Traffic modeling on the road of the fly over branch using totally asymmetric exclusion process a macroscopic approach

This research examines a dynamic model, namely the Totally Asymmetric Exclusion Process (TASEP). In addition, it discusses the boundary conditions and dynamic rules used in this modeling. TASEP has been applied in various fields, one of which is the traffic flow modeling. This system is modified into a unidirectional fly-over fork. Furthermore, a vehicle which travels through a road segment is modeled as a particle which jumps from one lattice to another. The continuity term which is to describe a particle dynamic in TASEP is solved by using the level set method. The profile of particle density and current density is affected by input rate and output rate of the particle jump.


Introduction
Transportation is an important means of supporting the development of a city. Therefore the need for transportation routes is increasing. However, it needs to be realized that this situation creates new problems surrounding the traffic density of roads. For example the city of Jakarta which has traffic problems (congestion). Currently, traffic flow research from a macroscopic perspective is being discussed. Basically, a macroscopic review looks at traffic globally. This research offers theoretical analysis and an extension of a mathematical model that can describe the dynamics of vehicle traffic on one-way roads as an effort to find solutions to the problems of traffic congestion and congestion. This mathematical model is known as a totally simple asymmetric exclusion process or better known as TASEP. TASEP is a non-equilibrium model in which particles with hard core interactions can jump left or right (not both) in a grid (path). TASEP is specified by the presence of dynamic rules and boundary conditions. From the 'traffic' of these particles, it can be obtained the average value of the position of the particles in occupying a grid or density, as well as the current density of the particles. Partial differential equations can be considered as an interface so that it can be worked with the level set method. The level set method is especially useful in calculating multivalued solutions. In their article developed a level set method for calculating the solution of a nonlinear equation of the order of one general form in any dimension of space. More specifically developed a level set method for calculating the velocity and multivalued electric fields for the 1-dimensional (1D) Euler-Poisson equation [1].

Fluid Dynamics (Macroscopic)
Dynamic, which means it can change depending on time. Macroscopic dynamic modeling describes the changes in traffic over time and space using differential equations. Traffic flow is often analogized as liquid or gas. Analytical approaches can still be used when observing a road section. But when the temporal and spatial interactions of traffic flows on the road network need to be evaluated, then the simulation method would be more appropriate [2]. The general term for a traffic flow model simulation is macroscopic simulation. Their use has grown widely, and has been facilitated by the extensive development of traffic measurement systems that have been installed in major cities and highways. An additional factor that helps gain the macroscopic

Traffic Flow Theory
This section would explain the flow of vehicles in the macroscopic model approach. The relationship between density, speed and flow can be analogous to road density flow. Based on these parameters, a traffic flow conservation law would be obtained which would be derived into a scalar function of a macroscopic model [3].

Notations and Definitions
Some of the notations that would be used in this paper are as follows.
• : Maximum traffic density . Based on data on the average length of cars, we set the value of  max to be 0.25 vehicles / meter [2]. B.

Theory Fundamental
There are three important factors if we discuss traffic flow macroscopically, namely vehicle flux, density, and velocity here, flux depends only on 2 independent variables, namely density and velocity. Assume a stable state (flux does not change along the way during the observation) and all vehicle speeds are considered constant. Therefore, the flux function can be rendered simpler because the dependence on distance, time, and interval is no longer valid in the stable state. The observations made by [4] show that one vehicle with another has a different tendency but approaches at a point and depends on the existing density.  Figure 1. Fundamental diagram. Combining all possible currents available to the equilibrium function, therefore the relationship between flux, density k, and velocity can be presented in a graphical form known as a fundamental diagram. Look at Figure 1. Some points that need to be considered in the diagram include:

Free flow of vehicles
When the road conditions are in a free state, the vehicle can increase its speed until it reaches the maximum. In this condition the density would be close to zero.

Jam Point
In this condition the vehicle can no longer accelerate, and the density increases until it reaches the maximum where the vehicle is stopped.

Figure 2. Illustration of Traffic Flow
Suppose that on a road there is a traffic flow moving at a constant speed ( 0 ), and has a density( 0 ), such that the distance between the vehicles is considered the same as shown in Figure 2   Assume that the number of vehicles passing through 0 a very small point in time ( )has not changed significantly, therefore ( , ) and ( , ) can be obtained by = 0 and = 0 approach. The number of vehicles passing the observer over a short distance, and can be estimated to be equivalent to ( , ), ( , ), , where the vehicle flux is given by (1.1)

Conservation Laws
Traffic modeling, whether consisting of a single equation or a system of equations, is all based on the laws of conservation of physics. It can be said to be constant if the number of particles does not change during the process. By applying this to a mathematical form, the pattern of density and velocity over the next several times would be possible to predict. In this case, the number of vehicles in road units is  Where is the free flow velocity and is the maximum density. For zero density, the model represents current velocity , so that when the density reaches its maximum no vehicles enter or exit 2.6 Model "exits and entrances" Assume no cars enter or leave except at the end of the road. This section would discuss the development of a traffic model that would consider incoming and outgoing cars in road segments, as illustrated in Figure 3.

Lighthill Whitham Rihards (LWR)
This section would show four different models of traffic flow in one dimension. The first model is a model with one equation, while the rest is a system with a two equation model. All models would be Described by partial differential equations and focused on mass conservation, which seek to capture the dynamic interactions found in traffic flow movements. In addition, this model offers a particular means of achieving velocity and density. As a consequence, the flow is not in harmony as a formula of a single equation. This section is the key distinction between scalar systems and model systems. In the first model used in explaining problems in vehicle flow, it is known as the LWR model. The LWR model is a scalar model, which is time dependent, non-linear, and includes hyperbolic differential equations. In this model, the density currents are kept in number, so that they are obtained Where ( ( , )) is the speed function given by (1.10). In the LWR model, speed depends only on density. As a result, the current is balanced when the velocitydensity function is used. When the traffic density is small enough, the vehicle speed would be relatively constant, so the model does not show any speed distribution for each vehicle. Therefore, this model cannot explain the properties to be observed in currents that are not so dense. However, the average velocity would probably solve this problem. On this side, the model is anisotropic as the principal of the observed vehicle flow, for example, the nature of the vehicle is largely influenced by the vehicle in front of it. In this case it can be found in the eigenvalue given by For the main line in the road segment and added roads in and out fork in the road segment. Each of the forked roads would be examined for 5 km for in and 5 km for out. This modeling would limit the vehicles traveling on the road segment and divided by three speeds and is illustrated in Figure 4.

Analysis of Transport Waves Using Level Set Method
In the first spot present the density profile figures with the comparative original and 6 minute location of cars with respect to some 10 km highway points. For Figure 4 (a) the maximum velocity is = 30 /ℎ , while for Figure 4 (b) = 60 /ℎ ,and = 90 /ℎ for Figure 4 (c). In each case, we find that the density of cars increases at the 5 ℎ km point and decreases at the 8 ℎ km point of our 10 highway. This particular condition occurs due to the influence of continuous inflow and outflow at 5 ℎ km and 8 ℎ km respectively. Although the overall velocity is 3 (a) It is = 30 /ℎ , which is very low so that the density of cars at the inflow position exceeds our fixed limit density of = 550 / , i.e. = > ,This state brings us to a congested traffic situation at the inflow spot. So in this situation, we should consider the inflow to be a local interruption in the flow of traffic. In Figure 4 (b) and Figure 4 (c), the median velocity of cars is equal to that of the case discussed in Figure  4 (a). In each of the following cases, the car density increases considerably at the inflow position but does not surpass the jam density, i.e. the density at the inflow position < . In each of these instances, however the inflow source word has no meaningful impact. Cars are free to drive at their optimal speed. We have also added the sink word at the 8 ℎ km point where some of the vehicles will exit our 10 highway. As a result, the vehicle density steadily decreases at the outflow position. As a result, drivers are able to maintain their own safe driving pace at the outflow position According to the wave diagram, it would be compared with the special case to the visual on the incoming and outgoing waves of vehicles on the road segment. In each case give a simulation of 6 minutes. In this case each simulation is given a different speed to see the results of the traffic waves that would be studied. Henceforth, it would be simulated at each current density, average speed, and traffic waves on the road segment that has crossed and would illustrate the solution diagram for the unidirectional forked road segment.  Figure 5 (a) and Figure 5 (b) display density profiles and velocity profiles at three different periods. Figure 5 (a) shows that the density of traffic at inflow location steadily increases the decreased wave height as time passes. But in the event of an outflow located at the 8th km location, the density of cars increases as time goes by. This particular situation is due to the continuous increase in the flow rate on the single-lane highway, but the number of cars on the single-lane highway. Leaving through the sink is constant. In Figure 5 (b), the average velocity of the cars decreases at a high density at the inflow source position at the 5th km and vice versa at the outflow position. Figure 5 (c) displays the flow profile over a period of minutes. a b c Figure 5. a) density profile on road segments, b) speed profiles of vehicles on road segments, c) transport wave profiles on road segments.