TRAFFIC FLOWS MODELLING BASED ON PROBABILITY OF COINCIDENCE OF CHAOTIC IMPULSES OF RANDOM DURATION AND RANDOM INTERVALS BETWEEN THEM

The features of applying the theory of chaotic electrical impulse flows in mathematical modelling of traffic flows have been considered. Both the moments of appearance of vehicles on the traffic lane and the moments of appearance of electrical impulse on the time axis are supposed to obey the Poisson law. It is assumed that the duration of an impulse is equivalent to the length of a vehicle, and the intervals between the impulses are equivalent to the intervals between the vehicles in the spatial-temporal axis.


Introduction
Computer simulation of traffic plays an important role in improving the transport situation on the roads. The application of various mathematical methods and modelling systems enables to improve the traffic flow capacity, reduce accident rate on the roads, and preserve the envi-ronment with minimal financial expenses.Road traffic is closely connected with the welfare and life of people. Therefore, the major road accidents are often covered in media, on television, on the Internet.

Analysis of Publications
According to the information of State Motor Vehicle Inspectorate of Ukraine, the most frequent reason for traffic accidents is related to the actions of road traffic participants, for which reason more than 90 % of accidents from their total number take place ( Fig. 1) [1,2]. Some of the most frequent violations connected with vehicles (except those involving pedestrians) are: exceeding the safe speed, violation of manoeuvring rules, non-observance of the distance and driving onto the oncoming lane of the road. Currently, in modelling traffic flows, the class of macroscopic models, when the motion of vehicles is likened to any physical flow (hydro-and gas-dynamic models) is also considered [3].

Purpose and Tasks
The aim of the work is mathematical modelling of transport flows on the basis of choosing the theory of chaotic electric impulse flows as a physical analogue of the transport flow.We will proceed from the assumption that violation of the manoeuvring rules, including driving onto the adjacent lane of traffic and onto the oncoming lane, will result in a collision of vehicles when they are in the cross section for a moment before the convergence or at the time of the convergence, that is, in the zone of compatible parallel movement in the cross section of the road (Fig. 2). We will assume that the moments of the appearance of vehicles on the lane of motion obey the Poisson law, that is, they meet the criteria of ordinarinessand not controlling the after-effect [4].
Let there be a transport flow with random lengths of vehicles and random intervals of d between them and known densities of probability   P d for the intervals between the vehicles and ( ) Q z for their lengths.
The distance between successive vehicles may be different. In some cases, it can slightly exceed the length of the vehicle, while in others it can be quite large. Similarly, the time interval between successive vehicles varies. This variability in time and space is the most notable characteristic of the traffic flow [5].
To know the intervals of time and distance between successive vehicles is sometimes more important than knowing the intensity or density of the flow, as it reflects the true nature of the traffic flow more fully. The intervals of time and distance between successive vehicles are those "bricks" on which the entire traffic flow is built [5]. Since vehicles are dynamic objects, which depending on the speed at each moment of time take a new position in the spatial-temporal plane [4] (traffic lane), this process can be represented as a random electric impulse flow on the time axis [6]. Therefore, it can be assumed that the duration of an impulse is equivalent to the length of a vehicle, and the intervals between the impulses are equivalent to the intervals between the vehicles.
To do this, we use the theory of random impulse flows. Let us consider the n probability of coincidence of vehicles of random lengths that go chaotic, and random intervals between them and the distribution of their coincidence [7].

Substantiation of mathematical modelling of traffic flows on the basis of choosing the theory of chaotic electric impulse flows as a physical analogue of the transport flow
Let the lengths of the vehicles i x (i 1,2)  in two lanes of motion ( 2) n  be distributed according to independent laws ) ( x are the lengths of the vehicles, which make a coincidence of with z duration.
As we know is the law of the distribution of the duration of coincidence of vehicles, which are on the first lane of motion; as the probability of coincidence of vehicles turns out to be higher with longer vehicles [4] and therefore the latter coincide on average more often than shorter vehicles. With the unchanged length 2 x of the vehicles on the second lane, the distribution law is where on the condition of normalization In the case of random magnitude, the law expressed by formula (2) should be considered as a conditional law of distribution An unconditional distribution law is found by averaging in the (2) probability C is the normalizing factor. Thus To determine the law let us consider two vehicles with the lengths of 1 x and 2 x correspondingly, (Fig. 2), and for certainty we will accept 2 1 x x  . , x x  , the random magnitude Z of coincidence duration changes according to trapezoid law ( ) z t taking the value  Taking into account that the probability of coincidence λ dp dt  , shown in Fig. 2, we find an integral distribution law Integrating and taking into account the fact that according to (5) 1 2 t z x   and 1 2 t z x   , and At the point where   H z is a solitary function; Function (6) can be considered as a conditional integral function of distribution Z relative to 1 X and 2 X , that is, Hence the conditional distribution law is where   δ z is the delta function. A similar ex- Substituting expressions (3), (4) and (7) into formula (1), we obtain Integrating, we define a two-dimensional distribution law Integrating again, we find a one-dimensional law of distribution of duration of vehicles' coincidence on two ( 2) n  lanes of motion The task of determining one-dimensional law   W z of the distribution of the duration of vehicles' coincidence by n lanes ( 2) n  is equivalent to the task of determining the law of distribution of the duration of the vehicles' coincidence by two lanes, one of which is the first given lane, and the other one is the equivalent lane, the length of the vehicle on which is equal to the length x , is expressed by the equality [7]    This probability can be considered as the probability of case 1 x A associated with the hypothesis 2 x , whose probability is equal to   2 2 2 f x dx . Therefore, the average (full) probability of coin-cidence of the vehicle with the length 1 x with a vehicle of any length (no matter which) moving along the second lane of motion, is found by averaging probability (12) by 2 x In turn, the probability expressed by the formula (13) can be considered as a conditional probability 1 ( / ) P A x of case A (the coincidence of two vehicles of the same or different length, moving along two adjacent lanes of the road) associated with the hypothesis 1 x (the one of occurrence of the vehicle with length 1 x on the first lane), whose probability is equal to 1 1 ( ) f x dx . Thus, the sought probability of coincidence of vehicles of random length on two lanes is The given considerations can be extended to any number of n lanes   1, 2, , i n   . For unchanged lengths of vehicles for each of the lanes, the probability of their coincidence (subject to the appearance of vehicles on one of the lanes) is expressed by equality [7] ' 1 The probability of coincidence of vehicles of random length along the n lanes is found by averaging probability ' n p by lengths i x on all lanes of motion.
Thus, as to the probability of coincidence, the lane of motion of a vehicle of a random length is equivalent to a lane of motion of a vehicle of a permanent length, which is equal to the mathematical expectation of the length of the vehicles on the lane of the road.
It can be said that the distribution of moments of occurrence of vehicles involved in the coincidence of vehicles is also subject to the Poisson law with average density n n p     .
Mathematical expectation 2 Z Z  of the duration of the coincidence of vehicles on two lanes of motion ( 2) n  can be found by formula (10). Let us look into more general considerations. By unchanged lengths of vehicles 1 x and 2 x the average duration of coincidence of vehicles on two lanes of motion is expressed by formula [3] By random values of 1 X and 2 X , the result expressed by the latter equality, must be averaged by lengths 1 x and 2 x of the vehicles involved in the coincidence of the two vehicles, that is * 5 m x x   , one may ignore the restrictions on the duration of the distribution law (22). This is illustrated by the relation of mathematical expectations   As we can see, the exponential law of the distribution of the vehicles' length by the lanes has stability: the duration of the coincidence is also distributed according to the exponential law.
Let us determine the probability of a collision of vehicles for lanes of I-IV categories of roads by violation of the rules of manoeuvring and driving the vehicle onto the adjacent or oncoming lane of motion. When the vehicle moves onto the oncoming traffic lane, the probability of vehicles' collision is denoted by p which depends on a number of parameters, including: flow density, vehicles' motion speed, condition of road surface, mental and physical state of the driver, his behaviour on the road, time of day, day of the week and other random factors.
We will mark the following cases for categorie-sII-IV of roads. Let 1 A be the case when a vehicle goes on the roadside with the control lost, and 2 A is the case when a vehicle goes onto the oncoming traffic lane. Then the probability of a collision will be equal to We will mark with F the case of a collision of vehicles at vehicle moving onto the oncoming traffic lane, and G  will be the case of vehicles collision for the II-IV categories of roads. Then Obviously, expression (2) is valid for the adjacent (second) lane of one traffic direction for the road of category I. Therefore, expression (2) will be written in the form of Let us determine the probability of vehicles' collision on the third lane of motion in the same direction. Let's mark the case of collision of vehicles on the third lane of traffic with 2 F . Then From where the probability of the collision will be equal to         Similarly, expressions (32) and (33) at the collision of vehicles on the fourth lane will be rewritten in the form of Then the probability of the case G will be equal to      