Introduction to the Crow-AMSAA Reliability Growth Model
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 t = 0 and let 0 < S1 < S2 < ... < Sn be the times that modifications are implemented on the equipment within the test phase. The failure intensity, λi, can be assumed constant between the times [Si-1,Si] when design changes are made on the system. Therefore, the number of failures, Ni, during the ith time period has the Poisson distribution with mean λi(Si-Si-1).
The constant failure intensity, λi, assumes that the times between successive failures for this interval follow the exponential distribution.
Let N(T) denote the cumulative number of failures in cumulative test time, T. If 0 < T < S1 then N(T) has the Poisson distribution with mean λ1T. If S1 < T < S2 then N(T) is the number of failures in the first interval plus the number of failures in the second interval, between S1 and T. Having failure rate λ1 for the first interval and failure intensity λ2 for the second interval, the mean of N(T), which is θ(T), is simply:
If the failure intensity is homogeneous (constant) over the test intervals, then N(T) follows a homogeneous Poisson process with mean λT. If the failure intensity is non-homogeneous, i.e. the failure intensity is not the same in the intervals [Si-1,Si] and [Si-2,Si-1], N(T) follows a non-homogeneous Poisson process. Therefore, the mean value function is:
Then for any T:
An integer-valued process [N(T), T > 0] is called a non-homogeneous Poisson process with an intensity function, ρ(T). If ΔT is infinitesimally small, then ρ(T)ΔT is approximately the probability of system failure in the interval (T, T + ΔT). The Crow-AMSAA model assumes that ρ(T) may be approximated by the Weibull failure rate function:
Therefore, if λ = 1/ηβ, the intensity function, ρ(T), or the instantaneous failure intensity, λiT, is defined as:
From Eqn. (1), the average number of failures by time T becomes:
The cumulative failure intensity, λc, is:
Therefore, the cumulative MTBFc is:
Parameter Estimation Using Maximum Likelihood
The probability density function (pdf) of the ith event given that the (i-1)th event occurred at Ti-1 is:
The likelihood function is:
where T* is the termination time and is given by:
Taking the natural log on both sides:
and differentiating with respect to λ yields:
Set equal to zero and solve for λ:
Now differentiate Eqn. 2 with respect to β:
Set equal to zero and solve for β:
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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