Growth processes are rather "noisy". In the development of this document, to determine the probable uncertainties for MTBF and growth estimates, Monte Carlo simulations were nun using the AMSAA model. For each trial, all failure times up to the 30th failure were recorded, and estimates of the growth rate and current MTBF were made at the 5th, 10th, 15th, 20th, 25th, and 30th failure. Figure C-1.A indicates the band containing 80 percent of the simulation results in terms of the ratio of estimated MTBF to true MTBF versus the number of failures. After 5 failures, 10 percent of the MTBF estimates would be expected to exceed the true value by factors greater than 2.6 and 10 percent would be less than 0.45 of the true value. Even at the 30th failure-implying a fairly lengthy test--the factors were approximately 1.4 and 0.7, respectively.
Again using the Monte Carlo simulations, Figures C-1,
B,C,and D reflect varying dispersions of growth estimates depending upon the
true growth rate. The greater the true growth rate, the smaller the band of
dispersion. Additionally, regardless of the true growth rate, as testing
continues and failures increase the dispersion band narrows. There are risks
of both gross overestimation and substantial underestimation of growth
rates--to the extreme of seemingly negative growth, especially in the case of
low true growth and limited test results. The point of Figure C-1
is that because of the considerable uncertainty, estimates of MTBF and growth rate are indicators only. Do not make hard decisions on the basis of these indicators alone - use engineering judgement.
While Figure C-1
illustrates how it is possible to observe a wide range of apparent MTBFs for a given true MTBF and, similarly, the range of apparent growth rates that could be observed, a program manager may well be interested in approaching the question from a different direction. That is: for a given estimated (or calculated, observed, etc.) growth rate, what is the possible range of the true MTBF and how does that range compare to the required MTBF?
Figure C-2.A is a graphic version of the confidence
interval tables found in MIL-HDBK-189 and assists in answering the manager's
question. Suppose that the required MTBF is 1000 hours and that after 5
failures the current estimate is 1500 hours. Figure C-2.B takes a "slice"
through the Figure C-2.A confidence band at the 5 failure point and shows the
results in MTBF terms rather than ratios. Even though the amount of test data
is small, the manager would justifiably have confidence in his progress: with
a range from 640 to 5080 hours, there is relatively little likelihood that the
true MTBF is under 1000 hours. The same cautions apply as stated above for