Reliability Data Analysis with Excel and Minitab Kenneth S. Stephens ASQ Quality Press Milwaukee, Wisconsin
Table of Contents CD-ROM Contents List of Figures and Tables xiii xvii Chapter 1 Introduction to Reliability and the Text 1 1.0 Introduction 1 2.0 Approach Taken by Text 3.0 Topic and Chapter Summary 5 2 Chapter 2 Basic Generic Reliability Relationships 9 I. 0 The Cumulative Distribution Function (CDF), F(x) 9 2.0 The Reliability Function, R{t) 12 3.0 The Failure Rate or Hazard Function, h(t) 15 4.0 The Cumulative Hazard Function, H(t) 17 5.0 The Average Failure Rate, AFR(rl5 f2) or AFR(t) 18 6.0 The Mean Time to Failure, MTTF 19 7.0 Hazard Function, h(t), Modeling 20 8.0 DFR,IFR, and Constant Failure Rates and the "Bathtub Curve" 21 9.0 Failure Rate or Rate of Occurrence of Failures (ROCOF) 22 10.0 Aspects of Reliability Data 10.1 Complete and Censored Data 24 10.2 Interval-Censored Data 27 10.3 Multi-Censored Data 28 10.4 Left-Censored Data 29 10.5 Reliability Testing 29 II. 0 Probability Distribution Functions 30 12.0 Generic Reliability Relationships 31 Chapter 3 Some Useful Discrete Distributions for Reliability Analysis... 1.0 Introduction 2.0 Discrete Probability Distribution Functions 24 33 33 34 2.1 Hypergeometric Distribution Function 34 2.2 Binomial Distribution Function 45 2.3 The/Binomial Function Hypergeometric Approximation 51 2.4 The Negative Binomial Distribution Function 52
viii Table ofcontents 2.5 The Geometric Distribution Function 2.6 The Poisson Distribution Function Chapter 4 Point and Interval Estimation for Discrete Distributions 79 79 1.0 Introduction 2.0 Hypergeometric and Negative Hypergeometric Distributions 81 2.1 Example 4-1 Hypergeometric Confidence Intervals 83 2.2 Example 4-2 Hypergeometric Confidence Interval Computations 2.3 Negative Hypergeometric 3.0 Binomial Distribution 3.1 Example 4-4 Confidence Limits and Intervals for Binomial Proportion 3.2 Example 4-5 Confidence Intervals for Binomial Proportion viaminitab 4.0 Negative Binomial Distribution 98 4.1 Example 4-6 Confidence Limits and Intervals for Negative Binomial 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 5-2 CDF Method Complete Data 121 3.2 Example 5-2 CDF Method Censored Data 122 3.3 Example 5-2 Cum Hazard Rate Method Complete 3.4 Example 5-2 Cum Hazard Rate Method 64 69 86 89 90 93 96 100 Data 123 Censored Data 125 3.5 Example 5-2 Readout or Interval Data Analysis via Least Squares Complete 125 3.6 Example 5-2 Readout or Interval Data Analysis via Least Squares Censored 126 3.7 Approximate Parameter Estimation from Graphs- -Exponential... 127 4.0 Normal and Lognormal Distributions 128 4.1 Normal Distribution 128 4.2 Lognormal Distribution 130 5.0 Weibull Distribution 133 5.1 Example 5-5 CDF Method Complete Data 134
Table of Contents ix 5.2 Example 5-5 CDF Method Censored Data 135 5.3 Example 5-7 CDF Method Interval Data with Truncation 135 5.4 Approximate Parameter Estimation from Graphs Weibull 136 6.0 Extreme-Value Distributions 137 6.1 Example 5-6 CDF Method Complete Data 138 7.0 Logistic and Log-Logistic Distributions 139 7.1 Example 5-7 CDF Method Complete 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 145 1.0 Introduction 145 2.0 The Exponential Distribution 146 2.1 The Exponential Average Failure Rate, AFRO,, h) or AFR(r) 148 2.2 Mean Time to Failure for the Exponential Model 148 2.3 Percentile Function and Median, Mean, and Variance 150 2.4 Lack of Memory for the Exponential Model 151 2.5 Failure Rate Scaling Units 152 2.6 The Exponential Distribution and System Reliability Closure Property 153 3.0 The Gamma Distribution 155 3.1 The Gamma, Poisson, and Negative Binomial Relationships 157 4.0 The Chi-square Distribution 159 4.1 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 Determination Planning 186 5.5 Exponential Distribution Analysis via Minitab 189 5.6 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 Reliability Closure 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
x Teblt of Contents 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 231 3.3 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 242 2.3 Logistic Distribution Reliability/Survival Example 243 3.0 The Log-Logistic Distribution 245 3.1 Mean, Median, Mode, Variance, and Percentile Function 248 3.2 Point and Interval Estimation for the Logistic Distribution 250 4.0 Logistic and Log-Logistic Distribution Simulation 251 Chapter 10 Systems Reliability 253 1.0 Introduction 253 2,0 Series Systems 254 3.0 Parallel or Active Redundancy Systems 256 3.1 Parallel or Active Redundancy Systems, k out of n 258 3.2 Parallel or Active Redundancy Shared Load Systems 259 4.0 Standby Redundancy Systems 260 5.0 Larger Systems Series and Parallel Subsystems Combined 264 6.0 More-Complex Systems 265 Chapter 11 Reliability of Repairable Systems 267 1.0 Introduction 267 2.0 Example 11-1 System Age-Inter-arrival Time-Data Analysis 269 3.0 Homogeneous Poisson Process 4.0 Non-Homogeneous Poisson Process, Power-Law Model, and Trend Tests 4.1 Power-Law Model ' 273 280 281
Table of Contents xi 4.2 Trend Tests 286 5.0 Summary 297 Chapter 12 Reliability Determination via Accelerated Life Testing 299 1.0 Introduction 299 2.0 Acceleration Factors and Relationships 301 2.1 Example 12-1 Multiple Accelerated Levels Analysis 304 3.0 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