# Decision Making of Software Release Time at Different Confidence Intervals with Ohba’s Inflection S-Shape Model

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Ohba’s DRGM with Stochastic Differential Equation

#### 3.1. Model Development

- $a$: the potential errors number are hidden in the system without any software debugging process;
- $m\left(t\right)$: the mean is calculated with the expected number of detected errors in the testing time range $\left(0,t\right)$;
- $\mathsf{\Phi}\left(t\right)$: the function of the residual error in a system at the testing time t and defined as $\mathsf{\Phi}\left(t\right)=a-m\left(t\right)$;
- $D\left(t\right)$: the error detection rate at testing time t;
- $\psi \left(t\right)$: the continuous-time stochastic process that indicates the magnitude of irregular fluctuations from the error detection rate $D\left(t\right)$;
- $\sigma $: the standard deviation of $\psi \left(t\right)$.

#### 3.2. Estimating Parameters

#### 3.3. Estimating Confidence Intervals of Mean and Software Reliability

#### 3.4. Model Validation

## 4. Decision with Confidence Levels

- (1)
- Setup cost ($StC$) for testing concerning necessary equipment and initial investment before the testing project begins;
- (2)
- Routine expense ($RtC\left({\theta}_{Rt},T\right)$) for testing including salary, insurance, rent, and so on during a planned testing period $\left[0,T\right]$. ${\theta}_{R}$ denotes the routine expense per unit time, and the routine expense is calculated by ${\theta}_{Rt}T$;
- (3)
- Debugging expense ($DC\left({\theta}_{E},{\xi}_{E},m\left(T\right)\right)$) for removing software errors during a planned testing period $\left[0,T\right]$. The estimation of the expense is related to the expense of omitting an error per unit time ${\theta}_{E}$ and the average required time to delete an error ${\xi}_{E}$. Therefore, the debugging expense is calculated by ${\rho}_{E}m\left(T\right){\xi}_{E}$;
- (4)
- The cost of risk of a software failure after its release ($RkC\left({\theta}_{Rk},R\left(x/T\right)\right)$ is estimated by ${\theta}_{Rk}\left(1-R\left(x/T\right)\right)$. The parameter ${\theta}_{Rk}$ is calculated by estimating how much risk cost for users or customers is caused by the 1% loss of software reliability at release time $T$;
- (5)
- Opportunity cost ($OpC\left({\theta}_{O},{\varpi}_{1},{\varpi}_{2},T\right)$), as tangible and intangible losses caused by postponing software release, is defined as ${\theta}_{O}{\left({\varpi}_{1}+T\right)}^{{\varpi}_{2}}$ in this study. ${\varpi}_{1}$ and ${\varpi}_{2}$ are parameters for the power-law function, estimated by marketing experts. ${\theta}_{O}$ denotes the scale for base opportunity cost;
- (6)
- Minimal requirement of software reliability ${R}_{0}$ is a standard indicator for the requirement of users or customers for which the operation of a software system must meet.

## 5. Computerized Implementation Architecture

#### 5.1. Model Development

#### 5.2. System Design and Operation

## 6. Discussions

## 7. Conclusions

- (1)
- High-performance computing capability is needed for numerical analyses to obtain the results in a tolerable period. In general, workstation-class computers are required to solve the problem of this study.
- (2)
- Change-point problems of SRGM cannot be solved by the proposed model. During debugging or testing, factors can be changed, possibly leading to an increase or decrease in the failure rate.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Confidence interval (95%) and the fitting result for Ohba’s inflection model (the proposed model).

Model | Calculation of Error Detection Rate and Mean Value Function |
---|---|

Goel and Okumoto’s model | $D\left(t\right)=\beta $ $m\left(t\right)=a\left(1-{e}^{-\beta t}\right)$ |

Delayed, S-shaped model (Yamada) | $D\left(t\right)=\frac{{\beta}^{2}t}{1+\beta t}$ $m\left(t\right)=a\left(1-\left(1+\beta t\right){e}^{-\beta t}\right)$ |

Musa’s exponential model | $D\left(t\right)=\frac{\gamma}{n\kappa}$ $m\left(t\right)=a\left(1-{e}^{-\left(\frac{\gamma t}{n\kappa}\right)}\right)$ |

Ohba’s inflection S-shaped model | $D\left(t\right)=\frac{\beta}{1+\gamma {e}^{-\beta t}}$ $m\left(t\right)=a\left(\frac{1-{e}^{-\beta t}}{1+\gamma {e}^{-\beta t}}\right)$ |

Dataset | Literature | Testing Dataset | Reference |
---|---|---|---|

(1) | Zhang and Pham (1998) | Failure dataset from Misra system | [25] |

(2) | Shyur (2003) | Failure dataset from Misra system | [26] |

(3) | Hossain and Dahiya (1993) | Failure dataset from NTDS system | [27] |

(4) | Pham and Zhang (2003) | Failure dataset from Tandem software | [9] |

(5) | Jeske and Zhang (2005) | Failure dataset from wireless data service system | [28] |

(6) | Zhang and Pham (2006) | Failure dataset from telecommunication system | [29] |

Testing Dataset | Goel and Okumoto Model | Yamada’s Delayed S-Shaped Model | Musa’s Exponential Model | Ohba Inflection S-Shaped Model (Proposed Model) |
---|---|---|---|---|

(1) | $\widehat{a}$= 135.891 $\widehat{\beta}$= 0.138 R ^{2} = 0.966$\widehat{\sigma}$= 0.079 | $\widehat{a}$= 136.710 $\widehat{\beta}$= 0.265 R ^{2} = 0.808$\widehat{\sigma}$= 0.128 | $\widehat{a}$= 135.96
$\widehat{\gamma}$= 3.731 n = 144.31 $\widehat{\kappa}$= 0.184 R ^{2} = 0.965$\widehat{\sigma}$= 0.080 | $\widehat{a}$= 135.96
$\widehat{\beta}$= 0.138 $\widehat{\gamma}$= 0.001 R ^{2} = 0.966$\widehat{\sigma}$= 0.079 |

(2) | $\widehat{a}$= 164.47 $\widehat{\beta}$= 0.063 R ^{2} = 0.976$\widehat{\sigma}$= 0.0273 | $\widehat{a}$= 148.19 $\widehat{\beta}$= 0.174546 R ^{2} = 0.948113$\widehat{\sigma}$= 0.0601241 | $\widehat{a}$= 165.61
$\widehat{\gamma}$= 1.677 n = 156.49 $\widehat{\kappa}$= 0.166 R ^{2} = 0.975$\widehat{\sigma}$= 0.0267 | $\widehat{a}$= 184.88
$\widehat{\beta}$= 0.071 $\widehat{\gamma}$= 0.556 R ^{2} = 0.990$\widehat{\sigma}$= 0.016 |

(3) | $\widehat{a}$= 31.19 $\widehat{\beta}$= 0.070 R ^{2} = 0.895$\widehat{\sigma}$= 0.052 | $\widehat{a}$= 25.68 $\widehat{\beta}$= 0.211924 R ^{2} = 0.964012$\widehat{\sigma}$= 0.061593 | $\widehat{a}$= 31.27
$\widehat{\gamma}$= 1.88 n = 25.43 $\widehat{\kappa}$= 1.029 R ^{2} = 0.894$\widehat{\sigma}$= 0.053 | $\widehat{a}$= 24.54
$\widehat{\beta}$= 0.248 $\widehat{\gamma}$= 4.78 R ^{2} = 0.964$\widehat{\sigma}$= 0.064 |

(4) | $\widehat{a}$= 122.64 $\widehat{\beta}$= 0.017 R ^{2} = 0.987$\widehat{\sigma}$= 0.011 | $\widehat{a}$= 101.90 $\widehat{\beta}$= 0.050708 R ^{2}= 0.947729$\widehat{\sigma}$= 0.040946 | $\widehat{a}$= 122.77
$\widehat{\gamma}$= 1.850 n = 106 $\widehat{\kappa}$= 1 R ^{2} = 0.988$\widehat{\sigma}$= 0.011 | $\widehat{a}$= 121.61
$\widehat{\beta}$= 0.020 $\widehat{\gamma}$= 0.275 R ^{2} = 0.990$\widehat{\sigma}$= 0.011 |

(5) | $\widehat{a}$= 22.86 $\widehat{\beta}$= 0.542 R ^{2}= 0.984$\widehat{\sigma}$= 0.108 | $\widehat{a}$= 21.76 $\widehat{\beta}$= 1.361881 R ^{2} = 0.964143$\widehat{\sigma}$= 0.343911 | $\widehat{a}$= 22.72
$\widehat{\gamma}$= 3.76511 n= 23.41 $\widehat{\kappa}$= 0.291 R ^{2} = 0.986$\widehat{\sigma}$= 0.106 | $\widehat{a}$= 21.88
$\widehat{\beta}$= 0.788 $\widehat{\gamma}$= 0.486 R ^{2} = 0.987$\widehat{\sigma}$= 0.210 |

(6) | $\widehat{a}$= 134.41 $\widehat{\beta}$= 0.098 R ^{2} = 0.864$\widehat{\sigma}$= 0.078 | $\widehat{a}$= 134.82 $\widehat{\beta}$= 0.246786 R ^{2} = 0.974$\widehat{\sigma}$= 0.033 | $\widehat{a}$= 133.19
$\widehat{\gamma}$= 1.062 N = 105 $\widehat{\kappa}$= 0.1 R ^{2} = 0.865$\widehat{\sigma}$= 0.081 | $\widehat{a}$= 111.68
$\widehat{\beta}$= 0.468 $\widehat{\gamma}$= 13.498 R ^{2} = 0.990$\widehat{\sigma}$= 0.038 |

**Table 4.**Values of $R\left(x/T\right),TC\left(T\right),{R}_{LB}^{CR}\left(x/T\right)$ and $T{C}_{LB}^{CR}\left(T\right)$ vs. testing time.

Average Case | Worst Case (Confidence Level = 0.95) | ||||
---|---|---|---|---|---|

T (months) | $\mathit{R}\left(\mathit{x}/\mathit{T}\right)$ | $\mathit{E}\left[\mathit{C}\left(\mathit{T}\right)\right]$ | T (Months) | ${\mathit{R}}_{\mathit{L}\mathit{B}}^{\mathit{C}\mathit{R}}\left(\mathit{x}/\mathit{T}\right)$ | ${\mathit{E}}_{\mathit{L}\mathit{B}}^{\mathit{C}\mathit{R}}\left[\mathit{C}\left(\mathit{T}\right)\right]$ |

3 | 0.830 | 281,040 | 3 | 0.723 | 306,685 |

3.05 | 0.842 | 279,303 | 3.05 | 0.740 | 303,711 |

3.1 | 0.854 | 277,754 | 3.1 | 0.757 | 300,946 |

3.15 | 0.864 | 276,385 | 3.15 | 0.772 | 298,387 |

3.2 | 0.874 | 275,186 | 3.2 | 0.787 | 296,029 |

3.25 | 0.883 | 274,149 | 3.25 | 0.801 | 293,867 |

3.3 | 0.891 | 273,264 | 3.3 | 0.814 | 291,896 |

3.35 | 0.899 | 272,524 | 3.35 | 0.826 | 290,109 |

3.4 | 0.907 | 271,921 | 3.4 | 0.838 | 288,499 |

3.45 | 0.913 | 271,446 | 3.45 | 0.849 | 287,060 |

3.5 | 0.920 | 271,091 | 3.5 | 0.859 | 285,783 |

3.55 | 0.926 | 270,850 | 3.55 | 0.868 | 284,663 |

3.6 | 0.931 | 270,716 | 3.6 | 0.877 | 283,691 |

3.65 | 0.936 | 270,682 * | 3.65 | 0.886 | 282,860 |

3.7 | 0.941 | 270,741 | 3.7 | 0.894 | 282,164 |

3.75 | 0.946 | 270,888 | 3.75 | 0.901 | 281,595 |

3.8 | 0.950 | 271,117 | 3.8 | 0.908 | 281,147 |

3.85 | 0.953 | 271,422 | 3.85 | 0.914 | 280,813 |

3.9 | 0.957 | 271,800 | 3.9 | 0.920 | 280,587 |

3.95 | 0.960 | 272,244 | 3.95 | 0.926 | 280,462 |

4 | 0.963 | 272,752 | 4 | 0.931 | 280,434 * |

4.05 | 0.966 | 273,318 | 4.05 | 0.936 | 280,496 |

4.1 | 0.971 | 273,938 | 4.1 | 0.941 | 280,642 |

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**MDPI and ACS Style**

Chang, T.-C.; Lin, Y.; Shi, K.; Meen, T.-H.
Decision Making of Software Release Time at Different Confidence Intervals with Ohba’s Inflection S-Shape Model. *Symmetry* **2022**, *14*, 593.
https://doi.org/10.3390/sym14030593

**AMA Style**

Chang T-C, Lin Y, Shi K, Meen T-H.
Decision Making of Software Release Time at Different Confidence Intervals with Ohba’s Inflection S-Shape Model. *Symmetry*. 2022; 14(3):593.
https://doi.org/10.3390/sym14030593

**Chicago/Turabian Style**

Chang, Ting-Cheng, Ying Lin, Kunquan Shi, and Teen-Hang Meen.
2022. "Decision Making of Software Release Time at Different Confidence Intervals with Ohba’s Inflection S-Shape Model" *Symmetry* 14, no. 3: 593.
https://doi.org/10.3390/sym14030593