Applied Reliability Page 1 APPLIED RELIABILITY. Techniques for Reliability Analysis

(1)

APPLIED

RELIABILITY

Techniques for Reliability

Analysis

with

Applied Reliability Tools (ART) (an EXCEL Add-In)

and

JMP® Software

AM216 Class 1 Notes

Santa Clara University

Copyright



David C. Trindade, Ph.D.

S

TAT-

T

ECH ®

(2)

Required Text

This material is based on the text:

APPLIED RELIABILITY

by Dr. P. A. Tobias & Dr. D. C. Trindade

2

nd

_{Edition Published in 1995}

CHAPMAN & HALL/CRC

Software Requirements

ART Excel Add-In

Available at

http://www.trindade.com/am216.htm

Access to Microsoft EXCEL (2003 or 2007)

Alternative (Open Software): OpenOffice at

http://www.openoffice.org/

JMP Recommended

Free 30 day trial at

www.jmp.com

Download at

http://www.onthehub.com/jmp/

(3)

Descriptive Statistics

– Variation

– Sample and Population – Random Sampling

– Types of Data

– Readout and Exact Data – Histograms

Reliability Terminology and Concepts

– Life Distributions – PDF

– CDF

– Reliability Function

– Hazard Rates (AFR and IFR) – Estimation

– Bathtub Curve – Failure Categories

(4)

Key Concept : Variation

• The objective in running experiments is to

extract useful information from data.

• Variation exists in all processes.

(5)

Variation Examples

Coin Toss

Signature

(6)

Descriptive Statistics

Terminology

Population :

The entire set or collection of measurements

of interest

Sample :

A subset of data from the population

Population

Sample

Probability

(7)

Probability Versus Inference

Probability

(Deduction from available information)

Example: I roll two dice.

What’s the probability that the sum of the two dice will be “7” ?

Statistical Inference

(Induction from observations)

Example: I randomly sample a hundred lines of code out of a program containing ten thousand lines and find six errors.

(8)

Random Sampling

What does “randomly sample” mean ?

Each measurement or data point in the population has an equal chance of being selected for the

sample.

(9)

Class Exercise

(10)

Class Project

Random Sampling Via EXCEL or

OpenOffice Spreadsheet

1. Assume we have n = 100 objects and we want to randomly choose 10.

2. Set up in Column A cells A1:A100 with numbers 1 through 100.

3. In Cell B1, type: =rand()

4. Use spreadsheet autofill to extend rand() function from cells B1 to B100. Recalculate several times using F9 key or hitting delete key in empty cell. 5. Highlight B1:B100. Do copy (Ctrl+C) and then Edit (Alt+E), paste special, numbers only, over cells B1:B100.

6. Highlight cells A1:B100. Do Data, Sort using column B, ascending.

(11)

Population, Sample,

Confidence

Population

• Large

• Contains unknown, fixed parameters (such as the average time to failure)

• Determining the exact values of the

interesting parameters may not be practical

Sample

• Typically limited, randomly sampled, finite members of the population

• Sample measurements are easier to obtain • Sample parameters estimate the respective

population parameters and change with different samples drawn

(12)

Class Exercise

Population / Sample

Write down an example of a population in your work :

What information would you like to know about this population ?

How would a sample be typically taken from this population? Is it random?

(13)

Types of Data: Categorical

Observations which are categorized by the

presence or absence of certain characteristics or qualities. Also, called qualitative data.

For example,

pass - fail, go - no go,

in spec - out of spec, mode of chip failure.

Ordinal categorical data has a meaningful ranking

or logical order, for example, ratings in a

questionnaire or classification by small-medium-large.

Nominal categorical data has no meaningful order,

(14)

Attribute Data

Quantitative Categorical Data

Counting the occurrences in specific categories

creates discrete quantitative categorical data, since only specific numbers are possible.

For example, the number of defects on a surface or the fraction of good dies on a wafer.

(15)

Types of Data: Continuous

For a continuous quantity, such as time, voltage, pressure, humidity, temperature, and so on, any value in an interval is possible.

In reliability work, we commonly refer to

(16)

What Type of Data Is?

Time to failure of a component ?

variables attributes ordinal nominal Number of failures in an interval of time ?

variables attributes ordinal nominal Brand of sputtering equipment ?

variables attributes ordinal nominal Serial number on capital equipment ?

variables attributes ordinal nominal

Size of an order of McDonald’s French fries? variables attributes

ordinal nominal Proportion of defective die on a wafer ?

variables attributes ordinal nominal Vendor source ?

variables attributes ordinal nominal Threshold voltage shift ?

variables attributes ordinal nominal Job classifications ?

(17)

Exact Times to Failure vs.

Readout or Interval Data

Two ways to obtain failure data:

1) Record the exact times of failure.

• Continuous monitoring system on

stressed components.

(18)

Readout or Interval Data

2) Record the number of failures or the

changes in variables data between periodic

readouts.

• Readout or interval data is very

common for components on stress.

• Remove the components from stress

for testing. Return unfailed units to

stress.

• There is additional handling of units for

testing and potential disturbance of

failure mechanisms (e.g., self-recovery)

by removal of stress.

• Good idea to use controls (unstressed

units) that are tested along with

(19)

Reliability Stress Test Example

We will use a large number (the population) of microprocessors for a new product.

We obtain a sample (random?) of 100

microprocessors that we stress dynamically

(operational voltages) at an elevated temperature (HTOL).

The stress is run until all components fail. The failure mode observed is an open circuit. The failure mechanism is electromigration resulting from high current flow in a line in the circuit

metallurgy.

(20)

Reporting the Sample Results

How do we analyze and report the results from the HTOL experiment on processors?

(21)

Averages and Ranges

Don’t Tell the Whole Story

All these distributions have the same mean and range !!!

(22)

Numerical Measures Need a

Distribution

(23)

Table 1.1

Measurement Data on a

Sample of 100 Fuses

Fuse Opening Value of Current in Amps

4.64 4.95 5.25 5.21 4.90 4.67 4.97 4.92 4.87 5.11 4.98 4.93 4.72 5.07 4.80 4.98 4.66 4.43 4.78 4.53 4.73 5.37 4.81 5.19 4.77 4.79 5.08 5.07 4.65 5.39 5.21 5.11 5.15 5.28 5.20 4.73 5.32 4.79 5.10 4.94 5.06 4.69 5.14 4.83 4.78 4.72 5.21 5.02 4.89 5.19 5.04 5.04 4.78 4.96 4.94 5.24 5.22 5.00 4.60 4.88 5.03 5.05 4.94 5.02 4.43 4.91 4.84 4.75 4.88 4.79 5.46 5.12 5.12 4.85 5.05 5.26 5.01 4.64 4.86 4.73 5.01 4.94 5.02 5.16 4.88 5.10 4.80 5.10 5.20 5.11 4.77 4.58 5.18 5.03 5.10 4.67 5.21 4.73 4.88 4.80

(24)

Descriptive Statistics

EXCEL Data Analysis Tools (DAT)

Data is entered as a single column.

In Data ribbon, click Data Analysis Add-In. Select Descriptive Statistics.

Enter selections.

Results are displayed.

(25)

Visualizing Data

Histograms

A histogram is a bar chart of a frequency table or frequency distribution of a sample.

(26)

Table 1.2

Frequency Table of Fuse Data

(27)

(28)

Histogram Using Data Analysis

Tools in EXCEL

Enter data into a column. Set up convenient bins to span data. Select DAT. Click Histogram.

Enter information and click boxes as shown.

(29)

Adjusting Bars on Histogram

Adjust bar spacing by clicking on chart bars, right-clicking, selecting Format Data Series, Series

(30)

Distribution Analysis in JMP

Enter data into a column of a data table. Then select Analyze, Distribution. Cast Fuse Data

column into “Y, Columns” role.

(31)

Histograms and Models

• This a histogram of variables data (the

current in amperes at which the fuse opens), which are continuous measurements.

• With enough data points, the histogram begins to look like a smooth curve

• The sample frequency distribution shown by the histogram is estimating a theoretical

model or equation for the distribution of

(32)

Probability Density Function

Population Model

The population model estimated by the sample frequency distribution is called the probability

density function or PDF and is denoted by f(t)

The PDF equation is model the which describes the

continuous distribution of the times to failure. The

area under the curve is normalized to 1.

The histogram estimates the population PDF curve.

(33)

JMP Fits PDF to Histogram

Click red triangle at Fuse Data.

(34)

Cumulative Data

An Alternative Way to Visualize

The cumulative frequency table accumulates the number of observations less than or equal to a given value.

Cumulative Frequency Table

Upper Cell Boundary Number of Observations

(UCB) Less Than or Equal To

UCB 4.495 2 4.595 4 ... ... ... ... 5.495 100

The graphical rendering is called a cumulative frequency plot.

(35)

Function

Population Model

The population model corresponding to the sample cumulative frequency distribution is called the

cumulative distribution function or CDF and is

denoted by F(t).

The CDF is related to the PDF via the equation :

F t

( )



t

f y dy

( )





(36)

Cumulative Frequency

Function Estimates CDF

(37)

CDF From PDF

• In percent, CDF goes from 0 to 100%

• In proportion, CDF goes from 0 to 1

• For variables data restricted to only

positive times, a CDF model is a possible

life distribution

(38)

Cumulative Distribution

Function

A Life Distribution

Interpretation 1

F(t) is the probability a unit randomly drawn

from the population fails by time t

For example, if F(500) = 10%, then the probability of a single (randomly drawn) unit failing by 500 hours is 10%.

Interpretation 2

F(t) is the fraction of all units in the population

which fail by time t

(39)

Interpretation of the CDF

F(t) = Probability of failure by time t

= Proportion of population that fails

by time t

Time(t)

f(t)

t

(40)

Class Project

CDF Interpretation

At 1500 hours the population CDF equals 0.16 or 16%. 1. How many failures do I expect at 1500 hours in a

random sample 100 units from this population, ?

2. What’s the probability that a single unit randomly sampled from the population will fail by 1500 hours?

3. If the population consists of one million units, how many units in the population fail by 1500 hours ?

4. What fraction of the population fails by 1500 hours ?

5.What’s the probability that no unit fails by 1500 hours if

(41)

The Reliability Function

• R(t) is called the reliability or survival

function. (Note: Some authors use S(t).)

• R(t) is the probability of surviving to time t.

• R(t) is also the fraction of survivors in the

population to time t

Since the probability of either surviving or failing

must equal one (a certainty), then,

R(t) + F(t) = 1

or

(42)

Reliability or Survival Plot

(43)

Empirical Distribution

Function (EDF)

If we have k measured values in a random sample of n units, instead of grouping data into intervals, we can construct an EDF by ordering the values from

(44)

Class Exercise

Constructing EDF in EXCEL

or OpenOffice Spreadsheet

Fuse Data (n = 100)

1. Enter label “Fuse Data” in cell A1, “Sorted Fuse Data” in cell B1, and “EDF” in cell C1.

2. Enter fuse data in Column A.

3. Highlight fuse data in A2:A101 and copy and paste to B2:B101. NOTE: Copy and Paste may be done with arrow cursor on highlighted boundary and Ctrl key.

4. With B2:B101 highlighted, select Data in menu, and choose Sort to sort data in ascending order.

5. In C2:C3, enter values 0.01, 0.02, 0.03. Highlight these three numbers. Place cursor arrow at right lower corner of highlighted region to change to a cross and autofill to C101.

6. Highlight B1:C101.

EXCEL: Select chart wizard and form a scatter plot with line.

SO: Select Insert Object, drag rectangle in sheet, Auto Format Chart, and form a scatter plot with line.

(45)

CDF in JMP

Click red triangle next to Fuse Data. Select CDF plot.

(46)

The Hazard Rate Concept

of a Life Distribution

The following American experience mortality table gives the proportion living as a function of age, starting from age 10 in increments of 10 years:

AGE 10 20 30 40 50 60 70 80 90 100 LIVING 1.000 .926 .854 .781 .698 .579 .386 .145 .008 .000 0 20 40 60 80 100 10 20 30 40 50 60 70 80 90 10 Time in Years Percent Alive 100

(47)

Creating a Histogram

American experience mortality table

AGE 10 20 30 40 50 60 70 80 90 100

LIVING 1.000 .926 .854 .781 .698 .579 .386 .145 .008 .000

To find the proportion of individuals who die during any ten year period, subtract applicable proportions. For example, during the interval 50 to 60 years,

0.698 - 0.579 = 0.119

or approximately 12% of those alive at age 10 die.

(48)

Life Distribution

Here is a histogram of the percent of

individuals alive at age 10 years who die in each subsequent ten year interval

0 5 10 15 20 25 Percent Dying 1 0 to 2 0 2 0 to 3 0 3 0 to 4 0 4 0 to 5 0 5 0 to 6 0 6 0 to 7 0 7 0 to 8 0 8 0 to 9 0 9 0 to 1 0 0

Ten Year Interval

(49)

The Average Hazard Rate

During an Interval

The percent dying is dropping during later intervals because there are very few people from age 10 alive at the beginning of those intervals.

To take into account the decreasing sample size, we use the concept of a hazard rate:

The ratio of the percent of people who die

during an interval

to the percent of people alive at the

beginning of the interval

divided by the

length of the interval

is the

average hazard (or failure) rate

(50)

Illustration of Hazard Rate

Calculation

American experience mortality table

AGE 10 20 30 40 50 60 70 80 90 100 LIVING 1.00 .926 .854 .781 .698 .579 .386 .145 .008 .000

Consider the interval 50 to 60 years

Roughly 70% survive to age 50 and 12% of those who started at age 10 die during the interval 50 to 60 years

So 12%/70% = 17% of those alive at age 50, the

beginning of the interval, die during the interval

Divide 17% by 10 years to get the average hazard

rate of 1.7% / yr during the ten year interval

(51)

The Hazard Rate Plot

0 2 4 6 8 10 12 15 25 35 45 55 65 75 85 95 Interval Midpoint in Years Percent

per Year

Plot the average failure rate during an interval (y) at the center of the interval (x) to obtain the

(average) hazard rate plot.

(52)

From Average to

Instantaneous Hazard Rate

The average failure rate measures the rate of failure

over a time interval for those units alive at the beginning of the interval.

By going to smaller and smaller time intervals, we approach the hazard rate at a point, that is, the

conditional rate of failure in the next instance of

time following t, given survival to t.

(53)

The Hazard Function

The Instantaneous Failure Rate (IFR)

We can show the IFR or hazard rate is :

)

(

)

(

)

(

1 )

(

)

(

t

R

t

f

t

F

t

f

t

h







F(t), f(t) or h(t) are informationally equivalent,

(54)

The Average Failure Rate

The average failure rate (AFR) between time t₁ and

time t₂ is given by

AFR t t

R t

t

( , )

₁ ₂

ln ( )

1

ln ( )

2 2 1





The average failure rate (AFR) over the interval 0 to t is

AFR t

R t

t

( )





ln ( )

For F(t) < 10% approximately, we can simplify the expression for the AFR in terms of the CDF

AFR t

F t

t

F t

t

(55)

The Average Failure Rate

One can also specify an AFR over a time

period, for example, between two times t

₁

and

time t

_2.

(56)

Example Supplier AFR

Requirements

0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100 TIme (Khrs) A F R (F IT S)

Time Interval

AFR

0 - 4,000 hrs

350 FITS

(57)

The Average Failure Rate

and CDF

To estimate the cumulative percent failures by time

t using the average failure rate, the formula is

For F(t) < 10% approximately, we can simplify the expression for the AFR in terms of the CDF

For small F(t) between time t₁ and time t₂

For small F(t) in the interval 0 to t

(58)

(59)

Percent Fallout from AFR

1. The average hazard rate (AFR) is specified as

0.1%/Khrs over the first 4,000 hours. What is the

expected % fallout after 4,000 hours?

Approximate Calculation

Estimated fallout =

Exact Calculation (ART)

Estimated fallout =

10%/Khrs over the first 4,000 hours. What is the

Estimated fallout =

(60)

Error in CDF Estimate from

AFR Approximation Formula

Error in CDF Estimate Using Approximate Formula

0% 1% 2% 3% 4% 5% 6% 7% 8% 0% 5% 10% 15% 20% 25% 30% 35% Exact CDF E rr o r (O v e re s ti m a ti o n )

(61)

AFR Calculations in EXCEL

Set up spreadsheet using formula as shown below.

(62)

Simple Estimates for CDF

and Reliability

A simple estimate of F(t) at the end of an interval is the total number of failures r by time t divide by the number of starting units

A simple estimate of R(t) at the end of an interval is the total number of survivors n - r by time t divide by the number of starting units

(63)

Ten units start test. Readouts occur at 24, 48, 168, and 500 hours. Number of failures at readouts are:

Failures 1 2 1 3

Readouts -- 24 -- 48 --- 168 --- 500

Estimate the CDF F(t) and the Reliability Function

(64)

Simple Estimates for PDF

and Hazard Rate

An estimate of the average f(t) during an interval is the number of failures during an interval divided by the number of units that started at time t = 0 divided by the time length of the interval

An estimate of the average h(t) during an interval is the number of failures during an interval divided by the number of surviving units starting the interval divided by the time length of the interval

(65)

(Continued)

Readouts -- 24 -- 48 --- 168 --- 500

Estimate the PDF f(t) and the average failure rate AFR h(t) during each interval

Time f(t) h(t)

0 to 24

24 to 48

48 to 168

(66)

IFR for Integrated Circuits

“

Bathtub Curve”

Early Fails Inherent Life Wearout

(67)

Failure Definition

An event or inoperable state in which

any equipment, or part of the equipment,

does not, or would not, perform as intended.

“Does not perform as intended” has subjectivity.

For example, if the performance is marginal, is

it a failure?

Is a device that is just outside of specification a

failure?

What if the device operates “as intended”

following a recoverable event?

(68)

Failure Categories

Catastrophic : Fails suddenly, unexpectedly,

and non-reversible; i.e., breaking, short, open,

etc.

Degradation : Output degrades below the

expected level, non-reversible; i.e., fatigue,

corrosion, wear-out

Intermittent : Flip-flopping performance

below and within the expected level randomly

at an unknown time and for an unknown

(69)

Failure Rate Units

Failure rates for components are often so

small that units of failures per hour are not

practical. For example, 1 failure in 100

units on test for 1,000 hours is roughly an

AFR of 0.00001 f/h.

Instead, by using suitable multiplication

factors, we can scale the failure rates.

- 10

5

_{for Percent per thousand hours}

(%/Khrs)

- 10

9

_{for FITS (nano-failures per hour or}

ppm per thousand hours)

(70)

Hazard Rates in FITS

There are two common views of the term

FITS.

1. For a constant hazard rate, for the

equivalent of a billion (10

9

_{) hours, e.g.,}

(71)

2. For nonconstant hazard rates, we can

use FITs as a convenient measure for the

instantaneous rate of failure at time t or the

average rate of failure over an interval of

time (AFR).

For example, consider a speedometer

reading of speed at time t or the average

reading of speed over a ten minute interval.

Note: Distinguish between point or interval rate estimates which can produce very different FITS values for nonconstant rates.

(72)

Table of Equivalent Failure

Rates

In Different Units

Failures

Per Hour % / K FITS

.00001 1.0 10,000

.000001 .1 1,000

.0000001 .01 100

.00000001 .001 10

.000000001 .0001 1

Failures per hour x 105 _{= % / K}

Failures per hour x 109 _{= FITS}

(73)

Class Project

Equivalent Failure Rates

Fill out the table below by converting two empty cells in each row into failure rate units equivalent to the units specified in that row :

UNITS

Failures / hr % / Khr FITS

200

0.00005

(74)

Converting Units in ART

Under Add-Ins, click ART. Select Unit Conversion.

(75)

Parameters of Distributions

Numerical Measures

Distributions may be characterized by descriptive numerical constants called

parameters.

Central Tendency (Location)

(76)

Parameters of Distributions

Numerical Descriptive Measures

The PDF and CDF equations

• describe the population distribution

• contain one or more parameters in a form that is not unique

These parameters typically have a convenient

interpretation as descriptive measures of the population. For example the PDF for the normal distribution has the equation :

f x

( )



1 e

 (x ) /

2

2 2 2





 

The parameters  and  can be shown to be

equal to the population mean and standard

(77)

Parameters

Statistics

In contrast to a population parameter which is fixed, a statistic is an expression whose value:

•depends on the sample measurements •changes with each sample drawn

•has its own sampling distribution

X X X X n X n n i i n  1  2  



1

(78)

Population

Sample Sample Sample Sample Sample Sample

Sampling Distribution of Means

(79)

The most important theorem in statistics.

For any population, the distribution of sample

averages will be approximately normal for large enough n.

The variance of the averages is equal to the

population variance of individual readings divided by the sample size for averages, that is,

Sampling Distribution of Means

The Central Limit Theorem

(80)

Sampling Distribution

Example

Class Exercise

Generate 500 random numbers in a spreadsheet.

Choose a fixed set of 500 points. Make a histogram of the data. What distribution best describes the results?

Using this data, calculate 100 averages based on a sample of size n = 5. Make a histogram of the

(81)

Censored Reliability Data

If we end the test at a time or failure count before

all units have failed, then there is no information

on the times to failure of censored units

Time Censored (Type I) Failure Censored (Type II)

We call such censoring, single censoring. In fact, reliability data may be multicensored.

Reliability data is usually ordered data.

Because of right censoring, reliability data comes from the early tail of the distribution.

(82)

Comparing Censored

Reliability Data

to Randomly Sampled Data

• Threshold data from ten randomly sampled units: 5.5, 8.2, 9.5, 1.4, 3.6, 4.7, 7.3, 6.2, 2.9, 4.1 mvolts

»Mean: 5.34 mV

»Range : (9.5-1.4) = 8.1 mV

• Failure data from ten randomly sampled units: (Total test time of 10 hrs)

1.9, 2.8, 3.3, 4.6, 5.7, 8.2 hrs

Four units still surviving (no failures) by10 hrs.

–What’s the mean time to failure of the ten units ? –What’s the range of failure times of the ten units ? –What’s the population model (PDF) for the data? • To get the answers, we need to assume or specify

the distribution.

(83)

(84)

What Type of Data Is?

Time to failure of a component ?

variables attributes ordinal nominal Number of failures in an interval of time ?

variables attributes ordinal nominal Brand of sputtering equipment ?

variables attributes ordinal nominal Serial number on capital equipment ?

Size of an order of McDonald’s French fries? variables attributes

ordinal nominal Proportion of defective die on a wafer ?

variables attributes ordinal nominal Vendor source ?

variables attributes ordinal nominal Threshold voltage shift ?

variables attributes ordinal nominal Job classifications ?

(85)

Class Project

CDF Interpretation

At 1500 hours the population CDF equals 0.16 or 16%. 1. How many failures do I expect at 1500 hours in a

random sample 100 units from this population, ?

100x0.16 = 16

2. What’s the probability that a single unit randomly

sampled from the population will fail by 1500 hours?

0.16 or 16 %

3. If the population consists of one million units, how many units in the population fail by 1500 hours ?

1,000,000x0.16 = 160,000

4. What fraction of the population fails by 1500 hours ?

0.16 or 16 %

5. What’s the probability that no unit fails by 1500 hours

if I randomly sample 10 units from the population?

Probability one unit survives is (1-0.16) = 0.84

(86)

Class Project

Percent Fallout from AFR

0.1%/Khrs over the first 4,000 hours. What is the

Estimated fallout = 4x0.001 = 0.004 = 0.4%

Estimated fallout = 1-exp(-4x0.001) = 1-0.996 = 0.004 = 0.4%

10%/Khrs over the first 4,000 hours. What is the

Estimated fallout = (10/105)x4000 = 0.40 or 40%

Estimated fallout =1 - exp{-(10/105)x4000} = 1 - exp(-0.4)

(87)

(Solution)

Readouts -- 24 -- 48 --- 168 --- 500

Estimate the CDF F(t) and the Reliability Function

(88)

Class Exercise

(89)

(Solution)

Readouts -- 24 -- 48 --- 168 --- 500

Estimate the PDF f(t) and the average failure rate AFR h(t) during each interval

Time f(t) h(t)

0 to 24 (1/10)/24=0.0042 (1/10)/24=0.0042

24 to 48 (2/10)/24=0.0084 (2/9)/24=0.0093

48 to 168 (1/10)/120=0.00083 (1/7)/120=0.0012

(90)

Class Exercise

(91)

Equivalent Failure Rates

Fill out the table below by converting two empty cells in each row into failure rate units equivalent to the units specified in that row :