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Through straightforward deduction procedure, we explicitly show analytical solutions for both Fukui-Ishibashi (FI) model and Quick-Start (QS) model, which are fundamental deterministic Cellular Automaton (CA), applied to traffic flow.

For recent years, simulation study on traffic flow has been attracted much attention of physicists. Among wide variety of approaches, Cellular Automaton (CA), where vehicles are treated as discrete self-drive particles in an entirely discrete spatiotemporal system, are most heavily used because of its flexibility as well as robustness to apply various practical problems.

There have been proposed many traffic CA models so far. For example, Rule-184 [

applied as a fundamental template model by many studies, considers random braking effect on the basis of FI model. Quick-start (QS) model [

By previous works, analytical solutions of Rule-184, ASEP and ZRP have been derived, since flux—density relation can be fixed deterministically [

The update rule of FI model is as follows;

where

and

flux maximum, is

Let us prove that the fundamental diagrams by FI model can be described as an asymmetrical tent-type poly- gonal line functions as below.

Proposition 1

When

with slope.

When

of flow state when

and it is also trivial;

where L is system length. Hence,

Therefore, flux

which explicitly implies a linear function of

complete jam state;

(QED)

According to Proposition 1, the fundamental diagram can be described as an asymmetric tent-type function, consisting of two liner functions. The above-deduced critical density and maximum flux is the vertex of this asymmetric tent-type function.

As the next step, we should discuss whether the function consisting of two linear lines can be expressed by a single expression

Proposition 2

What we expect is that the function consisting of two linear lines can be expressed by a single expression;

(

Meanwhile, two equations to touch at_{ }, shown in

as below;

Flow states explaining for FI model

Flow states explaining for FI model in case if

Asymmetric tent-type function

The expression of

By substituting Equations (1) & (2), we obtain:

When the branch of square root is taken into consideration, it is as follows;

(QED)

We know;

By substituting above conditions into Equations (4) & (5), we get;

By substituting these into Equation (2), we obtain the following.

As consequence,

Namely; we obtain;

By rearranging, analytical solution of FI model can be derived as follows;

where

The update rule of QS model is as follows;

If there is an empty site in forward S-sites then the focal vehicle moves.

Where S means the number of sites that a vehicle foresees for quick-start. It is obvious from

critical density is

model, Let us prove that the fundamental diagrams by QS model can be described as an asymmetrical tent-type poly-gonal line functions as below.

Proposition 3

When

with slope._{}

When

flow state when

Observing

and;

Therefore, flux

which explicitly implies a linear function of

The fundamental diagram by Equation (9)

Flow states explaining for QS

Flow states explaining for QS model l in case if

complete jam state;

(QED)

Proposition 3 enables us to draw that the fundamental diagram can be described as an asymmetric tent-type function, consisting of two liner functions. The above-deduced critical density and maximum flux is the vertex of this asymmetric tent-type function.

Like the case of FI model, as the next step for the discussion, we should note how Proposition 2 leads that the function consisting of two linear lines can be expressed by a single expression

Namely, we know;

By substituting above conditions into Equations (4) & (5), we get;

By substituting these into Equation (2), we obtain the following.

Therefore,

The fundamental diagram by Equation (13)

Finally, we get;

By rearranging, analytical solution of QS model is derived as follows;

We explicitly reported analytical solutions fort FI model and QS model.

Although the two are important models for depicting basic traffic features, analytical solutions for those two have not been known ever.

Helped by the fact that FI model and QS model never contain stochastic elements, our process to deduce was simple and straightforward.

This study was partially supported by a Grant-in-Aid for Scientific Research by the Japan Society for the Pro- motion of Science, awarded to Prof. Tanimoto (#25560165), Tateishi Science and Technology Foundation. We would like to express our gratitude to these funding sources.