A Unified Scheme of Some Nonhomogenous Poisson Process
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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
2
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.
3
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
4
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
7
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
10
Weighted arithmetic mean
Arithmetic mean
Weighted arithmetic mean
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Weighted geometric mean
Geometric mean
Weighted geometric mean
12
Weighted harmonic mean
Harmonic mean
Weighted harmonic mean
13
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.
14
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
15
Outline
Background and related work
NHPP model and three weighted means
A general discrete model
A general continuous model
Conclusion
16
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
21
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 work
NHPP model and three weighted means
A general discrete model
A general continuous model
Conclusion
27
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
28