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A factory has different categories of machines which require frequent adjustments and repair. Each category of machine fails uniformly after continuous operation and the failure profile of the different categories of machines is given by its mean time to failure (MTTF).

With the above information I need to assume a probability distribution for their failure profiles. So what are the meaningful/safe assumptions that I should make about the distribution?What actually do we know about machines when we know their MTTF and that they fail uniformly after continuous operation?

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  • $\begingroup$ you may assume constant hazard rate $\endgroup$
    – eldos
    Apr 1, 2014 at 16:22

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"Fail uniformly" probably means that they fail at a constant rate (with a different rate for each type of machine). Uniformly is not a good word in this context as it implies the machine failure time follows a uniform distribution (this would be roughly the same number of machines failing in every time period, which is not realistic).
Failing at a constant rate $\lambda$ per unit time happens when the number of fails X in the interval from 0 to t (so $\lambda$t fails on average) follows a Poisson distribution

$P(X = k) = \frac{{{(\lambda t)}^{k}}\exp (-\lambda t)}{k!}$

If roughly half of these machines have failed up to some time $t_0$ then half of the remaining will fail in the next interval up to $t_0$ and so on, decreasing all the time. If T is the time to failure of a machine, which is a random variable, then the probability that the machine is still working after time t is the same as the probability that X is 0 i.e. there has been no fail up to time t. Then

$P(T>t) = P(X = 0) = \exp (-\lambda t)$

This is the Reliability of the machine type, which is the proportion of machines which are still working after time t. The failure rate $\lambda$ is 1/MTTF for that type of machine. So for a given MTTF we can work out the proportion of machines which will fail up to time t using the reliability.
e.g If lightbulbs have a MTTF of 300hrs then the proportion which are still working after 200hrs is

$exp(-\frac{1}{300}*200) = 0.513$ or 51%

The proportion which have failed up to time t is 1 - Reliability. This can be written as an integral up to time t of the probability density for T. Differentiation will then give the density for the variable T, but the Reliability is a more meaningful thing to focus on for practical purposes.

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  • $\begingroup$ what about weibull distribution?? $\endgroup$
    – eldos
    Apr 1, 2014 at 16:22
  • $\begingroup$ The weibull distribution is used to model the reliability over the complete lifetime of a product. Typically, the early life and late life of a product is subject to higher (time dependent)failure rates than its so called normal life. I interpreted your question as asking about the normal life of the product, where the exponential distribution is appropriate. A suitably chosen Weibull distribution can also fit this normal life period. $\endgroup$
    – Paul
    Apr 1, 2014 at 16:51

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