The KIPS Transactions:PartD (정보처리학회논문지D)

Korea Information Processing Society (한국정보처리학회)
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The Bayesian Inference for Software Reliability Models Based on NHPP

NHPP에 기초한 소프트웨어 신뢰도 모형에 대한 베이지안 추론에 관한 연구

  • Published : 2002.06.01
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Abstract

Software reliability growth models are used in testing stages of software development to model the error content and time intervals between software failures. This paper presents a stochastic model for the software failure phenomenon based on a nonhomogeneous Poisson process(NHPP) and performs Bayesian inference using prior information. The failure process is analyzed to develop a suitable mean value function for the NHPP ; expressions are given for several performance measure. Actual software failure data are compared with several model on the constant reflecting the quality of testing. The performance measures and parametric inferences of the suggested models using Rayleigh distribution and Laplace distribution are discussed. The results of the suggested models are applied to real software failure data and compared with Goel model. Tools of parameter point inference and 95% credible intereval was used method of Gibbs sampling. In this paper, model selection using the sum of the squared errors was employed. The numerical example by NTDS data was illustrated.

본 논문에서는 비동질 포아송 과정에 기초한 소프트웨어 오류 현상에 대한 신뢰도 모형을 고려하고 사전정보를 이용한 베이지안 추론을 시행하였다. 고장 패턴은 NHPP에 대한 강도함수와 평균값 함수로서 나타낼 수 있다. 따라서 본 논문에서는 기존의 모형인 Goel이 제시한 모형과 신뢰성 분포로 많이 사용되는 와이블 분포의 특수형태인 레일리분포와 라플라스 분포를 이용한 모형을 제시하여 베이지안 추론을 시행하고 또, 효율적 모형을 위한 모형선택으로서 편차자승합을 이용하여 비교하였다. 모수의 추정을 위해서 마코브체인 몬테카를로 기법중에 하나인 깁스샘플링을 이용한 근사추정 기법이 사용되었다. 수치적인 예에서는 실측자료인 NTDS 자료를 이용하여 모수 및 신뢰도를 추정하였고 편차자승합을 이용한 모형비교의 결과를 나열하였다.

Keywords

References

  1. Charnes, A., Frome, E. L. and Yu, P. L., 'The Equivalence of Generalized Least Squares and Maximum Likelihood Es-timates in the Exponential Family,' Journal of the Amer-ican Statistical Association, Vol.7, No.353, pp.169-171, 1976 https://doi.org/10.2307/2285762
  2. Dey, D. K., and Lee, T. M., 'Bayes Compution for Life Testing and Reliability Estimation,' IEEE Transactions on Reliability, Vol.41, pp.621-626, 1992 https://doi.org/10.1109/24.249600
  3. Gelfand, A. E., and Smith, A. F. M., 'Sampling-Based Ap-proaches to Calculating Marginal Densities,' Journal of the American Statistical Association, 85, pp.398-409, 1990 https://doi.org/10.2307/2289776
  4. Gelfand, A. E., and Smith, A. F. M. and Lee, T-M., 'Bayes-ian analysis of constrained parameter and truncated data problems,' J. Amer. Statist. Assoc., 87, pp.523-532, 1992 https://doi.org/10.2307/2290286
  5. Gelman, A. E., and Rubin D., 'Inference from Iterative Sim-ulation Using Multiple Sequences,' Statistical Science, 7, pp.457-472, 1992 https://doi.org/10.1214/ss/1177011136
  6. Geman, S., and Geman, D., 'Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images,' IEEE Transactions on Pattern Analysis and Machine In-telligence, 6, pp.721-741, 1984 https://doi.org/10.1109/TPAMI.1984.4767596
  7. Goel, A. L., 'Software Reliability Model : Assumption, Lim-itations, And Applicability,' IEEE Trans. Software Eng., Vol.SE-11, No.12, pp.1411-1423, 1985 https://doi.org/10.1109/TSE.1985.232177
  8. Goel, A. L., and Okumoto, K., 'Time-Dependent Error-Detection Rate Model for Software Reliability and Other Performance Measures,' IEEE Trans. Rel., Vol.R-28, pp. 206-211, 1979 https://doi.org/10.1109/TR.1979.5220566
  9. Hossain, S. A. and Dahiya, R. C., 'Estimating the Parame-ters of a Non-homogeneous Poission-Process Model for Software Reliability,' IEEE Trans. Rel., Vol.R-42, No.4, pp.604-612, 1993 https://doi.org/10.1109/24.273589
  10. Johnson, N. L. and Kotz, S., 'Continuous Univariate Distri-butions 1,' A wiley-interscience publication, JOHN WILEY & SONS, pp.272-289, 1970
  11. Kuo, L., and Yang, T. Y., 'Bayesian Computation of Soft-ware Reliability,' Journal of Computational and Graphical Statistics, pp.65-82, 1995 https://doi.org/10.2307/1390628
  12. Lawless, J. F., 'Statistical Models and Methods for Life-time Data,' pp.494-500, 1981
  13. Musa, J. D., Iannino, A., and Okumoto, K., 'Software Re-liability : Measurement, Prediction, Application,' New York : McGraw Hill, 1987
  14. Pham, H. and Nordmann, L. and Zhang, X. 'A General Im-perfect-Software-Debugging Model with S-Shaped Fault-Detection Rate,' IEEE Trans. Rel., Vol.48, No.2, pp.169-175, 1999 https://doi.org/10.1109/24.784276
  15. Okumoto, K., 'A Statistical Method for Software Quality Control,' IEEE Trans. Software Eng., Vol.SE-11, No.12, pp.424-1431, 1985 https://doi.org/10.1109/TSE.1985.232178
  16. Raftery, A. E., 'Inference and Prediction for a General Order Statistic Model with Unknown Population Size,' Journal of the American Statistical Association, 82, pp.1163-1168, 1987 https://doi.org/10.2307/2289395
  17. Smith, A. F. M. and G. O. Roberts, 'Bayesian computation via the Gibbs Sampler and related Markov chain Monte Carlo methods,' Journal of the Royal Statistical Society, Ser. B, 55, pp.3-24, 1993
  18. Tanner, M., and Wong, W. 'The Calculation of Posterior Distributions by Date Augmentation (with discussion),' Journal of the American Statistical Association, 81, pp.82-86, 1987 https://doi.org/10.2307/2287970
  19. Yamada, S., Ohba, M. and Osaki, S., 'S-shaped Reliability Growth Modeling for Software Error Detection,' IEEE Trans. Rel., Vol.R-32, No.5, pp.475-478, 1983 https://doi.org/10.1109/TR.1983.5221735
  20. Yamada, S., Ohtera, H. and Narihisa, H., 'Software Reli-ability Growth Models with Testing-Effort,' IEEE Trans. Rel., Vol.R-35, No.1, pp.19-23, 1986
  21. Yamada, S. and Osaki, S., 'Software Reliability Growth Modeling : Models and application,' IEEE Trans. Software Eng., Vol.SE-11, No.12, pp.1431-1437, 1985 https://doi.org/10.1109/TSE.1985.232179
  22. 'USER'S MANUAL 'STAT/LIBRARY FORTRAN Sub-routines for statistical analysis,' IMSL, Vol.3, pp.1050-1054, 1987
  23. 간광연, 김희철, 이병수, 'NHPP에 기초한 소프트웨어 신뢰도 모형에 대한 비교연구', 정보처리학회논문지 D, 제8-D권, 제4호, pp.393-400, 2001
  24. 박만곤, '소프트웨어 신뢰도의 평가와 예측을 위한 베이지안 알고리즘', 정보처리학회논문지, Vol.1, No.1, pp.14-22, 1226-9182, 1994