On Mon, 4 Nov 2013 10:03:47 -0800 (PST), email@example.com wrote:
>When I read reliability lecture notes and reliability data information, I've seen references to the apparent failure rate of a population of components. Plotted against time, this failure rate looks like the step-function response of an underdamped second-order system. That is, from time zero (system is new), the curve accelerates upward from zero value and zero derivative until it peaks at t=MTTF, then is followed by decaying oscillations as parts are replaced and their ages become unsynchronized. At steady-state, the population failure rate is essentially constant (Poisson). >
What you describe as the result sounds like results you might get from simulation if you start out with an exponentially increasing function, or even better, with a typical mechanical failure curve that has a distinct "life-span" (and, as you say, skips past the fast-failure phase). Bathtub curve, minus the early failure rate.
(Low, then exponential.) The human rate-of-death is drops off from a peak at birth, and after the age 30, it doubles for every 9 (or so) years of age. It might stop increasing in the 80s or 90s. What underlies this is still a matter of interest.
(More narrowly specified than human life span?) Automobile axle ball-bearings that I read about 30 years ago had a more drastic curve of failure, with a sharp increase at 40,000 (?) miles. The study explained it in good detail. Micro- pitting is "uniform" at the start. Eventually, pits start to overlap, and the size of new pit and the amount of damage-per-hit increase. When my left front wheel (under the engine) froze up, I thought the mechanic had been trying to scam me about changing both front wheels; until the other one froze up a few hundred miles later.
How are curves obtained? Industries probably observed the picture before they described it mathematically. Ball-bearings. Light-bulbs. Lots of experience. Set in your numbers and simulate it.
>I'm trying to find a non-mathmetician's treatment of how this curve is obtained [ ideally online :) ] -- say, for a person with post-graduate engineering background. Actually, I haven't even been able to find a very mathematical treatment that explains how it is arrived at. Intuitively, I understand that there is a peak at t=MTTF, and the failures abate as parts begin to be replaced in earnest (though failures are still replaced before this). But I can't square this off with the probability density function (PDF) for failure in a Poisson process. Since it is an exponential decay, shouldn't we see heavy failures at t=0? Note that I am treating the components as governed by Poisson failures and hence assuming that those parts that die due to infant mortality have been vetted away by a burn-in stage before parts are released/sold for use. So the peaks in the population failure curve are due to synchronicity of component age, which diminish with time. > >I thought of using the CDF for a Poisson process rather than the PDF, since it grows with time, but I don't have an intuitive justification for that. Furthermore, the population failure rate accelerates from a zero derivative, which doesn't correspond to the Poisson CDF. > >I also thought of using the hazard function for a Poisson process, which constant. Not sure I that sheds light on the above "ringing" curve.