Mathematical methods and models of traffic flow management

. An overview of existing mathematical models of traffic flows is presented. Various macroscopic and microscopic mathematical models of traffic flows are considered, which make it possible to calculate the congestion of road transport networks. The basic principles of their application and significant differences between them are analyzed. Based on the considered mathematical models, a model of rebuilding vehicles between traffic lanes is proposed.


Introduction
Currently, despite the development of modern technologies, simple problems remain unresolved, which we meet literally every day. One of these problems is the management of the motor transport system and road networks of settlements.
Transport is almost the main basis of the life of a modern city, concerning absolutely every inhabitant of it, it is enough just to list its functions: starting with communication with vital points such as hospitals, police and fire stations, as well as the delivery of food and necessary goods, and ending with a simple trip to work, study or a place of rest, which it should take as little time as possible. Without a well-established road system, the life of the city is not possible now.
To understand the complexity of the task of optimizing traffic flows itself, it is worth listing a number of problems that periodically arise in the road network: laying new roads and corresponding changes in the distribution of the transport load of adjacent sectors [1], timely repair and expansion of the pavement with temporary difficulties during passage, setting a public transport schedule for the design sites of new business, public or residential centers [2], traffic optimization on existing sections [3], moreover, the list of these practical tasks can be continued for a long time.
If in small cities with fewer than fifty roads, the description of such a system does not yet seem difficult to implement, then for the megalopolis road network it becomes a serious task, which requires special control systems based on models and algorithms for mathematical modeling of traffic flows. At the same time, the models themselves must be sufficiently accurate and require taking into account a large number of factors, for example, such as weather conditions, time of day, flow density, the condition of the road surface and the vehicle itself, which makes calculations extremely difficult.
Hence, the relevance of the compilation and consideration of practical aspects, as well as the mathematical solution of the problems of modeling the traffic flows of the metropolis, which will be considered in this paper.

Materials and methods
Due to the growth of technological progress in large cities, there is a strong increase in the number of both personal cars and ground transport in general, which eventually leads to overcrowding of the transport network and threatens a complete transport collapse. In order to avoid such a problem, various mathematical models were invented that describe the behavior on the road of both individual cars and their groups, called traffic flow. Conventionally, such models can be divided into two types: macroscopic and microscopic [4]. Next, various types of models of these two types will be analyzed.

Macroscopic models
In macroscopic models, the main idea is to compare the transport flow with a liquid with a certain motivation. This idea is connected with the fact that the transport flow, as well as the liquid, has a relationship between the flow velocity and its density [5]. The traffic flow can be described using the conservation law and find a solution to the Cauchy problem corresponding to it [6]. The resulting generalized solution allows us to describe the states of the traffic flow and consider the transitions between them. The development of these models began with the Lighthill-Witham-Richards model, which will be analyzed further.

The Lighthill-Witham-Richards Model (LWR)
The first macroscopic model of the traffic flow can rightfully be considered the Lighthill-Witham-Richards model (abbreviated LWR) [7], which first appeared in the works of the 60s of the last century, which indicates the relative "youth" of these models from the point of view of history. The appearance of this transport model can be called accidental and initially it was associated with the solution of the equations of the conservation law for a compressible fluid [8]. In the future, this fluid began to be considered as a flow of motor vehicles, which led to the interpretation of this problem from the point of view of traffic flows.
Several assumptions are made for this model: 1. Let ( , ) flow rate, and ( , ) -its density. The equation of state establishes an unambiguous relationship between them; 2. The law of conservation of mass for the number of vehicles must be fulfilled. Next, let's look at the basic concepts introduced in the assumptions. Density ( , ) will be interpreted as the number of vehicles in a certain section at some point in time , and the argument will indicate the location of this section. In turn, ( , ) will mean the speed of these vehicles on the road interval in the area with coordinate at a given time .
It is worth considering that macroscopic models describe the behavior of the traffic flow in fairly large sections (starting from a hundred meters, that is, tuples of a large number of cars) [9]. If it is necessary to take into account the behavior of a particular car and study it in more detail, then microscopic models will be used, which will be discussed in this paper a little later.
In this case, it is possible to approximate the macroscopic model using microscopic ones [10]. In practice, it is customary to consider macroscopic models due to simplified calculations and simple algorithms for their study [11].
Assumption 1, concerning the equation of state, is expressed by the following condition, Since the speed of the vehicle decreases with an increase in the number of vehicles on a certain road interval, the value of the function ( ) decreases, which means ′( ) <0. We also denote by ( ) = ( ) the intensity of the flow of vehicles, in other words, the number of cars that pass a given section of the road at a certain point in time.
Assumption 2 will be expressed by the law of conservation of the number of vehicles [12]: Thus, for a rectangular contour Ω with parallel sides of the axes, it follows that ∫ Ω ( , ) − ( ( , )) = 0. Moreover, this relation will also be valid for an arbitrary piecewise-smooth contour Ω. If we add ( , ) to the equation for smoothness points, that is: initial condition of the Riemann type, then we obtain the Cauchy problem, which can be solved.
For example, in this way it is possible to determine how quickly the participants in the movement will understand that a traffic jam has formed along their route [13].
The LWR model, due to its great popularity, has given rise to several models with additional conditions, such as Payne's model.

Payne's model
Next, we consider another important model -the Payne model. The main difference of this model is that the speed at which the vehicle is moving will be regulated by some "desired" speed that the driver will strive for [14].
For this model, the equation will look like this: where the parameter describes the speed of the desire for the desired speed of the vehicle. Due to the fact that the traffic flow in this case has motivation, the main difference between the traffic flow and its hydrodynamic counterpart lies only in the right parts of these equations, which means that already known calculation and calculation algorithms can be used.
Unfortunately, this model has been seriously criticized; in the article [15], significant shortcomings were found, such as the appearance of a negative velocity and too close movement of cars to each other, leading to an inevitable emergency.
Over time, these problems were partially solved in [16], [17]. It is worth noting that at the moment there are more than a hundred macroscopic models, each of which makes its own adjustments or improvements to the models discussed above.
We now turn to the consideration of microscopic models that will be related to the already studied macroscopic models.

Microscopic models
In microscopic models, in contrast to macroscopic ones, the traffic flow considered in terms of interaction between several separate machines, not the whole group. The main interest of these models is the description of the behavior and movement of one particular car, depending on the rest of the vehicles in the stream. In this section, we will describe various models, including the optimal speed and leader-following models, as well as one of the most popular in recent times, the Treiber "intelligent driver" model».

Tribler's smart driver model
Since the models discussed above have received wide practical application, an attempt was made to combine both types of models into one general microscopic model. This is how a microscopic model of an intelligent driver appeared, described by the following equation for the acceleration of a vehicle: Among them, the Treiber model stands out: where -is a function that determines the distance that the driver is striving for 0 -is the optimal speed and -is a parameter responsible for the nature of the vehicle's acceleration.
This model is famous because the acceleration equation in it includes two terms: one of them describes the acceleration of the car when it is possible to freely move along the road lane, while the second term describes the forced braking when driving behind the car in front. Therefore, these terms can be attributed to the types of microscopic models described earlier in this work. The combination of these two models in one allows you to mathematically set the "reasonable" behavior of the motorist on the road and bring the simulated values as close as possible to the real ones.
The use of these models will help solve various real transport problems, that is, problems that arise in specific situations: resolving traffic jams [18], driving through a traffic light, changing lanes, avoiding obstacles, repairing roads, and so on.
Based on existing mathematical models and methods, a model for rebuilding vehicles between traffic lanes was implemented.

Results
Given the small width of the streets in the central part of the city, as an example take a typical two-lane unidirectional road. We will study the movement of vehicles on the road section when approaching to the traffic light. To do this, we introduce the Euler coordinate system along the carriageway parts in the direction of flow and the corresponding travel time . Middle flux density ( , ) can be calculated using the following formula: ( , ) = , (7) where -is the area occupied by cars, and -is the total area of the road section under consideration. Now we define = ℎ , where ℎ -is the width of one lane of the road given in the problem, -is the total number of vehicles on the section under consideration, and -is the average length of the car body. In turn = ℎ , where -is the length of the control section of the road. From here follows the formula: ( , ) = = ℎ ℎ = . It should be noted that the entered density has boundaries and varies in the range from 0 to 1 inclusive.
Let us assume that the movement of the traffic flow is regulated by a traffic light located at the end of the control section, as shown in Figure 1. Denote by the coordinate of the location of this traffic light. The main parameter of the traffic light is the duration of the phases of its signals.
To simplify the task, let's take a two-phase traffic light with two signal colors: green and red, and the corresponding durations of the work phases 1 and 2. A green signal will allow passage in both lanes, while a red signal will completely prohibit traffic. Let's assume that at the initial moment of time, the vehicles that must pass the traffic light in both rows are evenly distributed between the lanes. In this case, we define the flow density for each strip as ( ) = 1, + 2, , at the same time determines the coordinate on the lane, 1, -the density of vehicles that are going to continue driving straight without rebuilding, and 2, -the density of vehicles that will be rebuilt in the next row until the traffic light is reached. This phenomenon is often found in connection with the obligation of the driver to turn left or right from the corresponding extreme lane of traffic, unless other lanes for turning are prescribed by road signs. Then you can create balance equations for vehicles in adjacent lanes by setting them the indices and : for cars moving from the left row to the right and -from the right row to the left. Then the balance equations will have the form: 1, + ( 1, = , ( 2, , ), 2, + ( 2, = − , ( 2, , ), 2, + ( 2, = − , ( 2, , ). (12) Here, for , we denote the changing traffic flow from lane to lane . It is worth considering that rebuilding is not always possible at all. Therefore, it is necessary to define several conditions for the realignment of vehicles between lanes. If ℎ + ℎ > − , where ℎ -time spent rebuilding and ℎ -plus safe distance, the need to rebuild. This condition connects the possibility of rebuilding with the distance to the traffic light.
If the first condition is not met, then it is worth considering the second condition: ( > > ) > ℎ , where ℎ -is the boundary density of vehicles in the lane of the lane change. This condition connects the driver's actions with the observed density at the entrance to the traffic light.
In the case when the conditions are met, it is possible to determine , by the following formula, where the parameter ℎ determines the intensity of the flow: , = ( 2, (1− −)) ℎ ℎ . (13) In this case, the equations of speed change of vehicles moving along the lanes of the road can be written as follows: = 2 1, + 2, = 2 1, + 2, where is the velocity of propagation of disturbances. It is worth noting that this model of realignments between traffic lanes in traffic flow is correct both in the case of traffic with a small cluster of cars, and in a situation when a moving traffic jam occurs in front of a traffic light.
As a result, the following pattern is observed: the presence of rebuildings in the immediate vicinity of the traffic light leads to a drop in the throughput of the intersection, while early rebuildings will not greatly affect the traffic. Therefore, to solve this problem, it is worth prohibiting lane changes in the area close to the traffic lights, as well as warning drivers about possible maneuvers at the intersection in sufficient time with the help of prescriptive signs. Now that the issue of lane changes has been considered, it becomes clear that traffic jams in front of a traffic light can be avoided by crossing into an adjacent lane in advance. However, this rule does not allow you to avoid downtime at a red traffic light. That is why in the future it is required to solve the problem of setting the duration of the phases of the operation of traffic signals depending on the density of incoming flows.

Conclusion
As a result of this work, various types of mathematical models were studied that describe the behavior of the traffic flow, and a new model was proposed that sets the behavior of vehicles when passing through an intersection with a traffic light installed on it.
In conclusion, I would like to note that the relevance of using transport models in practice is steadily growing every year. Modern realities require a systematic approach to the process of transport planning and more coordinated interaction of various structures in the development and implementation of transport solutions. The constant increase in the complexity and complexity of transport systems, especially in large cities and metropolitan areas, the scale of the tasks that managers and designers face, the need for interrelated consideration of a huge number of factors -all this leads to the transition to new methods of transport planning using computer transport models and a comprehensive assessment of the consequences of measures to development of transport infrastructure.