How does machine age effect reliability?

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In a recent blog, I wrote about how to estimate the reliability of a machine in a production situation. Dave at Ramsoftuk rightly pointed out that the age of a machine at the time the reliability estimate is made can affect that estimate.

  by  kevindooley 

When a machine has achieved some operating time, it has experienced wear, so we would expect that the reliability would be affected. The reliability in this case is what’s called a “conditional probability”, usually written as R(T|t). This means the reliability at mission time t based on the condition that the machine has operated for time T at the time the reliability estimate is made.

So the question is:

Given that a piece of equipment has operated for T hours, what is the probability it will last for an additional t hours.

This is a conditional probability as:

R(T+t)=R(T)*R(T|t)

Or

R(T|t)=R(T+t)/R(T)

Now we need to decide what stage of the bathtub curve we’re in. We can do this with Weibull analysis. If we find that β (beta – the shape parameter in Wiebull) is <1, then we’re in the infant mortality stage. If β >1, then we’re in the wear out stage. If β=1, we’re in the chance failure or useful life stage.

Let’s look at the useful life stage first because that’s where most of our machinery should fall. This can be described by the exponential distribution. For the exponential distribution

R(t)=e^-λt

Then

R(T|t)= e^-λ(T+t)/e^-λT = (e^-λT*e^-λt)/e^-λT = e^-λt = R(t)

Therefore the prior age for equipment with an exponential distribution does not depend on its age. So the description in Measuring Reliability holds true regardless of the machines prior operating time.

For the other two cases, infant mortality and wear out, this is not true. Without duplicating the math, they both depend on β, which means they depend on T, the achieved life.

The bottom line? If you’re estimating the reliability of a machine in the useful life stage, just use R(t)=e^-λt. In the other stages, you’ll have to do a lot more calculations. Consult someone who understands them or learn how yourself (by far the best option). We’ll talk about that in a later blog.

While I like the math, how would you use this in a planning situation?