Paul
Posts:
425
Registered:
2/23/10


Failure rate of population of components: Underdamped response to step function
Posted:
Nov 4, 2013 1:03 PM


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 stepfunction response of an underdamped secondorder 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 steadystate, the population failure rate is essentially constant (Poisson).
I'm trying to find a nonmathmetician's treatment of how this curve is obtained [ ideally online :) ]  say, for a person with postgraduate 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 burnin 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.

