1 ole steuernagel and maria schilstra university of hertfordshire hatfield
TRANSCRIPT
1
Ole Steuernageland Maria Schilstra
University ofHertfordshireHatfield
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• Cellular Automaton ModellingCellular Automaton Modelling
• Typical behaviourTypical behaviour• Some semi-analytical resultsSome semi-analytical results
• Deriving the master equationDeriving the master equation
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Heavy Traffic
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Heavy traffic – solutions?
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No other distinguishing features:
same size, same acceleration, same behaviour…
Traffic flow modelled by point particles
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10 units Vmax
5 units Vmax
No other distinguishing features:
same size, same acceleration, same behaviour…
Cellular Automaton – Evolution Rules
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p
Heavy Traffic – Modelling with
Cellular automata
A la Nagel and Schreckenberg
http://www.traffic.uni-duisburg.de/model/index.html
Nagel.JP2.92.pdf.lnk
simulation.html
Heavy Traffic – Modelling with
Cellular automata
A la Nagel and Schreckenberg
Traffic – Modelling
)( max pJ H
= (remaining free road) (drive-off probability)
Acceleration Matrix A
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4
3
2
1
0
P
P
P
P
P
01000
01100
00110
00011
00001
4
3
2
1
0
P
P
P
P
P
4
3
2
1
0
P
P
P
P
P
A
)(
)(
)(
)(
)(
4
3
2
1
0
tP
tP
tP
tP
tP
11000
00100
00010
00001
00000
)1(
)1(
)1(
)1(
)1(
4
3
2
1
0
tP
tP
tP
tP
tP
4
3
2
1
0
)(
P
P
P
P
P
A1
Randomization Matrix R
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4
3
2
1
0
P
P
P
P
P
p
pp
pp
pp
p
0000
000
000
000
0000
4
3
2
1
0
P
P
P
P
P
4
3
2
1
0
P
P
P
P
P
R
)(
)(
)(
)(
)(
4
3
2
1
0
tP
tP
tP
tP
tP
p
pp
pp
pp
p
10000
1000
0100
0010
0001
)1(
)1(
)1(
)1(
)1(
4
3
2
1
0
tP
tP
tP
tP
tP
4
3
2
1
0
)(
P
P
P
P
P
R1
Slow-down Matrix S
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4
3
2
1
0
P
P
P
P
P
10000
1000
100
10
0
4
33
222
d
dd
ddd
dddd
4
3
2
1
0
P
P
P
P
P
4
3
2
1
0
P
P
P
P
P
S
)(
)(
)(
)(
)(
)(
4
3
2
1
0
tP
tP
tP
tP
tP
S1
)1(
)1(
)1(
)1(
)1(
4
3
2
1
0
tP
tP
tP
tP
tP 1 where d
Joint Transformation Matrix T
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)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)()()(
4
3
2
1
0
4
3
2
1
0
tP
tP
tP
tP
tP
tP
tP
tP
tP
tP
TA1S1R1
)1(
)1(
)1(
)1(
)1(
4
3
2
1
0
tP
tP
tP
tP
tP
SSSSP
P
P
P
P
P
P
P
P
P
4
3
2
1
0
4
3
2
1
0
:stateSteady T
SS
N
N
P
P
P
P
P
P
P
P
P
P
4
3
2
1
0
4
3
2
1
0
)0(
)0(
)0(
)0(
)0(
lim T
T contractive
Steady state of Joint Transformation
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)()()( A1S1R1T J
simulation
master equation
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Steady state of Joint Transformation )()()( A1S1R1T
simulation
master equation
Slow-down due to other vehicles
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Follower Leader
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Follower Leader
Slow-down due to other vehicles
10000
1000
100
10
0
4
33
222
d
dd
ddd
dddd
Slow Down Matrix S(P)
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4
3
2
1
0
P
P
P
P
P
FollowerLeader
3
04,
2
04,3
2
03,
1
04,2
1
03,2
1
02,
0
04,1
0
03,1
0
02,1
0
01,
0000
0000
)1(000
)1()1(00
)1()1()1(0
0
nn
gg
nn
gg
gg
nn
gg
gg
gg
nn
S
SS
SSS
SSSS
Leader
*0000
*000
*)1(*00
*)1(**0
***0
2
03,
1
03,2
0
03,1
0
nn
gg
gg
S
S
S
Slow Down Matrix S(P)
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**** 3P
g
llg P
0
3
04,
2
04,3
2
03,
1
04,2
1
03,2
1
02,
0
04,1
0
03,1
0
02,1
0
01,
0000
0000
)1(000
)1()1(00
)1()1()1(0
0
nn
gg
nn
gg
gg
nn
gg
gg
gg
nn
S
SS
SSS
SSSS
Slow Down Matrix S(P)
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43210 PPPPP
S(P) nonlinear in P
Steady state of Joint Transformation )()()( A1S1R1T
J
modified master equation
simulation
master equation
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Steady state of Joint Transformation )()()( A1S1R1T
modified master equation
simulation
master equation
Fundamental Diagrams
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J
p=0.1
p=0.5
p=0.9
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OUTLOOK:
• Insert real world numbers• Study effects of
lengthaccelerationlane biasnoisestructure formation
• Related fields?network traffic
OUTLOOK:
• Insert real world numbers• Study effects of
lengthaccelerationlane biasnoisestructure formation
• Related fields?network traffic
OUTLOOK ON NOISEMaria Schilstra’S recent simulations…
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