{"id":3807,"date":"2014-10-03T16:14:48","date_gmt":"2014-10-03T21:14:48","guid":{"rendered":"http:\/\/www.livingreliability.com\/en\/?p=3807"},"modified":"2025-11-06T05:33:56","modified_gmt":"2025-11-06T10:33:56","slug":"what-is-random-failure","status":"publish","type":"post","link":"https:\/\/www.livingreliability.com\/en\/posts\/what-is-random-failure\/","title":{"rendered":"Random failure and the MTTF"},"content":{"rendered":"

Nowlan and Heap said\u00a0that the (conditional) probability of\u00a0<\/em>occurrence of most failure modes is constant throughout the life of a\u00a0<\/em>component. They described this behavior as being that of “random” failure. What is the Mean Time to Failure (MTTF) of a randomly failing part?<\/em><\/p>\n

In cases of random failure the formula for a component’s reliability (survival) over time is one of\u00a0exponential decay [1<\/a>]<\/sup> expressed as:<\/p>\n

\"R\left ( t \right )=e^{-Lt}\"<\/p>\n

The relationship between the reliability R(t), and the failure rate function h(t) [2<\/a>]<\/sup> can be expressed by:<\/p>\n

\"h\left ( t \right )=f\left ( t \right )/R\left ( t \right )=\frac{dF(t)/dt}{R(t)}=\frac{d(1-R(t))/dt}{R(t)}=\frac{-dR(t)/dt}{R(t)}\"<\/p>\n

Substituting for R(t) in the case of exponential decay:<\/p>\n

\"h(t)=-\frac{de^{-Lt}/dt}{e^{-Lt}}=--\frac{Le^{-Lt}}{e^{-Lt}}=L\"<\/p>\n

That is to say, for components whose Reliability decreases exponentially with age, the failure rate is constant and equal to L.<\/p>\n

But what is L?<\/p>\n

By definition the mean time to failure (mttf) is the expected<\/em> time to failure:<\/p>\n

\"MTTF=\int_{0}^{\infty}tf(t)dt=\int_{0}^{\infty}R(t)dt=\int_{0}^{\infty}\frac{f(t)}{H(t)}dt\" [3<\/a>]<\/sup><\/p>\n

But for exponential decay we have shown that h(t)=L<\/em>, therefore<\/p>\n

\"MTTF=\frac{1}{L}\int_{0}^{\infty}f(t)dt=1/L=1/h(t)\"<\/p>\n

Hence for random failure: h(t) =L =1\/MTTF [4<\/a>]<\/sup><\/p>\n

and\u00a0\"R\left ( t \right )=e^{-Lt}=e^{-\frac{t}{MTTF}}\"<\/p>\n\n

    \n\t
  1. [1]<\/sup><\/strong>this was demonstrated\u00a0graphically in the article here<\/a>.↩<\/a><\/li>\n\t
  2. [2]<\/sup><\/strong>Or roughly speaking a discretized version of the failure rate, i.e. the Conditional Probability of Failure. For more on the difference between Hazard and Conditional Failure Probability see here<\/a> and here<\/a>↩<\/a><\/li>\n\t
  3. [3]<\/sup><\/strong>See here<\/a> why\u00a0MTTF is the area under the survival curve R(t)↩<\/a><\/li>\n\t
  4. [4]<\/sup><\/strong>Not exactly true if h(t) is taken as synonymous with\u00a0 the conditional failure probability. See here<\/a>.↩<\/a><\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

    Nowlan and Heap said\u00a0that the (conditional) probability of\u00a0occurrence of most failure modes is constant throughout the life of a\u00a0component. They described this behavior as being that of “random” failure. What is the Mean Time to Failure (MTTF) of a randomly failing part? In cases of random failure the formula for a component’s reliability (survival) over […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[89],"tags":[24],"class_list":["post-3807","post","type-post","status-publish","format-standard","hentry","category-theory-and-definitions","tag-conditional-failure-probability"],"_links":{"self":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3807","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/comments?post=3807"}],"version-history":[{"count":2,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3807\/revisions"}],"predecessor-version":[{"id":6471,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3807\/revisions\/6471"}],"wp:attachment":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/media?parent=3807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/categories?post=3807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/tags?post=3807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}