Editing Failure rate (section)

===Combinations of failure types===

If a complex system consists of many parts, and the failure of any single part means the failure of the entire system, then the total failure rate is simply the sum of the individual failure rates of its parts

:<math>\lambda_S = \lambda_{P1} + \lambda_{P2} + \ldots</math>

however, this assumes that the failure rate <math>\lambda(t)</math> is constant, and that the units are consistent (e.g. failures per million hours), and not expressed as a ratio or as probability densities. This is useful to estimate the failure rate of a system when individual components or subsystems have already been tested.<ref>
[http://www.weibull.com/hotwire/issue108/relbasics108.htm "Reliability Basics"].
2010.
</ref><ref>Vita Faraci.
[http://src.alionscience.com/pdf/1Q2006.pdf "Calculating Failure Rates of Series/Parallel Networks"] {{Webarchive|url=https://web.archive.org/web/20160303224453/http://src.alionscience.com/pdf/1Q2006.pdf |date=2016-03-03 }}.
2006.</ref>

Adding "redundant" components to eliminate a [[single point of failure]] may thus actually increase the failure rate, however reduces the "mission failure" rate, or the "mean time between critical failures" (MTBCF).<ref>
[https://www.quanterion.com/mission-reliability-and-logistics-reliability-a-design-paradox/ "Mission Reliability and Logistics Reliability: A Design Paradox"].
</ref>

Combining failure or hazard rates that are time-dependent is more complicated. For example, mixtures of Decreasing Failure Rate (DFR) variables are also DFR.<ref name="brown1980">{{Cite journal | last1 = Brown | first1 = M. | title = Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes | doi = 10.1214/aop/1176994773 | journal = The Annals of Probability | volume = 8 | issue = 2 | pages = 227–240 | jstor = 2243267| year = 1980 | doi-access = free }}</ref>  Mixtures of [[exponential distribution|exponentially distributed]] failure rates are [[Hyperexponential distribution|hyperexponentially distributed]].