{"id":3479,"date":"2014-06-14T14:19:43","date_gmt":"2014-06-14T19:19:43","guid":{"rendered":"http:\/\/www.livingreliability.com\/en\/?p=3479"},"modified":"2014-10-31T12:20:16","modified_gmt":"2014-10-31T17:20:16","slug":"expected-failure-time-for-an-item-whose-maintenance-policy-is-time-based","status":"publish","type":"post","link":"https:\/\/www.livingreliability.com\/en\/posts\/expected-failure-time-for-an-item-whose-maintenance-policy-is-time-based\/","title":{"rendered":"Expected failure time for an item whose maintenance policy is time-based"},"content":{"rendered":"

Let Tc<\/sub><\/em>\u00a0be the time of a cycle, tp<\/sub><\/em> the preventive maintenance time interval,\u00a0 and T<\/em> the failure time.<\/p>\n

The expected life cycle Tc<\/sub><\/em>\u00a0will be the planned maintenance time tp<\/sub><\/em> multiplied by the probability that planned maintenance does occur, plus the expected failure time (knowing that failure occurs before tp<\/sub><\/em>) multiplied by the probability that failure occurs before tp<\/sub><\/em>. This will be\u00a0expressed mathematically by:<\/p>\n

\"E\left ( T_{c} \right )=t_{p}P\left ( T> t_{p} \right )+E\left ( T|T\leq t_{p} \right )P\left ( T\leq t_{p} \right )\"\u00a0(Eqn. 1)<\/p>\n

The term \"\"<\/a> in Equation 1, is the expected time to failure, given that failure occurs prior to scheduled maintenance, under a policy where scheduled maintenance is carried out at time tp<\/sub><\/em> .\u00a0 We wish to show [1<\/a>]<\/sup> that it can be expressed as<\/p>\n

\"\"<\/a><\/p>\n

First, we recognize that the conditional distribution function of \u00a0\"\"<\/a> \u00a0is<\/p>\n

\"eq2\"<\/a><\/p>\n

 <\/p>\n

(Eq 2)<\/p>\n

In the first part of\u00a0 Equation 2, we have simply defined the distribution function of (T\u2264t given that T\u2264tp<\/sub>) as \"F_{c}\left ( t \right )=P\left ( T\leq t|T\leq t_{p}\right )\". We will call this conditional distribution function, \u201cFc<\/sub>(t)<\/em>\u201d. (Recall that this is the definition of a distribution function.)<\/p>\n

Now, moving towards the right in Equation 2, the top condition \u201c1, where t>tp<\/sub><\/em>\u201d is easy to understand. We know that failure will have occurred prior to tp<\/sub><\/em> (with 100% certainty) because T\u2264tp <\/sub><\/em>is our hypothesis in Fc<\/sub>(t)<\/em>.<\/p>\n

The bottom case\u00a0\"\frac{P\left ( \left ( T\leq t \right ) \right )}{P\left ( T\leq t_{p} \right )}\"\u00a0 requires us to know that the conditional probability P(A|B)<\/em> is \"\inline \frac{P\left ( A\cap B \right )}{P\left ( B \right )}\"\u00a0where A = T\u2264t and B = T\u2264tp<\/sub><\/p>\n

But we know (in\u00a0the bottom case) that the intersection of\u00a0 T\u2264t and T\u2264tp<\/sub> is T\u2264t because the probability that T\u2264t is entirely within the probability space of T\u2264tp<\/sub> (since t\u2264tp<\/sub> is the bottom case). Therefore the intersection of T\u2264t and T\u2264tp<\/sub> is actually T\u2264min(t,tp<\/sub>)=T\u2264t.\"VenneTt\"<\/a><\/p>\n

\"Venne<\/a>
Venne Diagram<\/figcaption><\/figure>\n

In the rightmost part of Equation 2 we apply the definition of F(t)<\/em> to the numerator and denominator. And, of course, we know that F(t) = 1-R(t).<\/p>\n

Then the conditional density function of \"\inline T|T\leq t_{p}\"\u00a0\u00a0is<\/p>\n

\"\inline f_{c}(t)=\begin{cases} 0 & \text{ if } t> t_{p} \hspace 2 or \hspace 2 t < 0\\ \frac{f\left ( t \right )}{1-R\left ( t_{p} \right )} & \text{ if } 0\leq t\leq t_{p} \end{cases}\"\u00a0(Eq. 3)<\/p>\n

We have used, in Equation 3, the fact that the density function is the first derivative of the distribution function. \u00a0 Therefore,<\/p>\n

\"\inline E(T|T\leq t_{p})=\int_{0}^{\infty }tf_{c}\left ( t \right )dt=\int_{0}^{t_{p}}\frac{tf\left ( t \right )}{1-R\left ( t_{p} \right )}dt=\frac{\int_{0}^{t_{p}}tf\left ( t \right )}{1-R\left ( t_{p} \right )}\"\u00a0(Eq 4)<\/p>\n

Here, in Equation 4, we have invoked the definition of \u201cExpectation\u201d as the integral of the product of t and the density function.<\/p>\n\n

    \n\t
  1. [1]<\/sup><\/strong>in order to derive the denominator of Eq 10\u00a0of the post “Real meaning of the six RCM curves<\/a>“↩<\/a><\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

    Let Tc\u00a0be the time of a cycle, tp the preventive maintenance time interval,\u00a0 and T the failure time. The expected life cycle Tc\u00a0will be the planned maintenance time tp multiplied by the probability that planned maintenance does occur, plus the expected failure time (knowing that failure occurs before tp) multiplied by the probability that failure […]<\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[89],"tags":[],"class_list":["post-3479","post","type-post","status-publish","format-standard","hentry","category-theory-and-definitions"],"_links":{"self":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3479","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/comments?post=3479"}],"version-history":[{"count":0,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3479\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/media?parent=3479"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/categories?post=3479"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/tags?post=3479"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}