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

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

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

Group Meeting 12/9/2006

C. Y. Huang, M. R. Lyu and S. Y. Kuo

IEEE Transactions on Software Engineering

29(3), March 2003

Outline

- Background and related work
- NHPP model and three weighted means
- A general discrete model
- A general continuous model
- Conclusion

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.

- 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

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

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

- Background and related work
- NHPP model and three weighted means
- A general discrete model
- A general continuous model
- Conclusion

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, ……

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:

NHPP models: examples

- Goel-Okumoto model
- Gompertz growth curve model
- Logistic growth curve model
- Yamada delayed S-shaped model

Weighted arithmetic mean

- Arithmetic mean
- Weighted arithmetic mean

Weighted geometric mean

- Geometric mean
- Weighted geometric mean

Weighted harmonic mean

- Harmonic mean
- Weighted harmonic mean

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.

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

- Background and related work
- NHPP model and three weighted means
- A general discrete model
- A general continuous model
- Conclusion

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

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

A more general case

- W is not a constant for all i w(i)
- Then

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:

- Background and related work
- NHPP model and three weighted means
- A general discrete model
- A general continuous model
- Conclusion

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 Δt0

A general continuous model

- Theorem 1:
g is a real-valued, strictly monotone, and differentiable function

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
- ……

Power transformation

- A parametric power transformation
- With the new g(x), several new SRGMs can be generated

- Background and related work
- NHPP model and three weighted means
- A general discrete model
- A general continuous model
- Conclusion

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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