{"id":5524,"date":"2015-10-24T15:43:19","date_gmt":"2015-10-24T20:43:19","guid":{"rendered":"http:\/\/www.livingreliability.com\/en\/?p=5524"},"modified":"2025-11-06T05:13:00","modified_gmt":"2025-11-06T10:13:00","slug":"failure-probability-prior-to-attaining-mttf","status":"publish","type":"post","link":"https:\/\/www.livingreliability.com\/en\/posts\/failure-probability-prior-to-attaining-mttf\/","title":{"rendered":"Failure probability prior to attaining MTTF"},"content":{"rendered":"

The article “What is the scale parameter<\/a>” showed that 63% of randomly failing items or items adhering to Weibull failure behavier will fail prior to attaining their MTTF. Generally speaking, the MTTF is not a good decision guide for the age replacement of an item. The following shows that for age dependent failure behavior (e.g. shape factor\u00a0 \u03b2 =2) over 50% of the items will fail prior to the MTTF.
\n<\/em><\/p>\n

The article “MTTF is the area under the reliability curve<\/a>” showed that:<\/p>\n

\"\int_{0}^{\infty}R(t)dt=\int_{0}^{\infty}t(f(t))dt=E(t)=MTTF\"<\/p>\n

Assuming\u00a0 Weibull behavior:<\/p>\n

(1) <\/span>   <\/span>\"\begin{align*} MTTF & =\int R\left ( t \right )dt \nonumber \\<\/p>\n

Substitute \u03b2=2 and \"a=\left ( \frac{1}{\eta } \right )^{\beta }\"<\/p>\n

(2) <\/span>   <\/span>\"\begin{align*} MTTF & =\int e^{-at^{2}}dt \nonumber \\<\/p>\n

Knowing that \"\int e^{-ax^{2 }}dx=\frac{1}{2}\left ( \frac{\pi }{a} \right )^{\frac{1}{2}}\", then<\/p>\n

(3) <\/span>   <\/span>\"\begin{align*} MTTF & = \frac{1}{2}\left ( \frac{\pi }{a} \right )^{\frac{1}{2}}\nonumber \\<\/p>\n

At t=MTTF = .89\u03b7<\/p>\n

(4) <\/span>   <\/span>\"\begin{align*} F(t) & = 1-e^{-\left ( \frac{t}{\eta } \right )^{\beta }}\nonumber \\<\/p>\n\n","protected":false},"excerpt":{"rendered":"

The article “What is the scale parameter” showed that 63% of randomly failing items or items adhering to Weibull failure behavier will fail prior to attaining their MTTF. Generally speaking, the MTTF is not a good decision guide for the age replacement of an item. The following shows that for age dependent failure behavior (e.g. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[89],"tags":[21],"class_list":["post-5524","post","type-post","status-publish","format-standard","hentry","category-theory-and-definitions","tag-shape-parameter"],"_links":{"self":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/5524","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=5524"}],"version-history":[{"count":1,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/5524\/revisions"}],"predecessor-version":[{"id":7668,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/5524\/revisions\/7668"}],"wp:attachment":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/media?parent=5524"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/categories?post=5524"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/tags?post=5524"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}