# Modeling of the Schemes for Organizing a Session of Person–System Interactions in the Information System for Critical Use Which Operates in a Wireless Communication Environment

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## Abstract

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## 1. Introduction

## 2. State of the Art

## 3. Materials and Methods

#### 3.1. Statement of Research

#### 3.2. Mathematical Model of a Person–System Interaction Session in Discrete Time

- According to Scheme 1, the session is activated to satisfy the request that is waiting for the longest: $i\left(n\right)=\mathrm{arg}\underset{1\le i\le {\xi}_{n}}{\mathrm{max}{A}_{in}}$. The bandwidth $C\left(H\right)$ is determined on the basis of Equation (1): $C\left({H}_{i\left(n\right)}\left(n\right)\right)={\displaystyle \sum _{k=1}^{m}{\mathrm{log}}_{2}\left(1+{\lambda}_{k}\frac{1}{{\sigma}^{2}}{\rho}_{k}\right)}$;
- According to Scheme 2, the session is activated to satisfy a request from the person-user $i\left(n\right)$, which corresponds to the largest declared bandwidth among the requests waiting in line: $i\left(n\right)=\mathrm{arg}\underset{1\le i\le {\xi}_{n}}{\mathrm{max}}C\left({H}_{i}\left(n\right)\right)$, where the bandwidth $C\left(H\right)$ is determined by Equation (1);
- According to Scheme 3, the sessions are activated to satisfy requests from person-users from subset ${N}^{\infty}\left(n\right)$, while providing the best bandwidth: ${C}_{SR}\left({H}_{k}\left(n\right)\right)=\underset{{\Sigma}_{j}>0,{\Sigma}_{j}\left(n\right){\left({\Sigma}_{j}\left(n\right)\right)}^{T}\le {P}_{T}}{\mathrm{max}}{\mathrm{log}}_{2}\left|\mathrm{I}+{\displaystyle \sum _{j}{H}_{j}\left(n\right){\Sigma}_{j}\left(n\right){{H}^{\prime}}_{j}\left(n\right)}\right|$, $k=1,2,\dots ,{\xi}_{n}$. Accordingly, the subset ${N}^{\infty}\left(n\right)$ is identified by the expression ${N}^{\infty}\left(n\right)=\mathrm{arg}\underset{P\left\{1,2,\dots ,{\xi}_{n}\right\}}{\mathrm{max}}{C}_{SR}\left({H}_{k}\left(n\right)\right)$, $k=1,2,\dots ,{\xi}_{n}$.

- The registration center server receives a request to activate a person–system interaction session. The probability of such an event is characterized by the stochastic parameter $a$, $0\le a\le 1$. Accordingly, the length of the queue of registered requests increases: ${\xi}_{n}={\xi}_{n-1}+1$. Additionally for the registered request: the final number of blocks of requested data is declared in the form of value of stochastic parameter ${F}_{{\xi}_{n}n}$; countdown of time of stay of request in the queue is considered by the value of parameter ${A}_{{\xi}_{n}n}$ begins (at the $n$th quantum of time ${A}_{{\xi}_{n}n}=1$); route matrix ${H}_{{\xi}_{n}}\left(n\right)$ is initiated by random values;
- Random values redefine route matrices ${H}_{i}\left(n\right)$ for active sessions of person–system interaction;
- According to the active scheme of organization of sessions of person–system or multi-person–system interactions, the values of the function $C\left(H\right)$ or ${C}_{SR}\left(H\right)$ are calculated for all registered requests;
- According to the used scheme of organization of sessions of person–system or multi-person–system interactions, the session of person–system interaction is activated for $i\left(n\right)$ or ${N}^{\infty}\left(n\right)$ request;
- For a newly activated person–system interaction session, the final number of blocks of requested data is recalculated: $i=i(n)\Rightarrow $${F}_{i\left(n\right)n}={F}_{i(n)n-1}-C\left({H}_{i\left(n\right)}\left(n\right)\right)\Delta t$, $i=k,k\in {N}^{\infty}\left(n\right)\Rightarrow $${F}_{kn}={F}_{kn-1}-C\left({H}_{k}\left(n\right)\right)\Delta t$;
- The time of each request in the queue is incremented: ${A}_{n}={A}_{n-1}+1$;
- The bandwidth for the newly activated person–system interaction session $C\left(H\right)$ is specified: the speed of loading the data block by the person-user terminal $T={A}_{in}$ is fixed, provided that the equality $\mathrm{max}\left(0,{F}_{in}\right)=0$ is not fulfilled;
- The person–system interaction sessions for which the equality $\mathrm{max}\left(0,{F}_{in}\right)=0$ holds are deactivated.

#### 3.3. Formalization in the Mathematical Apparatus of Queuing Systems of the Process of Person–System Interaction, Organized According to Schemes 1–3, in Discrete Time

#### 3.4. Formalization in the Mathematical Apparatus of Queuing Systems of the Process of Person–System Interaction, Organized According to Schemes 1–3, in a Continuous Time

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${M}_{T}$ | is a set of receiving–transmitting devices on the side of the registration center server; |

${M}_{R}$ | is a set of receiving–transmitting devices on side of the terminals of each person-user; |

$i$ | is an index of the person-user terminal; |

${y}_{i}$ | is a power that characterizes such a session on the terminal of person-user on the $i$ side; |

${H}_{i}$ | is a route matrix; |

$s$ | is a vector formed by data blocks of the information message, which is transmitted by the registration center server to the $i$th terminal; |

${v}_{i}$ | is a vector formed by random values that characterize the noise affecting each receiving–transmitting device of the $i$th terminal; |

$N$ | is a peak power with zero mathematical expectation and variance ${\sigma}^{2}={N}_{0}/2$; |

${N}_{0}$ | is the spectral density of noise power; |

$V$ | is a covariance matrix of noise; |

${\mathrm{I}}_{{M}_{R}}$ | is a unit matrix; |

${P}_{T}$ | is a peak power of the information signal for the receiving–transmitting devices of the registration center server; |

${P}_{SNR}$ | is a signal-to-noise ratio; |

${\lambda}_{k}$ | are the eigen values of the matrix ${H}_{i}$; |

${\rho}_{k}$ | is the power of signal transmission via the $k$th route; |

$\mu $ | is a constant, the value of which is chosen so that the equality $\sum _{k=1}^{m}{\rho}_{k}}={P}_{T$ is satisfied; |

${C}_{SR}$ | is the total bandwidth (Equation (2)); |

${\Sigma}_{k}$ | is a covariance matrix of the person–system interaction sessions; |

$T$ | is an average time it takes a person-user to download a finite amount of data; |

$\Delta t$ | is a quantization period; |

${X}_{n}$ | is a stochastic process that describe a session of multi-person–system interaction in discrete time; |

${\xi}_{n}$ | is a stochastic value that characterizes the number of registered requests for session activation; |

${F}_{n}$ | is a vector formed by the values of the final queue lengths from data blocks for active sessions; |

${A}_{n}$ | is a vector formed by the values of the waiting times for the registered requests for activation of the session at the beginning of the $n$th quantum of time, $n\ge 1$; |

$a$ | is a stochastic parameter that characterizes such an event: a registration center server receives a request to activate a person–system interaction session; |

$T$ | is a speed of loading the data block by the person-user terminal; |

${p}_{i}$ | are stationary probabilities; |

$\rho $ | is a workload of the registration center server of ISCU; |

$w\left(z\right)$ | is a generating function that characterizes a stationary distribution of the waiting time for the start of service of the first request in the group (start time of service of request); |

$N$ | is an average number of requests served by the registration center server of ISCU; |

$E\left[T\right]$ | is the average time that the group request spends on the registration center server of ISCU (duration of downloading the information package); |

$\widehat{b}$ | is a value of the bandwidth $C\left(H\right)$ of the communication equipment of the terminal of the person-user. |

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**Figure 1.**Basic scheme of organization of the session of person–system interaction at ${M}_{T}={M}_{R}=2$ (ISCU is an information system for critical use).

**Figure 2.**Example of application of the water filling principle to the basic scheme of organization of the session of person–system interaction at: (

**a**) ${\rho}_{1}+{\rho}_{2}={P}_{T}$ variant; (

**b**) ${\rho}_{1}={P}_{T}$ variant.

**Figure 3.**Schematic interpretation of bandwidth of a person–system interaction session in discrete time.

**Figure 4.**Structural model of a multi-person–system interaction session at ${M}_{T}={M}_{R}=2$, $i=\overline{1,N}$.

**Figure 6.**Dependence of average throughput on the number of requests and the scheme of organization of sessions of person–system interaction.

**Figure 7.**The dependence of the average duration of the person–system interaction session on the scheme involved for its organization and the intensity of the load on the registration center server, provided that: (

**a**) the volume of the information package $\widehat{c}$ is 1.5 kB; (

**b**) The volume of the information package $\widehat{c}$ is 2.2 kB.

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**MDPI and ACS Style**

Al-Ma’aitah, M.; Saad, A.; Alwadain, A. Modeling of the Schemes for Organizing a Session of Person–System Interactions in the Information System for Critical Use Which Operates in a Wireless Communication Environment. *Symmetry* **2021**, *13*, 391.
https://doi.org/10.3390/sym13030391

**AMA Style**

Al-Ma’aitah M, Saad A, Alwadain A. Modeling of the Schemes for Organizing a Session of Person–System Interactions in the Information System for Critical Use Which Operates in a Wireless Communication Environment. *Symmetry*. 2021; 13(3):391.
https://doi.org/10.3390/sym13030391

**Chicago/Turabian Style**

Al-Ma’aitah, Mohammed, Aldosary Saad, and Ayed Alwadain. 2021. "Modeling of the Schemes for Organizing a Session of Person–System Interactions in the Information System for Critical Use Which Operates in a Wireless Communication Environment" *Symmetry* 13, no. 3: 391.
https://doi.org/10.3390/sym13030391