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MIL-HDBK- 189C instantaneous failure rate, r(t), can be calculated for any time t within the test. Further, given a and X, it is possible to solve for t, the test time it will take to achieve a specific reliability. This assumes that the factors affecting reliability growth remain unchanged across the development. 5.2.6.2 Drawbacks to Duane's Method. Duane stated that a could be universally treated as being 0.5, the modal value within his database. This has since been shown to be unrealistic, as per Table II. All Duane MTBF growth curves pass through the origin of the graph on a linear-linear plot, imputing zero reliability at the start of test. The method is also a deterministic estimation of the regression, which makes no allowance for variation. Table II Historical Growth Rate Estimates System Type Mean/Median Range One Shot (Missiles) .046/0.47 0.27-0.64 Time or Distance Based 0.34/0.32 0.23-0.53 5.2.7 Development of AMSAA Crow Planning Model. Crow explored the advantages of using a NHPP with a Weibull intensity function to model several phenomena, including reliability growth. If system failure times follow the Duane Postulate, then they can be modeled as a NHPP with Weibull intensity function (i.e., based on the NHPP with power law mean value function). To make the transition from Duane's formulae to the Weibull intensity functional forms, ( has to be substituted for 1- a. Thus the parameters in the AMSAA Crow Planning Model are X and (, where ( determines the shape of the curve. The physical interpretation of ( (called the growth parameter) is the ratio of the average (cumulative) MTBF to the current (instantaneous) MTBF at time t. Even though Crow's growth parameter estimate is still interpreted as the estimate of the negative slope of a straight line on a log-log plot, the estimates of X and ( differ from Duane's procedures in that the estimation procedure is Maximum Likelihood Estimate (MLE), not least squares, thus each model's parameters correspond to different straight lines. The reliability planning curve may extend over all the test phases or just over one test phase. Typically a smooth growth curve is portrayed which represents the overall expected pattern growth over the test phases. As noted earlier, it can be modeled as a NHPP with power law mean value function (expected number of failures as a function of cumulative test time) E(N(t)) = which is comparable to the global pattern noted by Duane. Taking the derivative we obtain the idealized reliability growth pattern with failure intensity function p(t) given by p(t) = Apt1, 0 < ft <1. Thus, as with Duane, it has a singularity at t = 0. The methodology is based on first specifying an idealized curve that satisfies the expected number of failures at the end of each test phase with cumulative test times t1, t2, ..., tk. For planning purposes, the overall growth trend is represented only for t > t1. It simply utilizes a constant or average failure rate, 01 = M1, over the first test phase. The constant 01 is chosen such that the expected number of failures is satisfied for t = t1. Doing so, it follows that the MTBF growth trend for t > ti and 01 is given by, 32

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