1. 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 Presented by Teresa Cai Group Meeting 12/9/2006

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

3. 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. 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. 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. 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. 7 Outline Background and related work NHPP model and three weighted means A general discrete model A general continuous model Conclusion

8. 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. 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. 10 NHPP models: examples Goel-Okumoto model Gompertz growth curve model Logistic growth curve model Yamada delayed S-shaped model

11. 11 Weighted arithmetic mean Arithmetic mean Weighted arithmetic mean

12. 12 Weighted geometric mean Geometric mean Weighted geometric mean

13. 13 Weighted harmonic mean Harmonic mean Weighted harmonic mean

14. 14 Three weighted means Proposition 1: Let z 1, z 2 and z 3, 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)≤z 3 ≤ z 2 ≤ z 1 ≤ max(x,y) Where equality holds if and only if x=y.

15. 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. 16 Outline Background and related work NHPP model and three weighted means A general discrete model A general continuous model Conclusion

17. 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. 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. 19 A more general case W is not a constant for all i  w(i) Then

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

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

22. 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. 23 A general continuous model Theorem 1: g is a real-valued, strictly monotone, and differentiable function

24. 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. 25 Power transformation A parametric power transformation With the new g(x), several new SRGMs can be generated

26. 26

27. 27 Outline Background and related work NHPP model and three weighted means A general discrete model A general continuous model Conclusion

28. 28 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