# Application of Polling Scheduling in Mobile Edge Computing

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## Abstract

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## 1. Introduction

- We propose a two-level priority exhaustive service model, and the parameters of normal nodes are asymmetrical, corresponding to the 3rd section of this paper.
- We computed specific expressions for the model’s average queue length and polling period. We calculated the average delay with an approximate analysis of periodic query way, corresponding to this paper’s fourth and fifth sections.
- We designed a Monte Carlo experiment with a small error between the theoretical and experimental values for many repetitions, corresponding to the 7th section of this paper.

## 2. Literature Review

## 3. System Model

- Server initialization
- Edge servers use parallel scheduling to query the central node and node i
- Edge server service center node if the central node needs to process data
- Server servicing of node i
- Let $i=i+1$, and return to step (2) if $i\ne N$, else step (1)

## 4. System Model Analysis

#### 4.1. Variable Definitions

#### 4.2. System Conditions

#### 4.3. Mathematical Models

- Each information packet arrives at the nodes independently and Poisson distribution. The probability-generating function of normal node is $A\left(z\right)$, and the mean and variances are ${\lambda}_{\mathrm{i}}(i=1,2,\cdots ,N)=\lambda ={A}_{}^{\prime}\left(1\right)$ and ${\sigma}_{\lambda}^{2}={A}_{}^{\u2033}\left(1\right)+\lambda -{\lambda}_{}^{2}$. And the central node, the probability-generating function is ${A}_{h}\left(z\right)$, and the mean and variances are ${\lambda}_{h}={A}_{h}^{\prime}\left(1\right)$ and ${\sigma}_{\lambda h}^{2}={A}_{h}^{\u2033}\left(1\right)+{\lambda}_{h}-{\lambda}_{h}^{2}$
- The time for the server to serve any normal nodes is independent and Poisson distribution, and its probability-generating function is $B\left(z\right)$, the mean is $\beta ={B}_{}^{\prime}\left(1\right)$, and the variance is ${\sigma}_{\beta}^{2}={B}_{}^{\prime}\left(1\right)+\beta -{\beta}^{2}$. And the central node, the probability-generating function is ${B}_{h}\left(z\right)$, the mean is ${\beta}_{h}={B}_{h}^{\prime}\left(1\right)$, and the variance is ${\sigma}_{\beta h}^{2}={B}_{h}^{\u2033}\left(1\right)+{\beta}_{h}-{\beta}_{h}^{2}$
- The switch times of the server from node i to another node are independent and Poisson distribution. The probability-generating function is $R\left(z\right)$, the mean is $\gamma ={R}_{}^{\prime}\left(1\right)$, and its variance is ${\sigma}_{\gamma}^{2}={R}_{}^{\u2033}\left(1\right)+\gamma -{\gamma}^{2}$
- Each data is First Input First Output (FIFO)
- Sufficient node capacity, no data overflow [45]

## 5. Analysis of System Variables

#### 5.1. The Average Queue Length

#### 5.2. The Average Cycle

#### 5.3. The Average Delay

**Definition**

**1.**

## 6. Simulation

- Each information packet of the nodes is asymmetric
- Each information packet arrives at the nodes independently and Poisson process
- The experiment was repeated 300,000 times, and the mean was taken as the statistic.
- The system remains stable under the condition of $\sum _{i=1}^{N}}{\lambda}_{i}{\beta}_{i}+{\lambda}_{h}{\beta}_{h}={\displaystyle \sum _{i=1}^{N}}{\rho}_{i}+{\rho}_{h}<1$

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Literature | Asymmetrical | Machine Learning | Priority | Indicators of Measurement | Accurate Calculation |
---|---|---|---|---|---|

[10] | No | No | No | channel utilization, the system throughput | No |

[11] | No | Yes | No | Packet blockage rate and average delay | No |

[12] | No | No | No | Lyapunov function of queue backlog of operation functions and overall cost of communication and computing in the MEC | No |

[13] | No | Yes | Yes | Server utilization and number of users | No |

[14] | No | Yes | No | Server utilization | No |

[15] | No | Yes | No | Average delay and | No |

[16] | No | Yes | No | Average task queuing delay per time slot | No |

[17] | No | Yes | No | Total execution time and Average task failure | No |

Our work | Yes | No | Yes | Average queue length and average polling period, and average delay | Yes |

Variables | Definition |
---|---|

i or j or k | Normal nodes |

h | Central node |

${\mu}_{i}\left(n\right)$ | The server switching time from node i to other nodes |

${\nu}_{i}\left(n\right)$ | The time of service provided by server to node i |

${\nu}_{i}\left({n}^{*}\right)$ | The time of service provided by the server to the central node |

${\mu}_{j}\left({\mu}_{i}\right)$ | The amount of data entering the node j within time ${\mu}_{i}\left(n\right)$ |

${\mu}_{h}\left({\mu}_{i}\right)$ | The amount of data entering the central node within time ${\mu}_{i}\left(n\right)$ |

${\eta}_{j}\left({\nu}_{i}\right)$ | The amount of data entering the node j within time ${\nu}_{i}\left(n\right)$ |

${\eta}_{h}\left({\nu}_{i}\right)$ | The amount of data entering the node within time ${\nu}_{i}\left(n\right)$ |

${\eta}_{j}\left({\nu}_{h}\right)$ | The amount of data entering the node j within time ${\nu}_{i}\left({n}^{*}\right)$ |

${\eta}_{h}\left({\nu}_{h}\right)$ | The amount of data entering the central node within time ${\nu}_{i}\left({n}^{*}\right)$ |

i | ${\mathit{\lambda}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ |
---|---|---|---|

Node | Arrival Rate | Service Time | Switch Time |

1 | 0.01 | 4 | 2 |

2 | 0.006 | 5 | 3 |

3 | 0.001 | 3 | 1.5 |

4 | 0.005 | 2 | 1.7 |

5 | 0.002 | 1 | 1.9 |

h | 0.008 | 1 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Yang, Z.; Ding, H.
Application of Polling Scheduling in Mobile Edge Computing. *Axioms* **2023**, *12*, 709.
https://doi.org/10.3390/axioms12070709

**AMA Style**

Wang X, Yang Z, Ding H.
Application of Polling Scheduling in Mobile Edge Computing. *Axioms*. 2023; 12(7):709.
https://doi.org/10.3390/axioms12070709

**Chicago/Turabian Style**

Wang, Xiong, Zhijun Yang, and Hongwei Ding.
2023. "Application of Polling Scheduling in Mobile Edge Computing" *Axioms* 12, no. 7: 709.
https://doi.org/10.3390/axioms12070709