Mean time between failures

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#MTBF_and_MDT_for_networks_of_components"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>MTBF and MDT for networks of components</span> </div> </a> <ul id="toc-MTBF_and_MDT_for_networks_of_components-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variations_of_MTBF" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Variations_of_MTBF"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Variations of MTBF</span> </div> </a> <button aria-controls="toc-Variations_of_MTBF-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Variations of MTBF subsection</span> </button> <ul id="toc-Variations_of_MTBF-sublist" class="vector-toc-list"> <li id="toc-MTBF_considering_censoring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#MTBF_considering_censoring"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>MTBF considering censoring</span> </div> </a> <ul id="toc-MTBF_considering_censoring-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" 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Available in 27 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-27" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">27 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Gemiddelde_tyd_tussen_faling" title="Gemiddelde tyd tussen faling – Afrikaans" lang="af" hreflang="af" data-title="Gemiddelde tyd tussen faling" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Nasazl%C4%B1qlar_aras%C4%B1ndak%C4%B1_orta_zaman_(%C4%B0nformatika)" title="Nasazlıqlar arasındakı orta zaman (İnformatika) – Azerbaijani" lang="az" hreflang="az" data-title="Nasazlıqlar arasındakı orta zaman (İnformatika)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Temps_mitj%C3%A0_entre_errors" title="Temps mitjà entre errors – Catalan" lang="ca" hreflang="ca" data-title="Temps mitjà entre errors" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/St%C5%99edn%C3%AD_doba_mezi_poruchami" title="Střední doba mezi poruchami – Czech" lang="cs" hreflang="cs" data-title="Střední doba mezi poruchami" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Mean_time_between_failures" title="Mean time between failures – Danish" lang="da" hreflang="da" data-title="Mean time between failures" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mean_Time_Between_Failures" title="Mean Time Between Failures – German" lang="de" hreflang="de" data-title="Mean Time Between Failures" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tiempo_medio_entre_fallos" title="Tiempo medio entre fallos – Spanish" lang="es" hreflang="es" data-title="Tiempo medio entre fallos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%A7%D9%86%DA%AF%DB%8C%D9%86_%D8%B2%D9%85%D8%A7%D9%86_%D8%A8%DB%8C%D9%86_%D8%AE%D8%B1%D8%A7%D8%A8%DB%8C" title="میانگین زمان بین خرابی – Persian" lang="fa" hreflang="fa" data-title="میانگین زمان بین خرابی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Temps_moyen_entre_pannes" title="Temps moyen entre pannes – French" lang="fr" hreflang="fr" data-title="Temps moyen entre pannes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%EA%B7%A0_%EB%AC%B4%EA%B3%A0%EC%9E%A5_%EC%8B%9C%EA%B0%84" title="평균 무고장 시간 – Korean" lang="ko" hreflang="ko" data-title="평균 무고장 시간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%AB%E0%A4%B2%E0%A4%A4%E0%A4%BE_%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF_%E0%A4%B8%E0%A4%AE%E0%A4%AF" title="अन्तर्विफलता माध्य समय – Hindi" lang="hi" hreflang="hi" data-title="अन्तर्विफलता माध्य समय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Waktu_rata-rata_sebelum_kerusakan" title="Waktu rata-rata sebelum kerusakan – Indonesian" lang="id" hreflang="id" data-title="Waktu rata-rata sebelum kerusakan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tempo_medio_fra_i_guasti" title="Tempo medio fra i guasti – Italian" lang="it" hreflang="it" data-title="Tempo medio fra i guasti" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/MTBF" title="MTBF – Hebrew" lang="he" hreflang="he" data-title="MTBF" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D2%9B%D1%82%D0%B0%D1%83%D1%81%D1%8B%D0%B7_%D0%B6%D2%B1%D0%BC%D1%8B%D1%81%D1%82%D1%8B%D2%A3_%D0%BE%D1%80%D1%82%D0%B0%D1%88%D0%B0_%D1%83%D0%B0%D2%9B%D1%8B%D1%82%D1%8B" title="Тоқтаусыз жұмыстың орташа уақыты – Kazakh" lang="kk" hreflang="kk" data-title="Тоқтаусыз жұмыстың орташа уақыты" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/MTBF" title="MTBF – Hungarian" lang="hu" hreflang="hu" data-title="MTBF" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Mean_time_between_failures" title="Mean time between failures – Dutch" lang="nl" hreflang="nl" data-title="Mean time between failures" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E5%9D%87%E6%95%85%E9%9A%9C%E9%96%93%E9%9A%94" title="平均故障間隔 – Japanese" lang="ja" hreflang="ja" data-title="平均故障間隔" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/MTBF" title="MTBF – Polish" lang="pl" hreflang="pl" data-title="MTBF" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/MTBF" title="MTBF – Portuguese" lang="pt" hreflang="pt" data-title="MTBF" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%BE%D1%82%D0%BA%D0%B0%D0%B7" title="Наработка на отказ – Russian" lang="ru" hreflang="ru" data-title="Наработка на отказ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/MTBF" title="MTBF – Finnish" lang="fi" hreflang="fi" data-title="MTBF" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/MTBF" title="MTBF – Swedish" lang="sv" hreflang="sv" data-title="MTBF" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8B%E0%AE%B2%E0%AF%8D%E0%AE%B5%E0%AE%BF%E0%AE%95%E0%AE%B3%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%81_%E0%AE%87%E0%AE%9F%E0%AF%88%E0%AE%AF%E0%AE%BF%E0%AE%B2%E0%AE%BE%E0%AE%A9_%E0%AE%9A%E0%AE%B0%E0%AE%BE%E0%AE%9A%E0%AE%B0%E0%AE%BF_%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="தோல்விகளுக்கு இடையிலான சராசரி நேரம் – Tamil" lang="ta" hreflang="ta" data-title="தோல்விகளுக்கு இடையிலான சராசரி நேரம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B5%D1%80%D0%B5%D0%B4%D0%BD%D1%96%D0%B9_%D0%BD%D0%B0%D1%80%D0%BE%D0%B1%D1%96%D1%82%D0%BE%D0%BA_%D0%BC%D1%96%D0%B6_%D0%B2%D1%96%D0%B4%D0%BC%D0%BE%D0%B2%D0%B0%D0%BC%D0%B8" title="Середній наробіток між відмовами – Ukrainian" lang="uk" hreflang="uk" data-title="Середній наробіток між відмовами" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B9%B3%E5%9D%87%E6%95%85%E9%9A%9C%E9%96%93%E9%9A%94" title="平均故障間隔 – Cantonese" lang="yue" hreflang="yue" data-title="平均故障間隔" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E5%9D%87%E6%95%85%E9%9A%9C%E9%96%93%E9%9A%94" title="平均故障間隔 – Chinese" lang="zh" hreflang="zh" data-title="平均故障間隔" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q754940#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Mean_time_between_failures" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Mean_Time_Between_Failure&redirect=no" class="mw-redirect" title="Mean Time Between Failure">Mean Time Between Failure</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Predicted elapsed time between inherent failures of a system during operation</div> <p><b>Mean time between failures</b> (<b>MTBF</b>) is the predicted elapsed time between inherent <a href="/wiki/Failure" title="Failure">failures</a> of a mechanical or electronic system during normal system operation. MTBF can be calculated as the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> (average) time between failures of a system. The term is used for repairable systems while <b>mean time to failure</b> (<b>MTTF</b>) denotes the expected time to failure for a non-repairable system.<sup id="cite_ref-lienig_1-0" class="reference"><a href="#cite_note-lienig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The definition of MTBF depends on the definition of what is considered a failure. For complex, <a href="/wiki/Repairable" class="mw-redirect" title="Repairable">repairable</a> systems, failures are considered to be those out of design conditions which place the system out of service and into a state for repair. Failures which occur that can be left or maintained in an unrepaired condition, and do not place the system out of service, are not considered failures under this definition.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In addition, units that are taken down for routine scheduled maintenance or inventory control are not considered within the definition of failure.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The higher the MTBF, the longer a system is likely to work before failing. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mean time between failures (MTBF) describes the expected time between two failures for a repairable system. For example, three identical systems starting to function properly at time 0 are working until all of them fail. The first system fails after 100 hours, the second after 120 hours and the third after 130 hours. The MTBF of the systems is the average of the three failure times, which is 116.667 hours. If the systems were non-repairable, then their <a href="#Variations_of_MTBF">MTTF</a> would be 116.667 hours. </p><p>In general, MTBF is the "up-time" between two failure states of a repairable system during operation as outlined here: </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Time_between_failures.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Time_between_failures.svg/725px-Time_between_failures.svg.png" decoding="async" width="725" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Time_between_failures.svg/1088px-Time_between_failures.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Time_between_failures.svg/1450px-Time_between_failures.svg.png 2x" data-file-width="725" data-file-height="254" /></a></span> </p><p>For each observation, the "down time" is the instantaneous time it went down, which is after (i.e. greater than) the moment it went up, the "up time". The difference ("down time" minus "up time") is the amount of time it was operating between these two events. </p><p>By referring to the figure above, the MTBF of a component is the sum of the lengths of the operational periods divided by the number of observed failures: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MTBF}}={\frac {\sum {({\text{start of downtime}}-{\text{start of uptime}})}}{\text{number of failures}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MTBF</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>start of downtime</mtext> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>start of uptime</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> <mtext>number of failures</mtext> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MTBF}}={\frac {\sum {({\text{start of downtime}}-{\text{start of uptime}})}}{\text{number of failures}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139f9895b72e10b1323da596a51362f0c7548466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:51.765ex; height:5.843ex;" alt="{\displaystyle {\text{MTBF}}={\frac {\sum {({\text{start of downtime}}-{\text{start of uptime}})}}{\text{number of failures}}}.}"></span></dd></dl> <p>In a similar manner, mean down time (MDT) can be defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MDT}}={\frac {\sum {({\text{start of uptime}}-{\text{start of downtime}})}}{\text{number of failures}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MDT</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>start of uptime</mtext> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>start of downtime</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> <mtext>number of failures</mtext> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MDT}}={\frac {\sum {({\text{start of uptime}}-{\text{start of downtime}})}}{\text{number of failures}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b930406132c6461e156c077db9f1c6a67c953a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.377ex; height:5.843ex;" alt="{\displaystyle {\text{MDT}}={\frac {\sum {({\text{start of uptime}}-{\text{start of downtime}})}}{\text{number of failures}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Mathematical_description">Mathematical description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=2" title="Edit section: Mathematical description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The MTBF is the expected value of the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> indicating the time until failure. Thus, it can be written as<sup id="cite_ref-birolini_4-0" class="reference"><a href="#cite_note-birolini-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MTBF}}=\mathbb {E} \{T\}=\int _{0}^{\infty }tf_{T}(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MTBF</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>t</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MTBF}}=\mathbb {E} \{T\}=\int _{0}^{\infty }tf_{T}(t)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0c1f4c668eec095f22710f5a7d37d46fe4928d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.867ex; height:5.843ex;" alt="{\displaystyle {\text{MTBF}}=\mathbb {E} \{T\}=\int _{0}^{\infty }tf_{T}(t)\,dt}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{T}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{T}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29753bf567358f4c066ed5414c5899ed6dcad750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.177ex; height:2.843ex;" alt="{\displaystyle f_{T}(t)}"></span> is the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. Equivalently, the MTBF can be expressed in terms of the <a href="/wiki/Reliability_function" class="mw-redirect" title="Reliability function">reliability function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{T}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{T}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7904990fd1acf2d0ded346d1daecc9a5d679128d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.802ex; height:2.843ex;" alt="{\displaystyle R_{T}(t)}"></span> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MTBF}}=\int _{0}^{\infty }R(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MTBF</mtext> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>R</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MTBF}}=\int _{0}^{\infty }R(t)\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1fa2136352fd5d9c46562d9a9d9ec182e7fc72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.653ex; height:5.843ex;" alt="{\displaystyle {\text{MTBF}}=\int _{0}^{\infty }R(t)\,dt}"></span>.</dd></dl> <p>The MTBF and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> have units of time (e.g., hours). </p><p>Any practically-relevant calculation of the MTBF assumes that the system is working within its "useful life period", which is characterized by a relatively constant <a href="/wiki/Failure_rate" title="Failure rate">failure rate</a> (the middle part of the "<a href="/wiki/Bathtub_curve" title="Bathtub curve">bathtub curve</a>") when only random failures are occurring.<sup id="cite_ref-lienig_1-1" class="reference"><a href="#cite_note-lienig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In other words, it is assumed that the system has survived initial setup stresses and has not yet approached its expected end of life, both of which often increase the failure rate. </p><p>Assuming a constant failure rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> has an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>. Since the MTBF is the expected value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, it is given by the reciprocal of the failure rate of the system,<sup id="cite_ref-lienig_1-2" class="reference"><a href="#cite_note-lienig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-birolini_4-1" class="reference"><a href="#cite_note-birolini-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MTBF}}={\frac {1}{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MTBF</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MTBF}}={\frac {1}{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d9bf9e4529c82313bb6ed985b57f85e569e247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.262ex; height:5.343ex;" alt="{\displaystyle {\text{MTBF}}={\frac {1}{\lambda }}}"></span>.</dd></dl> <p>Once the MTBF of a system is known, and assuming a constant failure rate, the <a href="/wiki/Probability" title="Probability">probability</a> that any one particular system will be operational for a given duration can be inferred<sup id="cite_ref-lienig_1-3" class="reference"><a href="#cite_note-lienig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> from the <a href="/wiki/Reliability_function" class="mw-redirect" title="Reliability function">reliability function</a> of the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{T}(t)=e^{-\lambda t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{T}(t)=e^{-\lambda t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b325a6dad87bb0fcd291f8c3a6d72ba589525bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.047ex; height:3.176ex;" alt="{\displaystyle R_{T}(t)=e^{-\lambda t}}"></span>. In particular, the probability that a particular system will survive to its MTBF is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f5e6342b5d465a51daf323f383ed1c8e51f43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.408ex; height:2.843ex;" alt="{\displaystyle 1/e}"></span>, or about 37% (i.e., it will fail earlier with probability 63%).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Application">Application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=3" title="Edit section: Application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The MTBF value can be used as a system reliability parameter or to compare different systems or designs. This value should only be understood conditionally as the “mean lifetime” (an average value), and not as a quantitative identity between working and failed units.<sup id="cite_ref-lienig_1-4" class="reference"><a href="#cite_note-lienig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Since MTBF can be expressed as “average life (expectancy)”, many engineers assume that 50% of items will have failed by time <i>t</i> = MTBF. This inaccuracy can lead to bad design decisions. Furthermore, probabilistic failure prediction based on MTBF implies the total absence of systematic failures (i.e., a constant failure rate with only intrinsic, random failures), which is not easy to verify.<sup id="cite_ref-birolini_4-2" class="reference"><a href="#cite_note-birolini-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Assuming no systematic errors, the probability the system survives during a duration, T, is calculated as exp^(-T/MTBF). Hence the probability a system fails during a duration T, is given by 1 - exp^(-T/MTBF). </p><p>MTBF value prediction is an important element in the development of products. Reliability engineers and design engineers often use reliability software to calculate a product's MTBF according to various methods and standards (MIL-HDBK-217F, Telcordia SR332, Siemens SN 29500, FIDES, UTE 80-810 (RDF2000), etc.). The Mil-HDBK-217 reliability calculator manual in combination with RelCalc software (or other comparable tool) enables MTBF reliability rates to be predicted based on design. </p><p>A concept which is closely related to MTBF, and is important in the computations involving MTBF, is the <a href="/wiki/Mean_down_time" title="Mean down time">mean down time</a> (MDT). MDT can be defined as mean time which the system is down after the failure. Usually, MDT is considered different from MTTR (Mean Time To Repair); in particular, MDT usually includes organizational and logistical factors (such as business days or waiting for components to arrive) while MTTR is usually understood as more narrow and more technical. </p> <div class="mw-heading mw-heading2"><h2 id="Application_of_MTBF_in_manufacturing">Application of MTBF in manufacturing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=4" title="Edit section: Application of MTBF in manufacturing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>MTBF serves as a crucial metric for managing machinery and equipment reliability. Its application is particularly significant in the context of <a href="/wiki/Total_productive_maintenance" title="Total productive maintenance">total productive maintenance</a> (TPM), a comprehensive maintenance strategy aimed at maximizing equipment <a href="/wiki/Effectiveness" title="Effectiveness">effectiveness</a>. MTBF provides a quantitative measure of the time elapsed between failures of a system during normal operation, offering insights into the reliability and performance of manufacturing equipment.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>By integrating MTBF with TPM principles, manufacturers can achieve a more proactive maintenance approach. This synergy allows for the identification of patterns and potential failures before they occur, enabling preventive maintenance and reducing unplanned downtime. As a result, MTBF becomes a <a href="/wiki/Performance_indicator" title="Performance indicator">key performance indicator</a> (KPI) within TPM, guiding decisions on maintenance schedules, spare parts inventory, and ultimately, optimizing the lifespan and efficiency of machinery.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This strategic use of MTBF within TPM frameworks enhances overall production efficiency, reduces costs associated with breakdowns, and contributes to the continuous improvement of manufacturing processes. </p> <div class="mw-heading mw-heading2"><h2 id="MTBF_and_MDT_for_networks_of_components">MTBF and MDT for networks of components</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=5" title="Edit section: MTBF and MDT for networks of components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two components <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},c_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1},c_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ae7f7452e416f7b70940ca260b2582f10382cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.156ex; height:2.009ex;" alt="{\displaystyle c_{1},c_{2}}"></span> (for instance hard drives, servers, etc.) may be arranged in a network, in <a href="/wiki/Series_and_parallel_circuits" title="Series and parallel circuits"><i>series</i> or in <i>parallel</i></a>. 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 <i>either</i> causes the failure of the network, and that they are in parallel if only the failure of <i>both</i> 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:<sup id="cite_ref-auroraconsultingengineering_8-0" class="reference"><a href="#cite_note-auroraconsultingengineering-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-smith_9-0" class="reference"><a href="#cite_note-smith-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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})}}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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})}}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8dae8fddff71e04aebf8291c6da487df8ac920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:58.253ex; height:7.843ex;" alt="{\displaystyle {\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})}}\;,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1};c_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1};c_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7c07bf47ca5c643df4dcc754e8967c8f7260e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.156ex; height:2.009ex;" alt="{\displaystyle c_{1};c_{2}}"></span> is the network in which the components are arranged in series. </p><p>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:<sup id="cite_ref-auroraconsultingengineering_8-1" class="reference"><a href="#cite_note-auroraconsultingengineering-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-smith_9-1" class="reference"><a href="#cite_note-smith-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∥</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>PF</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>PF</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d2e5d033b397eff2629bd553865018dff9ca3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.478ex; margin-bottom: -0.194ex; width:72.623ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}{\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{aligned}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}\parallel c_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∥</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1}\parallel c_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0708cf10ec3ef3267c31e7791ba898a3e800f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.575ex; height:2.843ex;" alt="{\displaystyle c_{1}\parallel c_{2}}"></span> is the network in which the components are arranged in parallel, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PF(c,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PF(c,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206e68425a2322212832ffaba6ad27e13de27b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.176ex; height:2.843ex;" alt="{\displaystyle PF(c,t)}"></span> is the probability of failure of component <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> during "vulnerability window" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. </p><p>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. </p><p>Using similar logic, MDT for a system out of two serial components can be calculated as:<sup id="cite_ref-auroraconsultingengineering_8-2" class="reference"><a href="#cite_note-auroraconsultingengineering-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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})}}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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})}}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4c6297680a849c12dc1cbe6c2ff1a71b073410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.284ex; height:6.509ex;" alt="{\displaystyle {\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})}}\;,}"></span></dd></dl> <p>and for a system out of two parallel components MDT can be calculated as:<sup id="cite_ref-auroraconsultingengineering_8-3" class="reference"><a href="#cite_note-auroraconsultingengineering-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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})}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∥</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\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})}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d38fd53f1337aad2fce0dedf1c3c56a77666087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.591ex; height:6.509ex;" alt="{\displaystyle {\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})}}\;.}"></span></dd></dl> <p>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 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{mtbf}}(c_{1};\dots ;c_{n})=\left(\sum _{k=1}^{n}{\frac {1}{{\text{mtbf}}(c_{k})}}\right)^{-1}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <mo>…</mo> <mo>;</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mtbf</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{mtbf}}(c_{1};\dots ;c_{n})=\left(\sum _{k=1}^{n}{\frac {1}{{\text{mtbf}}(c_{k})}}\right)^{-1}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab197a8acb3b8223f8814760fdf543b1fa3e5163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.851ex; height:8.009ex;" alt="{\displaystyle {\text{mtbf}}(c_{1};\dots ;c_{n})=\left(\sum _{k=1}^{n}{\frac {1}{{\text{mtbf}}(c_{k})}}\right)^{-1}\;,}"></span></dd></dl> <p>which can be shown by induction,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and likewise </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{mdt}}(c_{1}\parallel \dots \parallel c_{n})=\left(\sum _{k=1}^{n}{\frac {1}{{\text{mdt}}(c_{k})}}\right)^{-1}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∥</mo> <mo>⋯</mo> <mo>∥</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mdt</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{mdt}}(c_{1}\parallel \dots \parallel c_{n})=\left(\sum _{k=1}^{n}{\frac {1}{{\text{mdt}}(c_{k})}}\right)^{-1}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ff49afa781ea4c6e829dbe58cd5d25072c1506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.878ex; height:8.009ex;" alt="{\displaystyle {\text{mdt}}(c_{1}\parallel \dots \parallel c_{n})=\left(\sum _{k=1}^{n}{\frac {1}{{\text{mdt}}(c_{k})}}\right)^{-1}\;,}"></span></dd></dl> <p>since the formula for the mdt of two components in parallel is identical to that of the mtbf for two components in series. </p> <div class="mw-heading mw-heading2"><h2 id="Variations_of_MTBF">Variations of MTBF</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=6" title="Edit section: Variations of MTBF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many variations of MTBF, such as <i>mean time between system aborts</i> (MTBSA), <i>mean time between critical failures</i> (MTBCF) or <i>mean time between unscheduled removal</i> (MTBUR). Such nomenclature is used when it is desirable to differentiate among types of failures, such as critical and non-critical failures. For example, in an automobile, the failure of the FM radio does not prevent the primary operation of the vehicle. </p><p>It is recommended to use <i>Mean time to failure</i> (MTTF) instead of MTBF in cases where a system is replaced after a failure ("non-repairable system"), since MTBF denotes time between failures in a system which can be repaired.<sup id="cite_ref-lienig_1-5" class="reference"><a href="#cite_note-lienig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/MTTFd" title="MTTFd">MTTFd</a> is an extension of MTTF, and is only concerned about failures which would result in a dangerous condition. It can be calculated as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{MTTF}}&\approx {\frac {B_{10}}{0.1n_{\text{onm}}}},\\[8pt]{\text{MTTFd}}&\approx {\frac {B_{10d}}{0.1n_{\text{op}}}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>MTTF</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mrow> <mn>0.1</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>onm</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>MTTFd</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mi>d</mi> </mrow> </msub> <mrow> <mn>0.1</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\text{MTTF}}&\approx {\frac {B_{10}}{0.1n_{\text{onm}}}},\\[8pt]{\text{MTTFd}}&\approx {\frac {B_{10d}}{0.1n_{\text{op}}}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469cc107083e433e1ee9aa68ed1dbb1416b1576c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:21.334ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}{\text{MTTF}}&\approx {\frac {B_{10}}{0.1n_{\text{onm}}}},\\[8pt]{\text{MTTFd}}&\approx {\frac {B_{10d}}{0.1n_{\text{op}}}},\end{aligned}}}"></span></dd></dl> <p>where <i>B</i><sub>10</sub> is the number of operations that a device will operate prior to 10% of a sample of those devices would fail and <i>n</i><sub>op</sub> is number of operations. <i>B</i><sub>10d</sub> is the same calculation, but where 10% of the sample would fail to danger. <i>n</i><sub>op</sub> is the number of operations/cycle in one year.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="MTBF_considering_censoring">MTBF considering censoring</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=7" title="Edit section: MTBF considering censoring"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In fact the MTBF counting only failures with at least some systems still operating that have not yet failed underestimates the MTBF by failing to include in the computations the partial lifetimes of the systems that have not yet failed. With such lifetimes, all we know is that the time to failure exceeds the time they've been running. This is called <a href="/wiki/Censoring_(statistics)" title="Censoring (statistics)">censoring</a>. In fact with a parametric model of the lifetime, the <a href="/wiki/Censoring_(statistics)#likelihood" title="Censoring (statistics)">likelihood for the experience on any given day is as follows</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\prod _{i}\lambda (u_{i})^{\delta _{i}}S(u_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>λ</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\prod _{i}\lambda (u_{i})^{\delta _{i}}S(u_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b2720a50439d54e2fd55e699d324806b405441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.357ex; height:5.509ex;" alt="{\displaystyle L=\prod _{i}\lambda (u_{i})^{\delta _{i}}S(u_{i})}"></span>,</dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span> is the failure time for failures and the censoring time for units that have not yet failed,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c5e905acee9cc0cf8bc01c08a4e876f43d5c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.832ex; height:2.676ex;" alt="{\displaystyle \delta _{i}}"></span> = 1 for failures and 0 for censoring times,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(u_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(u_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50c2895a637a2280e6013f2ed7f5bf5c21eb20df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.438ex; height:2.843ex;" alt="{\displaystyle S(u_{i})}"></span> = the probability that the lifetime exceeds <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span>, called the survival function, and</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (u_{i})=f(u)/S(u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (u_{i})=f(u)/S(u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9083021e377d685465ddd9f375bcf9a88d7d3991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.611ex; height:2.843ex;" alt="{\displaystyle \lambda (u_{i})=f(u)/S(u)}"></span> is called the <a href="/wiki/Failure_rate#hazard_function" title="Failure rate">hazard function</a>, the instantaneous force of mortality (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b282ccf3cf0f2b0b601e8629d61612d4951098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(u)}"></span> = the probability density function of the distribution).</dd></dl> <p>For a constant <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, the hazard, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, is constant. In this case, the MBTF is </p> <dl><dd>MTBF = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\hat {\lambda }}=\sum u_{i}/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∑</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\hat {\lambda }}=\sum u_{i}/k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ad89f8f3bf3179ea8156976e160fdfbc3ceff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.024ex; height:3.843ex;" alt="{\displaystyle 1/{\hat {\lambda }}=\sum u_{i}/k}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7bb058e73bb0c0545eed7bac5a93b7e1145709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.843ex;" alt="{\displaystyle {\hat {\lambda }}}"></span> is the maximum likelihood estimate of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, maximizing the likelihood given above and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\sum \sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo>∑</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\sum \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3116a670b5a892cb014fe21604e1108c1f8b16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.179ex; height:3.843ex;" alt="{\displaystyle k=\sum \sigma _{i}}"></span> is the number of uncensored observations. </p><p>We see that the difference between the MTBF considering only failures and the MTBF including censored observations is that the censoring times add to the numerator but not the denominator in computing the MTBF.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Annualized_failure_rate" title="Annualized failure rate">Annualized failure rate</a> – Probability that a device or component will fail during a year of use</li> <li><a href="/wiki/Failure_rate" title="Failure rate">Failure rate</a> – Frequency with which an engineered system or component fails</li> <li><a href="/wiki/Frames_per_stop" title="Frames per stop">Frames per stop</a> – Term of reliability in the bowling industry</li> <li><a href="/wiki/Mean_time_to_first_failure" title="Mean time to first failure">Mean time to first failure</a> – Average service life for non-repairable components</li> <li><a href="/wiki/Mean_time_to_repair" title="Mean time to repair">Mean time to repair</a> – Measure of the maintainability of repairable items</li> <li><a href="/wiki/Power-on_hours" title="Power-on hours">Power-on hours</a> – The length of time that electrical power is applied to a device</li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability engineering</a> – Sub-discipline of systems engineering that emphasizes dependability</li> <li><a href="/wiki/Residence_time_(statistics)" title="Residence time (statistics)">Residence time (statistics)</a> – Statistical parameter of random process evolution</li> <li><a href="/wiki/Bathtub_curve" title="Bathtub curve">Bathtub curve</a> – Curve for failure rates over time</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-lienig-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-lienig_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lienig_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-lienig_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-lienig_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-lienig_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-lienig_1-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJ._Lienig,_H._Bruemmer2017" class="citation book cs1">J. Lienig, H. Bruemmer (2017). "Reliability Analysis". <i>Fundamentals of Electronic Systems Design</i>. 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Retrieved <span class="nowrap">2016-01-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stephen+Foskett%2C+Pack+Rat&rft.atitle=Defining+Failure%3A+What+Is+MTTR%2C+MTTF%2C+and+MTBF%3F&rft.date=2011-07-06&rft_id=http%3A%2F%2Fblog.fosketts.net%2F2011%2F07%2F06%2Fdefining-failure-mttr-mttf-mtbf%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-birolini-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-birolini_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-birolini_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-birolini_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Alessandro Birolini: <i>Reliability Engineering: Theory and Practice</i>. Springer, Berlin 2013, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-39534-5" title="Special:BookSources/978-3-642-39534-5">978-3-642-39534-5</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.vicorpower.com/documents/quality/Rel_MTBF.pdf">"Reliability and MTBF Overview"</a> <span class="cs1-format">(PDF)</span>. Vicor Reliability Engineering<span class="reference-accessdate">. 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Retrieved <span class="nowrap">2024-02-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=total-manufacturing.com&rft.atitle=MTBF%3A+What+it+means+and+how+to+calculate+it&rft_id=https%3A%2F%2Ftotal-manufacturing.com%2Fmaintenance%2Ftpm-en%2Fmtbf%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhD2021" class="citation web cs1">PhD, Bartosz Misiurek (2021-11-22). <a rel="nofollow" class="external text" href="https://leancommunity.org/mtbf-mttr-and-mttf/">"MTBF MTTR MTTF: TPM Indicators"</a>. <i>Lean Community</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Lean+Community&rft.atitle=MTBF+MTTR+MTTF%3A+TPM+Indicators&rft.date=2021-11-22&rft.aulast=PhD&rft.aufirst=Bartosz+Misiurek&rft_id=https%3A%2F%2Fleancommunity.org%2Fmtbf-mttr-and-mttf%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-auroraconsultingengineering-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-auroraconsultingengineering_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-auroraconsultingengineering_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-auroraconsultingengineering_8-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-auroraconsultingengineering_8-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://auroraconsultingengineering.com/doc_files/Reliability_series_parallel.doc">"Reliability Characteristics for Two Subsystems in Series or Parallel or n Subsystems in m_out_of_n Arrangement (by Don L. Lin)"</a>. <i>auroraconsultingengineering.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=auroraconsultingengineering.com&rft.atitle=Reliability+Characteristics+for+Two+Subsystems+in+Series+or+Parallel+or+n+Subsystems+in+m_out_of_n+Arrangement+%28by+Don+L.+Lin%29&rft_id=http%3A%2F%2Fauroraconsultingengineering.com%2Fdoc_files%2FReliability_series_parallel.doc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-smith-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-smith_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-smith_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDr._David_J._Smith2011" class="citation book cs1">Dr. David J. Smith (2011). <i>Reliability, Maintainability and Risk</i> (eighth ed.). Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0080969022" title="Special:BookSources/978-0080969022"><bdi>978-0080969022</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reliability%2C+Maintainability+and+Risk&rft.edition=eighth&rft.pub=Butterworth-Heinemann&rft.date=2011&rft.isbn=978-0080969022&rft.au=Dr.+David+J.+Smith&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20021106143359/http://www.angelfire.com/ca/summers/Business/MTBFAllocAnalysis1.html">"MTBF Allocations Analysis1"</a>. <i><a href="/wiki/Angelfire" title="Angelfire">Angelfire</a></i>. Archived from <a rel="nofollow" class="external text" href="http://www.angelfire.com/ca/summers/Business/MTBFAllocAnalysis1.html">the original</a> on November 6, 2002<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-12-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Angelfire&rft.atitle=MTBF+Allocations+Analysis1&rft_id=http%3A%2F%2Fwww.angelfire.com%2Fca%2Fsummers%2FBusiness%2FMTBFAllocAnalysis1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.tuv.com/media/hungary/downloads_hu/B10d_EN.pdf">"B10d Assessment – Reliability Parameter for Electro-Mechanical Components"</a> <span class="cs1-format">(PDF)</span>. TUVRheinland<span class="reference-accessdate">. Retrieved <span class="nowrap">7 July</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=B10d+Assessment+%E2%80%93+Reliability+Parameter+for+Electro-Mechanical+Components&rft.pub=TUVRheinland&rft_id=https%3A%2F%2Fwww.tuv.com%2Fmedia%2Fhungary%2Fdownloads_hu%2FB10d_EN.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLu_Tian" class="citation cs2">Lu Tian, <a rel="nofollow" class="external text" href="https://web.stanford.edu/~lutian/coursepdf/unit2.pdf"><i>Likelihood Construction, Inference for Parametric Survival Distributions</i></a> <span class="cs1-format">(PDF)</span>, <a href="/wiki/WDQ_(identifier)" class="mw-redirect" title="WDQ (identifier)">Wikidata</a> <a href="https://www.wikidata.org/wiki/Q98961801" class="extiw" title="d:Q98961801">Q98961801</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Likelihood+Construction%2C+Inference+for+Parametric+Survival+Distributions&rft.au=Lu+Tian&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~lutian%2Fcoursepdf%2Funit2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mean_time_between_failures&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.eventhelix.com/RealtimeMantra/FaultHandling/reliability_availability_basics.htm">"Reliability and Availability Basics"</a>. EventHelix.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Reliability+and+Availability+Basics&rft.pub=EventHelix&rft_id=http%3A%2F%2Fwww.eventhelix.com%2FRealtimeMantra%2FFaultHandling%2Freliability_availability_basics.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpeaks2005" class="citation web cs1">Speaks, Scott (2005). <a rel="nofollow" class="external text" href="http://www.vicorpower.com/documents/quality/Rel_MTBF.pdf">"Reliability and MTBF Overview"</a> <span class="cs1-format">(PDF)</span>. Vicor Reliability Engineering.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Reliability+and+MTBF+Overview&rft.pub=Vicor+Reliability+Engineering&rft.date=2005&rft.aulast=Speaks&rft.aufirst=Scott&rft_id=http%3A%2F%2Fwww.vicorpower.com%2Fdocuments%2Fquality%2FRel_MTBF.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath498/kmath498.htm">"Failure Rates, MTBF, and All That"</a>. MathPages.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Failure+Rates%2C+MTBF%2C+and+All+That&rft.pub=MathPages&rft_id=http%3A%2F%2Fwww.mathpages.com%2Fhome%2Fkmath498%2Fkmath498.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.roadtoreliability.com/mtbf-mean-time-between-failure/">"Simple Guide to MTBF: What It Is and When To use It"</a>. Road to Reliability. 10 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Simple+Guide+to+MTBF%3A+What+It+Is+and+When+To+use+It&rft.pub=Road+to+Reliability&rft.date=2021-12-10&rft_id=https%3A%2F%2Fwww.roadtoreliability.com%2Fmtbf-mean-time-between-failure%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMean+time+between+failures" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nexgenam.com/blog/what-is-mean-time-to-failure-mttf/">"What is Mean Time to Failure and How Do We Calculate?"</a>. 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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electricity distribution</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ASUI" title="ASUI">ASUI</a></li> <li><a href="/wiki/Average_Service_Availability_Index" title="Average Service Availability Index">ASAI</a></li> <li><a href="/wiki/CAIDI" title="CAIDI">CAIDI</a>/<a href="/wiki/SAIDI" title="SAIDI">SAIDI</a></li> <li><a href="/wiki/CAIFI" title="CAIFI">CAIFI</a>/<a href="/wiki/SAIFI" title="SAIFI">SAIFI</a></li> <li><a href="/wiki/MAIFI" title="MAIFI">MAIFI</a></li> <li><a href="/wiki/CTAIDI" title="CTAIDI">CTAIDI</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electricity generation</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/LOLP" class="mw-redirect" 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Mean time between failures - Wikipedia