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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?


  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.

    [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.

    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]


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