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A Parametric Empirical Bayes Model to predict Software Reliability Growth
A Parametric Empirical Bayes Model to
predict Software Reliability Growth
Néstor R. Barraza
nbarraza@untref.edu.ar
March - 2015
Universidad Nacional de Tres de Febrero

Contents
1 Software Reliability

Contents
2 Software Reliability Growth Models

Contents
3 Poisson Processes

Contents
3 Poisson Processes
4 Empirical Bayes Estimator

Contents
3 Poisson Processes
5 Simulation

Contents
3 Poisson Processes
5 Simulation
6 Conclusions

Contents
3 Poisson Processes
5 Simulation
6 Conclusions
7 Bibliography

Software Reliability
Software Development (waterfall, agile sprint)
Quality:
Standards
ISO-IEC 9126 (1991)
ISO-IEC 14598 (1995)
ISO-IEC 15504 (1998)
CMMI (1991)
Analysis
Reliability
Metrics

Software Reliability (waterfall, agile sprint).
Models
Growth Models
Markov Chains
Clusters
...

Software Reliability Growth

Software Reliability Growth Models
Independent Increments Stochastic process P(N(t) = n).
Remaining number of failures µ(t, s) = E[N(s)|N(t)].

Non homogeneous Poisson Process
P((N(s) − N(t)) = n) =
µ(t, s)n
n!
e−µ(t,s)
µ(t, 0) = a(1 − e−bt
) Goel − Okumoto
µ(t, 0) =
1
θ
ln(µ0θt + 1) Musa − Okumoto

Other Models
Model µ(t, 0)
Delayed S-shaped a(1 − (1 + b t)e−b t )
Log Power α lnβ
(1 + t)
Gomperts a(bct
)
Yamada Exponential a(1 − e−b c (1−e(−d t)))

Poisson Processes
Compound Poisson Process
P(N(t) = n) =
m
k=1
(λ t)k
k!
e−λt
f∗k
(X1 + X2 + · · · + Xk = n)
E[N(t)] = λ t E[X]

Empirical Bayes Estimator
Estimate a binary probability based on a previous number of
occurrences. Applications:

Speech recognition

Speech recognition
Word processing

Speech recognition
Word processing
Arithmetic compression

Speech recognition
Word processing
Arithmetic compression
Reliability

Try to estimate the probability of success θ, given n samples with
r successful events.

Maximum likelihood: ˆθ = r
n

n → Problems when r = 0

n → Problems when r = 0
Smoothing ˆθ = r+a
n+b

Family of estimators

Laplace law of succession ˆθ = r+1
n+2

n+2
Lidstone law ˆθ = r+b
n+s b

n+2
n+s b
Good-Turing ˆθ = r+1
n
Nr
Nr+1

n+2
n+s b
n
Nr
Nr+1
Discount ˆθ = r−b
n

n+2
n+s b
n
Nr
Nr+1
Discount ˆθ = r−b
n
New approach, to introduce a non linear function of
r: ˆθ = f(r)
n

P(r/λ) =
λr
r!
e−λ
. (1)
θ =
λ
n
(2)

ˆθ =
ˆλ
n
=
E[λ|r]
n
(3)
ˆθ =
1
n
λλr
r! e−λdS(λ)
λr
r! e−λdS(λ)
=
r + 1
n
P(r + 1)
P(r)
(4)

θ: Probability of failure occurrence
r : The number of failures up to time n
P(r): Prior probability of the number of failures
Reliability Growth → A decreasing probability of failure θ
ˆθ = r+1
n
> 1
P(r+1)
P(r)
n
r
r
P(r)

ˆθ = r+1
n
> 1> 1
P(r+1)
P(r)
P(r+1)
P(r)
n
r
r
P(r)

ˆθ = r+1
n
> 1< 1
P(r+1)
P(r)
P(r+1)
P(r)
n
r
r
P(r)

Simulation
Simulation procedure
s ← initial count of seconds
r ← initial number of failures
for i = s to the total testing time do
p =
r + 1
i
P(r + 1)
P(r)
{Estimate the probability of failure}
GENERATE a failure with probability p
if a failure arrived then
r ← r + 1
end if
print number of arrived failures r and seconds i
end for
(Failure generation code)

Simulation
5,000,000 10,000,000 15,000,000
10
20
30
40
secs
Number of failures

Simulation
5,000,000 10,000,000 15,000,000
10
20
30
40
secs
Number of failures
ˆθ0 = r0+1
n0
P(r0+1)
P(r0)

Simulation
5,000,000 10,000,000 15,000,000
20
40
60
80
100
120
secs
Number of failures
Actual Data
Simulation

Simulation
5,000,000 10,000,000 15,000,000
20
40
60
80
100
120
secs
Number of failures
Actual Data
SimulationSimulation

Simulation
Same Mixing Distribution
Different Starting Points
5,000,000 10,000,000 15,000,000
20
40
60
80
100
120
secs
Number of failures

Simulation
SS1 Project
0 20 40 60 80 100 120
0
2
4
6
·105
Failure Number
MeanTimeBetweenFailures(Secs)
+ + Actual Data
ML
Our Model

Simulation
SS1 Project
0 20 40 60 80 100 120
0
2
4
6
·105
Failure Number
+ + Actual Data
MLML
Our Model

Simulation
SS1 Project
0 20 40 60 80 100 120
0
2
4
6
·105
Failure Number
+ + Actual Data
MLML
Our ModelOur Model

Simulation
Naval Tactical Data System
5 10 15 20 25 30
0
50
100
Failure Number
MeanTimeBetweenFailures(Days)
Actual Data

Simulation
5 10 15 20 25 30
0
50
100
Failure Number
Actual Data
Mazzuchi Model I

Simulation
5 10 15 20 25 30
0
50
100
Failure Number
Actual Data
Mazzuchi Model I
Mazzuchi Model II

Simulation
5 10 15 20 25 30
0
50
100
Failure Number
Actual Data
Mazzuchi Model I
Mazzuchi Model II
Our Model

Conclusions
Conclusions

Conclusions
Conclusions
A new method in order to predict software failures
production was proposed.

Conclusions
Conclusions
This method is based on The Empirical Bayes estimator
usually used in other areas of Engineering.

Conclusions
Conclusions
In order to model reliability growth, a specially smoothing
nonlinear factor was introduced in the estimate.

Conclusions
Conclusions
This factor comes from a mixed Poisson distribution.

Conclusions
Conclusions
This factor comes from a mixed Poisson distribution.
A simulation of the cumulative failures curve using a
Poisson mixed by a Generalized Inverse Gaussian
probability density function has been performed with good
agreement with reported data.

Bibliography
Bibliography

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