Abstract

Chris and Fred discuss what ‘distribution parameters’ mean when it comes to random processes. Specifically failure random processes. This is an interesting podcast in response to a question from one of our listeners – which are podcasts we love!

Key Points

Join Chris and Fred as they discuss a question directed to us by a listener. In fact they were two questions – as follows:

Think of probability distributions and the sequence you define your observation points. Neither the distribution type nor the parameters change e.g. when you reverse the sequence or change the order. It’s ambiguous to me, because if I have higher rate of failures in the past but better conditions now, I’d like to see it in my parameters and shape. Otherwise, how can I rely on e.g. beta in my Weibull distribution?   2) How may I determine the rate of events (say rate of TTR, TTF, or any other parameter) when my distribution is not Weibull? Which parameter should I use? Let me appreciate your time & willingness to help in advance. Keyvan.

Just for the uninitiated, a Weibull distribution is a type of probability distribution that is used a lot in reliability engineering. The ‘beta’ refers to what we call a shape parameter, which describes the nature in which failure occurs.

Topics include:

• Order of data shouldn’t matter … if we are looking at time to failure. The first step of any random data analysis is to order the data from smallest to largest. Unless … we are talking about a ‘renewal process.’ This is where you might have a single machine that works until it fails, and then it is repaired, and it keeps working. In which case … the order of data does matter. In a renewal process, one machine might have lots of times to failure (noting it gets repaired). This means that we can’t use probability distributions to describe single times to failure (like a Weibull distribution).
• But what if it is a renewal process? Then we can perhaps examine monthly failure rates, or the Mean Cumulative Function (MCF) to identify trends over time. And by trends, we mean failure rate behaviors that show wear-in, wear-out or something in between. If you are really interested in getting to the bottom of what is going on, then research this thing called the nonhomogenous Poisson process.
• OK … so what if it is simple ‘time to failure?’ Well before we talk about ‘betas,’ we need to confirm the Weibull distribution is an appropriate model. It is sometimes useful to break down failure into failure modes. If there are different failure modes, they might be modelled by different Weibull distributions. For example, if a system fails due to wear-in around half the time, and wear-out the other half, then fitting a Weibull distribution might try and ‘average’ the two and (incorrectly) conclude the system has a constant or non-changing failure rates. So always confirm you have the right model.

Enjoy an episode of Speaking of Reliability. Where you can join friends as they discuss reliability topics. Join us as we discuss topics ranging from design for reliability techniques to field data analysis approaches.

SOR 667 Interpreting Distribution ParametersChristopher Jackson

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