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Reliability Basics | ||||||||

In "Reliability Analysis for Complex, Repairable Systems" (1974), Dr. Larry H. Crow noted that the Duane model could be stochastically represented as a Weibull process, allowing for statistical procedures to be used in the application of this model in reliability growth. This statistical extension became what is known as the Crow-AMSAA (N.H.P.P.) model. This method was first employed by the U.S. Army Materiel Systems Analysis Activity (AMSAA). It is frequently used on systems when usage is measured on a continuous scale. It can also be applied for high reliability, a large number of trials and one-shot items. Test programs are generally conducted on a phase by phase basis. The Crow-AMSAA model is designed for tracking the reliability within a test phase and not across test phases.
The Crow-AMSAA model
is based on the empirical relationship developed by J. T. Duane. It is
equivalent to a non-homogeneous Poisson process (N.H.P.P.) model with a
Weibull intensity function. While the test phases within a reliability
program can be of equal or unequal length, the Crow-AMSAA model consists of
looking at the reliability growth within a particular phase. Assume that a
particular phase of the program begins at λ,
can be assumed constant between the times [_{i}S]
when design changes are made on the system. Therefore, the number of
failures, _{i-1},S_{i}N, during the _{i}i time period
has the Poisson distribution with mean ^{th}λ(_{i}S-_{i}S)._{i-1}The constant failure intensity,
Let N(T) has the Poisson distribution with mean
λ.
If _{1}TS then _{1 }< T < S_{2}N(T) is the number
of failures in the first interval plus the number of failures in the second
interval, between S and _{1}T. Having failure rate
λ
for the first interval and failure intensity
_{1}λ
for the second interval, the mean of _{2}N(T), which is
θ(T),
is simply:If the failure intensity is homogeneous
(constant) over the test intervals, then S], _{i-2},S_{i-1}N(T) follows a
non-homogeneous Poisson process. Therefore, the mean value function is:
where:
Then for any
An integer-valued process [
Therefore, if
ρ(T),
or the instantaneous failure intensity, λ,
is defined as:_{i}T
From Eqn. (1), the average number of failures
by time
The cumulative failure intensity,
Therefore, the cumulative MTBF
The probability density function ( (i-1)
event occurred at ^{th}T is:_{i-1}
The likelihood function is:
where
Taking the natural log on both sides:
and differentiating with
respect to
Set equal to zero and
solve for
Now differentiate Eqn. 2
with respect to
Set equal to zero and
solve for
Additional information on the Crow-AMSAA model and ReliaSoft's RGA software, which supports this and other reliability growth analysis models, can be found at http://RGA.ReliaSoft.com/. | ||||||||

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