Application of EM Algorithm to NHPP-Based Software Reliability Assessment with Generalized Failure Count Data
Abstract
:1. Introduction
2. NHPP-Based SRMs
2.1. Model Description
- Assumption A: A software failure occurs at a random time. The probability distribution of all failure times are identical and mutually independent.
- Assumption B: The number of inherent software faults causing failures is finite.
- Assumption C: The number of inherent faults is unknown, but prior information is given by a Poisson distribution.
2.2. Parameter Estimation
3. EM Algorithms for NHPP-Based SRMs
3.1. EM Algorithm
3.2. EM Algorithm for NHPP-Based SRMs
- EM-step formula for tnorm:
- EM-step formula for lnorm:
- The EM-step formula for tnorm
- The EM-step formula for lnorm
- The EM-step formula for txvmax and txvmin
- The EM-step formula for lxvmax and lxvmin
4. Numerical Example
- Data label: SS1A
- Working days: 151
- The number of failures: 112
- LOC: Hundreds of thousands
- Software type: Operations
- Data label: SS1B
- Working days: 663
- The number of failures: 375
- LOC: Hundreds of thousands
- Software type: Operations
- Data label: SS1C
- Working days: 472
- The number of failures: 277
- LOC: Hundreds of thousands
- Software type: Operations
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Derivation of Equation (12)
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Model | Failure Time Distribution |
---|---|
exp | Exponential distribution [2] |
gamma | Gamma distribution [17,18] |
pareto | Pareto type-II distribution [50] |
tnorm | Truncated normal distribution [34] |
lnorm | Log-normal distribution [34] |
tlogis | Truncated logistic distribution [15] |
llogis | Log-logistic distribution [52] |
txvmax | Truncated extreme-value distribution (max) [35] |
lxvmax | Log-extreme-value distribution (max) [35] |
txvmin | Truncated extreme-value distribution (min) [35] |
lxvmin | Log-extreme-value distribution (min) [35,51] |
Model | EM | Nelder-Mead | BFGS |
---|---|---|---|
exp | 100 | 19 | 19 |
gamma | 100 | 34 | 34 |
pareto | 100 | 57 | 57 |
tnorm | 100 | 88 | 88 |
lnorm | 100 | 98 | 98 |
tlogis | 100 | 100 | 100 |
llogis | 100 | 100 | 100 |
txvmax | 100 | 100 | 100 |
lxvmax | 99 | 99 | 99 |
txvmin | 64 | 65 | 65 |
lxvmin | 92 | 92 | 92 |
Model | EM | Nelder-Mead | BFGS |
---|---|---|---|
exp | 100 | 7 | 7 |
gamma | 100 | 7 | 6 |
pareto | 100 | 18 | 18 |
tnorm | 100 | 98 | 98 |
lnorm | 87 | 87 | 87 |
tlogis | 100 | 100 | 100 |
llogis | 87 | 87 | 87 |
txvmax | 99 | 99 | 99 |
lxvmax | 0 | 0 | 0 |
txvmin | 89 | 89 | 89 |
lxvmin | 100 | 41 | 41 |
Model | EM | Nelder-Mead | BFGS |
---|---|---|---|
exp | 100 | 9 | 8 |
gamma | 100 | 13 | 12 |
pareto | 100 | 47 | 47 |
tnorm | 100 | 98 | 98 |
lnorm | 96 | 97 | 97 |
tlogis | 100 | 100 | 100 |
llogis | 96 | 96 | 96 |
txvmax | 99 | 99 | 99 |
lxvmax | 0 | 0 | 0 |
txvmin | 89 | 89 | 89 |
lxvmin | 100 | 43 | 43 |
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Okamura, H.; Dohi, T. Application of EM Algorithm to NHPP-Based Software Reliability Assessment with Generalized Failure Count Data. Mathematics 2021, 9, 985. https://doi.org/10.3390/math9090985
Okamura H, Dohi T. Application of EM Algorithm to NHPP-Based Software Reliability Assessment with Generalized Failure Count Data. Mathematics. 2021; 9(9):985. https://doi.org/10.3390/math9090985
Chicago/Turabian StyleOkamura, Hiroyuki, and Tadashi Dohi. 2021. "Application of EM Algorithm to NHPP-Based Software Reliability Assessment with Generalized Failure Count Data" Mathematics 9, no. 9: 985. https://doi.org/10.3390/math9090985