Date: Nov 11, 2013 5:47 PM
Author: Richard Ulrich
Subject: Re: Failure rate of population of components: Underdamped response to step function

On Sun, 10 Nov 2013 10:50:58 -0800 (PST),
>I think I should have been clearer about the fact that I'm not trying
>to model second order systems. Rather, I'm trying to find a reference
>for the failure rate with time of an ensemble of parts with Poisson
>failure rates, each of which are replaced upon failure. I assumed
>(perhaps wrongly) that it is well-known and iconic, since it shows up
>in reliability material that I alluded to in my original post.

I've looked back at your original post. You do mention there,
correctly, that there is a curve that becomes asymptotically
"Poison" in distribution of failures counted in small time intervals;
the failures eventually tend to occur uniformly.

I think you have a serious misunderstanding of the vocabulary,
and of the point being made in your source.

EVERY curve with a continuous, increasing failure rate is
going to have some a defined MTTF. This is not peculiar to
some single failure curve. If you replace every failure as it
occurs with a "new" part, the curve is going to be "damped"
when you look at later peaks, and the curve is going to
evolve toward a uniform rate of failures as the starting points
become heterogeneous. It is "uniform rate" that invokes
"Poisson" as one of the possible descriptors.

Rich Ulrich