UAV Flight Computer: Reading Growth From Two Data Points
A flight-control computer for a UAV has accumulated 14,000 h of test across 20 units. The first 4,000 h produced 9 failures; after two design changes, the next 10,000 h produced 7 failures. The program requires an instantaneous MTBF of 3,000 h at 30,000 cumulative hours before low-rate initial production. The reliability manager fits a Crow-AMSAA model, reports beta = 0.46, and concludes that growth is healthy and the requirement will be met. Two of the seven late failures were repeats of earlier modes; one corrective action was a supplier screening change that removed a defective component lot but did not alter the design. Test funding currently ends at 20,000 h, and the customer wants a one-sided 90% confidence statement rather than a point projection.
1. Reproduce the growth calculation: estimate beta, the instantaneous MTBF at 14,000 h, and the projected instantaneous MTBF at 30,000 h. Is the 3,000 h requirement met on this projection?
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Cumulative failures: N(4,000) = 9, N(14,000) = 16.
beta = ln(16/9) / ln(14,000/4,000) = 0.5754 / 1.2528 = 0.459
beta < 1 indicates growth, with growth rate 1 - beta = 0.54.
Instantaneous MTBF at 14,000 h: M_i(T) = T / (beta x N(T)) = 14,000 / (0.459 x 16) = 14,000 / 7.34 = 1,906 h
Note this is not the cumulative MTBF of 14,000/16 = 875 h; the requirement is stated on the instantaneous value, which is the right choice because it reflects current, not average historic, performance.
Projection to 30,000 h uses the same shape parameter: N(30,000) = 16 x (30,000/14,000)^0.459 = 16 x 1.419 = 22.7 M_i(30,000) = 30,000 / (0.459 x 22.7) = 30,000 / 10.42 = 2,880 h
Equivalently, M_i scales as (30,000/14,000)^(1-beta) = 1.51, so 1,906 x 1.51 = 2,878 h.
On the point projection the requirement is not met: 2,880 h against 3,000 h, about 4% short. The manager's conclusion of success is wrong on his own model. And the point projection is the most favourable number available; the confidence bound will be materially lower.
2. Critique the validity of the fit and the projection. What would you require before accepting a 90% confidence statement at 30,000 h?
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The fit is not a fit. Two interval counts define a straight line through two points; beta is a two-point interpolation with no degrees of freedom left to test anything. To compute a confidence interval on beta you need the individual failure times, and they are not in evidence.
Substantive concerns:
- Trend. A Laplace or Cramer-von Mises test on the individual times is needed to show that the process is still improving rather than flattening. With 7 failures in the last 10,000 h (interval MTBF 1,429 h) and 9 in the first 4,000 h (444 h), improvement is plausible but not established.
- Repeat modes. Two of the seven late failures repeat earlier modes. A test-fix-find-test model (Crow with a fix effectiveness factor) discounts fixes that did not actually reduce the failure intensity. A repeat mode is evidence that the fix effectiveness for that mode is near zero, not evidence of growth.
- Screening is not a design fix. Removing a defective component lot changes the population, not the design. That portion of the apparent growth may not extrapolate to a production population built with controlled suppliers. It should be treated as a manufacturing correction, with its effect stated separately.
- Extrapolation range. Data end at 14,000 h, funding ends at 20,000 h, and the claim is at 30,000 h, more than twice the observed range, on the assumption that the same failure modes and the same fix rate continue.
- Confidence. With approximately 23 projected failures, a one-sided 90% lower bound on MTBF is roughly 60-70% of the point estimate, so on the order of 1,800-2,100 h, well below 3,000 h.
Requirements before any acceptance statement: individual failure times in a FRACAS with mode, cause, and corrective action; a trend test; the test-fix-find-test fit with fix effectiveness factors; and a one-sided 90% bound computed at 30,000 h. If the bound cannot reach 3,000 h, the options are more test time, further design action, or a renegotiated requirement stated as MTBF at a specified confidence.
3. You have funding to 20,000 h only. Propose a plan that maximizes the chance of a defensible 3,000 h claim, and state the decision rule you would agree with the customer now.
Read the worked answer
Plan for the remaining 6,000 h:
- Fix before testing, not after. Close out the two repeat modes and the highest-intensity new modes first; testing with known ineffective fixes burns hours and corrupts the growth estimate.
- Record every failure individually with time, mode, cause, corrective action, and whether the fix was verified. This is the single highest-value change; without it no confidence statement is possible.
- Classify corrections. Separate design changes (which contribute to growth) from screening, process, and supplier corrections. Report the growth projection both with and without the manufacturing corrections, so the customer can see how much of the improvement depends on supplier control.
- Pre-agree the model and the confidence method with the customer: Crow-AMSAA test-fix-find-test, one-sided 90% bound on instantaneous MTBF at 30,000 h, stated assumptions about continuing fix effectiveness.
- Consider a focused reliability test on the dominant remaining modes rather than uniform extra hours, since 6,000 h spread across an entire system rarely accumulates enough failures on any one mode to prove a fix.
Decision rule, agreed in advance: at 20,000 h, compute the one-sided 90% lower bound on projected instantaneous MTBF at 30,000 h. If the bound is at or above 3,000 h, proceed to LRIP with a field monitoring plan. If the point estimate is above 3,000 h but the bound is not, proceed only with a documented growth commitment and a gated follow-up test. If the point estimate is below 3,000 h, do not proceed; the shortfall is a design problem, and additional test hours at the current fix rate will not close it in the available time.