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.