Talk:Mean time between failures

MTBF of components in series / parallel; sources

I propose to add formulae to compute the MTBF of systems from the MTBFs of its components (arranged in series or in parallel). I know the formulae, and they are available at several places, such as forum threads [1] or student notes [2], and many others, but I haven't found a good, clean, reliable source. If someone has a good source for such things, please share. (This is on my todo list, but I'll probably leave this be for a few days or weeks before editing the article.) — Gamall Wednesday Ida (t · c) 06:00, 20 December 2016 (UTC)

I have added the formulae in a new section. I still haven't found any really good ref. This is not my domain -- just something I encountered in a course when I was a student, and haven't used since -- so I don't intend to dig much further. If someone with working knowledge of those issues is motivated to improve this article, it would be great. I think a synthetic article explaining the relationships between MTBF, MTTR, MDT, availability, and whatnot would be an extremely interesting thing to have, but I'm not the one to write it. — Gamall Wednesday Ida (t · c) 16:35, 23 December 2016 (UTC)

mtbf(c1||c2) = mtbf(c1)*mtbf(c2) would mean that at every attempt to calculate anything practical - we'll be multiplying hours by hours and getting hours-squared (ouch!). I know that this kind of formula is in circulation (and under certain assumptions it will stand), but I honestly prefer much more obvious and straightforward formula from [5] (auroraconsulting) - mtbf(c1||c2) = mtbf(c1)*mtbf(c2)/(mdt(c1) + mdt(c2)). Ipsign (talk) 15:43, 27 December 2016 (UTC)

Then if you feel so inclined, by all means be bold and add this formula, discuss the assumptions under which it my strangely-dimensioned one may stand, etc. (You might then want to define MDT somewhere properly, I see no proper mathematical model or definition anywhere on WP.) Note the talk archives: the definitions of the quantities involved have been subject to some controversy in past years. As for me, I have no special insight into the topic, and due to RL constraints shall have little time or joie de vivre to invest in WP content for some time. My addition was something of a drive-by, hoping to wake up more knowledgeable editors here. — Gamall Wednesday Ida (t · c) 16:24, 27 December 2016 (UTC)

"discuss the assumptions under which it my strangely-dimensioned one may stand, etc." - this is going to be difficult (my understanding that it is similar to the strange-dimensional math systems dealing with Maxwell equations via postulating that c === 1 and removing it from equations completely - then playing math games within this strange-dimension system ad infinitum - and then restoring missing c's using dimensional analysis when converting back to real-world; while it works - it is not exactly easy to explain). Do you mind if I just replace this strangely-dimensioned formula with the more obvious one from [5] (saying all the necessary words to explain it)? This whole thing is not exactly my specialty either (though defining MDT in a reasonable-for-this-article-manner won't be a problem, and justifying the formula from [5] using basic probability theory is also rather obvious) - but dimensional analysis _is_ my hobby :-). Whenever I see something like "liters-cubed" (or seconds-to-100th-power) - I get nervous ;-). Ipsign (talk) 16:57, 27 December 2016 (UTC)

I understand your nervosity. Like I said, be bold! This article has been in a horrid state for years, any insight you can add into it is quite welcome in my book. I acknowledge the dimension as a serious problem in the formula I gave. When I come back, if I absolutely want to put it in again, it will be incumbent on me to find decent arguments for it. Thanks, and happy editing! — Gamall Wednesday Ida (t · c) 17:16, 27 December 2016 (UTC)

Phew, done - whatever I could do at the moment. Now the formulae do need to know MDTs to calculate MTBFs in general case (and vice versa), but that's a fact of life (the faster you repair parallel components - the less chances are for the whole system to fail, and IMO all the math models which assume that the time is discreet and everything is fixed on the next time slot - are significantly flawed exactly as they ignore this all-important observation). Also - now the formulae are perfectly consistent from dimensional analysis point of view :-). Ipsign (talk) 06:57, 28 December 2016 (UTC)

Thanks for your contribution, looks interesting. I'd begun to do some trivial copy-editing, but then I ran into something that made me nervous, this time: failure of any component leads to the failure of the whole system, so probability of the failure of the whole system within a given interval is a sum of failure probabilities of the components.. You can only sum probabilities like that for mutually exclusive events. If I have two systems with independent 3/4 chance of failing, I don't get a 1.5 proba of failure. Do you intend to go back on this part? — Gamall Wednesday Ida (t · c) 07:13, 28 December 2016 (UTC)

"You can only sum probabilities like that for mutually exclusive events.". Strictly speaking - yes, but as soon as we agree on an assumption that failures are very rare (formally - with probabilities under consideration being << 1), adding probabilities becomes a valid approximation (give or take a small-o). Probably this assumption of failures being rare needs to be articulated somewhere - feel free to add it :-). Without this assumption, the whole thing would become way too complicated :-(. Ipsign (talk) 07:20, 28 December 2016 (UTC)

added the assumption myself. Ipsign (talk) 07:26, 28 December 2016 (UTC)

I see. How naughty of you to make such assumptions without declaring them in triplicate :P When I have more time and energy I'll have to go over that stuff more carefully. When I'm retired, let's say. (Would it really be so complicated without the assumption?). Anyway, thanks again, and cheers. — Gamall Wednesday Ida (t · c) 07:31, 28 December 2016 (UTC)

"How naughty of you to make such assumptions without declaring them in triplicate :P" - guilty as charged ;-). The whole MTBF field is ridden with various assumptions, the problem is which ones are realistic, and which are not. IMO, the set of assumptions currently in the article is reasonable enough for practical use (and MTBFs, as least as I see them, are all about practice). "Would it really be so complicated without the assumption?" - I am not really a math guy, but just yesterday I asked a mathematician friend of mine to take a look at MTBF maths, and his starting comments were like "oh, here we'd have a factorial, but as long as X<<Y it doesn't really matter much and approximation will stand". It is certainly possible to write it down strictly, but I have doubts that I'll be able to do it myself without making some subtle mistakes (probability in general is a Big Can of Worms :-( ). Ipsign (talk) 07:45, 28 December 2016 (UTC)

Ideally there would be a source laying it all out clearly (Def. Lemma. Thm), but that doesn't seem to exist -- it would have been found years ago and avoided the disputes I've archived. Some degree of WP:OR may be unavoidable here in order to have a good article. I for one am not tackling it anytime soon. (I haven't done probas in a decade). I know someone who might have good sources, but I'll wait for a deadline-free period before contacting him, so I can reliably put some effort in the exchange. — Gamall Wednesday Ida (t · c) 09:20, 28 December 2016 (UTC)