Crow-AMSAA (N.H.P.P.)

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 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 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.

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 associate 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:

Reliability of the configuration currently on test
Reliability of the configuration on test at the end of the test phase
Expected reliability if the test time for the test phase is extended
Growth rate
Available confidence intervals
Applicable goodness-of-fit tests

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 N(t) be the cumulative number of failures observed in cumulative test time t 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 [0, T] 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:

(1)

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 N(T) experienced during the testing over [0, T] is random. The expected number of failures N(T) 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 E[N(T)] to be linear when plotted on ln-ln scale. The Crow-AMSAA model lets . The probability that the number of failures N(T) will be equal to n under growth is then given by the Poisson distribution.

This is the general growth situation, and the number of failures, N(T), 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)

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 T. is the shape parameter that characterizes the shape of the graph of the intensity function.

The total number of failures, N(T), is a random variable with Poisson distribution. Therefore, the probability that exactly n failures occur by time T is:

The number of failures occurring in the interval from T1 to T2 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 T0, the failure intensity is . If improvements are not made to the system after time T0, it is assumed that failures would continue occurring at the constant rate . Future failures would then follow an exponential distribution with mean . The instantaneous MTBF of the system at time T is:

This chapter includes the following sections:

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