Cellular Automaton and traffic Flow 1
Now that we have all the necessary theoretical tools, we proceed to realize a model traffic on computers. There is however one more step to achieve that end. That is the discretization of continuous quantities.
As is well known, the mainsteam of modern computers is of digital sorts, and internally, only discrete numberes "1, 2, 3 ..." are processed. The continuous analog numbers could be treated at user programming level by software, but there would be considerable performace degradation.
This naturally suggest us to build a model of traffic flow, from the outset, utilizing discrete numbers only. The continuous quantities such as the position and velocity of a car is to be approximated by discontinuous integers. This is exactly what is called discretization.
In dicretized model, the motion of cars on a ring road is described in the following manner:
|
* A road is treated as a one dimensional array of "cells" which is numbered sequentially, c = 0, 1, 2, ... , L.
* The cells at both edges c = 0 and c = L are treated as the same object.
* M cars are all placed at one of L cells.
* The motion of cars are considered as the change of cells they reside at distretized time step t = 0, 1, 2, 3, ..., Tx.
|
(3.1)
|
For now, we only consider a single-lane traffic on which there is no passing. For this we add:
|
* At any given time step, only one car occupies a cell, and there is no spatial reordering of cars.
|
(3.2)
|
This assumption (3.2) shall be later abondoned for the treatment of two-lane traffic.
We sequentially number cars in spatial order toward the direction of the movement of cars which we assume to be unidirectional. Indexing the cars with i = 1, 2, ..., M, we describe the traffic flow as follows:
|
* Let the array element X[i] denote the position, i.e., the cell number occupied by the i-th car.
* Let the array element V[i] denote the velocity of i-th car.
* The dynamics of the traffic flow is in deciding V[i]s from X[i]s.
* The temporal evolution at a time step is described by the replacement, X[i] to X[i]+V[i]
|
(3.3)
|
This is the framework of discretized models of traffic flow.
|
Go to
Research Page