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Topic: Failure rate of population of components: Underdamped response to
step function

Replies: 15   Last Post: Nov 18, 2013 10:15 AM

 Messages: [ Previous | Next ]
 Paul Posts: 517 Registered: 2/23/10
Re: Failure rate of population of components: Underdamped response to
step function

Posted: Nov 10, 2013 1:50 PM

On Thursday, November 7, 2013 2:13:06 PM UTC-5, Rich Ulrich wrote:
> ...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.

If you mean that you're referring to a probabilistic treatment of a
single part rather than a rate for an ensemble of identical parts,
then we are talking about different creatures, albeit related.

>>On Wednesday, November 6, 2013 12:24:16 PM UTC-5, Rich Ulrich wrote:
>>
>> 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.

Hmmm, I thought that would be hostile to autowrapping. A quick google
shows this to be correct
(http://www.softdevlabs.com/personal/Usenet101.html) but limiting the
width is still advised because proper treatment of text flow in quotes
is (still!) a rarity. My best attempt at treading the middle ground
has been to wrap quoted text, but not unquoted text. However, this
doesn't solve the flow problem for people who reply because my
unquoted text suddenly becomes quoted text (of very long lines). So
I'll limit all text to 70 character lines. This will still break the
flow for deeply quoted text. I might bring it down to 60 in the
future. Thanks for that.

> 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

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. My
reference to underdamped response to step function is simply to say
that its the only way I know how to descibe its general appearance. I
should add that in diagrams that I saw, it takes noticably longer for
the function to increase in value and slop to its inflection point.

>>> Are you indeed plotting events?
>>
>> No, population failure rate.

>
> Well, basically events-per-fixed-unit of time, so... the same.

I would have to agree, to a degree. However, I was trying to
distinguish between counting functions versus counting functions that
normalize to elapsed time. I've seen counting functions that don't
normalize to time in the characterization of random processes.

> >>> 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....

> >>
> >> 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.
>
> 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.

Yes, for sure. I ensemble failure rate looks like a 2nd order
response to a step function in that there are transients from time
zero, followed by a decaying oscillation that settles on steady-state.
Steady-state corresponds to a Poisson governed process, but I was
interested in how the transient behaviour was derived.

Date Subject Author
11/4/13 Paul
11/4/13 Richard Ulrich
11/5/13 Paul
11/5/13 Richard Ulrich
11/6/13 Paul
11/6/13 Richard Ulrich
11/7/13 Paul
11/7/13 Richard Ulrich
11/10/13 Paul
11/11/13 Richard Ulrich
11/15/13 mr.fred.ma@gmail.com
11/15/13 Richard Ulrich
11/15/13 Paul
11/17/13 Richard Ulrich
11/18/13 Paul
11/5/13 Dan Heyman