# Construction and Analysis of Queuing and Reliability Models Using Random Graphs

## Abstract

**:**

## 1. Introduction

## 2. Logical-Probabilistic Models of Complex Systems Constructed on the Modular Principle and Their Reliability

## 3. Disconnection Probability of Planar Graph with High Reliable Edges

**Numerical example.**Consider the graph in Figure 2.

## 4. Construction and Calculation of Queuing Networks with Failures

## 5. Limit Distribution in Queuing System $M\left|M\right|1|\infty $ with Randomly Varying Intensity of Input Flow

## 6. Convergence to the Limit Distribution in the Barabasi-Albert Model of a Growing Random Network

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 7. Discussion

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Representation of a logical function $A({x}_{1},...,{x}_{n})$ in the form of a directed tree.

**Figure 3.**Undirected graph $\Gamma $ representing network G with failures at critical levels ${N}_{1},{N}_{2}$ for numbers of customers in nodes $1,2$.

**Figure 5.**Undirected graph $\Gamma ,$ representing network G with failures at critical level N for total number of customers in the network $G$.

**Table 1.**Results of ${P}_{MC},\phantom{\rule{0.277778em}{0ex}}{P}_{h},\phantom{\rule{0.277778em}{0ex}}{\Delta}_{h}$ calculations.

h | ${\mathit{P}}_{\mathit{MC}}$ | ${\mathit{P}}_{\mathit{h}}$ | ${\mathbf{\Delta}}_{\mathit{h}}$ |
---|---|---|---|

0.01 | 0.000023 | 0.00002233 | 0.029 |

0.02 | 0.000179 | 0.000181 | 0.011 |

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**MDPI and ACS Style**

Tsitsiashvili, G.
Construction and Analysis of Queuing and Reliability Models Using Random Graphs. *Mathematics* **2021**, *9*, 2511.
https://doi.org/10.3390/math9192511

**AMA Style**

Tsitsiashvili G.
Construction and Analysis of Queuing and Reliability Models Using Random Graphs. *Mathematics*. 2021; 9(19):2511.
https://doi.org/10.3390/math9192511

**Chicago/Turabian Style**

Tsitsiashvili, Gurami.
2021. "Construction and Analysis of Queuing and Reliability Models Using Random Graphs" *Mathematics* 9, no. 19: 2511.
https://doi.org/10.3390/math9192511