A mathematical model of the operation process of a radio communication network based on ip technologies in the conditions of information impact during the transmission of a non-repetitive data stream

. In this article, using the mathematical modeling of the operation process of the radio communication network based on IP technologies under the conditions of information impact when transmitting a non-repetitive data stream, the mathematical expectation of the packet transmission time between the relevant nodes, the function of the safe and successful transmission time of data packets, the reliability of the radio communication network based on IP technologies in railway transport increase, increase the functionality of the radio communication system and build a single technological railway radio communication network.


Introduction
Currently, it is important to abandon analog devices and create digital systems in the transmission of information all over the world.Technological networks for fast and reliable management of operational works in railway transport ensure increased safety during the movement of trains.Great attention is paid to the improvement of rapid technological communication (RTC) networks in railway transport [1][2][3][4][5][6][7][8][9][10].
Certain results have been achieved in the application of radio communication systems to railway transport, particularly in the development of them, using Internet Protocol (IP) technologies.Mathematical modeling of the operation process of the radio communication network based on IP technologies is considered an important issue.Today, the requirements for operational and technological communication networks remain.In order to calculate the reliability of information in the conditions of information influence, the operation of the radio communication network based on the logical-probabilistic model and the introduction of systems that implement them play an important role in solving such problems.

Method
Statement of a specific problem: Assume that the goal of the network jammer (NJ) is to block the equipment of the corresponding nodes of the IP radio network, which the NJ performs and disrupts their operation and correspondingly and probabilistically.If the operation of the nodes is broken, then the packet received at the entrance of the communication channel is transmitted in a time determined by the technical transmission speed and the amount of transmitted data , these are .When transmitting packets of arbitrary size , it is a random variable distributed according to the law.If a failure occurs, the network is random time ; is recovered and the incoming data packet is retransmitted with the distribution function (DF) recovery time.We describe the operation process of the IP radio communication network described in the problem statement in the form of a stochastic network, (Fig. 2): (2) Where is: ; -if it is implemented through mathematical assumptions (expectations) of intensive recovery and NJ movement, then accordingly: probability of recovery of network elements when retransmitting packets and implementation of the next NJ.
To determine the mathematical expectation and transmission time distribution function, it is required to calculate the value of the derivative of the polynomial and denominator of the point image.(2) 0 s It is determined by the mathematical expectation of packet transmission time under NJ conditions (5): (5) By changing it as follows: (6) We get the expression (5) in the following form: Accordingly, the variance is determined by this formula: In order to obtain the distribution function of the transmission time of the packet, it is necessary to perform an inverse transformation, allowing to calculate the original from its image.
To illustrate the above method of determining the time distribution function for bringing data packets into an IP radio network, we consider the following problem [11][12][13][14][15][16].
The solution: Let us imagine the operation process of an IP radio network when a network destructor implements two types of NJ in the form of a stochastic network (Fig. 1).
It can be assumed, ; ; .Using Mason's equation for closed graphs, we construct the equivalent function of a stochastic network: Where is: ; ; ; ; We express the denominator of the equivalent function in the canonical form, which allows us to pass to the Heaviside expansion for the case of simple poles ; ; h T the average time of a successful packet transmission is: The propagation time is a probability density function (11) while the integral function of the probability distribution density of the transmission time: . Calculations were performed according to formulas (10)(11)(12), the results of which are presented in Figure 3 as a family of distribution functions [17][18][19][20][21][22][23][24][25][26].
This was assumed during the calculations: -The average transmission time -After a successful implementation by NJ, the average recovery time of the IP radio network changes accordingly and within and ; -The average execution time by NJ is equal to and respectively; -The probability of a successful implementation by NJ takes values in the range.In the model characterized by the recovery time under the conditions of NJ implementation, the role of the effectiveness of the information security management system of the IP-radio network increases significantly after NJ.Thus, a slight increase 0, 5 i P in the recovery time with the probability of a successful implementation by NJ leads to a sharp increase in the average time of packet transmission: The time of successful delivery of information in the network depends on the ability of NJ to influence network elements.Thus, we can expect a more than five-fold increase in the average time to bring data to the network if the effect is not less (bad) 0, 7 i P than the one that is able to successfully implement.In the general case, the distribution law of successful packet transmission time is hyperexponential.It can be approximated with sufficient accuracy by the incomplete Gamma function.Calculations show that the resulting distribution has significant rightsided asymmetry ( 1, 9 K asymmetric coefficient), to a not so great peak ( 5, 7 E coefficient of kurtosis) and the flow of successfully transmitted packets is not the same ( 1, 3 K coefficient of variation).Transferred successfully it will be necessary to wait for the parameter of the flow of packets, t and is not constant over time. ( that is, the intensity of successful packet transmission does not exceed the intensity defined as the reciprocal of the average time of successful transmission of packets.This shows that the flow of successfully transmitted packets is not simple, and therefore the problem of estimating the time of data delivery in IP radio networks for the situation where the incoming flow is not small, but corresponds to the actual interaction of the corresponding pairs, is updated in the network. Thus, this developed model ensures that results are obtained that are not inconsistent with logic, are sensitive to changes in parameters, and are efficient.In the event of an IP radio failure, the network at random times ; 1, i n recovery time with DF ( ) i t is recovered and the incoming data packet is retransmitted.

Method and materials
The flow of incoming packets is characterized by Weibull or Paretto distributions.destructive effects on the elements of the network can occur both during the transmission of packets and during the pauses between them.The number of waiting places is determined by the size of the collector (nakopitel)..

It is required h
T to determine the mathematical expectation of the packet transmission time between the relevant nodes and the distribution function of the time of successful transmission of data packets under the conditions of NJ probability.

The solution:
Let's present the process described above as a stochastic network (Fig. 4).

Fig. 4. Stochastic network of transmission process
Traditionally, the transfer process is divided into a sub-process of waiting and a direct transfer (service) process.A process in a transmission queue is characterized by a random ( ) t waiting time with a sub-waiting time distribution function (DF).
Service Process DF Service Time between nodes corresponding to The service delivery time distribution function was obtained in the works of the authors and has the following form: (14) Where is: The work shows that the distribution function (1) can be approximated by an incomplete Gamma function, that is: (18) where: if the scale and shape of the incomplete Gamma-function are parametric, then ; . The error of this approximation does not exceed 10%, which is shown in Figure 5. Thus, with an accuracy sufficient for engineering calculations, the image of the DF according to Laplace-Stiltes can be defined as the image of an incomplete ( ) r H t Gamma- function, i.e.It allows to describe the process of transmitting these packets in the form of an enlarged stochastic network (Fig. 6), with the equivalent function Q(S)).

Fig. 6. Magnified stochastic network of the transmission process
In determining , we rely on the following well-known expressions low packet loss rates (less than 10-3) with a finite and infinite buffered gross service system has the equivalence property.
Based on the "Law of Conservation of Queued Work", the priority queued work is equal to the work accumulated in the non-priority gross service system with a constant and total load; The length of the service (for transmission) queue and the waiting time in the queue do not depend on the laws of distribution of minutes between the arrival of packets and the service time, but are determined only by the first two.
Expressions to determine based on results: The first minute, that is, the average waiting time of the r-th packet service priority: here: Average waiting time for serving packets without prioritization, taking into account the variability of the time distribution between packet arrival and service time: T T mathematical expectation of packet transmission time under DF conditions with infrequent subsequent input flow (6).
In turn, the change in waiting time is equal to: here: -secondary wait time without priority: C the coefficient of variation of the time between the arrival of packets and the time of service (transmission) of packets.
In contrast, it is proposed to approximate the DF waiting time with an incomplete Gamma-function that fits the results well.Therefore, the representation of DF according to Laplace-Stilt'es: here: .
The probability of losing packets with different priorities is as follows: To determine the DF delay time, it is necessary to determine the coefficient of variation of the incoming stream, which is mainly determined by the Paretto expression, which depends on the type of model for the distribution of the duration of the time intervals between the packets in the incoming stream: Expressions for calculating the n-coefficient depend significantly on the parameter ar in the form of a Weibull or Paretto distribution.
According to the results of simulation modeling, an empirical formula for calculating the Hearst Hw coefficient was derived: The Hearst coefficient in the Weibull distribution is defined by the following expression: 1.2exp( 9 ) 0.51 r Hw Here, in the Weibull distribution, r -parameter is numerically equal: In the Pareto distribution, r -parameter is defined by the following expression:      (28) Calculations according to the formula were carried out, the results are presented in Figure 8.

Conclusion
Thus, all the necessary quantities needed to calculate the distribution function are calculated.It follows from the calculations that the developed models gave logical results that did not contradict each other.In the model characterized by the recovery time under the conditions of implementation by NJ, the role of the effectiveness of the information security management system of the IP-radio network increased significantly after NJ.The time of successful delivery of information in the network depends on the ability of NJ to influence network elements.Thus, if the effect is able to successfully implement if not less (or worse) than , we can expect to increase the average time to fetch data to the network by more than five times.In the general case, the distribution law of successful packet transmission time is hyperexponential.It can be approximated with sufficient accuracy by the incomplete Gamma function.Calculations show that the resulting distribution has significant right-sided asymmetry ( asymmetric coefficient), to a small peak ( kurtosis coefficient) and the flow of successfully transmitted packets are not the same ( 1, 3 K coefficient of variation).This shows that the flow of successfully transmitted packets is not simple, and therefore the problem of estimating the time of information delivery in IP radio networks for the situation where the incoming flow is not small, but corresponds to the actual interaction of the corresponding pairs, has been updated in the network.
Thus, these developed models have ensured that the results are non-contradictory, sensitive to parameter changes and efficient..

Fig. 1 .
Fig. 1.IP radio network state of stochastic operation process when implementing two-stage type of NJThe incoming stream of data packets is rarely consecutive, and during the transmission of packets sent by the NJ, there may be pauses between them.The number of places to wait for a transfer is unlimited.It is required to determine the mathematical expectation ( ) F t

Fig. 2 .
Fig. 2. NJ in the network of stochastic processes under the conditions of n-level attempts Operation of the IP radio network In a stochastic network, the following are defined: ( ) s and ( ) i s The Laplace- Stieljes transformations of the corresponding distribution functions (1) mean DF of the packet transmission time without considering NJ, which are "ideal conditions" and the recovery of TF time after an n-step attempt of NJ.

Fig. 3 .
Fig. 3. shows the family of extension time distribution functionsmIP-radio -packets in the implementation of two types of NJ Statement of a specific problem: let's say it's an IP radio network over which data packets are transmitted.At the stage of organization and planning of information exchange, transmission routes are determined, each of them, in general, m nodes and 1 m consists of plots.It transmits a stream of size-sorted 0 E packets arriving in a memory buffer along the route.Depending on the service class (Quality of Service -QoS), 1, i R packets are prioritized, r -packet flow intensity, r is equal to the total intensity of packet arrival (transmission) in the case of priority.IP-The radio network operates in NJ conditions, each of which determines the performance of the network elements i P ; 1, i n likely to break.If the network is not interrupted, then the first packets to be transmitted are of random size and determined by the transmission rate equal to can be successfully transmitted to the reporter in a random time, where is the service intensity of the packet.rth is characterized by the priority and distribution function (DF) transmission time.In a stochastic network, it is defined as follows:-Laplace-Stiltes transform, that is, the distribution function of the packet transmission time of the r-th category without taking into account NJ: 1 ,..., n P P -Probability of exposure to NJ; -Laplace-Stiltes transform, that is, the distribution function of the network recovery time after the i-th NJ;-intensity of incoming and outgoing packet flows; t -Laplace-Stiltes transform, that is, the distribution function of the message transmission waiting time of the r-th category of urgency.
here: G(*) Gamma-function.Usually, the self-similar incoming flow is characterized by Hurst's n-coefficient.E3S Web of Conferences 420, 03022 (2023) https://doi.org/10.1051/e3sconf/202342003022EBWFF 2023 the necessary quantities needed to calculate the distribution function are determined.Let's consider the following problem statement to illustrate the workings of the model.Statement of a specific problem: imagine an IP radio network over which data packets are transmitted.At the stage of organization and planning of information exchange, transmission routes are determined, each of them, in general, n nodes and 1 n consists of plots.According to the route 0 E transmits a stream of sorted packets arriving in a memory buffer.. Packets are prioritized based on Quality of Service (QoS) r is the priority packet flow intensity, the total packet arrival intensity is equal radio network works in the conditions of NJ, each of which includes network elements; i P ; 1, 2 i likely to break.If the network is uninterrupted, then with probability r-priority packets will be successfully transmitted to the reporter in a random time with random size and R transmission rate equal to r V volume and R is equal to the transmission rate , where is / r r R V r-characterized by the service rate and distribution function (DF) of packet priority ( ) r B t .If there is a failure, the IP radio network recovers with DF at random time , 1, 2 i and DF ( ) i t the incoming data packet is retransmitted.The flow of incoming packets is characterized by Weibull or Pareto distributions.Destructive effects on the network elements of the network can occur both during the transmission of packets and during the pauses between them.The number of waiting places is determined by the storage capacity.It is required to determine the mathematical expectation of the packet transmission time between the corresponding ( r T )nodes and the distribution function of the time of successful transmission of data packets under NJ conditions.The solution: Let's imagine a scaled stochastic network defined as

Fig. 7 .
Fig. 7. Extended stochastic network of the transmission process.Here

4
of the packet Tr, given above;

4
difference.On the other hand, the change of TF in the communication line of the transmission waiting time is determined as follows: equivalent function of the stochastic network (Fig.2.3) is equal to the following: the distribution function, it will have the following form:

Fig. 8 .
Fig. 8.A family of functions for distribution of the time of successful transmission of packets different values of the probability of successful execution at and from NJ.