Ole Steuernagel
and Maria Schilstra
UH
University of
Hertfordshire
Hatfield
1
1
UH
1
UH
Heavy Traffic
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Heavy traffic – solutions?
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Traffic flow modelled by point particles
10 units Vmax
5 units Vmax
No other distinguishing features:
same size, same acceleration, same behaviour…
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Cellular Automaton
–
Evolution Rules
p
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Heavy Traffic – Modelling with
Cellular automata
A la Nagel and Schreckenberg
http://www.traffic.uni-duisburg.de/model/index.html
Heavy Traffic – Modelling with
Cellular automata
A la Nagel and Schreckenberg
Traffic – Modelling
J H    (max  p)
= (remaining free road) (drive-off probability)
Acceleration Matrix
 P0 
 
 P1 
  P2 
 
 P3 
P 
 4
 P0 
 
 P1 
 A   P2  
 
 P3 
P 
 4
A
1

1
0

0
0

 P0 
 P0 (t  1) 
0





 P1 
 P1 (t  1) 
1
 P (t  1)   (1  A)   P    0
 2

 2

 P3 (t  1) 
 P3 
0
 P (t  1) 
P 
0
 4


 4
0   P0 
  
 1 0 0 0   P1 
1  1 0 0    P2 
  
0 1  1 0   P3 
P 

0 0 1 0  4 
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0   P0 (t ) 

 
0   P1 (t ) 
0    P2 (t ) 

 
0   P3 (t ) 
 P (t ) 

1  4 
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Randomization Matrix
 P0 
 
 P1 
  P2 
 
 P3 
P 
 4
R
 P0 
 P0 
0 p 0 0 0 
 
 


 P1 
 P1 
0  p p 0 0 
 R   P2    0 0  p p 0    P2 
 
 


 P3 
 P3 
0 0 0  p p 
P 
P 
0 0 0 0  p
 4
 4


 P0 (t  1) 


 P1 (t  1) 
 P (t  1)   (1  R)
 2

 P3 (t  1) 
 P (t  1) 
 4

0
0
0   P0 (t ) 
 P0 
1 p

 

 
0
0   P1 (t ) 
 P1 
0 1 p p
  P2    0 0 1  p p
0    P2 (t ) 

 

 
0 1  p p   P3 (t ) 
 P3 
0 0
 P (t ) 
P 
0 0

0
0 1 p  4 

 4
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Slow-down Matrix
S



   P0 
 P0 
 P0   0
 
  
  
d
 d   P1 
 P1 
 P1   0 d  1  d
  P2   S   P2    0
0
d 2  1  d 2  d 2    P2 
 
  



3
3
P
P
0
0
d  1  d   P3 
 3
 3 0
P 
P  
P 
4

0
0
0
d  1  4 
 4
 4 0
 P0 (t  1) 
 P0 (t ) 




 P1 (t  1) 
 P1 (t ) 
 P (t  1)   (1  S )   P (t ) 
 2

 2 
 P3 (t  1) 
 P3 (t ) 
 P (t  1) 
 P (t ) 
 4

 4 
where d  1  
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Joint Transformation Matrix
T
 P0 (t ) 
 P0 (t ) 
 P0 (t  1) 






 P1 (t ) 
 P1 (t ) 
 P1 (t  1) 
 P (t  1)   (1  R )  (1  S)  (1  A )   P (t )   T   P (t ) 
 2 
 2 
 2

 P3 (t ) 
 P3 (t ) 
 P3 (t  1) 
 P (t ) 
 P (t ) 
 P (t  1) 
 4 
 4 
 4

 P0 
 P0 
 
 
 P1 
 P1  T contractive
 P0 (0)   P0 

  
Steady state :  P2   T   P2 
 
 
 P1 (0)   P1 
 P3 
 P3 
N
 P ( 0)    P 
T

lim
2
2
P 
P 




N 
 4  SS
 4  SS
 P3 (0) 
 P ( 0) 
 4 
 P3 
P 
 4  SS
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Steady state of Joint Transformation
T  (1  R )  (1  S)  (1  A)
J
simulation
master equation
U H
Steady state of Joint Transformation
T  (1  R )  (1  S)  (1  A)

simulation
master equation

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Slow-down due to other vehicles
Follower
Leader
UH
Slow-down due to other vehicles
Follower
Leader
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Slow Down Matrix
S(P)
0
0
0
 00  0 

 0 
  P0 

0
0 
0
0   S n ,1  1  (1  S g , 2 )  1  (1  S g ,3 )  1  (1  S g , 4 )   
g 0 d
 0 n0d  1 g 0 1  d g 10  d
  P1 
1
0
0
  S n2, 2  2  (1  S g ,32)  2  (1  S2g , 4 )   
0 nd0  1 g 0 2 d
g
0 d
0
 P2
2
 



3S
3 )
0
0
0


(
1

S


n
,
3
3
 0
0
0
dn0  1 g0 d g ,4   P3 
3




4

0 0
00
  Sn , 41   P4 
 00 0 0
d


n 0
Leader
Follower
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Slow Down Matrix
*
0

0

0


0

0
g 
*
*
S(P)
*
P3
*
0
*
0
*
*
 1  (1  S g , 3 )
g 0
1
0
*
 2  (1  S g , 3 )
g 0
2
0
g
0
0P0

l
  S n,3
n 0
0
*

*

*


*
*

l 0
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Slow Down Matrix
P
0
P1
P2
S(P) nonlinear in P
S(P)
P3
P4 
0
0
0
0
0



0
0
0
0
 0   S n ,1  1  (1  S g , 2 )  1  (1  S g , 3 )  1  (1  S g , 4 ) 
n 0
g 0
g 0
g 0


1
1
1
0
0
  Sn,2
 2  (1  S g ,3 )  2  (1  S g , 4 ) 
n 0
g 0
g 0


2
2


0
0
  S n,3
 3  (1  S g , 4 ) 
0
n 0
g 0
3


0
0
0
  Sn,4 
0


n 0
UH
Steady state of Joint Transformation
T  (1  R )  (1  S)  (1  A)
J
simulation
modified master
equation
master equation

Steady state of Joint Transformation
T  (1  R )  (1  S)  (1  A)

simulation
modified master
equation
master equation

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Fundamental Diagrams
J
p=0.1
p=0.5
p=0.9

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OUTLOOK:
• Insert real world numbers
• Study effects of
length
acceleration
lane bias
noise
structure formation
• Related fields?
network traffic
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OUTLOOK ON NOISE
Maria Schilstra’S recent simulations…
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