Transcript A Unified Scheme of Some Nonhomogenous Poisson Process

Presented by Teresa Cai
Group Meeting 12/9/2006
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
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), 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
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 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 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(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 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+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
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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
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