# A Comprehensive Analysis of Proportional Intensity-Based Software Reliability Models with Covariates

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. NHPP-Based Software Reliability Modeling

- (i)
- $N\left(0\right)=0$;
- (ii)
- $\left\{N\right(t),t\ge 0\}$ has independent increment;
- (iii)
- $\mathrm{Pr}\left\{N\right(t+\Delta t)-N(t)\ge 2\}=o(\Delta t)$;
- (iv)
- $\mathrm{Pr}\left\{N\right(t+\Delta t)-N(t)=1\}=\lambda \left(t\right)\Delta t+o(\Delta t)$,

- (i)
- Software faults are detected at independent and identically distributed (i.i.d.) random times with the non-degenerate cumulative distribution function (CDF), $F(t;\mathbf{\alpha})$, where $\mathbf{\alpha}$ is a free parameter vector.
- (ii)
- The total number of software faults remaining in software before testing, say, at time $t=0$, is a Poisson random variable with parameter $\omega \phantom{\rule{3.33333pt}{0ex}}(>0)$.

## 4. Proportional Intensity Model

#### 4.1. Model Description

#### 4.2. Maximum Likelihood Estimation

## 5. Numerical Examples

#### 5.1. Goodness-of-Fit Performance

#### 5.2. Predictive Performance

**Case I:**- All the test/development metric data are completely known through the testing phase in advance, so the software testing expenditures are exactly given in the testing.
**Case II:**- The test/development metrics data do not change from the observation point in the future.
**Case III:**- The test/development metrics data experienced in the future are regarded as independent random variables and predictable by any statistical method.

#### 5.3. Software Reliability Assessment

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Behavior of the predicted cumulative number of software faults with PI-SRMs and common NHPP-based SRM in GDS1 (50% observation point).

**Figure 4.**Behavior of the predicted cumulative number of software faults with PI-SRMs and common NHPP-based SRM in GDS1 (80% observation point).

Models | $\mathit{\lambda}(\mathit{t};\mathit{\theta})\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}\mathit{F}(\mathit{t};\mathit{\alpha})$ |
---|---|

Exponential distribution (exp) [4] | $\lambda (t;\mathbf{\theta})=\omega b{e}^{-bt}$ $F(t;\mathbf{\alpha})=1-{e}^{-bt}$ |

Gamma distribution (gamma) [5,6] | $\lambda (t;\mathbf{\theta})=\omega \frac{{e}^{-\frac{t}{c}}{\left(\frac{t}{c}\right)}^{b-1}}{c\Gamma \left(b\right)}$ $F(t;\mathbf{\alpha})={\int}_{0}^{t}\frac{{c}^{b}{s}^{b-1}{e}^{-cs}}{\Gamma \left(b\right)}ds$ |

Pareto distribution (pareto) [7] | $\lambda (t;\mathbf{\theta})=\frac{\omega bc{\left(\frac{c}{c+t}\right)}^{b-1}}{{(c+t)}^{2}}$ $F(t;\mathbf{\alpha})=1-{\left(\frac{b}{t+b}\right)}^{c}$ |

Truncated normal distribution (tnorm) [10] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{{(c-t)}^{2}}{2{b}^{2}}}}{\sqrt{2\pi}b\left(1-\frac{1}{2}erfc\left(\frac{c}{\sqrt{2}b}\right)\right)}$ $F(t;\mathbf{\alpha})=\frac{1}{\sqrt{2\pi}b}{\int}_{-\infty}^{t}{e}^{-\frac{{(s-c)}^{2}}{2{b}^{2}}}ds$ |

Log-normal distribution (lnorm) [10,11] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{{(c-log\left(t\right))}^{2}}{2{b}^{2}}}}{\sqrt{2\pi}btt}$ $F(t;\mathbf{\alpha})=\frac{1}{\sqrt{2\pi}b}{\int}_{-\infty}^{t}{e}^{-\frac{{(s-c)}^{2}}{2{b}^{2}}}ds$ |

Truncated logistic distribution (tlogist) [8] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{t-c}{b}}}{b\left(1-\frac{1}{{e}^{c/b}+1}\right){\left({e}^{-\frac{t-c}{b}}+1\right)}^{2}}$ $F(t;\mathbf{\alpha})=\frac{1-{e}^{-bt}}{1+c{e}^{-bt}}$ |

Log-logistic distribution (llogist) [9] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{log\left(t\right)-c}{b}}}{bt{\left({e}^{-\frac{log\left(t\right)-c}{b}}+1\right)}^{2}}$ $F(t;\mathbf{\alpha})=\frac{{\left(bt\right)}^{c}}{1+{\left(bt\right)}^{c}}$ |

Truncated extreme-value maximum distribution (txvmax) [12] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{t-c}{b}-{e}^{-\frac{t-c}{b}}}}{b\left(1-{e}^{-{e}^{c/b}}\right)}$ $F(t;\mathbf{\alpha})={e}^{-{e}^{-\frac{t-c}{b}}}$ |

Log-extreme-value max maximum distribution (lxvmax) [12] | $\lambda (t;\mathbf{\theta})=\frac{\omega c{e}^{-{\left(\frac{t}{b}\right)}^{-c}}{\left(\frac{t}{b}\right)}^{-c-1}}{b}$ $F(t;\mathbf{\alpha})={e}^{-{\left(\frac{t}{b}\right)}^{-c}}$ |

Truncated extreme-value minimum distribution (txvmin) [12] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{-c-t}{b}-{e}^{-\frac{-c-t}{b}}+{e}^{c/b}}}{b}$ $F(t;\mathbf{\alpha})={e}^{-{e}^{-\frac{t-c}{b}}}$ |

Log-extreme-value minimum distribution (lxvmin) [46] | $\lambda (t;\mathbf{\theta})=\frac{\omega {e}^{-\frac{-c-log\left(t\right)}{b}-{e}^{-\frac{-c-log\left(t\right)}{b}}}}{bt}$ $F(t;\mathbf{\alpha})={e}^{-{e}^{-}\frac{t-c}{b}}$ |

Data | No. Faults | Testing Days |
---|---|---|

GDS1 | 136 | 21 |

GDS2 | 54 | 17 |

GDS3 | 38 | 14 |

GDS4 | 53 | 16 |

Metrics Data: | Failure identification work, Execution time, Computer time-failure identification. |

$\mathit{g}({\mathit{x}}_{\mathbf{kl}};\mathit{\beta})(\mathit{l}=1,2,3)$ | |
---|---|

Combination I | $exp({\beta}_{0}+{x}_{k1}{\beta}_{1})$ |

Combination II | $exp({\beta}_{0}+{x}_{k2}{\beta}_{2})$ |

Combination III | $exp({\beta}_{0}+{x}_{k3}{\beta}_{3})$ |

Combination IV | $exp({\beta}_{0}+{x}_{k1}{\beta}_{1}+{x}_{k2}{\beta}_{2})$ |

Combination V | $exp({\beta}_{0}+{x}_{k1}{\beta}_{1}+{x}_{k3}{\beta}_{3})$ |

Combination VI | $exp({\beta}_{0}+{x}_{k2}{\beta}_{2}+{x}_{k3}{\beta}_{3})$ |

Combination VII | $exp({\beta}_{0}+{x}_{k1}{\beta}_{1}+{x}_{k2}{\beta}_{2}+{x}_{k3}{\beta}_{3})$ |

${x}_{k1}:$ Execution time, ${x}_{k2}:$ Failure identification work. ${x}_{k3}:$ Computer time-failure identification. |

(i) Best proportional intensity model (cumulative metrics data) | ||||
---|---|---|---|---|

Model | AIC | MSE | $\widehat{\mathbf{\beta}}$ | |

GDS1 | tlogist-VI | 110.114 | 0.470 | $\widehat{{\beta}_{0}}=-2.5903,\widehat{{\beta}_{2}}=-0.0805,\widehat{{\beta}_{3}}=0.0277$ |

GDS2 | tlogist-III | 69.785 | 0.282 | $\widehat{{\beta}_{0}}=1.7326,\widehat{{\beta}_{3}}=0.1406$ |

GDS3 | txvmin-II | 57.281 | 0.289 | $\widehat{{\beta}_{0}}=-3.7048,\widehat{{\beta}_{2}}=1.2197$ |

GDS4 | exp-I | 81.059 | 0.612 | $\widehat{{\beta}_{0}}=4.6132,\widehat{{\beta}_{1}}=-0.1659$ |

(ii) Best proportional intensity model (non-cumulative metrics data) | ||||

GDS1 | txvmin-II | 109.015 | 0.721 | $\widehat{{\beta}_{0}}=2.9503,\widehat{{\beta}_{2}}=0.0206$ |

GDS2 | llogist-II | 67.352 | 0.261 | $\widehat{{\beta}_{0}}=-0.4155,\widehat{{\beta}_{2}}=0.0447$ |

GDS3 | gamma-II | 50.696 | 0.221 | $\widehat{{\beta}_{0}}=0.6061,\widehat{{\beta}_{2}}=1.1493$ |

GDS4 | exp-VI | 81.131 | 0.450 | $\widehat{{\beta}_{0}}=3.8840,\widehat{{\beta}_{2}}=-0.2963,\widehat{{\beta}_{3}}=0.8060$ |

(iii) Best SRATS (no metrics data) | ||||

GDS1 | tlogist | 116.891 | 0.820 | - |

GDS2 | llogist | 73.053 | 0.501 | - |

GDS3 | lxvmax | 61.694 | 0.481 | - |

GDS4 | txvmin | 79.761 | 0.530 | - |

GDS1 | ||
---|---|---|

Best model | PMSE | |

Case I (cumulative) | tlogist-III | 6.409 |

Case I (non-cumulative) | tlogist-II | 4.014 |

Case II (cumulative) | lxvmax-II | 2.160 |

Case II (non-cumulative) | txvmax-IV | 4.931 |

Case III (cumulative): Linear regression | exp-IV | 4.146 |

Case III (cumulative): Exponential regression | txvmin-V | 19.213 |

Case III (non-cumulative): Linear regression | txvmax-II | 3.916 |

SRATS | tnorm | 3.408 |

Best model | PMSE | |

Case I (cumulative) | tlogist-II | 0.816 |

Case I (non-cumulative) | tnorm-III | 0.799 |

Case II (cumulative) | gamma-II | 0.742 |

Case II (non-cumulative) | txvmax-II | 0.407 |

Case III (cumulative): Linear regression | tlogist-IV | 0.616 |

Case III (cumulative): Exponential regression | tnorm-III | 1.644 |

Case III (non-cumulative): Linear regression | tlogist-IV | 0.780 |

SRATS | tlogist | 1.769 |

GDS3 | ||

Best model | PMSE | |

Case I (cumulative) | tlogist-II | 2.676 |

Case I (non-cumulative) | txvmax-III | 0.481 |

Case II (cumulative) | exp-VII | 0.467 |

Case II (non-cumulative) | pareto-VI | 1.506 |

Case III (cumulative): Linear regression | llogist-II | 0.748 |

Case III (cumulative): Exponential regression | lxvmax-VI | 1.842 |

Case III (non-cumulative): Linear regression | lxvmax-VII | 1.769 |

SRATS | exp | 1.836 |

GDS4 | ||

Best model | PMSE | |

Case I (cumulative) | tlogist-III | 2.088 |

Case I (non-cumulative) | pareto-II | 1.506 |

Case II (cumulative) | exp-I | 0.495 |

Case II (non-cumulative) | tnorm-VI | 0.425 |

Case III (cumulative): Linear regression | txvmax-VI | 1.139 |

Case III (cumulative): Exponential regression | exp-II | 0.688 |

Case III (non-cumulative): Linear regression | lxvmin-I | 0.703 |

SRATS | tlogist | 1.754 |

GDS1 | ||
---|---|---|

Best model | PMSE | |

Case I (cumulative) | tnorm-II | 2.482 |

Case I (non-cumulative) | txvmax-III | 1.768 |

Case II (cumulative) | txvmax-VII | 2.142 |

Case II (non-cumulative) | txvmax-V | 2.903 |

Case III (cumulative): Linear regression | tnorm-II | 1.033 |

Case III (cumulative): Exponential regression | tlogist-VII | 3.159 |

SRATS | txvmin | 1.218 |

GDS2 | ||

Best model | PMSE | |

Case I (cumulative) | pareto-IV | 0.488 |

Case I (non-cumulative) | gamma-V | 0.277 |

Case II (cumulative) | lnorm-VII | 0.399 |

Case II (non-cumulative) | pareto-I | 0.466 |

Case III (cumulative): Linear regression | exp-IV | 0.455 |

Case III (cumulative): Exponential regression | llogist-VI | 0.499 |

Case III (non-cumulative): Linear regression | llogist-IV | 0.508 |

SRATS | lnorm | 0.531 |

GDS3 | ||

Best model | PMSE | |

Case I (cumulative) | tnorm-II | 0.326 |

Case I (non-cumulative) | txvmax-II | 0.150 |

Case II (cumulative) | txvmax-IV | 0.330 |

Case II (non-cumulative) | lxvmax-II | 0.982 |

Case III (cumulative): Linear regression | lxvmin-I | 0.340 |

Case III (cumulative): Exponential regression | txvmin-VI | 1.484 |

Case III (non-cumulative): Linear regression | pareto-III | 0.293 |

SRATS | exp | 0.295 |

GDS4 | ||

Best model | PMSE | |

Case I (cumulative) | exp-I | 0.213 |

Case I (non-cumulative) | lxvmin-V | 0.227 |

Case II (cumulative) | tnorm-IV | 0.220 |

Case II (non-cumulative) | tnorm-II | 0.206 |

Case III (cumulative): Linear regression | tlogist-II | 0.207 |

Case III (cumulative): Exponential regression | lxvmax-III | 0.273 |

Case III (non-cumulative): Linear regression | tlogist-VII | 0.220 |

SRATS | gamma | 0.230 |

(i) Best proportional intensity model (cumulative metrics data) | ||
---|---|---|

Model | Reliability | |

GDS1 | tlogist-VI | 2.969 × 10${}^{-2}$ |

GDS2 | tlogist-III | 9.260 × 10${}^{-1}$ |

GDS3 | txvmin-II | 9.998 × 10${}^{-1}$ |

GDS4 | exp-I | 5.455 × 10${}^{-3}$ |

(ii) Best proportional intensity model (non-cumulative metrics data) | ||

GDS1 | txvmin-II | 4.393 × 10${}^{-\mathbf{1}}$ |

GDS2 | llogist-II | 1.984 × 10${}^{-2}$ |

GDS3 | gamma-II | 2.945 × 10${}^{-1}$ |

GDS4 | exp-VI | 4.324 × 10${}^{-1}$ |

(iii) Best SRATS (no metrics data) | ||

GDS1 | tlogist | 6.977 × 10${}^{-5}$ |

GDS2 | llogist | 4.152 × 10${}^{-3}$ |

GDS3 | lxvmax | 7.236 × 10${}^{-5}$ |

GDS4 | txvmin | 9.559 × 10${}^{-1}$ |

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**MDPI and ACS Style**

Li, S.; Dohi, T.; Okamura, H.
A Comprehensive Analysis of Proportional Intensity-Based Software Reliability Models with Covariates. *Electronics* **2022**, *11*, 2353.
https://doi.org/10.3390/electronics11152353

**AMA Style**

Li S, Dohi T, Okamura H.
A Comprehensive Analysis of Proportional Intensity-Based Software Reliability Models with Covariates. *Electronics*. 2022; 11(15):2353.
https://doi.org/10.3390/electronics11152353

**Chicago/Turabian Style**

Li, Siqiao, Tadashi Dohi, and Hiroyuki Okamura.
2022. "A Comprehensive Analysis of Proportional Intensity-Based Software Reliability Models with Covariates" *Electronics* 11, no. 15: 2353.
https://doi.org/10.3390/electronics11152353