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Crow Extended Model

Crow Extended - Continuous Evaluation Model

Crow-AMSAA (NHPP) Model

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 $N(t)$ be the cumulative number of failures observed in cumulative test time $t$ and let $\rho (t)$ be the failure intensity for the Crow-AMSAA model. Under the NHPP model, $\rho (t)\Delta t$ is approximately the probably of a failure occurring over the interval $[t,t+\Delta t]$ for small $\Delta t$. In addition, the expected number of failures experienced over the test interval $[0,T]$ under the Crow-AMSAA model is given by:

MATH

The Crow-AMSAA model assumes that $\rho (T)$ may be approximated by the Weibull failure rate function: MATH Therefore, if MATH the intensity function, $\rho (T),$ or the instantaneous failure intensity, $\lambda _{i}(T)$, is defined as: MATH

In the special case of exponential failure times there is no growth and the failure intensity, $\rho (t)$, is equal to $\lambda $. In this case, the expected number of failures is given by:

MATH

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:

MATH (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 $\lambda T$ and is given by:

MATH

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 MATH. The probability that the number of failures, $N(T),$ will be equal to $n$ under growth is then given by the Poisson distribution:

MATH

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 $\beta =1$.

The cumulative failure rate, $\lambda _{c}$, is: MATH (2) The cumulative $MTBF_{c}$ is: MATH (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 $MTBF_{c}$ is equal to: MATHAnd the Duane cumulative failure rate, $\lambda _{c}$, is: MATHThus a relationship between Crow-AMSAA parameters and Duane parameters can be developed, such that: MATH

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, $\beta =1$, then MATH, a constant. For $\beta >1$, $\lambda _{i}(T)$ is increasing. This indicates a deterioration in system reliability. For $\beta <1$, $\lambda _{i}(T)$ 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 $\lambda $ is called a scale parameter because it depends upon the unit of measurement chosen for $T$. $\beta $ 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: MATHThe number of failures occurring in the interval from $T_{1}$ to $T_{2}$ is a random variable having a Poisson distribution with mean: MATHThe 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 $T_{0}$, the failure intensity is MATH. If improvements are not made to the system after time $T_{0}$, it is assumed that failures would continue to occur at the constant rate MATH. Future failures would then follow an exponential distribution with mean MATH. The instantaneous $MTBF$ of the system at time $T$ is: MATH

Applicability of the Crow-AMSAA Model

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 ($\beta <1$ indicates that there is growth in MTBF, $\beta =1$ indicates a constant MTBF and $\beta >1$ 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.