# Extreme Path Delay Estimation of Critical Paths in Within-Die Process Fluctuations Using Multi-Parameter Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. Methods

#### 3.1. The Pearson Distribution Family

#### 3.2. The Metalog Distribution Family

#### 3.3. Comparison of the Distributions

#### 3.4. Goodness-of-Fit Statistics

#### 3.5. WID Maximum Critical Path Delay Distribution

## 4. Simulations and Results

#### 4.1. Origins of the Data

- Dataset 1 (DS 1) = Hold SS 0.51 V ${\mathrm{RC}}_{\mathrm{worst}}$;
- Dataset 2 (DS 2) = ${\mathrm{Setup}}_{\mathrm{win}}$ SS 0.51 V ${\mathrm{RC}}_{\mathrm{worst}}$;
- Dataset 3 (DS 3) = ${\mathrm{Setup}}_{\mathrm{critical}\phantom{\rule{4.pt}{0ex}}\mathrm{path}}$ SS 0.51 V ${\mathrm{RC}}_{\mathrm{worst}}$.

#### 4.2. Goodness-of-Fit Statistics of the Data

#### 4.3. Singular Path Delay Confidence Intervals

#### 4.4. Multiple Critical Path Delay Distributions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Pearson Type IV Distribution

## References

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**Figure 1.**Probability distributions fitted to simulated path delay data and their respective $(1-p)$-quantiles.

**Figure 2.**Histogram of the normalized average path delays of $5\times {10}^{4}$ setup paths from STA.

**Figure 3.**Heat map of the path correlation of ${10}^{4}$ setup paths from STA showing large sparsity and therefore small correlation between the paths.

**Figure 4.**Maximum critical path delay densities for different values of N in DS 3, when each path has one of the three distributions. (

**a**) Normal distribution. (

**b**) Pearson type IV distribution. (

**c**) Six-term metalog distribution.

KS | CM | AD | ||

Normal | 0.039 | 0.585 | 4.395 | |

DS 1 | Pearson IV | 0.027 | 0.096 | 0.586 |

Metalog 6 | 0.020 | 0.055 | 0.460 | |

Normal | 0.061 | 0.992 | 6.016 | |

DS 2 | Pearson IV | 0.016 | 0.028 | 0.198 |

Metalog 6 | 0.011 | 0.014 | 0.102 | |

Normal | 0.022 | 0.107 | 0.918 | |

DS 3 | Pearson IV | 0.017 | 0.051 | 0.370 |

Metalog 6 | 0.016 | 0.025 | 0.211 |

**Table 2.**99.7% confidence intervals of the $(1-p)$-quantiles (p-quantiles) for DS 2–3 (DS 1), in nanoseconds.

$p=1.35\times {10}^{-3}$ | $p=3.17\times {10}^{-5}$ | ||

Normal | (5.77, 6.06) | (5.04, 5.43) | |

DS 1 | Pearson IV | (6.02, 6.52) | (5.04, 6.23) |

Metalog 6 | (5.90, 6.61) | (4.71, 6.36) | |

Normal | (77.16, 79.48) | (81.07, 84.01) | |

DS 2 | Pearson IV | (80.48, 89.28) | (87.63, 115.41) |

Metalog 6 | (78.62, 85.13) | (82.96, 96.79) | |

Normal | (91.85, 93.57) | (94.91, 97.13) | |

DS 3 | Pearson IV | (92.37, 97.15) | (95.82, 107.94) |

Metalog 6 | (92.43, 98.09) | (96.12, 108.12) |

**Table 3.**The maximum (minimum) path delays of $N={10}^{2}$-independent critical paths corresponding to the $(1-p)$-quantile (p-quantile) for DS 2–3 (DS 1), in nanoseconds.

$p=1.35\times {10}^{-3}$ | $p=3.17\times {10}^{-5}$ | ||

Normal | 8.98 | 8.82 | |

DS 1 | Pearson IV | 9.04 | 8.83 |

Metalog 6 | 9.00 | 8.81 | |

Normal | 83.41 | 86.72 | |

DS 2 | Pearson IV | 100.77 | 118.30 |

Metalog 6 | 92.60 | 101.04 | |

Normal | 96.65 | 99.24 | |

DS 3 | Pearson IV | 101.99 | 108.37 |

Metalog 6 | 103.28 | 109.87 |

**Table 4.**The maximum (minimum) path delays of $N={10}^{3}$-independent critical paths corresponding to the $(1-p)$-quantile (p-quantile) for DS 2–3 (DS 1), in nanoseconds.

$p=1.35\times {10}^{-3}$ | $p=3.17\times {10}^{-5}$ | ||

Normal | 9.63 | 9.52 | |

DS 1 | Pearson IV | 10.04 | 9.85 |

Metalog 6 | 9.97 | 9.78 | |

Normal | 85.50 | 88.53 | |

DS 2 | Pearson IV | 111.00 | 131.49 |

Metalog 6 | 97.78 | 106.17 | |

Normal | 98.29 | 100.65 | |

DS 3 | Pearson IV | 105.84 | 112.60 |

Metalog 6 | 107.32 | 113.87 |

**Table 5.**The maximum (minimum) path delays of $N={10}^{4}$-independent critical paths corresponding to the $(1-p)$-quantile (p-quantile) for DS 2–3 (DS 1), in nanoseconds.

$p=1.35\times {10}^{-3}$ | $p=3.17\times {10}^{-5}$ | ||

Normal | 10.12 | 10.03 | |

DS 1 | Pearson IV | 11.05 | 10.85 |

Metalog 6 | 10.91 | 10.73 | |

Normal | 87.40 | 90.21 | |

DS 2 | Pearson IV | 122.94 | 147.00 |

Metalog 6 | 102.96 | 110.87 | |

Normal | 99.77 | 101.96 | |

DS 3 | Pearson IV | 109.90 | 117.11 |

Metalog 6 | 111.36 | 117.55 |

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**MDPI and ACS Style**

Runolinna, M.; Turnquist, M.; Teittinen, J.; Ilmonen, P.; Koskinen, L.
Extreme Path Delay Estimation of Critical Paths in Within-Die Process Fluctuations Using Multi-Parameter Distributions. *J. Low Power Electron. Appl.* **2023**, *13*, 22.
https://doi.org/10.3390/jlpea13010022

**AMA Style**

Runolinna M, Turnquist M, Teittinen J, Ilmonen P, Koskinen L.
Extreme Path Delay Estimation of Critical Paths in Within-Die Process Fluctuations Using Multi-Parameter Distributions. *Journal of Low Power Electronics and Applications*. 2023; 13(1):22.
https://doi.org/10.3390/jlpea13010022

**Chicago/Turabian Style**

Runolinna, Miikka, Matthew Turnquist, Jukka Teittinen, Pauliina Ilmonen, and Lauri Koskinen.
2023. "Extreme Path Delay Estimation of Critical Paths in Within-Die Process Fluctuations Using Multi-Parameter Distributions" *Journal of Low Power Electronics and Applications* 13, no. 1: 22.
https://doi.org/10.3390/jlpea13010022