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A Unified Scheme of Some Nonhomogenous Poisson Process Models for Software
Reliability Estimation
C. Y. Huang, M. R. Lyu and S. Y. KuoIEEE Transactions on Software Engineering
29(3), March 2003
Presented by Teresa Cai
Group Meeting 12/9/2006
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Outline
Background and related workNHPP model and three weighted meansA general discrete modelA general continuous modelConclusion
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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.
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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
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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 nonidentical
ly distributed exponential random variables
Trachtenberg (1990) General theory: failure rates = average size of remainin
g faults* apparent fault density * software workload
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Contributions of this paper
Relax some assumptionsDefine a general mean based on three
weighted means: weighted arithmetic means Weighted geometric means Weighted harmonic means
Propose a new general NHPP model
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Outline
Background and related workNHPP model and three weighted meansA general discrete modelA general continuous modelConclusion
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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, ……
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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 even
tually
Failure intensity function at testing time t:
Reliability:
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NHPP models: examples
Goel-Okumoto model
Gompertz growth curve model
Logistic growth curve model
Yamada delayed S-shaped model
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Weighted arithmetic mean
Arithmetic mean
Weighted arithmetic mean
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Weighted geometric mean
Geometric mean
Weighted geometric mean
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Weighted harmonic mean
Harmonic mean
Weighted harmonic mean
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Three weighted means
Proposition 1: Let z1, z2 and z3, respectively, be the weighted ar
ithmetic, 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.
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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
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Outline
Background and related workNHPP model and three weighted meansA general discrete modelA general continuous modelConclusion
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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
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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
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A more general case
W is not a constant for all i w(i)Then
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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:
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Outline
Background and related workNHPP model and three weighted meansA general discrete modelA general continuous modelConclusion
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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
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A general continuous model
Theorem 1:
g is a real-valued, strictly monotone, and differentiable function
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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 ……
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Power transformation
A parametric power transformation
With the new g(x), several new SRGMs can be generated
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Outline
Background and related workNHPP model and three weighted meansA general discrete modelA general continuous modelConclusion
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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