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
Presented by Teresa Cai Group Meeting 12/9/2006

• C. Y. Huang, M. R. Lyu and S. Y. Kuo
• IEEE Transactions on Software Engineering
• 29(3), March 2003

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Outline

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

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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
  nonidentically distributed exponential random
  variables
• Trachtenberg (1990)
• General theory failure rates average size of
  remaining faults apparent fault density
  software workload

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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

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Outline

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

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Nonhomogeneous Poisson Process (NHPP) Model

• An SRGM based on an NHPP with the mean value
  function m(t)
• N(t), tgt0 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 eventually
• 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 arithmetic, the weighted geometric, and
  the weighted harmonic means of two nonnegative
  real numbers z and y with weights w and 1- w,
  where 0lt w lt1. Then
• min(x,y)z3 z2 z1 max(x,y)
• Where equality holds if and only if xy.

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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)), 0ltwlt1
• Where g-1 is the inverse function of g

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Outline

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

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A General discrete model

• Testing time t ? test run i
• Suppose m(i1) is equal to the quasi arithmetic
  mean of m(i) and a with weights w and 1-w
• Then
• where am(?) 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, uiexp-bic, w(i)exp-bic-(i-1)c
• Delayed S-shaped model

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Outline

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

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A general continuous model

• Let m(t?t) be equal to the quasi arithmetic
  means of m(t) and a with weights w(t,?t) and
  1-w(t,?t), we have
• where b(t)(1-w(t,?t))/?t as ?t?0

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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 work
• NHPP model and three weighted means
• A general discrete model
• A general continuous model
• Conclusion

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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