On Thu, 7 Nov 2013 08:57:56 -0800 (PST), firstname.lastname@example.org wrote:
>Sorry, but I left out a crucial detail when I started this thread. I'm not just referring to a population of parts. I'm taking about a population of parts that are replaced upon failing. Makes a big difference, I know.
No, you did not leave that out. My simple model refers to 2nd, 3rd, etc., failures, based on replacing particular units while keeping count of the lineage so that you can *say*, 2nd, 3rd, etc.
> >On Wednesday, November 6, 2013 12:24:16 PM UTC-5, Rich Ulrich wrote: >>I wrote: >>> So the fact that the 2nd order response to a step function starts >>> at zero with zero slope was a bit of a mystery. >> >> I have no idea what "step function" you have in mind, or what you >> mean by "2nd order response." > >http://hydraulicspneumatics.com/site-files/hydraulicspneumatics.com/files/uploads/2008/02/2008.06-Motion2.png > >Check out the curve for zeta (squiggle) = 0.25. It's not exactly like that, but that's sort of the idea. Initially zero value and zero slope, accelerates up to a peak at t=MTBF, then oscillates in a decaying fashion. (I tried to shorten the URL, but our site firewall is aggressive and TinyURL isn't considered harmless-looking enough).
My reader handles long URLs. I'd like it if you would shorten your regular lines, though, to make them easier to read.
If that curve initially has zero slope, then that is a plot of damping a sine wave. My background is clinical research and epidemiology, where we don't have sinusoidal curves for failures. Or worry about damping them.
As fair as I know, those concerns are the bailiwick of physicists or engineers. I think I would expect the equations to be infinite sums using sines and cosines, and I can't help you further with those.
I still have no idea of why you asked (SUBJECT) about a step function. If the modeling nvolves a sine wave, your "population of components" is more abstract than I would attribute to a "population of components."
If you state what your subject actually is, perhaps someone will have more advice, or advice about where to look.
> >> Are you indeed plotting events? > >No, population failure rate.
Well, basically events-per-fixed-unit of time, so... the same.
> >>> I confided with a colleague, who didn't have a reference. However, >>> he described that it is related to conditional probabilities (if I >>> understood correctly) which to me means Bayes or Total Probability >>> Law. I have yet to sit down and suss it out, but he indicated that >>> one has to convolve t1, t2, t3, etc., where they are decaying >>> exponential PDFs. I'll chime in if I get to that point. As a >>> citation reference, I'm going to cite the manuals where I found the >>> curves in (for now). >> >> The PDF of what ... is "decaying exponential[ly]"? > >The PDF for the MTBF of a part is a decaying exponential. Basically, Poisson governed failure.
Peaks and dips are decaying exponentially TO an asymptote at the steady-state rate. If that is really so.
The steady-state (for your model) exists after the starting points have become randomized, so that new failures are "uniformly random across time" and can be described as Poisson for the counts in short intervals. I don't know if you are misusing Poisson. To me, your emphasis seems inappropriate. You only observe "Poisson governed failure" at the steady state.