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Unavailability/probabilities over a period of time

  1. Mar 10, 2009 #1
    Hey, can anyone give me some help?

    I'm trying to calculate the time unavailable of a system due to a component failure over a certain period of time. I know the probability of failure of this component, the time to be considered (assume 1 year) and how long the component is unavailable for due to a failure.

    How can i work out how long this component causes the system to be unavailable for?

    Thanks,

    --notap
     
  2. jcsd
  3. Mar 10, 2009 #2
    Disclaimer: If you are using this for engineering applications please consult a proper text in saftey instrumented systems as I'm doing this off the top of my head.

    Anyway, we need to start with something.

    Let:
    [tex] P(\Delta t) [/tex] be the probability of the component not failing in a time period of [tex]\Delta t[/tex].

    There are [tex]{365 \over \Delta t}[/tex] time steps in one year.

    What we want to find is the expected time the system will be unavailable. That is we are computing the expectation value.
    http://en.wikipedia.org/wiki/Expected_value

    If the system fails at time t. It is unavailable for f(t)=365-t.

    To compute the probability of failure we are summing each possible failure time by the probability it will fail at that time. In sumation form this is written as:

    [tex]<f(t)>=\sum_{n=1}^{365 / \Delta t}}f(n \Delta t)(1-P(\Delta t)^n)[/tex]

    This can also be written in integral form (Proof left as exercise).

    [tex]<f(t)>=\int_{0}^{365}}f(t)exp(-t \lambda)[/tex]

    where:

    [tex]\lambda=\mathop{\lim }\limits_{\Delta t \to 0} {1-P(\Delta t) \over \Delta t}[/tex]
     
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