```Date: Nov 4, 2013 1:03 PM
Author: Paul
Subject: Failure rate of population of components: Underdamped response to<br> step function

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).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.
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