# Are Infinite-Failure NHPP-Based Software Reliability Models Useful?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Non-Homogeneous Poisson processes

#### 2.1. Preliminary

- NHPP has independent increments, so the number of occurrences in a specific time interval depends on only the current time $t$ and not on the past history of the process, which is also known as the Markov property.
- The initial state of the process is given by $N\left(0\right)=0$.
- The occurrence probability of one event in a given time period $\left[t,t+\mathsf{\Delta}t\right)$ for an NHPP is defined by $\mathrm{Pr}\{N\left(t+\mathsf{\Delta}t\right)-N\left(t\right)=1\}=o\left(\mathsf{\Delta}t\right)+\mathsf{\lambda}\left(t\right)\mathsf{\Delta}t$. $\mathsf{\lambda}\left(t\right)$ is an absolutely continuous function, and is named the intensity function of NHPP. $\mathsf{\Delta}t$ is recognized as an infinitesimal period of time.
- NHPP has negligible probability for two or more events occurring in $\left[t,t+\mathsf{\Delta}t\right)$, i.e., $\mathrm{Pr}\{N\left(t+\mathsf{\Delta}t\right)-N\left(t\right)\ge 2\}=o\left(\mathsf{\Delta}t\right)$, where $\underset{\mathsf{\Delta}t\to 0}{\mathrm{lim}}\frac{o\left(\mathsf{\Delta}t\right)}{\mathsf{\Delta}t}=0$ and $o\left(\mathsf{\Delta}t\right)$ is the higher-order term of $\mathsf{\Delta}t$.
- As a typical Markov process, the Kolmogorov forward equations of NHPP can be written as$$\frac{d}{dt}{P}_{0}\left(t\right)=-\mathsf{\lambda}\left(t;\theta \right){P}_{0}\left(t\right),$$$$\frac{d}{dt}{P}_{n}\left(t\right)=\mathsf{\lambda}\left(t;\mathit{\theta}\right){P}_{n-1}\left(t\right)-\mathsf{\lambda}\left(t;\mathit{\theta}\right){P}_{n}\left(t\right),n=1,2,\cdots ,$$$${P}_{n}\left(t\right)=\mathrm{exp}\left(-M\left(t;\mathit{\theta}\right)\right)\frac{{\left\{M\left(t;\mathit{\theta}\right)\right\}}^{n}}{n!},n=0,1,2,\dots .$$

#### 2.2. NHPP-Based SRMs

#### 2.2.1. Finite-Failure (Type-I) NHPP-Based SRMs

#### 2.2.2. Infinite-Failure (Type-II) NHPP-Based SRMs

#### 2.3. Parameter Estimation

#### 2.3.1. Software Fault-Count Time-Domain Data

#### 2.3.2. Software Fault-Count Time-Interval Data (Group Data)

## 3. Performance Comparison

#### 3.1. Datasets

#### 3.2. Goodness-of-Fit Performance

#### 3.3. Predictive Performance

#### 3.4. Software Reliability Assessment

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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SRM & Time Distribution | $\mathit{F}\left(\mathit{t};\mathit{\alpha}\right)$ | $\mathit{M}\left(\mathit{t};\mathit{\theta}\right)$ |
---|---|---|

Exp [2] (Exponential distribution) | $1-\mathrm{exp}\left(-{\mu}_{1}t\right)$ | ${\mu}_{0}F\left(t;\mathit{\alpha}\right)$ |

Gamma [9,10] (Gamma distribution) | $\underset{0}{\overset{t}{{\displaystyle \int}}}\frac{{\mu}_{2}^{{\mu}_{1}}{s}^{{\mu}_{2}-1}\mathrm{exp}\left(-{\mu}_{2}s\right)}{\mathsf{\Gamma}\left({\mu}_{1}\right)}ds$ | ${\mu}_{0}F\left(t;\mathit{\alpha}\right)$ |

Pareto [11] (Pareto distribution) | $1-{\left(\frac{{\mu}_{1}}{t+{\mu}_{1}}\right)}^{{\mu}_{2}}$ | ${\mu}_{0}F\left(t;\mathit{\alpha}\right)$ |

Tnorm [3] (Truncated normal distribution) | $\frac{1}{\sqrt{2\mathsf{\pi}{\mu}_{1}}}\underset{-\infty}{\overset{t}{{\displaystyle \int}}}\mathrm{exp}\left(-\frac{{\left(s-{\mu}_{2}\right)}^{2}}{2{\mu}_{1}^{2}}\right)ds$ | ${\mu}_{0}\frac{F\left(t;\mathit{\alpha}\right)-F\left(0;\mathit{\alpha}\right)}{1-F\left(0;\mathit{\alpha}\right)}$ |

Tlogist [5] (Truncated logistic distribution) | $\frac{1-\mathrm{exp}\left(-{\mu}_{1}t\right)}{1+{\mu}_{2}\mathrm{exp}\left(-{\mu}_{2}t\right)}$ | ${\mu}_{0}\frac{F\left(t;\mathit{\alpha}\right)-F\left(0;\mathit{\alpha}\right)}{1-F\left(0;\mathit{\alpha}\right)}$ |

Txvmax [8] (Truncated extreme-value maximum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | ${\mu}_{0}\frac{F\left(t;\mathit{\alpha}\right)-F\left(0;\mathit{\alpha}\right)}{1-F\left(0;\mathit{\alpha}\right)}$ |

Txvmin [8] (Truncated extreme-value minimum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | ${\mu}_{0}\frac{F\left(0;\mathit{\alpha}\right)-F\left(t;\mathit{\alpha}\right)}{F\left(0;\mathit{\alpha}\right)}$ |

Lnorm [3,4] (Log-normal distribution) | $\frac{1}{\sqrt{2\pi {\mu}_{1}}}\underset{-\infty}{\overset{t}{{\displaystyle \int}}}\mathrm{exp}\left(-\frac{{\left(s-{\mu}_{2}\right)}^{2}}{2{\mu}_{1}^{2}}\right)ds$ | ${\mu}_{0}F\left(\mathrm{ln}t;\mathit{\alpha}\right)$ |

Llogist [6] (Log-logistic distribution) | $\frac{1-\mathrm{exp}\left(-{\mu}_{1}t\right)}{1+{\mu}_{2}\mathrm{exp}\left(-{\mu}_{2}t\right)}$ | ${\mu}_{0}F\left(\mathrm{ln}t;\mathit{\alpha}\right)$ |

Lxvmax [8] (Log-extreme-value maximum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | ${\mu}_{0}F\left(\mathrm{ln}t;\mathit{\alpha}\right)$ |

Lxvmin [7] (Log-extreme-value minimum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | ${\mu}_{0}\left(1-F\left(-\mathrm{ln}t;\mathit{\alpha}\right)\right)$ |

SRM & Time Distribution | $\mathit{F}\left(\mathit{t};\mathit{\alpha}\right)$ | $\mathit{M}\left(\mathit{t};\mathit{\theta}\right)$ |
---|---|---|

Exp (HPP) (Exponential distribution) | $1-\mathrm{exp}\left(-{\mu}_{1}t\right)$ | ${\mu}_{1}t$ |

Gamma (Gamma distribution) | $\underset{0}{\overset{t}{{\displaystyle \int}}}\frac{{\mu}_{2}^{{\mu}_{1}}{s}^{{\mu}_{2}-1}\mathrm{exp}\left(-{\mu}_{2}s\right)}{\mathsf{\Gamma}\left({\mu}_{1}\right)}ds$ | $\mathrm{ln}\left(\mathsf{\Gamma}\left({\mu}_{1}\right)\right)-\mathrm{ln}\left(\mathsf{\Gamma}\left({\mu}_{1},\frac{t}{{\mu}_{2}}\right)\right)$ |

Pareto (Musa-Okumoto) [15,16] (Pareto distribution) | $1-{\left(\frac{{\mu}_{1}}{t+{\mu}_{1}}\right)}^{{\mu}_{2}}$ | $-{\mu}_{2}\left(\mathrm{ln}\left({\mu}_{1}\right)-\mathrm{ln}\left({\mu}_{1}+t\right)\right)$ |

Tnorm (Truncated normal distribution) | $\frac{1}{\sqrt{2\pi {\mu}_{1}}}\underset{-\infty}{\overset{t}{{\displaystyle \int}}}\mathrm{exp}\left(-\frac{{\left(s-{\mu}_{2}\right)}^{2}}{2{\mu}_{1}^{2}}\right)ds$ | $\mathrm{ln}\left(\mathrm{erf}\left(\frac{{\mu}_{2}}{\sqrt{2{\mu}_{1}}}\right)+1\right)-\mathrm{ln}\left(\mathrm{erf}\left(\frac{{\mu}_{2}-t}{\sqrt{2{\mu}_{1}}}\right)+1\right)$ |

Tlogist (Truncated logistic distribution) | $\frac{1-\mathrm{exp}\left(-{\mu}_{1}t\right)}{1+{\mu}_{2}\mathrm{exp}\left(-{\mu}_{2}t\right)}$ | $\mathrm{ln}\left(\mathrm{exp}\left({\mu}_{2}/{\mu}_{1}\right)+\mathrm{exp}\left(t/{\mu}_{1}\right)\right)-\mathrm{ln}\left(\mathrm{exp}\left({\mu}_{2}/{\mu}_{1}\right)+1\right)$ |

Txvmax (Truncated extreme-value maximum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | $\mathrm{ln}\left(1-\mathrm{exp}\left(-\mathrm{exp}\left({\mu}_{2}/{\mu}_{1}\right)\right)\right)-\mathrm{ln}\left(1-\mathrm{exp}\left(-\mathrm{exp}\left(\frac{{\mu}_{2}-t}{{\mu}_{1}}\right)\right)\right)$ |

Cox-Lewis [22] (Truncated extreme-value minimum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | $-\mathrm{ln}\left(\mathrm{exp}\left(-\mathrm{exp}\left({\mu}_{2}/{\mu}_{1}\right)\left(\mathrm{exp}\left(t/{\mu}_{1}\right)-1\right)\right)\right)$ |

Lnorm (Log-normal distribution) | $\frac{1}{\sqrt{2\pi {\mu}_{1}}}\underset{-\infty}{\overset{t}{{\displaystyle \int}}}\mathrm{exp}\left(-\frac{{\left(s-{\mu}_{2}\right)}^{2}}{2{\mu}_{1}^{2}}\right)ds$ | $\mathrm{ln}\left(2\right)-\mathrm{ln}\left(\mathrm{erf}\left(\frac{{\mu}_{2}-\mathrm{ln}\left(t\right)}{\sqrt{2}{\mu}_{1}}\right)+1\right)$ |

Llogist (Log-logistic distribution) | $\frac{1-\mathrm{exp}\left(-{\mu}_{1}t\right)}{1+{\mu}_{2}\mathrm{exp}\left(-{\mu}_{2}t\right)}$ | $\mathrm{ln}\left(\mathrm{exp}\left({\mu}_{2}/{\mu}_{1}\right)+{t}^{1/{\mu}_{1}}\right)-{\mu}_{2}/{\mu}_{1}$ |

Lxvmax (Log-extreme-value maximum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | $-\mathrm{ln}\left(1-\mathrm{exp}\left(-\mathrm{exp}\left(\frac{{\mu}_{2}-\mathrm{ln}\left(t\right)}{{\mu}_{1}}\right)\right)\right)$ |

Power-law [12,13,14] (Log-extreme-value minimum distribution) | $\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{t-{\mu}_{2}}{{\mu}_{1}}\right)\right)$ | ${\mu}_{2}/{\mu}_{1}{t}^{1/{\mu}_{1}}$ |

Data Source | Nature of System | Testing Length (CPU Time) | Numbers of Detected Faults | |
---|---|---|---|---|

TDDS1 | SYS2 [23] | Real-time command and control system | 108708 | 54 |

TDDS2 | S10 [23] | Real-time command and control system | 233700 | 38 |

TDDS3 | SYS3 [23] | Military application | 67362 | 38 |

TDDS4 | S27 [23] | Single-user workstation | 4312598 | 41 |

TDDS5 | SYS4 [23] | Operating system | 52422 | 53 |

TDDS6 | Project J5 [18] | Real-time command and control system | 5090 | 73 |

TDDS7 | S17 [23] | Single-user workstation | 19572126 | 101 |

TDDS8 | SYS1 [23] | Single-user workstation | 88682 | 136 |

Data Source | Nature of System | Testing Length (Week) | Numbers of Detected Faults | |
---|---|---|---|---|

TIDS1 | SYS2 [23] | Real-time command and control system | 17 | 54 |

TIDS2 | NASA-supported project [24] | Inertial navigating system | 14 | 9 |

TIDS3 | SYS3 [23] | Military application | 14 | 38 |

TIDS4 | DS3 [25] | Embedded application for printer | 30 | 52 |

TIDS5 | DS2 [25] | Embedded application for printer | 33 | 58 |

TIDS6 | Release 3 [26] | Tandem software system | 12 | 61 |

TIDS7 | DS1 [25] | Embedded application for printer | 20 | 66 |

TIDS8 | Release 2 [26] | Tandem software system | 19 | 120 |

Type-I NHPP | Type-II NHPP | |||||
---|---|---|---|---|---|---|

Best SRM | AIC | MSE | Best SRM | AIC | MSE | |

TDDS1 | Lxvmax | 896.666 | 1.950 | Musa-Okumoto | 895.305 | 2.315 |

TDDS2 | Lxvmax | 721.928 | 1.442 | Cox-Lewis | 726.052 | 2.803 |

TDDS3 | Lxvmax | 598.131 | 1.705 | Musa-Okumoto | 596.501 | 1.809 |

TDDS4 | Lxvmax | 1008.220 | 5.970 | Musa-Okumoto | 1007.100 | 7.039 |

TDDS5 | Txvmin | 759.579 | 3.747 | Cox-Lewis | 759.948 | 5.509 |

TDDS6 | Exp | 757.869 | 18.985 | Power-law | 757.031 | 19.315 |

TDDS7 | Pareto | 2504.170 | 47.404 | Musa-Okumoto | 2503.370 | 63.699 |

TDDS8 | Lxvmin | 1938.160 | 6.570 | Musa-Okumoto | 1939.600 | 8.052 |

Type-I NHPP | Type-II NHPP | |||||
---|---|---|---|---|---|---|

Best SRM | AIC | MSE | Best SRM | AIC | MSE | |

TIDS1 | Llogist | 73.053 | 4.115 | Tlogist | 85.339 | 48.269 |

TIDS2 | Exp | 29.911 | 0.118 | Exp | 27.753 | 0.186 |

TIDS3 | Lxvmax | 61.694 | 3.239 | Llogist | 60.674 | 3.557 |

TIDS4 | Llogist | 117.470 | 9.408 | Llogist | 148.438 | 45.178 |

TIDS5 | Txvmin | 123.265 | 2.122 | Tlogist | 138.029 | 24.847 |

TIDS6 | Tlogist | 51.052 | 1.968 | Cox-Lewis | 63.556 | 27.199 |

TIDS7 | Lxvmax | 108.831 | 22.514 | Llogist | 107.211 | 24.394 |

TIDS8 | Tnorm | 87.267 | 6.151 | Cox-Lewis | 91.919 | 31.232 |

20% Observation Point | ||||
---|---|---|---|---|

Type-I NHPP | Type-II NHPP | |||

Best SRM | PMSE | Best SRM | PMSE | |

TDDS1 | Lxvmax | 5.073 | Musa-Okumoto | 6.420 |

TDDS2 | Txvmin | 83.964 | Llogist | 79.614 |

TDDS3 | Tnorm | 42.104 | Musa-Okumoto | 145.648 |

TDDS4 | Lxvmax | 32.217 | Llogist | 207.592 |

TDDS5 | Lnorm | 56.477 | Musa-Okumoto | 198.490 |

TDDS6 | Exp | 9177.670 | Tlogist | 467.320 |

TDDS7 | Lxvmax | 1852.520 | Lnorm | 1474.020 |

TDDS8 | Lxvmax | 32.131 | Power-law | 1417.110 |

50% Observation Point | ||||

Type-I NHPP | Type-II NHPP | |||

Best SRM | PMSE | Best SRM | PMSE | |

TDDS1 | Pareto | 6.118 | Musa-Okumoto | 6.420 |

TDDS2 | Lxvmax | 10.493 | Llogist | 30.944 |

TDDS3 | Txvmin | 5.874 | Llogist | 11.747 |

TDDS4 | Exp | 4480.620 | Llogist | 18.425 |

TDDS5 | Tlogist | 103.504 | Cox-Lewis | 106.282 |

TDDS6 | Llogist | 193.903 | Tlogist | 77.498 |

TDDS7 | Txvmin | 3569.230 | Musa-Okumoto | 45.344 |

TDDS8 | Pareto | 11.712 | Musa-Okumoto | 10.283 |

80% Observation Point | ||||

Type-I NHPP | Type-II NHPP | |||

Best SRM | PMSE | Best SRM | PMSE | |

TDDS1 | Lxvmax | 5.772 | Power-law | 3.432 |

TDDS2 | Lxvmax | 2.041 | Lxvmax | 3.697 |

TDDS3 | Lxvmax | 0.588 | Musa-Okumoto | 0.819 |

TDDS4 | Txvmin | 6.875 | Power-law | 4.291 |

TDDS5 | Txvmin | 4.253 | Cox-Lewis | 4.258 |

TDDS6 | Lxvmax | 21.715 | Power-law | 51.677 |

TDDS7 | Lxvmax | 57.901 | Power-law | 9.268 |

TDDS8 | Lxvmax | 9.419 | Power-law | 819.992 |

20% Observation Point | ||||
---|---|---|---|---|

Type-I NHPP | Type-II NHPP | |||

Best SRM | PMSE | Best SRM | PMSE | |

TIDS1 | Gamma | 220.732 | Power-law | 218.763 |

TIDS2 | Pareto | 2.628 | Musa-Okumoto | 2.625 |

TIDS3 | Lxvmax | 29.244 | Llogist | 47.377 |

TIDS4 | Txvmin | 448.935 | Cox-Lewis | 423.360 |

TIDS5 | Exp | 387.694 | Cox-Lewis | 67.730 |

TIDS6 | Exp | 142.854 | Tlogist | 86.083 |

TIDS7 | Tlogist | 98.903 | Llogist | 25.613 |

TIDS8 | Gamma | 820.049 | Gamma | 171.702 |

50% Observation Point | ||||

Type-I NHPP | Type-II NHPP | |||

Best SRM | PMSE | Best SRM | PMSE | |

TIDS1 | Txvmin | 96.992 | Musa-Okumoto | 159.545 |

TIDS2 | Exp | 0.344 | Musa-Okumoto | 0.347 |

TIDS3 | Txvmin | 30.786 | Power-law | 3.722 |

TIDS4 | Txvmin | 29.097 | Llogist | 156.329 |

TIDS5 | Lxvmax | 22.894 | Gamma | 27.045 |

TIDS6 | Exp | 101.303 | Musa-Okumoto | 101.258 |

TIDS7 | Pareto | 365.493 | Gamma | 18.825 |

TIDS8 | Lxvmax | 564.782 | Gamma | 849.736 |

80% Observation Point | ||||

Type-I NHPP | Type-II NHPP | |||

Best SRM | PMSE | Best SRM | PMSE | |

TIDS1 | Lnorm | 1.762 | Llogist | 8.736 |

TIDS2 | Tnorm | 0.224 | Lxvmax | 0.090 |

TIDS3 | Exp | 0.464 | Cox-Lewis | 0.464 |

TIDS4 | Tnorm | 0.864 | Llogist | 6.333 |

TIDS5 | Txvmin | 6.118 | Llogist | 17.300 |

TIDS6 | Lxvmax | 1.850 | Llogist | 18.985 |

TIDS7 | Lnorm | 3.432 | Llogist | 6.144 |

TIDS8 | Tnorm | 0.331 | Cox-Lewis | 41.228 |

Type-I NHPP | Type-II NHPP | |||
---|---|---|---|---|

Best SRM | Reliability | Best SRM | Reliability | |

TDDS1 | Lxvmax | $2.631\times {10}^{-6}$ | Musa-Okumoto | $2.674\times {10}^{-6}$ |

TDDS2 | Lxvmax | $3.283\times {10}^{-4}$ | Cox-Lewis | $4.694\times {10}^{-8}$ |

TDDS3 | Lxvmax | $3.687\times {10}^{-3}$ | Musa-Okumoto | $3.751\times {10}^{-7}$ |

TDDS4 | Lxvmax | $2.453\times {10}^{-4}$ | Musa-Okumoto | $2.398\times {10}^{-4}$ |

TDDS5 | Txvmin | $4.573\times {10}^{-1}$ | Cox-Lewis | $3.231\times {10}^{-3}$ |

TDDS6 | Exp | $1.035\times {10}^{-5}$ | Power-law | $2.596\times {10}^{-8}$ |

TDDS7 | Pareto | $8.971\times {10}^{-6}$ | Musa-Okumoto | $7.736\times {10}^{-6}$ |

TDDS8 | Lxvmin | $4.592\times {10}^{-5}$ | Musa-Okumoto | $2.516\times {10}^{-10}$ |

Type-I NHPP | Type-II NHPP | |||
---|---|---|---|---|

Best SRM | Reliability | Best SRM | Reliability | |

TIDS1 | Llogist | $4.152\times {10}^{-3}$ | Tlogist | $2.217\times {10}^{-25}$ |

TIDS2 | Exp | $9.832\times {10}^{-4}$ | Exp | $1.234\times {10}^{-4}$ |

TIDS3 | Lxvmax | $7.236\times {10}^{-5}$ | Llogist | $6.264\times {10}^{-5}$ |

TIDS4 | Llogist | $6.373\times {10}^{-1}$ | Llogist | $4.052\times {10}^{-10}$ |

TIDS5 | Txvmin | $9.633\times {10}^{-1}$ | Tlogist | $1.280\times {10}^{-27}$ |

TIDS6 | Tlogist | $2.816\times {10}^{-1}$ | Cox-Lewis | $3.221\times {10}^{-27}$ |

TIDS7 | Lxvmax | $1.939\times {10}^{-7}$ | Llogist | $3.892\times {10}^{-7}$ |

TIDS8 | Tnorm | $3.865\times {10}^{-2}$ | Cox-Lewis | $2.203\times {10}^{-23}$ |

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**MDPI and ACS Style**

Li, S.; Dohi, T.; Okamura, H.
Are Infinite-Failure NHPP-Based Software Reliability Models Useful? *Software* **2023**, *2*, 1-18.
https://doi.org/10.3390/software2010001

**AMA Style**

Li S, Dohi T, Okamura H.
Are Infinite-Failure NHPP-Based Software Reliability Models Useful? *Software*. 2023; 2(1):1-18.
https://doi.org/10.3390/software2010001

**Chicago/Turabian Style**

Li, Siqiao, Tadashi Dohi, and Hiroyuki Okamura.
2023. "Are Infinite-Failure NHPP-Based Software Reliability Models Useful?" *Software* 2, no. 1: 1-18.
https://doi.org/10.3390/software2010001