͠ͷͨΊͷҠಈ͸‫ڞ‬൪໨ͷं·Ͱͷۭͷηϧͷ਺ +1 Λ gn = xn+1 − xn௨Ͱ͋ΔɻυΠπ‫Ͱܕ‬͸ɺ௥Ճͱͯ͠௥ӽंઢΛ૸

ͱ͢Δɻ͜ͷ࣌ɺҎԼͷࡾͭͷखॱͰ଎౓Λߋ৽͢Δɻશंͷ଎౓ͷߋ৽ͷ‫ʹޙ‬ɺंͷҐஔΛߋ৽͢Δɻ Δ௿଎ंΛൈ͔ͳ͍ͨΊͷҠಈ͕͋Δɻ௥ӽंઢ͔Β૸ߦंઢ΁ͷҠಈͰ͸ɺ௥ӽंઢΛ૸Γଓ͚ͳ͍Ճ଎ɿ ࠷ߴ଎౓ vmax ·ͰͷൣғͰ 1 Ճ଎͢ΔɻͨΊͷ૸ߦंઢ΁ͷҠಈ͸‫ڞ‬௨Ͱ͋Δɻ೔ຊ‫Ͱܕ‬͸ɺ

vn = min (vn + 1, vmax )௥ӽंઢΛ૸Δ௿଎ंΛ௥͍ӽ͢͜ͱ͕Ͱ͖ΔɻҰ(1)2ද 1: ೔ຊ‫ͱܕ‬υΠπ‫ܕ‬ͷඞཁੑͷ৚݅೔ຊ૸ߦंઢ͔Β௥͍ӽ͠௥ӽंઢ΁ͷंઢมߋ௥ӽंઢ͔ΒυΠπ௥͍ӽ͠
ࣜ (4)·ͨ͸ࣜ (8)૸ߦंઢ͔Βͷ௥͍ӽ͠཈ࢭ‫ن‬ଇ௥ӽंઢΛ૸ߦ͠ଓ͚ͳ͍૸ߦंઢ΁ͷंઢมߋ௥ӽंઢΛ૸ߦ͠ଓ͚ͳ͍

·ͨ͸

ࣜ (7)

͔ͭ௥͍ӽࣜ͠ (9)૸ߦंઢ͔Βͷ௥͍ӽ͠཈ࢭ‫ن‬ଇํɺυΠπ‫Ͱܕ‬͸ɺઌߦं͕஗͍৔߹ʹ͸૸ߦंઢ

Ͱ͋Δɻ͜ͷ৚݅͸ɺೋͭͷλΠϓͷ‫ن‬ଇͰ‫ڞ‬௨Ͱ΁ҠಈͰ͖ͳ͍ɻ

∗͋Δɻࣜ (6) ͷ vmax͸ྡ઀ंઢͷ࠷ߴ଎౓Λද͢ɻ∗͸ɺंઢຖʹ࠷ߴ଎౓͕ҟͳΔ৔߹΋ߟྀͯ͠ɺvmax૸ߦंઢ͔Β௥ӽंઢ΁ͷҠಈͷ৔߹͸௥ӽंઢͷf࠷ߴ଎౓ vmaxɺ௥ӽंઢ͔Β૸ߦंઢ΁ͷҠಈͷ৔s߹͸૸ߦंઢͷ࠷ߴ଎౓ vmax
ͱͳΔɻप‫ڥظ‬ք৚݅ͷԼͰɺ֤ंઢͷηϧ਺=10000 ͱ͠ɺҟͳΔंͷ୆਺ͰγϛϡϨʔγϣϯΛߦͬͨɻंͷॳ‫ظ‬഑ஔ͸ϥϯμϜͱ͠ɺ૯ྲྀྔͱंઢར༻཰Λ‫ܭ‬ଌ͢Δ·Ͱͷ؇࿨࣌ؒ=1000ɺࢹք d = 16 ͱ͢Δɻ

ਤ 2: ंઢมߋ‫ن‬ଇͷॾྔɻ‫ؙ‬͸ंΛද͠ɺ੺͍‫ؙ‬͸஫໨͍ͯ͠ΔंΛද͍ͯ͠Δɻ

4೔ຊ‫ܕ‬ͷೋंઢ࿏Ϟσϧਤ 2 ͸ंઢมߋ‫ن‬ଇͷॾྔͷઆ໌Ͱ͋Δɻ஫໨͠೔ຊ‫ܕ‬ͷंઢมߋ‫ن‬ଇͱྲྀྔ‫ٯ‬సͷؔ܎Λௐ΂Δɻ͍ͯΔंͷ଎౓Λ v ɺ஫໨͍ͯ͠ΔंͱಉҰंઢͷ೔ຊ‫ܕ‬ͷंઢมߋ‫ن‬ଇͰ͸ɺલड़ʢද 1ʣͷΑ͏ʹɺઌߦंͱͷ‫཭ڑ‬Λ ∆xp (લઅͷ gn ͱಉ͡)ɺಉҰं૸ߦंઢ͔Β௥ӽंઢ΁Ҡಈ͢ΔͨΊͷඞཁੑ͸ɺ௥͍ӽ͠ͷΈ (ࣜ (4)) Ͱ͋Δɻ͔͠͠ɺ௥ӽंઢ͔ઢͷઌߦंͷ଎౓Λ vp ɺ஫໨͍ͯ͠Δंͱྡ઀ंઢ

Β૸ߦंઢ΁ͷҠಈ͢ΔͨΊͷඞཁੑ͸ɺࣜ (5) Ͱͷઌߦंͱͷ‫཭ڑ‬Λ ∆xnp ɺྡ઀ंઢͷઌߦंͷ଎ද͞ΕΔ௥ӽंઢΛ૸Γଓ͚ͳ͍ͱ͍͏‫ڞ‬௨෦෼ʹ౓Λ vnp ɺ஫໨͍ͯ͠Δंͱྡ઀ंઢͰ‫ํޙ‬Λ૸ߦՃ͑ͯɺ૸ߦंઢଆ͔Β௥ӽंઢΛ૸Δ௿଎ंΛ௥͍ͯ͠Δंͱͷ‫཭ڑ‬Λ ∆xnb ͱ͢Δɻंઢมߋ‫ن‬ଇͷඞཁੑʹؔ͢Δ‫ڞ‬௨෦෼Λߟ͑Δɻ ͍ӽ͢৚͕݅෇Ճ͞ΕΔ (ࣜ (7))ɻ

૸ߦंઢ͔Β௥ӽंઢ΁ͷҠಈ͕ඞཁͱͳΔͷ͸ɺಉҰंઢͷઌߦंͷ଎౓͕ࣗ਎ΑΓ஗͘ɺ͔ͭྡ઀ंઢͷઌߦं͕ಉҰंઢͷઌߦंΑΓ଎͍৔߹Ͱ͋Δ:vp ≤ v ͔ͭ vp < vnpvnp > v ·ͨ͸ (vp ≤ v ͔ͭ vp < vnp )(7)͜ͷंઢมߋ‫ن‬ଇΛ‫ʹج‬ɺ૸ߦंઢͷ࠷ߴ଎౓
sfvmaxΛ 5 ͱ͠ɺ௥ӽंઢͷ࠷ߴ଎౓ vmax
Λύϥ

(4)

ϝʔλͱͯ͠มԽͤͨ͞ɻ·ͨɺલઅͰड़΂ͨࢹքd ͷ஋Λม͑ͯγϛϡϨʔγϣϯΛߦͬͨɻҰํɺ௥ӽंઢ͔Β૸ߦंઢ΁ͷҠಈ͕ඞཁͱͳΔਤ 3 ͸ɺ೔ຊ‫ܕ‬ͷंઢมߋ‫ن‬ଇʹ͓͚Δɺ૯ྲྀྔͷ͸ɺྡ઀ंઢͷઌߦं͕ࣗ਎ΑΓ଎͍৔߹Ͱ͋Δ:

ʹର͢Δ૸ߦंઢͷंઢར༻཰ͷ݁ՌͰ͋Δɻਤ 3vnp > v(5)͸ɺ֤ंઢͷ࠷ߴ଎౓͕౳͍͠৔߹ͱҟͳΔ৔߹ͷͳ͓ɺઌߦं·Ͱͷ‫ࢹ͕཭ڑ‬ք d ΑΓେ͖͍৔߹ɺ
ઌߦंͷ଎౓Λແ‫ݶ‬େͱ൑அ͢Δɻsfస͸ൃੜ͠ͳ͍ɻ͔͠͠ɺvmax< vmaxͷ৔߹Ͱ͸ɺྲྀྔ‫ٯ‬స͕ൃੜͨ͠ɻΑͬͯɺྲྀྔ‫ٯ‬స͸ंઢͷ࠷҆શͳंઢมߋͷҝʹຬͨ͞ΕΔ΂͖৚݅͸∗∆xnp > v ͔ͭ ∆xnb > vmaxsf
ൺֱ݁ՌͰ͋Δɻ vmax
= vmaxͷ৔߹Ͱ͸ɺྲྀྔ‫ٯ‬ߴ଎౓ͷࠩʹΑͬͯൃੜ͍ͯ͠Δ͜ͱ͕Θ͔Δɻ

(6)

3

f
vmax =5f
vmax =6

flow ratio

flow ratio

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00

0.2

0.4

0.6

0.8

1

1.2

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

with rule
without rule00.2

total flow [cars per time step]ਤ 3: ೔ຊ‫ܕ‬ͷंઢมߋ‫ن‬ଇʹ͓͚Δɺ֤ंઢͷ࠷ߴ଎౓͕౳͍͠৔߹ͱҟͳΔ৔߹ͷൺֱ݁ՌͰ͋
Δɻ֤ंઢͷ࠷ߴ଎౓͕ҟͳΔ৔߹ʹྲྀྔ‫ٯ‬స͕
ൃੜͨ͠ɻ

5

0.4

0.6

0.8

1

1.2

total flow [cars per time step]ਤ 4: υΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇʹ͓͚Δɺ૸ߦंઢ͔Βͷ௥͍ӽ͠཈੍‫ن‬ଇͷద༻༗ແͷൺֱ݁ՌͰ͋Δɻ཈੍‫ن‬ଇ͕͋Δ৔߹͸ྲྀྔ‫ٯ‬స͕ൃੜͨ͠ɻ6υΠπ‫ܕ‬ͷೋंઢ࿏Ϟσϧ·ͱΊຊߘͰ͸ɺNagel-Schreckenberg ϞσϧΛ༻͍ͯɺ࣍ʹɺυΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇͱྲྀྔ‫ٯ‬సͷؔ܎೔ຊ‫ͱܕ‬υΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇΛ࣋ͭೋंઢ࿏ͷΛௐ΂ΔɻυΠπ‫Ͱܕ‬͸ɺ௥ӽंઢ্ͷ஗͍ंʹରNS ϞσϧΛߏ੒͠ɺྲྀྔ‫ٯ‬సʹ஫໨ͯ݁͠Ռͷൺ͢Δ (૸ߦंઢ͔Βͷ) ௥͍ӽ͠Λ཈੍͢Δ‫ن‬ଇֱ͕͋ΛߦͬͨɻΔɻͭ·Γɺ௥ӽंઢΛઌߦ͢Δ஗͍ं͕͍Δ৔߹ɺ೔ຊ‫ܕ‬ͷंઢมߋ‫ن‬ଇͷ৔߹͸ɺྲྀྔ‫ٯ‬స͸ंઢ૸ߦंઢΛ૸ߦ͢Δं͸௥ӽंઢ΁Ҡಈ͢Δɻ͜ͷͷ࠷ߴ଎౓ͷࠩʹΑͬͯൃੜ͍ͯ͠Δ͜ͱ͕Θ͔ͬ͜ͱΑΓɺ૸ߦंઢʹۭ͖Λ࡞Γɺ௥ӽंઢͷ௿଎ͨɻυΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇͷ৔߹͸ɺ૸ߦंઢ͔ंΛ҆શʹ૸ߦंઢ΁໭͢ɻΒͷ௥͍ӽ͠཈੍‫ن‬ଇ͕ྲྀྔ‫ٯ‬సʹӨ‫͍ͯ͠ڹ‬Δ͜υΠπ‫ܕ‬ͷंઢมߋ‫ن‬ଇͰ͸ɺલड़ʢද 1ʣͷΑ͏ͱ͕Θ͔ͬͨɻंઢมߋͷඇܾఆੑ΍ύϥϝʔλͷʹɺ૸ߦंઢ͔Β௥ӽंઢ΁Ҡಈ͢ΔͨΊͷඞཁੑӨ‫͍ͯͭʹڹ‬͸ɺߨԋʹͯٞ࿦͢Δɻ͸ɺࣜ (4) Ͱද͞ΕΔ௥͍ӽ͠ͷ৚݅ʹՃ͑ͯɺ૸

ࢀߟจ‫ݙ‬ߦंઢ͔Βͷ௥͍ӽ͠཈੍‫ن‬ଇ͕௥Ճ͞ΕΔ:(vp ≤ v ͔ͭ vp < vnp ) ·ͨ͸ vnp ≤ v[1] K. Nagel, D. E. Wolf, P. Wagner, and P. Si-(8)mon, Phys. Rev. E58 (1998) 1425.

௥ӽंઢ͔Β૸ߦंઢ΁ͷҠಈ͢ΔͨΊͷඞཁੑ৚[2] S. Tadaki, K. Nishinari, M. Kikuchi, Y.

݅΋·ͨɺ௥ӽंઢΛ૸Γଓ͚ͳ͍ͱ͍͏‫ڞ‬௨෦෼Sugiyama and S. Yukawa, J. Phys. Soc. Japan(ࣜ (5)) ʹՃ͑ͯɺ૸ߦंઢ͔Βͷ௥͍ӽ͠཈੍‫ن‬

71 (2002) 2326.

ଇ͕෇Ճ͞ΕΔ:[3] S.. Tadaki, K. Nishinari, M. Kikuchi, Y.
vnp > v ͔ͭ vp > v(9)Sugiyama and S. Yukawa, Physica A315(2002) 156.

૸ߦंઢ͔Βͷ௥͍ӽ͠཈੍‫ن‬ଇ͕ྲྀྔ‫ٯ‬సʹ‫ٴ‬[4] S. Tadaki et al. Traﬃc and Granular Flow ’01΅͢Ө‫ڹ‬Λ֬ೝ͢ΔͨΊɺ͜ͷ‫ن‬ଇ͕ͳ͍৔߹ͱͷsൺֱΛߦͬͨɻ૸ߦंઢͷ࠷ߴ଎౓ vmax
ͱ௥ӽंfઢͷ࠷ߴ଎౓ vmaxͷ஋͸ɺಉ͡஋ 5 ʹઃఆͨ͠ɻ

(Springer, 2003) 211.
[5] K. Nagel and M. Schreckenberg, J. Phys. Iਤ 4 ͸ɺ૸ߦंઢ͔Βͷ௥͍ӽ͠཈੍‫ن‬ଇͷద༻

France, 2 (1992) 2221.

༗ແͷൺֱ݁ՌͰ͋Δɻਤ 4 ΑΓɺ཈੍‫ن‬ଇ͕ద༻͞Ε͍ͯΔ৔߹͸ྲྀྔ‫ٯ‬స͕ൃੜͨ͠ɻͦ͏Ͱͳ͍

[6] ಓ࿏ަ௨๏ʢ1960 ೥ 6 ݄ 25 ೔๏཯ୈඦ‫߸ޒ‬ʣ,৔߹͸ྲྀྔ‫ٯ‬స͕ൃੜ͠ͳ͔ͬͨɻΑͬͯɺ͜ͷ‫ن‬

http://www.japaneselawtranslation.go.jpଇ͕ྲྀྔ‫ٯ‬సʹ‫د‬༩͍ͯ͠Δ͜ͱ͕Θ͔Δɻ

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2017 ೥ 11 ݄ 2 ೔֬ೝ4