# A Measurement-Based Frame-Level Error Model for Evaluation of Industrial Wireless Sensor Networks

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Transmission Quality Measurement and Analysis

_{i}is the average RSSI value in a certain hour i, $\overline{x}$ is the average of x

_{i}(i = 0, 1, …, 23), y

_{i}is the FDR in a certain hour i, and $\overline{y}$ is the average of y

_{i}(i = 0, 1, …, 23).

## 4. Proposed Error Model

^{n}minutes of the sixth hour were used to retrain the three models, where n is an integer ranging from 0 to 5. The authenticity of the resulting error models was quantified by computing the Kullback-Leibler (K-L) divergence [17] of the synthesized data relative to the original records, i.e.,

_{i}is the K-L divergence of the independent model and D

_{m}is the K-L divergence of the two-state Markov model or second-order Markov model. In other words, a value of R less than 1 indicates that the Markov model (two-state or second-order) outperforms the independent model, and vice versa. Figure 17 shows the value of R for the correct-frame bursts in each hour of the experimental period. For the fourth and sixth hours, both Markov models outperform the independent model. Furthermore, the two models provide a comparable performance to the independent model in all of the other hours. It is additionally noted that the second-order Markov model significantly outperforms the two-state Markov model in the sixth hour. Figure 18 presents the equivalent performance results for the error-frame bursts. For ease of presentation, the y-axis is plotted with a base 10 logarithmic scale. Hence, a value of R less than 0 indicates that the Markov models outperform the independent model, and vice versa. The results show that the second-order Markov model generally outperforms the other two models, particularly in the third and seventh hour.

## 5. Overestimation Errors of Independent Model

^{4}= 0.991. In other words, the transmission reliability evaluated by the independent error model (0.991) overestimates the actual transmission reliability (0.9).

## 6. Simulation Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Average Received Signal Strength Indicator (RSSI) value in each hour of the experimental measurement period.

**Figure 7.**Autocorrelation coefficients of received data in first minute (21:45) of trace obtained in the sixth hour.

**Figure 8.**Autocorrelation coefficients of received data in time interval of 20/512 s within first minutes (21:45) of trace obtained in the sixth hour.

**Figure 9.**Training results for the two-state Markov model using: (

**a**) records in the first minute of the sixth hour, and (

**b**) records in the first minute of the tenth hour.

**Figure 10.**Training results for the second-order Markov model using: (

**a**) records in the first minute of the sixth hour, and (

**b**) records in the first minute of the tenth hour.

**Figure 11.**Cumulative Distribution Functions (CDFs) of correct-frame burst length derived using different error models based on recorded trace in the first minute of the sixth hour.

**Figure 12.**CDFs of correct-frame burst length derived using different error models based on recorded trace in the first minute of the tenth hour.

**Figure 13.**CDFs of error-frame burst length derived using different error models based on recorded trace in the first minute of the sixth hour.

**Figure 14.**CDFs of error-frame burst length derived using different error models based on recorded trace in the first minute of the tenth hour.

**Figure 15.**Kullback-Leibler (K-L) divergence between correct-frame bursts of original records and those of synthetic data.

**Figure 16.**K-L divergence between error-frame bursts of original records and those of synthetic data.

**Figure 17.**K-L divergence ratios of Markov error models for correct-frame bursts in each hour of the experimental period.

**Figure 18.**K-L divergence ratios of Markov error models for error-frame bursts in each hour of the experimental period (note that the K-L divergence ratio is plotted using a logarithm base 10 scale for ease of presentation).

Fraction | p000 | p010 | p100 | p110 | Our TR^{1} | FDR | Original TR | Equation (6) TR | AI^{2} |
---|---|---|---|---|---|---|---|---|---|

1 | 0.860 | 0.595 | 0.746 | 0.379 | 0.913 | 0.753 | 0.975 | 0.996 | 0.068 |

0.9 | 0.774 | 0.536 | 0.671 | 0.341 | 0.820 | 0.618 | 0.892 | 0.979 | 0.088 |

0.8 | 0.688 | 0.476 | 0.596 | 0.303 | 0.716 | 0.515 | 0.804 | 0.945 | 0.123 |

0.7 | 0.602 | 0.417 | 0.522 | 0.265 | 0.606 | 0.415 | 0.667 | 0.883 | 0.100 |

0.6 | 0.516 | 0.357 | 0.447 | 0.227 | 0.494 | 0.327 | 0.546 | 0.795 | 0.105 |

^{1}TR denotes Transmission Reliability and

^{2}AI denotes Accuracy Improvement.

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Yu, Y.-S.; Chen, Y.-S.
A Measurement-Based Frame-Level Error Model for Evaluation of Industrial Wireless Sensor Networks. *Sensors* **2020**, *20*, 3978.
https://doi.org/10.3390/s20143978

**AMA Style**

Yu Y-S, Chen Y-S.
A Measurement-Based Frame-Level Error Model for Evaluation of Industrial Wireless Sensor Networks. *Sensors*. 2020; 20(14):3978.
https://doi.org/10.3390/s20143978

**Chicago/Turabian Style**

Yu, Yun-Shuai, and Yeong-Sheng Chen.
2020. "A Measurement-Based Frame-Level Error Model for Evaluation of Industrial Wireless Sensor Networks" *Sensors* 20, no. 14: 3978.
https://doi.org/10.3390/s20143978