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This chapter includes the following sections:
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 (NHPP) model. This method was first developed at 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 the analysis of one shot items when there is high reliability and large number of trials.
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. A development testing program may consist of several separate test phases. If corrective actions are introduced during a particular test phase then this type of testing and the associated data are appropriate for analysis by the Crow-AMSAA model. The model analyzes the reliability growth progress within each test phase and can aid in determining the following:
The reliability growth pattern for the Crow-AMSAA model is exactly the same pattern as for the Duane postulate. That is, the cumulative number of failures is linear when plotted on ln-ln scale. Unlike the Duane postulate, the Crow-AMSAA model is statistically based. Under the Duane postulate, the failure rate is linear on ln-ln scale. However for the Crow-AMSAA model statistical structure, the failure intensity of the underlying non-homogeneous Poisson process (NHPP) is linear when plotted on ln-ln scale.
Let
be the cumulative number of failures observed in cumulative test time
and let
be the failure intensity for the Crow-AMSAA model. Under the NHPP model,
is approximately the probably of a failure occurring over the interval
for small
.
In addition, the expected number of failures experienced over the test
interval
under the Crow-AMSAA model is given by:
![]()
The Crow-AMSAA model assumes that
may be approximated
by the Weibull failure rate function:
Therefore, if
the intensity function,
or the instantaneous failure intensity,
,
is defined as: ![]()
In the special case of exponential failure times there is no growth
and the failure intensity,
, is equal to
. In this case, the
expected number of failures is given by:

In order for the plot to be linear when plotted on ln-ln scale under the general reliability growth case, the following must hold true where the expected number of failures is equal to:
To put a statistical structure on the reliability growth process, consider
again the special case of no growth. In this case the number of failures,
experienced during the testing over
is random. The expected
number of failures,
,
is said to follow the homogeneous (constant) Poisson process with mean
and is given by:
![]()
The Crow-AMSAA generalizes this "no growth" case to allow
for reliability growth due to corrective actions. This generalization
keeps the Poisson distribution for the number of failures but allows for
the the expected number of failures,
, to be linear when
plotted on ln-ln scale. The Crow-AMSAA model lets
. The probability that the number
of failures,
will be equal to
under growth is then given by the Poisson distribution:
![]()
This is the general growth situation and the number of failures,
,
follows a non-homogeneous Poisson process. The exponential, "no growth"
homogeneous Poisson process is a special case of the non-homogeneous Crow-AMSAA
model. This is reflected in the Crow-AMSAA model parameter where
.
The cumulative failure rate,
, is:
(2) The cumulative
is:
(3)
As mentioned above, the local pattern for reliability growth within
a test phase is the same as the growth pattern observed by Duane, discussed
in the Duane chapter of
this on-line reference. The Duane
is equal to:
And
the Duane cumulative failure rate,
, is:
Thus
a relationship between Crow-AMSAA parameters and Duane parameters can
be developed, such that: 
Note that these relationships are not absolute. They change according
to how the parameters (slopes, intercepts, etc.) are defined when the
analysis of the data is performed. For the exponential case,
, then
, a constant. For
,
is increasing. This indicates a deterioration in system reliability. For
,
is decreasing. This is indicative of reliability growth. Note that the
model assumes a Poisson process with Weibull intensity function, not the
Weibull distribution. Therefore, statistical procedures
for the Weibull distribution do not apply for this model. The parameter
is called a scale parameter because it depends upon the unit of measurement
chosen for
.
is the shape parameter that characterizes the shape of the graph of the
intensity function.
The total number of failures,
, is a random variable
with Poisson distribution. Therefore, the probability that exactly
failures occur by time
is:
The number of
failures occurring in the interval from
to
is a random variable
having a Poisson distribution with mean:
The number of failures in any interval is statistically
independent of the number of failures in any interval that does not overlap
the first interval. At time
, the failure intensity is
. If improvements are not made
to the system after time
, it is assumed that failures would continue to
occur at the constant rate
.
Future failures would then follow an exponential distribution with mean
.
The instantaneous
of the system at time
is: ![]()
The Duane and Crow-AMSAA
models are the most frequently used reliability
growth models. Their relationship comes from the fact that both make
use of the underlying observed linear relationship between the logarithm
of cumulative MTBF and cumulative test time. However, the Duane model
does not provide a capability to test whether the change in MTBF observed
over time is significantly different from what might be seen due to random
error between phases. The Crow-AMSAA model allows for such assessments.
Also, the Crow-AMSAA allows for development of hypothesis testing procedures
to determine growth presence in the data where (
indicates
that there is growth in MTBF,
indicates
a constant MTBF and
indicates a decreasing MTBF). Additionally, the Crow-AMSAA model views
the process of reliability growth as probabilistic, while the Duane model
views the process as deterministic.