reliability distribution

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Technical Support Document

Distribution Models for Reliability Data

IntroductionWhen performing reliability analysis, you must choose a distribution to model your data. The more closely thedistribution fits your data, the more likely the reliability statistics will accurately describe the performance of

your product. In addition, a well-fitting model can be used to make reasonable projections when extrapolatingbeyond the range of data.

To choose a distribution, rely on your practical knowledge and experience with the product performance. Askyourself the following questions:

Do the data follow a symmetric distribution? Are they skewed in one direction?

Is the failure rate constant? Increasing? Decreasing?

What distribution has worked in the past for this situation?

You can also evaluate the fit of your data using Minitabs Distribution ID plot (Stat > Reliability/Survival >

Distribution Analysis(Right-Censoringor Arbitrary Censoring). The output displays probability plots formany types of distributions commonly used in reliability analysis, including:

Weibull distribution

Exponential distribution

Lognormal distribution

Smallest Extreme Value distribution

Normal distribution

Frequently, you can model a given set of data with more than one distribution, or with a distribution that isdefined by one, two, or three parameters. For example, distributions that can model each type of data are

summarized below:

Left-skewed dataIn many cases, you can fit theWeibullorsmallest extreme valuedistribution.

Symmetric dataYou can generally obtain a good fit with aWeibullorlognormaldistribution. In somecases, you can fit thenormaldistribution (depending on the heaviness of the tails) and obtain similarresults.

Right-skewed dataYou can often fit either theWeibullor thelognormaldistribution and obtain a goodfit to the data.

A given set of data can sometimes be modeled using either 2 or 3 parameters. A 3-parameter model mayprovide a better fit for some data, but may also result in overfitting the model. In general, experts generallyadvise choosing the simplest model that works.

In this paper, well discuss the characteristics of each distribution and provide some common applications.


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The Weibull DistributionThe Weibull distribution is the most commonly used distributions for modeling reliability data. This distribution

is easy to interpret and extremely versatile. By adjusting the value of its shape parameter, , you can model thecharacteristics of many different life distributions, as shown in the table below.

Shape

parameter

Probability

Distribution Function

(PDF)

Hazard function

(failure rate)

Type of Product Failure

0 < < 1

Exponentially decreasing

from infinity

Initially high failure rate

that decreases over time

(first part of bathtub

shaped hazard function).

Early failures, also known as

Infant mortality, because

they occur in initial period of

product life

These failures may

necessitate a product burn-in period to reduce risk of

initial failure.

= 1

Exponentially decreasing

from 1/

scale parameter)

Constant failure rate over

life of product

Random failures, multiple-

cause failures

Models useful life of

product

= 1.5

Rises to peak then

decreases

Increasing failure rate, with

most rapid increase initially

Early wear-out failure


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= 2

Rayleigh distribution Linearly increasing failure

rate

Risk of wear-out failure

increases steadily over the

life of the product

3 4

Bell-shaped Rapidly increasing

Rapid wear-out failures

Models the final period of

product life, when mostfailures occur

> 10

Similar to extreme valuedistribution

Very rapidly increasing

Very rapid wear-out failures

Models the final period of

product life, when most

failures occur

Applications of the Weibull Distribution

The Weibull distribution can model data that are right-skewed, left-skewed, or symmetric. For this reason, the

distribution is used to evaluate reliability across diverse applications, including vacuum tubes, capacitors, ballbearings, relays, and material strengths. The Weibull distribution can also model a hazard function that is

decreasing, increasing or constant, allowing it to describe any phase of an items lifetime.

Increasing hazard functionAn increasing hazard function is probably the most common scenario, where items are more likely to fail with

time. For example, many mechanical items that are prone to stress or fatigue will have an increased risk offailure over the lifetime of the product. Engineers might use a test to simulate wear-out stress, such as using amachine to simulate extended usage of a light bulb over time and then recording the time until a failure occurs.

A Weibull distribution is often used to model this type of wear-out failure.

Decreasing hazard functionA decreasing hazard function indicates failures that are more likely to occur early in the life of a product. Oneexample is products or parts composed of metals that harden with use and thus grow stronger as time passes.

Another example is bugs in a computer program, which may be likely to appear initially but then decrease as

time passes. Often, this type of data can be modeled using aWeibull distribution with a shape parameter less

than 1.


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Constant hazard functionA constant hazard function indicates failures that are equally likely to occur at any time in the products life.

This relatively constant period of low failure risk characterizes the middle portion of the Bathtub Curve (see

below). This function can also be modeled using theExponential distribution.

Bathtub-shaped hazard functionIf the hazard function is bathtub-shaped, the risk of failure is high at the start, decreases rapidly, levels off and

remains fairly constant, and then increases rapidly at the end of the products life. Televisions and handheld

calculators are two products that commonly exhibit a bathtub-shaped hazard function. Another example is a

microprocessor, which may fail soon after being put into a computer system.

Figure 1:Bathtub shaped hazard function

The Weibull distribution can be used to model every phase of the bathtub-shaped hazard function. In the early

failure period, the risk of failure is initially high, then decreases rapidly and levels off. Typically, the decreasing

risk occurs over a period of several weeks to a few months. This early infant mortality phase can be modeledusinga Weibull shape parameter, , between 0 and 1.

Products that are initially prone to manufacturing, design, or component defects often have a high risk of failure

at the onset. Defects of this type include poor soldering of parts, faulty mechanical attachments, weak wire

bonding, and poor die attachment. To reduce the risk of their infant mortality, these products often requireinitial stress testing such as burn-in (to stress devices under constant operating conditions); power cycling (to

stress devices under the surges of turn-on and turn-off); and temperature cycling (to mechanically andelectrically stress devices over temperature extremes).

After the early high-risk period is passed, the product is then likely to fail only from normal wear-out, near the

end of its expected life, at which time the risk of failure sharply increases. This last stage of product life can be

modeled using aWeibull shape parameter, , greater than 3.

Rayleigh distributionWhen the Weibull distribution has a shape parameter of 2, it is known as the Rayleigh distribution. Thisdistribution is often used to describe measurement data in the field of communications engineering, such as


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measurements for input return loss, modulation side-band injection, carrier suppression, and RF fading. Thisdistribution is also commonly used in the life testing of electrovacuum devices.

Weakest link modelThe Weibull distribution can also model a life distribution with many identical and independent processes

leading to failure, in which the first to reach a critical stage determines the failure time. Extreme value theoryserves as the basis for this weakest link model, where many flaws compete to be the eventual site of failure.

Because the Weibull distribution can be theoretically derived from theSmallest Extreme Value Distribution,it

can also provide an effective model for weakest-link applications such as capacitor, ball bearing, relay and

material strength failures. However, if the variable of interest can take negative values, the smallest extremevalue distribution is preferable, because the Weibull distribution can only model positive values due to its lower

bound of 0.

In short, the Weibull distribution is extremely versatile and can be used to model a wide range of reliability

data. Use this distribution to answer questions such as:

What percent of items are expected to fail during the burn-in period? E.g. What percent of fuses areexpected to fail during the 8-hr burn-in period?

How many warranty claims can be expected during the useful life phase? E.g. How many warrantyclaims do you expect to see over the 50,000 mile useful life of this tire?

When is rapid wear-out expected to occur? E.g. When should maintenance be regularly scheduled toprevent engines from entering their wear-out phase?

The Weibull distribution may not work as effectively for product failures that are caused by chemical reaction

or a degradation process like corrosion, which can occur with semiconductor failures. Typically, these types ofsituations are usually modeled using thelognormal distribution.

Example 1: CapacitorsCapacitors were tested at high stress to obtain failure data (in hours). The lifetime data were modeled by a

Weibull distribution.

280024002000160012008004000

6

5

4

3

2

1

0

Hours to Failure

Frequency

Capacitor Failure Times


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Example 2: Tensile strengthEngineers tested and recorded tensile strength from a sample of alloys. Distribution of the data is skewed to left

and modeled with a 2-parameter Weibull.

8886848280

25

20

15

10

5

0

Strength (ksi)

Frequency

Histogram of Strength (ksi)

The Exponential DistributionThe exponential distribution is a simple distribution with only one parameter and is commonly used to modelreliability data. This distribution is actually a special case of the Weibull distribution (seeexample of Weibull

with = 1).

An important property of the exponential distribution is that it is memoryless. The memoryless propertyindicates that the remaining life of a component is independent of its current age. For example, a system that

experiences wear and tear and therefore becomes more likely to fail later in its life is not memoryless.Therefore, this distribution should be used when the failure rate is constant over the entire life of the product.

The number of failures per unit in time is typically expressed in failures per unit of time, such as percent offailure per thousand-hour units.

Probability Distribution Function

Skewed right


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Risk of failure is constant

Hazard function

Applications of the Exponential DistributionThe exponential distribution provides a good model for a product or item that is just as likely to fail at any time,

regardless of whether it is brand new, a year old, or several years old. In other words, the item should not age or

wear out over its expected application.

For this reason, the exponential distribution is often used to model electronic components that typically do not

wear out until long after the expected life of the product in which they are installed. Examples includecomponents of high-quality integrated circuits, such as diodes, transistors, resistors, and capacitors.

The exponential distribution is also considered an excellent model for the long, flat(relatively constant) period

of low failure risk that characterizes the middle portion of theBathtub Curve.This phase corresponds with theuseful life of the product and is known as the intrinsic failure portion of the curve.

However, the exponential distribution should not be used to model mechanical or electric components that areexpected to show fatigue, corrosion, or wear before the expected life of the product is complete, such as ball

bearings, or certain laser or filament devices.

Example: TransistorsAn electronic component is known to have a constant failure rate over the expected life of a product. Engineers

record the time to failure of the component under normal operating conditions.


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6404803201600

7

6

5

4

3

2

1

0

Hours

Frequency

Component failure

Example 2: Filaments

A light bulb company makes an incandescent filament that is not expected to wear out during an extended

period of normal use. They want to guarantee it for 10 years of operation. Engineers stress the bulbs to simulatelong-term use and record the months until failure for each bulb.

10008006004002000

50

40

30

20

10

0

Month to Failure

Percent

Bulb Failure


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The Lognormal DistributionThe lognormal distribution is a flexible distribution that is closely related to the normal distribution. This

distribution can be particularly useful for modeling data that are roughly symmetric or skewed to the right.

Probability Density Function

Skewed right

Hazard Function

Increases to a maximum then decreases

Like the Weibull distribution, the lognormal distribution can take on markedly different appearances dependingon its shape parameter.

X

De

nsity

0.2

0.5

1

5

Scale

Lognormal distributions

In fact, the lognormal model and the Weibull model may sometimes fit a given set of life test data equally well.However, there is one important difference to consider. When using these distributions to extrapolate beyond


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the range of sample data, the lognormal will predict lower average failure rates at earlier times than the Weibulldistribution.

Applications of the Lognormal DistributionThe lognormal distribution has been called the most commonly used life distribution model for many high-

technology applications. The distribution is based on the multiplicative growth model, which means that at any

instant of time, the process undergoes a random increase of degradation that is proportional to its present state.The multiplicative effect of all these random independent growths accumulates to trigger failure. Therefore, the

distribution is often used to model parts or components that fail primarily due to stress or fatigue, including:

Failure due to chemical reactions or degradation, such as corrosion, migration, or diffusion, which iscommon with semiconductor failure

Time to fracture in metals subject to the growth of fatigue cracks

Electronic components that exhibit decreased risk of failure after a certain time

However, if components are not expected to fail until well after the technological life of the product in which

they are installed is complete (that is, the failure rate a component is constant over its expected lifetime), anexponential distributionmay be more appropriate.

Example 1: Electronic Components

Engineers record the time to failure of an electronic component under normal operating conditions. Thecomponent shows a decreased risk of failure over time.

1351201059075604530

12

10

8

6

4

2

0

Time to failure

Frequency

Component life


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Example 2: Diesel Generator FansTime until failure was tracked over the life of diesel generator fans.

120001000080006000400020000

16

14

12

10

8

6

4

2

0

Hours

Frequency

Histogram of Hours

Smallest Extreme Value DistributionThe smallest extreme value distribution is a limiting distribution for the minimum of a very large collection of

random observations from the same arbitrary distribution. Extreme value theory is a useful model in situations

where many identical and independent processes can lead to failure and the first one to fail determines thefailure time. In this scenario, sometimes referred to as the worst or weakest link, the distribution is typically

skewed to the left.


X

Density

5

6

7

Scale

Smallest Extreme Value Distribution

The relationship between the extreme value distributions and theWeibull distributionis similar to that between

thenormalandlognormaldistributions. Specifically, the log (base e) of a variable following a Weibulldistribution has a smallest extreme value distribution.

Despite this equivalence, the distributions are not strictly interchangeable in their applications. The National

Institute of Standards and Technology recommends trying the extreme value distribution in any modeling


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application for which the variable of interest is the minimum of many random factors, all of which can takepositive or negative values.

The hazard function of the smallest extreme value distribution shows a risk of failure that is exponentiallyincreasing.

This hazard function suggests that the smallest extreme value distribution is suitable for modeling the life of a

product that experiences very rapid wear-out after a certain age. This includes the final stage of theBathtubCurve,known as the wear-out period.

Applications of the Smallest Extreme Value DistributionThe smallest extreme value distribution is often appropriate for product failures related to load and strength.

The extreme value distribution is used to model minimum values. When using this distribution, you are

typically not concerned with the distribution of variables that describes most of the population but only with the

extreme values that can lead to failure. In other words, you are investigating imperfections in certain materialsthat can cause nonuniform stress under a load. The strength of the material is therefore related to the effect of

the imperfection causing the greatest reduction in strength (the weakest link).

One common application is capacitor dielectric breakdown, where many flaws compete to be the eventual site

of failure. Another example is semiconductor wire bonds, which typically do not fracture or overheat undernormal operating conditions, unless they are subject to extreme electrical load or extremely low bond strength.

Similarly, coolant tubes have a minimum thickness to provide sufficient heat transfer to coolant liquid. But a

failure occurs if the hot combustion gases burn pinholes through any point on the tubes.

Use the smallest extreme value distribution to answer questions such as:

Which material can withstand the largest load?

How many items are expected to break during the warranty period?

What is the minimum force needed to break a pouch when multiple strength tests are performed on differentsection of each part?

Which brand of cable is better able to withstand a load of 1,000 pounds?


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Example 1: Wire StrengthWire samples of equal length are tested for breaking strength.

8072645648

4

3

2

1

0

Strength

Frequency

Histogram of Strength

Example 2: Alloy SpecimensEngineers subject an alloy specimen to a total of 300,000 cycles and measure the number of cycles until failure.

280240200160120

14

12

10

8

6

4

2

0

Number of Cycles Until Failure

Frequency

The Normal DistributionThis well-known distribution, which is symmetric and bell-shaped, is widely used in statistics. Industrialapplications often generate data that are normally distributed.

Historically, the normal distribution has not been as commonly used to model reliability data as otherdistributions, partially because its left tail extends to negative infinity, which could result in erroneous modeling

of negative times to failure. Most reliability data are modeled using distributions for positive random variables,

such as exponential, Weibull, gamma, and lognormal. Fewer applications use the normal distribution as a model


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for product life. However, if the mean of the data is greater than 0 and its variation relatively low, the normaldistribution can be useful for modeling certain types of life data.


Symmetric and bell-shaped

The risk of failure is strictly increasing

Hazard Function

Note that the normal distribution closely approximatesthe Weibull distribution when 3 < < 4.

Applications of the Normal DistributionThe normal distribution can sometimes be used to model the life of consumable items, in which the risk of

failure is always increasing. Electric filament devices, such as incandescent light bulbs and toaster heating

elements, have been given as examples of items whose failure data may follow a normal distribution. The

strength of a wire bond in integrated circuits is another example.


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Example 1: Beverage Shelf LifeTo evaluate the shelf life of beverages, analysts record the number of days before a bottled beverage discolors.

16014012010080

9

8

7

6

5

4

3

2

1

0

Shelf Life

Frequency

Histogram of Shelf Life

Example 2: Toaster ReliabilityEngineers perform life tests on a toaster with a new component design.

10509007506004503001500

20

15

10

5

0

Hours

Frequency

Histogram of Hours


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References

National Institute of Standard and Technology Engineering Statistics Handbook. Chapter 8: Assessing Product

Reliability. Available online at:http://itl.nist.gov/div898/handbook/toolaids/pff/8-apr.pdf

Patrick O Connor. Practical Reliability Engineering. 3rdedition. New York: John Wiley and Sons, 1991.

Russell J Hoppenstein. Statistical reliability analysis on Rayleigh probability distributions. RF design tutorial.

October 2000. Available at:

http://s3.amazonaws.com/zanran_storage/www.frbb.utn.edu.ar/ContentPages/42682870.pdf

Lawrence M. Leemis. Reliability: Probabilistic Models and Statistical Methods. Upper Saddle River, NJ:

Prentice Hall, 1995.

David K Lloyd and Myron Lipow. Reliability: Management, Methods, and Mathematics. Englewood Cliffs, NJ:Prentice-Hall, 1962.

William Q. Meeker and Luis A. Escobar. Statistical Methods for Reliability Data. New York: John Wiley andSons, 1998.

Paul A. Tobias and David C. Trinidade. Applied Reliability. 2nd

edition. New York: Chapman and Hall/CRC,1995.

Raytheon Reliability Analysis Laboratory. Finding Defects Using Burn-in Analysis. Joe Dzekevich. Availableat:https://www.reliabilityanalysislab.com/0208_BurnIn.asp
http://itl.nist.gov/div898/handbook/toolaids/pff/8-apr.pdfhttp://itl.nist.gov/div898/handbook/toolaids/pff/8-apr.pdfhttp://itl.nist.gov/div898/handbook/toolaids/pff/8-apr.pdfhttp://s3.amazonaws.com/zanran_storage/www.frbb.utn.edu.ar/ContentPages/42682870.pdfhttp://s3.amazonaws.com/zanran_storage/www.frbb.utn.edu.ar/ContentPages/42682870.pdfhttps://www.reliabilityanalysislab.com/0208_BurnIn.asphttps://www.reliabilityanalysislab.com/0208_BurnIn.asphttps://www.reliabilityanalysislab.com/0208_BurnIn.asphttps://www.reliabilityanalysislab.com/0208_BurnIn.asphttp://s3.amazonaws.com/zanran_storage/www.frbb.utn.edu.ar/ContentPages/42682870.pdfhttp://itl.nist.gov/div898/handbook/toolaids/pff/8-apr.pdf

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