# Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems

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## Abstract

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^{−3}, which is well within acceptable limits for engineering purposes. Primarily designed for radio channels in cyber-physical systems, the model’s applicability extends to wireless communications, providing a valuable tool for evaluating channel efficiency and security in the face of increasing cyber threats.

## 1. Introduction

#### 1.1. Relevance

#### 1.2. Previous Surveys

- By integrating the Gamma distribution and a logical “AND” node, the model offers a more flexible and comprehensive approach to understanding and predicting the behavior of radio communication networks.
- The application of the Gamma distribution allows for a more accurate description of time-related processes in data transmission, thereby enhancing the model’s utility in engineering calculations.
- The model is designed to be practically applicable, particularly in railway CPSs, where it can help optimize network performance and resilience against disruptions, including cyberattacks.
- The new approach reduces the computational burden by requiring fewer series members than traditional methods while maintaining high accuracy, as evidenced by a maximum absolute error within acceptable engineering limits.

## 2. Mathematical Model and Methodology

#### 2.1. Descriptive Model

#### 2.2. Problem Statement

#### 2.3. Solution

- $v=\frac{{M}_{1h}}{{D}_{h}}$—scale parameter;
- $\varsigma =\frac{{M}_{1h}^{2}}{{D}_{h}}$—shape parameter.

- $\sigma =\frac{{M}_{1a}}{{D}_{a}}$—scale parameter;
- $\rho =\frac{{M}_{1a}^{2}}{{D}_{a}}$—shape parameter.

- ${M}_{1b}={(-1)}^{1}\frac{d}{ds}{\left(\frac{b\left(s\right)}{b\left(0\right)}\right)}_{s=0}$; ${M}_{2b}={(-1)}^{2}\frac{{d}^{2}}{{ds}^{2}}{\left(\frac{b\left(s\right)}{b\left(0\right)}\right)}_{s=0}$; ${D}_{b}={M}_{2b}-{M}_{1b}^{2}$.
- $\theta =\frac{{M}_{1b}}{{D}_{b}}$—scale parameter;
- $\zeta =\frac{{M}_{1b}^{2}}{{D}_{b}}$—shape parameter.

- $H{A}_{1}(s,N)=\mathsf{\Gamma}(\rho +\zeta ){\displaystyle \sum _{k=0}^{N}\left[\frac{{\sigma}^{k}{(\sigma +\theta )}^{\rho +\zeta +k}}{{(s+\sigma +\theta )}^{\rho +\zeta +k}{(\sigma +\theta )}^{\rho +\zeta +k}}\frac{f(\rho +\zeta +k-1,k)}{f(\rho +k,k+1)}\right]}$
- $H{A}_{2}(s,N)=\mathsf{\Gamma}(\rho +\zeta ){\displaystyle \sum _{k=0}^{N}\left[\frac{{\theta}^{k}{(\sigma +\theta )}^{\rho +\zeta +k}}{{(s+\sigma +\theta )}^{\rho +\zeta +k}{(\sigma +\theta )}^{\rho +\zeta +k}}\frac{f(\rho +\zeta +k-1,k)}{f(\zeta +k,k+1)}\right]}$
- $f(x,k)=\left|\begin{array}{lll}{\displaystyle \prod _{j=0}^{k-1}(x-j)},& at& k0\\ 1,& at& k=0\end{array}\right.$—descending factorial.

- $H{F}_{1}(t,N)=\mathsf{\Gamma}(\rho +\zeta ){\displaystyle \sum _{k=0}^{N}\left[\frac{{\sigma}^{k}\gamma \left[(\sigma +\theta )t,\rho +\zeta +k\right]}{{(\sigma +\theta )}^{\rho +\zeta +k}}\frac{f(\rho +\zeta +k-1,k)}{f(\rho +k,k+1)}\right]}$;
- $H{F}_{2}(t,N)=\mathsf{\Gamma}(\rho +\zeta ){\displaystyle \sum _{k=0}^{N}\left[\frac{{\theta}^{k}\gamma \left[(\sigma +\theta )t,\rho +\zeta +k\right]}{{(\sigma +\theta )}^{\rho +\zeta +k}}\frac{f(\rho +\zeta +k-1,k)}{f(\zeta +k,k+1)}\right]}$.

- ${\lambda}_{k}=\frac{{T}_{k}}{{D}_{k}}$; ${\alpha}_{k}=\frac{{\left({T}_{k}\right)}^{2}}{{D}_{k}}$—k-th scale and form parameters.
- ${T}_{k}=\frac{\rho +\varsigma +k}{\sigma +\theta}+{M}_{1h}$; ${D}_{k}=\frac{\left(\rho +\zeta +k\right)+(\rho +\zeta +k+1)}{{\left(\sigma +\theta \right)}^{2}}+{D}_{h}$.

## 3. Results and Discussion

#### 3.1. Calculation Example

#### 3.2. Analysis of the Results Obtained

^{−3}, which is entirely satisfactory for engineering calculations.

## 4. Conclusions

^{−3}, which is well within acceptable limits for engineering purposes. Additionally, our analysis underscores the necessity of developing effective early detection methods for cyber impacts, crucial for preventing disruptions in data transmission over the radio channel.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Variable | Definition |

${P}_{ca}$ | Probability of a cyberattack being carried out by a perpetrator |

$1-{P}_{ca}$ | Probability of no cyberattack being carried out by a perpetrator |

${t}_{h}$ | Random time of the connection establishment process |

${t}_{b}$ | Random time of the connection maintenance process |

${t}_{a}$ | Random time of the command transmission process |

${t}_{b1}$ | Random time of successful data transmission in the absence of a cyberattack |

${t}_{d1}$ | Random time of cyberattack neutralization |

${t}_{c}$ | Random duration of the command to maintain an established connection |

$B\left(t\right)$ | Distribution function of the connection maintenance process |

$A\left(t\right)$ | Distribution function of the command transmission process |

${B}_{1}\left(t\right)$ | Distribution function of data transmission in the absence of a cyberattack |

$H\left(t\right)$ | Distribution function of the connection establishment process |

${D}_{1}\left(t\right)$ | Distribution function of the cyberattack neutralization process |

C$\left(t\right)$ | Distribution function of the process of maintaining an established connection command |

$R$ | Data transmission rate |

$V$ | Volume of data transmitted |

h(s) | Laplace–Stieltjes transform of the probability distribution function of connection establishment time |

b(s) | Laplace–Stieltjes transform of the probability distribution function of connection maintenance time |

a(s) | Laplace–Stieltjes transform of the probability distribution function of command transmission time |

Q(s) | Final equivalent function |

F(x) | Distribution function of data transmission and connection maintenance processes |

f(x) | Density function of data transmission and connection maintenance processes |

HA(s,N) | Graph of the density function of data transmission and connection maintenance processes |

HF(t,N) | Distribution function of data transmission and connection maintenance processes obtained through series expansion |

QA(s,N) | Equivalent function of a stochastic network considering series expansion |

QF(t,N) | Distribution function of the duration of successful data transmission via radio channel |

T(N) | Average time of successful data transmission via radio channel |

N | Number of series terms |

${T}_{r}$ | Average time of data delivery |

${\lambda}_{1}\left(t,N\right)$ | Actual channel throughput |

${T}_{neutr}$ | Average time of the cyberattack neutralization process |

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**Figure 3.**An aggregated stochastic network of the data transmission process over a mobile radio communication channel of a standard.

**Figure 6.**The probability distribution function for the time taken for successful data transmission over the CPS radio network.

**Figure 7.**The probability distribution function for the time of successful data transmission, assuming completely reliable components of the radio channel and the absence of informational interference.

**Figure 8.**The probability distribution function for the duration of successful data transmission under different probabilities of a cyber attack by an adversary.

**Figure 9.**Graph depicting the relationship between the intensity of successfully transmitted data packets at various values of the probability of a computer attack by the attacker.

**Figure 10.**The probability distribution function for the time of successful data delivery in the context of cyberattacks by an attacker, varying according to different timeframes for their detection and neutralization.

**Figure 11.**Distribution functions of the time of realization of the logical “AND” node obtained using Formulas (13) and (20).

**Figure 12.**Graphs illustrating the dependence of the absolute error in determining the distribution function $F\left(t\right)$ using the series $HF(t,N)$.

**Figure 13.**Dependency of the absolute error in the approximation of the distribution function ${F}_{\gamma}(t)$ with a Gamma distribution.

Reception time for KVS frames | ${t}_{B}=0.35\left(\mathrm{s}\right)$ |

Reception time for UPS, KPS and DISTANCE | ${t}_{r}=0.34\left(\mathrm{s}\right)$ |

UI adjustment time | ${t}_{i}=0.56\left(\mathrm{s}\right)$ |

Probability values | ${P}_{1}={P}_{2}={P}_{3}=0.999$ |

Packet transmission time | ${t}_{b1}=0.95\left(\mathrm{s}\right)$ |

Cyberattack neutralization time | ${t}_{d1}=1\left(\mathrm{s}\right)$ |

Probability of a cyberattack by the attacker | ${P}_{ca}=0.01$ |

Connection maintenance time | ${t}_{c}=0.1\left(\mathrm{s}\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Makhmudov, F.; Privalov, A.; Privalov, A.; Kazakevich, E.; Bekbaev, G.; Boldinov, A.; Kim, K.H.; Im-Cho, Y.
Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems. *Mathematics* **2024**, *12*, 1452.
https://doi.org/10.3390/math12101452

**AMA Style**

Makhmudov F, Privalov A, Privalov A, Kazakevich E, Bekbaev G, Boldinov A, Kim KH, Im-Cho Y.
Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems. *Mathematics*. 2024; 12(10):1452.
https://doi.org/10.3390/math12101452

**Chicago/Turabian Style**

Makhmudov, Fazliddin, Andrey Privalov, Alexander Privalov, Elena Kazakevich, Gamzatdin Bekbaev, Alexey Boldinov, Kyung Hoon Kim, and Young Im-Cho.
2024. "Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems" *Mathematics* 12, no. 10: 1452.
https://doi.org/10.3390/math12101452