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Improved Algorithm for Non-Homogeneous Poisson Process Software Reliability Growth Models Incorporating Testing-Effort

Volume 15, Number 5, May 2019, pp. 1265-1272
DOI: 10.23940/ijpe.19.05.p1.12651272

Vidhyashree Nagarajua, Thierry Wandjib, and Lance Fiondellaa

aUniversity of Massachusetts Dartmouth, North Dartmouth, 02747, USA
bNaval Air Systems Command, Patuxent River, 20670, USA

 

(Submitted on September 26, 2017; Revised on February 19, 2018; Accepted on March 26, 2018)

Abstract:

Critical systems are becoming increasingly software intensive, necessitating reliable software to ensure proper operation. Non-homogeneous Poisson process software reliability growth models are commonly used to characterize fault detection as a function of testing time, which enables quantitative assessment of software reliability. Many early models assumed that the testing-effort was constant throughout software testing. To remove this assumption, researchers have proposed models incorporating testing-effort, yet this significantly increases model complexity to the degree that most previous studies utilized a two-step procedure involving least squares estimation (LSE) and algorithms, including Newton's method to estimate the parameters of a testing-effort model. This approach may limit the quality of the model fit achieved. Moreover, the research trend over the past 30 years has been to propose progressively more complex models, sacrificing practical considerations such as predictive accuracy. This paper proposes a two-step procedure that utilizes the expectation conditional maximization (ECM) algorithm, referred to as the ECM/ECM approach, to obtain the parameter estimates of a software reliability growth model incorporating testing-effort. The results of the proposed approach are compared to past methods as well as a simpler model that does not consider testing-effort to assess whether the additional complexity introduced by testing-effort functions compromises predictive accuracy. Our results indicate that the ECM/ECM approach achieves a better goodness of fit with respect to four measures, including three predictive measures. In some cases, the simpler model omitting testing-effort outperforms methods considering testing-effort. These results suggest that the proposed ECM/ECM approach can achieve better parameter estimates than the previously proposed LSE/MLE approach and that algorithms to improve fit and predictive accuracy may better serve users of software reliability models.

 

References: 27

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    14. S. Yamada, H. Ohtera, and H. Narihisa, Software Reliability Growth Models with Testing-Effort, IEEE Transactions on Reliability, Vol. R-35, No. 1, pp. 19-23, 1986
    15. D. Kleinbaum, L. Kupper, A. Nizham, and K. Muller, Applied Regression Analysis and other Multivariable Methods, 4th Edition, Thomson Learning Inc. Brooks/Cole, Australia, 2008
    16. V. Nagaraju, L. Fiondella, P. Zeephongsekul, C. Jayasinghe, and T. Wandji, Performance Optimized Expectation Conditional Maximization Algorithms for Nonhomogeneous Poisson Process Software Reliability Models, IEEE Transactions on Reliability, Vol. 66, No. 3, pp. 722-734, 2017
    17. V. Nagaraju, A. K. Murthy, L. Fiondella, P. Zeephongsekul, and T. Wandji, Expectation Conditional Maximization Algorithms for Failure Count Non-Homogeneous Poisson Process Software Reliability Models, in Proceedings of ISSAT International Conference on Reliability and Quality in Design, 2016
    18. L. Fiondella and S. Gokhale, Software Reliability Model with Bathtub-Shaped Fault Detection Rate, in Proceedings of Annual Reliability and Maintainability Symposium, Orlando, FL, 2011
    19. K. Sharma, R. Garg, C. Nagpal, and R. Garg, Selection of Optimal Software Reliability Growth Models using a Distance based Approach, IEEE Transactions on Reliability, Vol. 59, No. 2, pp. 266-276, 2010
    20. A. Goel and K. Okumoto, Time-Dependent Error-Detection Rate Model for Software Reliability and other Performance Measures, IEEE Transactions on Reliability, Vol. 28, No. 3, pp. 206-211, 1979
    21. L. Putnam, A General Empirical Solution to the Macro Software Sizing and Estimating Problem, IEEE Transactions on Software Engineering, Vol. SE-4, No. 4, pp. 345-361, 1978
    22. R. Burden and J. Faires, Numerical Analysis, 8th Edition, Brooks/Cole, Belmont, CA, 2004
    23. H. Okamura, Y. Watanabe, and T. Dohi, An Iterative Scheme for Maximum Likelihood Estimation in Software Reliability Modeling, in Proceedings of International Symposium on Software Reliability Engineering, 2003
    24. P. Zeephongsekul, C. Jayasinghe, L. Fiondella, and V. Nagaraju, Maximum Likelihood Estimation of Parameters of NHPP Software Reliability Models using EM and ECM Algorithms, IEEE Transactions on Reliability, Vol. 65, No. 3, pp. 1571-1583, 2016
    25. V. Nagaraju, T. Wandji, and L. Fiondella, An Implicit Expectation Conditional Maximization Algorithm for Non-Homogeneous Poisson Process Software Reliability Models, in Proceedings of Conference on Applied Statistics in Defense, Fairfax, VA, 2016
    26. A. Dempster, N. Laird, and D. Rubin, Maximum Likelihood from Incomplete Data via the EM Algorithm, Journal of the Royal Statistical Society: Series B, Vol. 39, No. 1, pp. 1-38, 1977
    27. M. Zhao and M. Xing, Robustness of Optimum Software Release Policies, in Proceedings of IEEE International Symposium on Software Reliability Engineering, pp. 218-225, 1993

     

     

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    1.        S. Ross, Introduction to Probability Models, 8th Edition, Academic Press, New York, 2003

    2.        M. Lyu, Handbook of Software Reliability Engineering, McGraw-Hill, New York, 1996

    3.        L. Leemis, Reliability: Probabilistic Models and Statistical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1995

    4.        J. Musa, A Theory of Software Reliability and its Application, IEEE Transactions on Software Engineering, Vol. SE-1, No. 3, pp. 312-327, 1975

    5.        S. Gokhale, P. Marinos, and K. Trivedi, Important Milestones in Software Reliability Modeling, Communications in Reliability, Maintainability, and Serviceability, 1996

    6.        B. Boehm, Software Engineering Economics, Prentice-Hall, Englewood Cliffs, NJ, 1981

    7.        J. Musa and K. Okumoto, A Logarithmic Poisson Execution Time Model for Software Reliability Measurement, in Proceedings of International Conference on Software Engineering, 1984

    8.        M. Trachtenberg, A General Theory of Software-Reliability Modeling, IEEE Transactions on Reliability, Vol. 39, No. 1, pp. 92-96, 1990

    9.        W. Everett, An Extended Execution Time Software Reliability Model, in Proceedings of International Symposium on Software Reliability Engineering, pp. 4-13, 1992

    10.     J. Tian, P. Lu, and J. Palma, Test-Execution-based Reliability Measurement and Modeling for Large Commercial Software, IEEE Transactions on Software Engineering, Vol. 21, No. 5, pp. 405-414, 1995

    11.     W. Brooks and R. Motley, Analysis of Discrete Software Reliability Models (RADC-TR-80-84), 1980

    12.     M. Ohba, Software Reliability Analysis Models, IBM Journal of Research and Development, Vol. 28, No. 4, pp. 428-443, 1984

    13.     Y. Tohma, R. Jacoby, Y. Murata, and M. Yamamoto, Hyper-Geometric Distribution Model to Estimate the Number of Residual Software Faults, in Proceedings of Computer Software and Application Conference, 1989

    14.     S. Yamada, H. Ohtera, and H. Narihisa, Software Reliability Growth Models with Testing-Effort, IEEE Transactions on Reliability, Vol. R-35, No. 1, pp. 19-23, 1986

    15.     D. Kleinbaum, L. Kupper, A. Nizham, and K. Muller, Applied Regression Analysis and other Multivariable Methods, 4th Edition, Thomson Learning Inc. Brooks/Cole, Australia, 2008

    16.     V. Nagaraju, L. Fiondella, P. Zeephongsekul, C. Jayasinghe, and T. Wandji, Performance Optimized Expectation Conditional Maximization Algorithms for Nonhomogeneous Poisson Process Software Reliability Models, IEEE Transactions on Reliability, Vol. 66, No. 3, pp. 722-734, 2017

    17.     V. Nagaraju, A. K. Murthy, L. Fiondella, P. Zeephongsekul, and T. Wandji, Expectation Conditional Maximization Algorithms for Failure Count Non-Homogeneous Poisson Process Software Reliability Models, in Proceedings of ISSAT International Conference on Reliability and Quality in Design, 2016

    18.     L. Fiondella and S. Gokhale, Software Reliability Model with Bathtub-Shaped Fault Detection Rate, in Proceedings of Annual Reliability and Maintainability Symposium, Orlando, FL, 2011

    19.     K. Sharma, R. Garg, C. Nagpal, and R. Garg, Selection of Optimal Software Reliability Growth Models using a Distance based Approach, IEEE Transactions on Reliability, Vol. 59, No. 2, pp. 266-276, 2010

    20.     A. Goel and K. Okumoto, Time-Dependent Error-Detection Rate Model for Software Reliability and other Performance Measures, IEEE Transactions on Reliability, Vol. 28, No. 3, pp. 206-211, 1979

    21.     L. Putnam, A General Empirical Solution to the Macro Software Sizing and Estimating Problem, IEEE Transactions on Software Engineering, Vol. SE-4, No. 4, pp. 345-361, 1978

    22.     R. Burden and J. Faires, Numerical Analysis, 8th Edition, Brooks/Cole, Belmont, CA, 2004

    23.     H. Okamura, Y. Watanabe, and T. Dohi, An Iterative Scheme for Maximum Likelihood Estimation in Software Reliability Modeling, in Proceedings of International Symposium on Software Reliability Engineering, 2003

    24.     P. Zeephongsekul, C. Jayasinghe, L. Fiondella, and V. Nagaraju, Maximum Likelihood Estimation of Parameters of NHPP Software Reliability Models using EM and ECM Algorithms, IEEE Transactions on Reliability, Vol. 65, No. 3, pp. 1571-1583, 2016

    25.     V. Nagaraju, T. Wandji, and L. Fiondella, An Implicit Expectation Conditional Maximization Algorithm for Non-Homogeneous Poisson Process Software Reliability Models, in Proceedings of Conference on Applied Statistics in Defense, Fairfax, VA, 2016

    26.     A. Dempster, N. Laird, and D. Rubin, Maximum Likelihood from Incomplete Data via the EM Algorithm, Journal of the Royal Statistical Society: Series B, Vol. 39, No. 1, pp. 1-38, 1977

    27.     M. Zhao and M. Xing, Robustness of Optimum Software Release Policies, in Proceedings of IEEE International Symposium on Software Reliability Engineering, pp. 218-225, 1993

     
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