As one of the four key functions related to reliability engineering, the reliability function is often confused or misunderstood. Let’s spend a few minutes exploring the reliability function and how to use it.

Reliability generally means that the product or item is durable. This involves the passing of time and the continued useful functioning of the product. Reliability has a careful definition that includes function, environment, probability, and duration. It is the probability element of reliability that is where the reliability function comes to play.

The reliability function mathematically defines the probability over the duration. It is a function of time (or cycles, or miles, or whatever unit of time passing makes sense). It is a coupling of probability and time. Always.

The function starts with all items working or 100% functioning at time zero. Time zero is when the unit is placed into service. We often describe the idea with 100 units started at time zero and they are all working. And over time, eventually, all the units fail. When the last unit fails, the reliability function is 0%. We can also say the reliability function ranges from 1 to 0.

A common question that the reliability function answers is “how many units will survive over the warranty period?”. If a product operates 24/7, it will operate 8760 hours in a year. R(t) becomes R(8760 hours) and depending on the particular life distribution the formula differs. The result, in this case, provides the probability of units surviving 8760 hours. Let’s say this is 0.78. The 0.78 (78%) means that 78 out 100 are expected to survive 8760 hours. It also can mean that one unit has a 78% chance of surviving out to 8760 hours.

78 out of 100 customers in the above example are expected to experience product failure. That may or may not be acceptable. That is a business decision.

Now back to the idea of reliability and some of the confusion. Reliability is not just the probability of success or failure rate. It involves the other elements of the definition of reliability, function, environment and duration, too. Simply using MTBF (a representation of failure rate) does not include duration and implies function and environment. To avoid confusion, I recommend always thinking about all four elements and specifically stating the couplet of probability and duration as the reliability function suggests, a function that describes the probability of success as a function of time, R(t).

Rather than setting a single product reliability goal, I recommend setting three. One for the early failure period, R(one month), and, the warranty period, R(1 year), and, for the expected useful or design life, R(5 years). The examples here, one month, 1 year, and five years vary depending on the technology, market, customer expectations, and business objectives. The goals provide an envelope to describe the reliability objectives. It also provides three points, possibly tied to business and customer needs, that describe a curve related to the reliability function.

As the product is developed and tested, and eventually fielded, the information should be compared to the reliability goals. The data creates a mathematical description of the time to failure or reliability function. From this function we can calculate the number of units that survived a period of time of interest.

Related:

The Four Functions (article)

Reliability Goal (article)

Series reliability question (article)

Tim says

Hi Fred,

In order to understand and use the info above.

In your example you mention nothing about failure distribution.

Is this what you try to accomplish using three reliability goals (in your text the reliability function) ?

Also how did you got the information on the 78% chance of surviving?

Is this by testing?

Field information?

By having the failure information of the 78% chance of surviving, shouldn’t it be possible to derive a failure distribution and base the failures in time on this?

And (don’t shoot me now) if you assume exponential distribution, you can still use MTBF as a metric or not?

R(t)= e-Lambda x t

Lambda = 1/MTBF

Fred Schenkelberg says

Hi Tim,

Thanks for the comment and questions.

The reliability function is the complement of the CDF which is the integral of the PDF (or failure distribution… ) If you know one function you can determine others, hazard, PDF, CDF, and Reliability.

The 78% is just an example, completely made up. The distribution could be determined many ways, here, I’m talking about how to interpret a specific example.

Having only a single point along the distribution is not enough unless you assume a specific distribution and remaining parameters. Having a few or many time to failure points is better. Having a known time to failure model helps.

I would not recommend assuming the exponential distribution unless you have compelling information that it applies (which it very rarely does). If the exponential applies, then MTBF on it’s own is sufficient to know the reliability function.

Tim, thanks for reading the article closely – good questions.

Cheers,

Fred

Hilaire Perera says

Fred,

http://reliawiki.org/index.php/The_Weibull_Distribution

Weibull distribution uses an exponential function. It is a versatile distribution that can take on the characteristics of other types of distributions based on the value of the Shape Parameter.

Mandar Gokhale says

Fred,

Is this statement inverted: “78 out of 100 customers in the above example are expected to experience product failure. ”

Shouldn’t it be

“22 (100-78) out of 100 customers in the above example are expected to experience product failure within 1 year warranty period. ”

— Mandar

Jason says

Hi Fred

By saying duration, you mean the use of censored data?

Fred Schenkelberg says

Hi Jason,

No, not really. Duration is a span of time, some count of cycles, or similar manner of measuring the passage of time.

Censored data is a phrase used to describe items that we have some knowledge about, and they haven’t failed as of yet. We know they have worked till now and haven’t failed. There are regression and statistical techniques to account for this knowledge. Often used for test or field data where not every item has failed.

Cheers,

Fred