Reliability Data Analysiswith Excel and Minitab

Kenneth S. Stephens

ASQ Quality Press

Milwaukee, Wisconsin

Table of Contents

CD-ROM Contents xiiiList of Figures and Tables xvii

Chapter 1 Introduction to Reliability and the Text 11.0 Introduction 12.0 Approach Taken by Text 23.0 Topic and Chapter Summary 5

Chapter 2 Basic Generic Reliability Relationships 9I.0 The Cumulative Distribution Function (CDF), F(x) 92.0 The Reliability Function, R{t) 123.0 The Failure Rate or Hazard Function, h(t) 154.0 The Cumulative Hazard Function, H(t) 175.0 The Average Failure Rate, AFR(rl5 f2) or AFR(t) 186.0 The Mean Time to Failure, MTTF 197.0 Hazard Function, h(t), Modeling 208.0 DFR,IFR, and Constant Failure Rates and the "Bathtub Curve" 219.0 Failure Rate or Rate of Occurrence of Failures (ROCOF) 2210.0 Aspects of Reliability Data 24

10.1 Complete and Censored Data 2410.2 Interval-Censored Data 2710.3 Multi-Censored Data 2810.4 Left-Censored Data 2910.5 Reliability Testing 29

II.0 Probability Distribution Functions 3012.0 Generic Reliability Relationships 31

Chapter 3 Some Useful Discrete Distributions for Reliability Analysis ... 331.0 Introduction

332.0 Discrete Probability Distribution Functions 34

2.1 Hypergeometric Distribution Function 342.2 Binomial Distribution Function 452.3 The/Binomial FunctionHypergeometric Approximation 512.4 The Negative Binomial Distribution Function 52

viii Table ofContents

2.5 The Geometric Distribution Function64

2.6 The Poisson Distribution Function69

Chapter 4 Point and Interval Estimation for DiscreteDistributions 79

1.0 Introduction79

2.0 Hypergeometric and Negative Hypergeometric Distributions81

2.1 Example 4-1 Hypergeometric Confidence Intervals83

2.2 Example 4-2 Hypergeometric Confidence Interval

Computations86

2.3 Negative Hypergeometric 89

3.0 Binomial Distribution90

3.1 Example 4-4 Confidence Limits and Intervals for Binomial

Proportion 93

3.2 Example 4-5 Confidence Intervals for Binomial Proportion

viaMinitab 96

4.0 Negative Binomial Distribution 98

4.1 Example 4-6 Confidence Limits and Intervals for Negative

Binomial 100

4.2 Example 4-7 Confidence Intervals for Negative Binomial

Proportion via Minitab 102

5.0 Poisson Distribution 103

5.1 Example 4-8 Poisson Confidence Limits 106

5.2 Example 4-9 Confidence Intervals for Poisson via Minitab 107

Chapter 5 Linear Rectification of Reliability Models For Least

Squares Estimation of Parameters 109

1.0 Introduction 109

2.0 Least Squares Procedure 109

2.1 Estimating F(t) 112

3.0 Exponential Distribution 120

3.1 Example 5-2CDF MethodComplete Data 121

3.2 Example 5-2CDF MethodCensored Data 122

3.3 Example 5-2Cum Hazard Rate MethodComplete Data 123

3.4 Example 5-2Cum Hazard Rate Method

Censored Data 125

3.5 Example 5-2Readout or Interval Data Analysis via Least

SquaresComplete 125

3.6 Example 5-2Readout or Interval Data Analysisvia Least SquaresCensored 126

3.7 Approximate Parameter Estimation from Graphs--Exponential ... 1274.0 Normal and Lognormal Distributions 128

4.1 Normal Distribution 128

4.2 Lognormal Distribution 1305.0 Weibull Distribution 133

5.1 Example 5-5CDF MethodComplete Data 134

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5.2 Example 5-5CDF MethodCensored Data 1355.3 Example 5-7CDF MethodInterval Data with Truncation 1355.4 Approximate Parameter Estimation from GraphsWeibull 136

6.0 Extreme-Value Distributions 1376.1 Example 5-6CDF MethodComplete Data 138

7.0 Logistic and Log-Logistic Distributions 1397.1 Example 5-7CDF MethodComplete Data 140

8.0 Minitab's Simulation Capabilities 141

8.1 Change of Variable and the Probability Integral Transformation ... 142

Chapter 6 Exponential, Gamma, and Chi-Square (#2) Distributions 1451.0 Introduction 145

2.0 The Exponential Distribution 146

2.1 The Exponential Average Failure Rate, AFRO,, h) or AFR(r) 1482.2 Mean Time to Failure for the Exponential Model 1482.3 Percentile Function and Median, Mean, and Variance 150

2.4 Lack of Memory for the Exponential Model 1512.5 Failure Rate Scaling Units 1522.6 The Exponential Distribution and System ReliabilityClosure

Property 1533.0 The Gamma Distribution 155

3.1 The Gamma, Poisson, and Negative Binomial Relationships 157

4.0 The Chi-square Distribution 1594.1 Goodness-of-Fit Tests 162

5.0 Point and Interval Estimation for the Exponential Distribution 173

5.1 Confidence Limits and Intervals for Parameters X and 6 175

5.2 Confidence Limits and Intervals for Reliability 181

5.3 The Case of Zero Failures 182

5.4 Test Characteristics DeterminationPlanning 186

5.5 Exponential Distribution Analysis via Minitab 1895.6 Exponential Distribution Simulation 190

Chapter 7 Weibull and Extreme-Value Distributions 191

1.0 Introduction 191

2.0 The Weibull Distribution 192

2.1 The Weibull Average Failure Rate, AFR(? t2) or AFR(D 197

2.2 Mean Time to Failure (MTTF) for the Weibull Distribution 197

2.3 Percentile Function, Median, and Variance 198

2.4 The Weibull Distribution and System ReliabilityClosure

Property 199

2.5 Point and Interval Estimation for the Weibull Distribution 201

2.6 Weibull Distribution Reliability/Survival Examples 203

3.0 The Smallest Extreme-Value Distribution 206

3.1 The Principal Functions 207

3.2 The SEV Average Failure Rate, AFRO,, h) or AFR(7") 209

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3.3 Mean, Median, Mode, Variance, and Percentile Function 210

3.4 SEV Distribution Estimation and Examples 211

4.0 Weibull and SEV Distribution Simulation 213

Chapter 8 Normal and Lognormal Distributions 215

1.0 Introduction 215

2.0 The Normal Distribution 216

2.1 Mean, Median, Mode, Variance, and Percentile Function 219

2.2 The Central Limit Theorem and Its Practical Consequences 220

2.3 Strength-Stress Analysis via Normal Distributions 222

2.4 Point and Interval Estimation for the Normal Distribution 224

2.5 Normal Distribution Reliability/Survival Example 225

3.0 The Lognormal Distribution 226

3.1 Mean, Median, Mode, Variance, and Percentile Function 230

3.2 Point and Interval Estimation for the Lognormal Distribution 2313.3 Lognormal Distribution Reliability/Survival Example 232

4.0 Normal and Lognormal Distribution Simulation 234

Chapter 9 Logistic and Log-Logistic Distributions 237

1.0 Introduction 237

2.0 The Logistic Distribution 238

2.1 Mean, Median, Mode, Variance, and Percentile Function 241

2.2 Point and Interval Estimation for the Logistic Distribution 2422.3 Logistic Distribution Reliability/Survival Example 243

3.0 The Log-Logistic Distribution 2453.1 Mean, Median, Mode, Variance, and Percentile Function 2483.2 Point and Interval Estimation for the Logistic Distribution 250

4.0 Logistic and Log-Logistic Distribution Simulation 251

Chapter 10 Systems Reliability 2531.0 Introduction 2532,0 Series Systems 2543.0 Parallel or Active Redundancy Systems 256

3.1 Parallel or Active Redundancy Systems, k out of n 2583.2 Parallel or Active Redundancy Shared Load Systems 259

4.0 Standby Redundancy Systems 2605.0 Larger SystemsSeries and Parallel Subsystems Combined 2646.0 More-Complex Systems 265

Chapter 11 Reliability of Repairable Systems 2671.0 Introduction 2672.0 Example 11-1 System Age-Inter-arrival Time-Data Analysis 2693.0 Homogeneous Poisson Process 2734.0 Non-Homogeneous Poisson Process, Power-Law Model, and

Trend Tests'

2804.1 Power-Law Model 281

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4.2 Trend Tests 286

5.0 Summary 297

Chapter 12 Reliability Determination via Accelerated Life Testing 2991.0 Introduction 299

2.0 Acceleration Factors and Relationships 301

2.1 Example 12-1 Multiple Accelerated Levels Analysis 3043.0 Arrhenius Model 307

4.0 Inverse Power Law Model 311

5.0 The General Eyring Model and Useful Two-Stress Forms 315

6.0 Summary 319

References 321

Index 331