_{1}

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Dynamics is studied for one-dimensional single-lane traffic flow by means of an extended optimal-velocity model with continuously varied bottleneck strength for nonlinear roads. Two phases exist in this model such as free flow and wide moving jam states in the systems having relatively small values of the bottleneck strength parameter. In addition to the two phases, locally congested phaseappears as the strength becomes prominent. Jam formation occurs with the similar mechanism to the boomerang effect as well as the pinch one in it. Wide scattering of the flow-density relation in fundamental diagram is found in the congested phase.

Study of phase transition behavior in vehicular dynamics is one of the interesting themes in traffic systems. Kerner has proposed the three-phase traffic theory in describing the nature of the dynamics [

Recently, critical discussions to the theory have been expressed by Schönhof and Helbing [

As shown in the above place, Schönhof and Helbing questioned the three-phase traffic theory [

There have been various models proposed in simulating traffic dynamics in highways. The optimal-velocity (OV) model, proposed by Bando et al. [

We find complex geometry in actual highways, for instance, such as horizontal curves, uphill gradients, downhill slopes, ramps, and other structures. Drivers should incessantly adjust their speed in responding to road shape. There has been negative correlation between actual operating speed of vehicles and road curvature at each measuring point on highways [

We introduce the effect of bottleneck strength into the OV model. First, in the original OV model, the equation of motion for each vehicle is given as [

where

Recently it was applied to the system with the effects of gravitational force on road or highway tollgates [

Let us imagine vehicle movements, for instance, on horizontally curved roads. We have introduced the function

where

when road shape is continuously periodic like a sine curve, the road curvature is expressed by

where

where

We have simulated the time development of positions and velocities of vehicles with Equation (1). In this paper the function

As shown in

The different behavior has been found for larger values of the parameter

Fundamental diagram as the relation between flow and density of the system. Open triangles are for the system with, open circles for, filled triangles for, and filled circles for

the density region between

In summary of this section, we observed the existence of two states in the fundamental diagram of the systems for relatively small values as

In order to know dynamic features of vehicles, we have obtained the time development of positions of all vehicles constructed in the system. First we examined the case of

The other patterns have been found for different values of

We have examined the development at

Time developments of positions of all vehicles in the system. (a) for and; (b) for and; (c) for and; (d) for and; (e) for and; and (f) for and. Time range is between and for these figures

Let us examine the case for

From

As shown in Figures 2(a)-(f) there have been various congestion patterns in the present model. In this section we examine jam formation process in detail. In clarifying the jamming mechanism, it is effective to refer to the three dimensional diagram of time development of local density of the system [

We have recognized the formation of regularly arranged narrow jams as shown in

We have observed another pattern concerning narrow jam formation.

Kerner has suggested that the general pattern is the important concept as the jam formation mechanism on highways. The pattern is as follows [

Three dimensional projections of time developments of local density as a function of position. (a) for and, (b) for with, and (c) for with

into a high density state. Then local perturbation grows and it forms narrow jams. The self-compressed state is called as the pinch effect. The wide jams are formed by merging and dissolution of the jams.

We see the process of wide jam formation from the pinch region.

Wide scattering of flow-density relation with time has been a characteristic feature of the SF state [

Time developments of positions of vehicles in the system. The vehicles whose velocity is less than 1.1 are depicted. (a) for as and (b) for as. Time range is between and for these figures

In actual measurement, the q-d relation on a fixed position on highways was evaluated as one minute interval [_{p}.

In this paper we have found various patterns of congested states as shown in Figures 2(a)-(f). Schönhof and Helbing have classified the states into five patterns such as MLC, OCT, PLC, SGW, and HCT [

(a) Variation of local flow-densityrelation at the position for the system with and. Abscissa is for density and ordinate for flow at the position. Small dots are depicted with time interval. (b) Variation of time-averaged local flow-density relation at the same position. The values of and are and. Time average is taken over for both of density and flow. Succeeded points in time are connected with lines

serious accidents with lane closing [

Vehicular dynamics has been simulated for one-dimensional single-lane traffic flow by means of the optimal-velocity model with continuously varied bottleneck strength for nonlinear lanes. The strength is adjusted with the value of the parameter

Two phases exist in the systems for relatively small values of the parameter as

Locally defined flow-density relation without time average describes a round triangle shape with tapering vertices. This is almost similar to the shape schematically illustrated by Kerner as the area of the SF state in the fundamental diagram. The present model succeeds to reproduce the widely scattering behavior in the time averaged flow-density relation. This is identical with the fundamental diagrams actually observed in highways.