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Theyproposed that the time-dependent failure ratefollows an exponential distribution.

The instantaneous failure rate function is given by

Where y = 0, 1, 2, 3,..m(t)= expected number of failure observed by time

t.

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Where

m (t) = a(1-e-bt) and

m(t) = the expected number of failure observed by time.

(t)= Failure density.

b= Fault detection rate per fault.

a= expected number of failure.

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Comparing with the exponential model thecumulative distribution function F(t) and probability

density function f(t) is given by,

CDF = F(t)= m (t) = a(1-e-bt)

PDF=f(t) =

The parameter a and b corresponds to k and .

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The model curve can be obtain by fitting the actualdata and the number of faults remaining in the system

with the help of cumulative distribution

function(CDF).

Goel proposed a generalization of the NHPP model

by allowing one more parameter in the CDF function.

Such model is known as Goel generalize

nonhomogeneous Poisson process model.

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For this generalize model the PDF and CDF will be,

PDF =

CDF =

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Musa- Okumoto Logarithmic Poisson

Execution Time Model (M-O Model):

The model observes number of failures by a certain

time t.

It attempts to consider that later fixes have a smallereffect on the softwares reliability than earlier one.

Also the process modeled the number of failures in

specified execution time interval (Instead of calendar

time).

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The model therefore consists of two components.

1) The execution time components.

2) The calendar time components.

The mean value function or the cumulative

distribution function is given by

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Where

= Initial failure intensity.

= Rate of reduction in the failure intensityper

failure.

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The Delayed S Models

Based on the failure analysis ofproducts.

The model observed that a significant delay can occur

between the time of the first failure observation and

the time of reporting the failure.

In this model the observed growth curve of thecumulative number of detected defects is S- shaped.

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The mean value function or the cumulativedistribution function to reflect the delay in failure

reporting is given by,

Where

t = Time = Error detection rate.

k= Total number of defects or total cumulative defect

rate.