Standard Model of Nagel and Schreckenberg 1
Now we come back to the main subject of traffic flow. It is two german theoretical physisits Nagel and Schreckenberg who came up with a neat cellular automata model for traffic flow. They made following observation on the driver's behavior on a single-lane road
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1 : A driver increases the velocity of his car until it reaches the allowed speed limit.
2 : He adjusts his velocity to avoid the collision when he sees a preceeding car.
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(4.1a)
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They also tried to take into account the "accidental" elements in car driving
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3 : A driver ocasionally decelerates his car in seemingly random fasshion with various reasons unconnected to the existense of a preceeding car.
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(4.1b)
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The cause of this random breaking could be the curved road, narrowed road, spotting of unusual event at roadside, sppotting of police surveillance, music change in iPod, conversation with his date, sudden coughing, or just a purecaprice.
They modelled these behavior in the following rules for the discretized diescription of the car movement:
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* A car increases its velocity by one unit unless it exceeds the give speed limit U.
* If a car with current velocity is projected to cause collision with the preceeding car, it decreases its velocity down to the point of avoiding the collision.
* A car decreases its velocity by one unit at random ocasion with a given probability R.
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(4.2)
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The remaining task is to implement these rules on the temporal evolution of the arrays of position X[i] and of velocity V[i].
Additing the rule to change the positions X[i] after the determination of the velocitis V[i], we obtain the following four rules as the updating scheme ( X[i] \to X'[i] , V[i] \to V'[i] )
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1) V[i] \to V'[i] = min( U, V[i]+1 )
2) if ( X[i]+V[i] >= X[i+1] ) then V[i] \to V'[i] = X[i+1]-X[i]-1
3) V[i] \to V'[i] = max( 0, V[i]-1 ) with probability R
4) X[i] \to X'[i] = X[i]+V'[i]
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(4.3)
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Here, max(A,B) and min(A,B) respectively stand for smaller and larger one among A and B. If we apply the above rule for all the cars i = 1, 2, ..., M, the evelution of the system in one time step is described. This is the famed cellular automata traffic model of Nagel and Schreckenber.
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