Applied Reliability Page 1 APPLIED RELIABILITY. Techniques for Reliability Analysis

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1 Applied Reliability Page 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. STAT-TECH Spring 2010

2 Applied Reliability Page 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 Access to Microsoft EXCEL (2003 or 2007) Alternative (Open Software): OpenOffice at JMP Recommended Free 30 day trial at Download at 6 month license: $ month license: $49.95

3 Applied Reliability Page 3 Applied Reliability Class 1 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 Failure Units (%/KHr and FITS) Parameters of Distributions Censored Data

4 Applied Reliability Page 4 Key Concept : Variation The objective in running experiments is to extract useful information from data. Variation exists in all processes. Statistics is concerned with variation. How do we visualize variation?

5 Applied Reliability Page 5 Variation Examples Coin Toss Signature Product Performance

6 Applied Reliability Page 6 Descriptive Statistics Terminology Population : The entire set or collection of measurements of interest Sample : A subset of data from the population Population Statistical Inference Probability Sample

7 Applied Reliability Page 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. What s the estimated number of errors in the total program? How sure am I about the estimate? What possible factors can influence the results?

8 Applied Reliability Page 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. Samples not randomly drawn may be biased and correct inference about the population may not be possible.

9 Applied Reliability Page 9 Class Exercise Pick a Random Number

10 Applied Reliability Page 10 Class Project Random Sampling Via EXCEL or OpenOffice Spreadsheet 1. Assume we have n = 100 objects and we want to randomly choose Set up in Column A cells A1:A100 with numbers 1 through 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:B Highlight cells A1:B100. Do Data, Sort using column B, ascending. 7. Use first 10 numbers in cells A1:A10 for random selection among n objects.

11 Applied Reliability Page 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 The larger the sample size, the more confidence in the estimates

12 Applied Reliability Page 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? What form does the data take: count, proportion, or measurement? Why?

13 Applied Reliability Page 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-mediumlarge. Nominal categorical data has no meaningful order, for example, failure mode causes (corrosion, electromigration, oxide breakdown, etc.

14 Applied Reliability Page 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. Such data is referred to as attributes data.

15 Applied Reliability Page 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 continuous quantitative data as variables data.

16 Applied Reliability Page 16 Class Project Select appropriate boxes: 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? variables attributes ordinal nominal

17 Applied Reliability Page 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. Highly useful information but cost to obtain can be high.

18 Applied Reliability Page 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 stressed units at each readout.

19 Applied Reliability Page 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. The exact time at which each microprocessor fails is recorded, thereby producing 100 times to failure.

20 Applied Reliability Page 20 Reporting the Sample Results How do we analyze and report the results from the HTOL experiment on processors? Frequently, an average or mean time to failure is used, possibly with a measure of spread such as a range (high low) or standard deviation.

21 Applied Reliability Page 21 Averages and Ranges Don t Tell the Whole Story All these distributions have the same mean and range!!!

22 Applied Reliability Page 22 Numerical Measures Need a Distribution All these distributions have the same mean and standard deviation!!!

23 Applied Reliability Page 23 Table 1.1 Measurement Data on a Sample of 100 Fuses Fuse Opening Value of Current in Amps What does this table tell you?

24 Applied Reliability Page 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. We need a way to visualize the results.

25 Number in Cell Applied Reliability Page 25 Visualizing Data Histograms A histogram is a bar chart of a frequency table or frequency distribution of a sample. Raw Sample (Fuse) Data 4.64, 4.98, 4.73, 5.21,..., 5.11, 4.80 Frequency Table Cell Boundaries to to Number in Cell Total Count 100 Histogram of Measurements Cell Boundaries

26 Applied Reliability Page 26 Table 1.2 Frequency Table of Fuse Data Cell Boundaries Number In Cell to to to to to to to to to to to Total Count 100

27 Number of Percentage Applied Reliability Page 27 Histogram of Measured Values Cell Midpoint

28 Applied Reliability Page 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. Results

29 Applied Reliability Page 29 Adjusting Bars on Histogram Adjust bar spacing by clicking on chart bars, rightclicking, selecting Format Data Series, Series Options, and adjusting Gap Width.

30 Applied Reliability Page 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. JMP output includes descriptive statistics.

31 Applied Reliability Page 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 the population values

32 Number in Cell Applied Reliability Page 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 Possible Model Cell Boundaries

33 Applied Reliability Page 33 JMP Fits PDF to Histogram Click red triangle at Fuse Data. Output includes fit and estimates.

34 Cumulative Percent Applied Reliability Page 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 (UCB) UCB Number of Observations Less Than or Equal To The graphical rendering is called a cumulative frequency plot % 100.0% 80.0% 60.0% 40.0% 20.0% 0.0% Measured Value

35 Applied Reliability Page 35 Cumulative Distribution 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 : t F( t) f ( y) dy Thus, F(t) represents the area under the PDF curve f(t) to the left of t. f(t) Area = F(t 1 ) Area = F(t 2 ) t 1 t 2 Time(t)

36 Applied Reliability Page 36 Cumulative Frequency Function Estimates CDF Just as the histogram estimates the population PDF curve, so does the sample cumulative frequency plot estimate the population CDF curve

37 Applied Reliability Page 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 PDF can be obtained from CDF and vice versa - informationally equivalent

38 Applied Reliability Page 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 For example, if F(500) = 10%, then the 10% of the units in the population fail by 500 hours.

39 Applied Reliability Page 39 Interpretation of the CDF f(t) Area = F(t) t Time(t) F(t) = Probability of failure by time t = Proportion of population that fails by time t

40 Applied Reliability Page 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 I randomly sample 10 units from the population?

41 Applied Reliability Page 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 F(t) = 1 - R(t)

42 Applied Reliability Page 42 Reliability or Survival Plot Example

43 Applied Reliability Page 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 smallest to largest and graphing using 1/n, 2/n, 3/n,..., k/n for the plotting positions. EDF estimates the population CDF using all measured values.

44 Applied Reliability Page 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, Highlight these three numbers. Place cursor arrow at right lower corner of highlighted region to change to a cross and autofill to C 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. Modify chart as desired.

45 Applied Reliability Page 45 CDF in JMP Click red triangle next to Fuse Data. Select CDF plot. CDF plot is displayed.

46 Applied Reliability Page 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 LIVING The survival curve shows, at the end of each ten year period, the percent of those still alive. Percent Alive Time in Years

47 Applied Reliability Page 47 Creating a Histogram American experience mortality table AGE LIVING 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, = or approximately 12% of those alive at age 10 die. Let s make a histogram of the percent of individuals alive at age 10 years who die in each subsequent ten year interval.

48 10 to to to to to to to to to 100 Applied Reliability Page 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 25 Percent Dying Ten Year Interval Why is the rate of dying dropping in later intervals?

49 Applied Reliability Page 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 during that interval

50 Applied Reliability Page 50 Illustration of Hazard Rate Calculation American experience mortality table AGE LIVING 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 running from age 50 to 60 years

51 Applied Reliability Page 51 The Hazard Rate Plot Plot the average failure rate during an interval (y) at the center of the interval (x) to obtain the (average) hazard rate plot. Percent per Year Interval Midpoint in Years So as one gets older, the rate of dying does increase.

52 Applied Reliability Page 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. We often use the equivalent term instantaneous failure rate (IFR).

53 Applied Reliability Page 53 The Hazard Function The Instantaneous Failure Rate (IFR) We can show the IFR or hazard rate is : h( t) f ( t) 1 F( t) f ( t) R( t) F(t), f(t) or h(t) are informationally equivalent, that is, having any one allows us to calculate the other two.

54 Applied Reliability Page 54 The Average Failure Rate The average failure rate (AFR) between time t 1 and time t 2 is given by AFR( t, t ) 1 2 ln R( t1 ) ln R( t2 ) t t 2 1 The average failure rate (AFR) over the interval 0 to t is AFR( t) R t ln ( ) t For F(t) < 10% approximately, we can simplify the expression for the AFR in terms of the CDF AFR( t) ln[ F( t)] F( t) 1 t t

55 Applied Reliability Page 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. Some OEMs described their requirements in terms of AFRs over different intervals of time.

56 AFR (FITS) Applied Reliability Page 56 Example Supplier AFR Requirements Time Interval AFR 0-4,000 hrs 350 FITS 4,000-30,000 hrs 150 FITS 30, ,000 hrs 25 FITS TIme (Khrs)

57 Applied Reliability Page 57 The Average Failure Rate and CDF To estimate the cumulative percent failures by time t using the average failure rate, the formula is F( t) 1 e t AFR t ( ) For F(t) < 10% approximately, we can simplify the expression for the AFR in terms of the CDF AFR( t) ln[ F( t)] F( t) 1 t t For small F(t) in the interval 0 to t F( t) t AFR( t) For small F(t) between time t 1 and time t 2 F( t ) F( t ) ( t t ) AFR( t, t )

58 Applied Reliability Page 58 Translating Supplier AFR Requirements Into CDF

59 Applied Reliability Page 59 Class Project 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 = 2. The average hazard rate (AFR) is specified as 10%/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 =

60 Error (Overestimation) Applied Reliability Page 60 Error in CDF Estimate from AFR Approximation Formula Note error using approximate calculation increases to sizable amount as CDF estimate becomes greater than 10%. Error in CDF Estimate Using Approximate Formula 8% 7% 6% 5% 4% 3% 2% 1% 0% 0% 5% 10% 15% 20% 25% 30% 35% Exact CDF

61 Applied Reliability Page 61 AFR Calculations in EXCEL Set up spreadsheet using formula as shown below. Enter values for evaluation.

62 Applied Reliability Page 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 Fˆ( t ) r n 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 Rˆ( t ) n n r

63 Applied Reliability Page 63 Class Exercise Ten units start test. Readouts occur at 24, 48, 168, and 500 hours. Number of failures at readouts are: Failures Readouts Estimate the CDF F(t) and the Reliability Function R(t) at each readout Time F(t) R(t)

64 Applied Reliability Page 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 r ˆ( n r ) ht r f ˆ( t ) n t t

65 Applied Reliability Page 65 Class Exercise (Continued) Ten units start test. Readouts occur at 24, 48, 168, and 500 hours. Number of failures at readouts are: Failures Readouts Estimate the PDF f(t) and the average failure rate AFR h(t) during each interval Time f(t) h(t) 0 to to to to 500

66 Applied Reliability Page 66 IFR for Integrated Circuits Bathtub Curve IFR Early Fails Inherent Life Wearout Time

67 Applied Reliability Page 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? We must be careful and precise with our definition of failure!

68 Applied Reliability Page 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 reason

69 Applied Reliability Page 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 f/h. Instead, by using suitable multiplication factors, we can scale the failure rates for Percent per thousand hours (%/Khrs) for FITS (nano-failures per hour or ppm per thousand hours) The word FITS is short for Failure Units or Failures In Time.

70 Applied Reliability Page 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., 1,000,000 units for 1,000 hours or 100,000 units for 10,000 hours, FITs is a prediction for the number of failures.

71 Applied Reliability Page 71 Hazard Rates in FITS 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. We ll use FITS primarily as a rate measured in units of PPM / K hrs or nano-failures per hour.

72 Applied Reliability Page 72 Table of Equivalent Failure Rates In Different Units Failures Per Hour % / K FITS , , Failures per hour x 10 5 = % / K Failures per hour x 10 9 = FITS % / K x 10 4 = FITS

73 Applied Reliability Page 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

74 Applied Reliability Page 74 Converting Units in ART Under Add-Ins, click ART. Select Unit Conversion. Choose units for conversion and input value. Click Calculate.

75 Applied Reliability Page 75 Parameters of Distributions Numerical Measures Distributions may be characterized by descriptive numerical constants called parameters. Central Tendency (Location) Mean or Average Median Mode Spread (Dispersion) Range Standard Deviation Variance Interquartile Range

76 Applied Reliability Page 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) ( e x ) / 2 The parameters and can be shown to be equal to the population mean and standard deviation, respectively.

77 Applied Reliability Page 77 Sample Estimates of Parameters Statistics A statistic estimates a population parameter. For example, the statistic for the sample average is: X X X X 1 2 n i 1 n n n X i 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

78 Applied Reliability Page 78 Sampling Distribution of Means X k X k 1 Sample Sample n n Population n Sample X 4, 2 n n n Sample Sample Sample X 3 X 1 X 2

79 Applied Reliability Page 79 Sampling Distribution of Means The Central Limit Theorem The most important theorem in statistics. For any population, the distribution of sample averages enough n. X i will be approximately normal for large The variance of the averages X 2 is equal to the population variance of individual readings X 2 divided by the sample size for averages, that is, 2 X 2 X. n

80 Applied Reliability Page 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 distribution of the averages. Compare to the original distribution of the data.

81 Applied Reliability Page 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. Different from uncensored, randomly drawn data.

82 Applied Reliability Page 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 : ( ) = 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. Reliability distributions will be covered in next class.

83 Applied Reliability Page 83 Appendix

84 Applied Reliability Page 84 Class Project Select appropriate boxes: 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? variables attributes ordinal nominal

85 Applied Reliability Page 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 = 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, 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 Probability all ten survive is = or 17.5 %

86 Applied Reliability Page 86 Class Project 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 = 4x0.001 = = 0.4% Exact Calculation (ART) Estimated fallout = 1-exp(-4x0.001) = = = 0.4% 2. The average hazard rate (AFR) is specified as 10%/Khrs over the first 4,000 hours. What is the expected % fallout after 4,000 hours? Approximate Calculation Estimated fallout = (10/105)x4000 = 0.40 or 40% Exact Calculation (ART) Estimated fallout =1 - exp{-(10/105)x4000} = 1 - exp(-0.4) = = or 33.0%

87 Applied Reliability Page 87 Class Exercise (Solution) Ten units start test. Readouts occur at 24, 48, 168, and 500 hours. Number of failures at readouts are: Failures Readouts Estimate the CDF F(t) and the Reliability Function R(t) at each readout Time F(t) R(t) 24 1/10= = /10= = /10= = /10= = 0.3

88 Applied Reliability Page 88 Class Exercise Step-Plot of CDF and Reliability Estimates F(t) R(t) Time

89 Applied Reliability Page 89 Class Exercise (Solution) Ten units start test. Readouts occur at 24, 48, 168, and 500 hours. Number of failures at readouts are: Failures Readouts 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= (1/10)/24= to 48 (2/10)/24= (2/9)/24= to 168 (1/10)/120= (1/7)/120= to 500 (3/10)/332= (3/6)/332=0.0015

90 Applied Reliability Page 90 Class Exercise Step-Plot of PDF and Hazard Rate Estimates f(t) h(t) Time

91 Applied Reliability Page 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 : UNITS Failures / hr % / Khr FITS , ,000