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