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A Unified Scheme of Some Nonhomogenous Poisson Process Models for Software Reliability Estimation

Presented by Teresa Cai Group Meeting 12/9/2006. A Unified Scheme of Some Nonhomogenous Poisson Process Models for Software Reliability Estimation. C. Y. Huang, M. R. Lyu and S. Y. Kuo IEEE Transactions on Software Engineering 29(3), March 2003. Outline . Background and related work

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A Unified Scheme of Some Nonhomogenous Poisson Process Models for Software Reliability Estimation

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  1. Presented by Teresa Cai Group Meeting 12/9/2006 A Unified Scheme of Some Nonhomogenous Poisson Process Models for Software Reliability Estimation C. Y. Huang, M. R. Lyu and S. Y. Kuo IEEE Transactions on Software Engineering 29(3), March 2003

  2. Outline • Background and related work • NHPP model and three weighted means • A general discrete model • A general continuous model • Conclusion

  3. Software reliability growth modeling (SRGM) • To model past failure data to predict future behavior Failure rate: the probability that a failure occurs in a certain time period.

  4. SRGM: some examples • Nonhomogeneous Poisson Process (NHPP) model • S-shaped reliability growth model • Musa-Okumoto Logarithmic Poisson model μ(t) is the mean value of cumulative number of failures by time t

  5. Unification schemes for SRGMs • Langberg and Singpurwalla (1985) • Bayesian Network • Specific prior distribution • Miller (1986) • Exponential Order Statistic models (EOS) • Failure time: order statistics of independent nonidentically distributed exponential random variables • Trachtenberg (1990) • General theory: failure rates = average size of remaining faults* apparent fault density * software workload

  6. Contributions of this paper • Relax some assumptions • Define a general mean based on three weighted means: • weighted arithmetic means • Weighted geometric means • Weighted harmonic means • Propose a new general NHPP model

  7. Outline • Background and related work • NHPP model and three weighted means • A general discrete model • A general continuous model • Conclusion

  8. Nonhomogeneous Poisson Process (NHPP) Model • An SRGM based on an NHPP with the mean value function m(t): • {N(t), t>=0}: a counting process representing the cumulative number of faults detected by the time t • N = 0, 1, 2, ……

  9. NHPP Model • M(t): • expected cumulative number of faults detected by time t • Nondecreasing • m()=a: the expected total number of faults to be detected eventually • Failure intensity function at testing time t: • Reliability:

  10. NHPP models: examples • Goel-Okumoto model • Gompertz growth curve model • Logistic growth curve model • Yamada delayed S-shaped model

  11. Weighted arithmetic mean • Arithmetic mean • Weighted arithmetic mean

  12. Weighted geometric mean • Geometric mean • Weighted geometric mean

  13. Weighted harmonic mean • Harmonic mean • Weighted harmonic mean

  14. Three weighted means • Proposition 1: Let z1, z2 and z3, respectively, be the weighted arithmetic, the weighted geometric, and the weighted harmonic means of two nonnegative real numbers z and y with weights w and 1- w, where 0< w <1. Then min(x,y)≤z3≤ z2≤ z1≤ max(x,y) Where equality holds if and only if x=y.

  15. A more general mean • Definition 1: Let g be a real-valued and strictly monotone function. Let x and y be two nonnegative real numbers. The quasi arithmetic mean z of x and y with weights w and 1-w is defined as z = g-1(wg(x)+(1-w)g(y)), 0<w<1 Where g-1 is the inverse function of g

  16. Outline • Background and related work • NHPP model and three weighted means • A general discrete model • A general continuous model • Conclusion

  17. A General discrete model • Testing time t  test run i • Suppose m(i+1) is equal to the quasi arithmetic mean of m(i) and a with weights w and 1-w • Then where a=m(): the expected number of faults to be detected eventually

  18. Special cases of the general model • g(x)=x: Goel-Okumoto model • g(x)=lnx: Gompertz growth curve • g(x)=1/x: logistic growth model

  19. A more general case • W is not a constant for all i  w(i) • Then

  20. Generalized NHPP model • Generalized Goel NHPP model: g(x)=x, ui=exp[-bic], w(i)=exp{-b[ic-(i-1)c]} • Delayed S-shaped model:

  21. Outline • Background and related work • NHPP model and three weighted means • A general discrete model • A general continuous model • Conclusion

  22. A general continuous model • Let m(t+Δt) be equal to the quasi arithmetic means of m(t) and a with weights w(t,Δt) and 1-w(t,Δt), we have where b(t)=(1-w(t,Δt))/Δt as Δt0

  23. A general continuous model • Theorem 1: g is a real-valued, strictly monotone, and differentiable function

  24. A general continuous model • Take different g(x) and b(t), various existing models can be derived, such as: • Goel_Okumoto model • Gompertz Growth Curve • Logistic Growth Curve • ……

  25. Power transformation • A parametric power transformation • With the new g(x), several new SRGMs can be generated

  26. Outline • Background and related work • NHPP model and three weighted means • A general discrete model • A general continuous model • Conclusion

  27. Conclusion • Integrate the concept of weighted arithmetic mean, weighted geometric mean, weighted harmonic mean, and a more general mean • Show several existing SRGMs based on NHPP can be derived • Propose a more general NHPP model using power transformation

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