Editing Mean time between failures (section)

==MTBF and MDT for networks of components==

Two components <math>c_1,c_2</math> (for instance hard drives, servers, etc.) may be arranged in a network, in [[Series and parallel circuits|''series'' or in ''parallel'']]. The terminology is here used by close analogy to electrical circuits, but has a slightly different meaning. We say that the two components are in series if the failure of ''either'' causes the failure of the network, and that they are in parallel if only the failure of ''both'' causes the network to fail. The MTBF of the resulting two-component network with repairable components can be computed according to the following formulae, assuming that the MTBF of both individual components is known:<ref name="auroraconsultingengineering">{{Cite web|url=http://auroraconsultingengineering.com/doc_files/Reliability_series_parallel.doc|title=Reliability Characteristics for Two Subsystems in Series or Parallel or n Subsystems in m_out_of_n Arrangement (by Don L. Lin)|website=auroraconsultingengineering.com}}</ref><ref name="smith">{{Cite book|author=Dr. David J. Smith|title=Reliability, Maintainability and Risk|edition=eighth|isbn=978-0080969022|year=2011|publisher=Butterworth-Heinemann }}</ref>

:<math>\text{mtbf}(c_1 ; c_2) = \frac{1}{\frac{1}{\text{mtbf}(c_1)} + \frac{1}{\text{mtbf}(c_2)}} = \frac{\text{mtbf}(c_1)\times \text{mtbf}(c_2)}  {\text{mtbf}(c_1) + \text{mtbf}(c_2)}\;,</math>
where <math>c_1 ; c_2</math> is the network in which the components are arranged in series.

For the network containing parallel repairable components, to find out the MTBF of the whole system, in addition to component MTBFs, it is also necessary to know their respective MDTs. Then, assuming that MDTs are negligible compared to MTBFs (which usually stands in practice), the MTBF for the parallel system consisting from two parallel repairable components can be written as follows:<ref name="auroraconsultingengineering" /><ref name="smith" />

<math>

\begin{align}\text{mtbf}(c_1 \parallel c_2) &= \frac{1}{\frac{1}{\text{mtbf}(c_1)}\times\text{PF}(c_2,\text{mdt}(c_1))+\frac{1}{\text{mtbf}(c_2)}\times\text{PF}(c_1,\text{mdt}(c_2))} 
\\[1em]&= \frac{1}{\frac{1}{\text{mtbf}(c_1)}\times\frac{\text{mdt}(c_1)}{\text{mtbf}(c_2)}+\frac{1}{\text{mtbf}(c_2)}\times\frac{\text{mdt}(c_2)}{\text{mtbf}(c_1)}} 
\\[1em]&= \frac{\text{mtbf}(c_1)\times \text{mtbf}(c_2)}  {\text{mdt}(c_1) + \text{mdt}(c_2)}\;,

\end{align}
</math>

where <math>c_1 \parallel c_2</math> is the network in which the components are arranged in parallel, and <math>PF(c,t)</math> is the probability of failure of component <math>c</math> during "vulnerability window" <math>t</math>.

Intuitively, both these formulae can be explained from the point of view of failure probabilities. First of all, let's note that the probability of a system failing within a certain timeframe is the inverse of its MTBF. Then, when considering series of components, failure of any component leads to the failure of the whole system, so (assuming that failure probabilities are small, which is usually the case) probability of the failure of the whole system within a given interval can be approximated as a sum of failure probabilities of the components. With parallel components the situation is a bit more complicated: the whole system will fail if and only if after one of the components fails, the other component fails while the first component is being repaired; this is where MDT comes into play: the faster the first component is repaired, the less is the "vulnerability window" for the other component to fail.

Using similar logic, MDT for a system out of two serial components can be calculated as:<ref name="auroraconsultingengineering" />
:<math>\text{mdt}(c_1 ; c_2) = \frac{\text{mtbf}(c_1)\times \text{mdt}(c_2) + \text{mtbf}(c_2)\times \text{mdt}(c_1)}  {\text{mtbf}(c_1) + \text{mtbf}(c_2)}\;,</math>
and for a system out of two parallel components MDT can be calculated as:<ref name="auroraconsultingengineering" />
:<math>\text{mdt}(c_1 \parallel c_2) = \frac{\text{mdt}(c_1)\times \text{mdt}(c_2)}  {\text{mdt}(c_1) + \text{mdt}(c_2)}\;.</math>

Through successive application of these four formulae, the MTBF and MDT of any network of repairable components can be computed, provided that the MTBF and MDT is known for each component. In a special but all-important case of several serial components, MTBF calculation can be easily generalised into 
:<math>\text{mtbf}(c_1;\dots; c_n) = \left(\sum_{k=1}^n \frac 1{\text{mtbf}(c_k)}\right)^{-1}\;,</math>
which can be shown by induction,<ref>{{Cite web|url=http://www.angelfire.com/ca/summers/Business/MTBFAllocAnalysis1.html|archive-url=https://web.archive.org/web/20021106143359/http://www.angelfire.com/ca/summers/Business/MTBFAllocAnalysis1.html|url-status=dead|archive-date=November 6, 2002|title=MTBF Allocations Analysis1|website=[[Angelfire]]|access-date=2016-12-23}}</ref> and likewise
:<math>\text{mdt}(c_1\parallel\dots\parallel c_n) = \left(\sum_{k=1}^n \frac 1{\text{mdt}(c_k)}\right)^{-1}\;,</math>
since the formula for the mdt of two components in parallel is identical to that of the mtbf for two components in series.