Advances in Stochastic Models for Reliability, Quality and Safety Waltraud Kahle Elart von Collani Jürgen Franz Uwe Jensen Editors Birkhäuser Boston Basel Berlin
Preface List of Contributors List of Tables List of Figures xv xvii xxi xxv PART I: LIFETIME ANALYSIS 1. The Generalized Linnik Distributions Gerd Christoph and Karina Schreiber 3 1.1 Introductory Example and Preliminaries, 3 1.2 Assembling Discrete Linnik and Discrete Stable Distributions, 5 1.3 Calculation of Probabilities, 8 1.4 Characterization via Survival Distributions, 11 1.5 Asymptotic Behaviour, 15 References, 17 2. Acceptance Regions and Their Application in Lifetime Estimation Klaus Dräger 19 2.1 Confidence Bounds Based on Acceptance Regions, 20 2.1.1 Basic notations, 20 2.1.2 Confidence bound and system of acceptance regions, 20 2.1.3 The algorithm "System of lower /^-acceptance regions", 22 2.1.4 Quality of lower confidence bounds, 24 2.1.5 Optimality of the algorithm, 25 2.1.6 Quick determination vs. good quality, 27
2.2 Confidence Bound for the Expectation of a Weibull Distribution, 31 2.2.1 The model, 31 2.2.2 Applying the algorithm "System of lower /3-acceptance regions", 32 2.2.3 Quality of lower confidence bounds for the expectation, 34 References, 35 On Statistics in Failure-Repair Models Under Censoring Jürgen Franz 3.1 Introduction, 37 3.2 Survival Data Analysis Under Censoring, 38 3.3 Nonparametric Estimators for Rr(t), 40 3.4 The Failure-Repair Model Under Censoring, 43 3.4.1 The general model, 43 3.4.2 Model under Koziol-Green assumption, 46 References, 50 Parameter Estimation in Renewal Processes with Imperfect Repair Sofiane Gasmi and Waltraud Kahle 4.1 Introduction, 53 4.2 A General Model, 54 4.3 Specifications, 55 4.4 Parameter Estimation in the General Model, 57 4.4.1 Estimation of the parameters of failure intensity, 58 4.4.2 A simple model for estimating the degree of repair, 62 References, 65 Investigation of Convergence Rates in Risk Theory in the Presence of Heavy Tails Simone Liebner 5.1 A Model in Risk Theory, 67 5.2 Limit Theorem, 71 5.3 Rates of Convergence, 72 References, 79 Least Squares and Minimum Distance Estimation in the Three-Parameter Weibull and Frechet Models with Applications to River Drain Data Robert Offinger 6.1 Introduction, 81
vn 6.2 Least Squares and Minimum Distance Methods, 82 6.2.1 General, 82 6.2.2 Least Squares and minimum distance estimators for the three-parameter Weibull model, 84 6.3 Modelling of River Drain Data, 89 6.3.1 The data, 89 6.3.2 General, 92 6.3.3 Analysis of river Danube data, 92 6.3.4 Analysis of river Main data, 93 References, 96 PART II: RELIABILITY ANALYSIS 7. Maximum Likelihood Estimation With Different Sequential fc-out-of-n Systems Erhard Cramer and Udo Kamps 101 7.1 Introduction, 101 7.2 Sequential fc-out-of-n Systems With Unknown Model Parameters, 104 7.3 Estimation in Specific Distributions, 106 7.4 Sequential A;-out-of-n Systems With Known Model Parameters and Underlying One-Parameter Exponential Family, 107 7.5 Example: Sequential 2-out-of-4 System, 109 References, 110 8. Stochastic Models for the Return of Used Devices Berthold Heiligers and Jürgen Ruf 113 8.1 Introduction, 113 8.2 Additive Models for Returns, 114 8.3 Model Fit, 119 References, 123 9. Some Remarks on Dependent Censoring in Complex Systems Tina Herberts and Uwe Jensen 125 9.1 Introduction and Summary, 125 9.2 Dependence of the Components Within a Parallel System, 128 9.2.1 Estimation of F by means of Fjk, 130 9.2.2 Estimation of F by means of the Kaplan-Meier estimators of Fj, 132 9.2.3 Estimation of F by means of multivariate Kaplan-Keier estimators, 133
9.3 Dependence of the Lifelengths and their Censoring Variables, 135 References, 137 Parameter Estimation in Damage Processes: Dependent Observations of Damage Increments and First Passage Time Waltraud Kahle and Axel Lehmann 139 10.1 Introduction, 139 10.2 The Likelihood Function if Both Damage Increments and Failure Time are Observed, 141 10.3 An Example, 147 10.4 Appendix: Proofof Lemma 10.2.1, 151 References, 152 Boundary Crossing Probabilities of Poisson Counting Processes with General Boundaries Axel Lehmann 153 11.1 Introduction, 153 11.2 The Homogeneous Poisson Process, 156 11.2.1 Upper boundary case, 157 11.2.2 Lower boundary case, 160 11.2.3 Two boundary case, 161 11.3 The Nonhomogeneous Poisson Process, 163 11.4 A Special Mixed Poisson Process, 164 References, 165 Optimal Sequential Estimation for Markov-Additive Processes Ryszard Magiera 167 12.1 Introduction, 168 12.2 The Model and Sampling Times, 170 12.3 Efficient Sequential Procedures, 171 12.4 Minimax Sequential Procedures, 174 References, 180 Some Models Describing Damage Processes and Resulting First Passage Times Heide Wendt 183 13.1 Introduction, 183 13.2 Basic Definitions, 184
ix 13.3 System Failure Time in the Case of Independent Marking, 185 13.3.1 ML-Estimates for parameters in the distribution of the System failure time, 190 References, 194 14. Absorption Probabilities of a Brownian Motion in a Triangulär Domain Erik Zierke 197 14.1 Introduction, 197 14.2 A Random Walk Result and Some Used Limit Theorems, 198 14.3 The Case of Equal Drifts, 202 14.4 The Case of Opposite Drifts, 205 14.5 Discussion of the Results, 206 Appendix, 207 References, 209 PART III: NETWORK ANALYSIS 15. A Simple Algorithm for Calculating Approximately the Reliability of Almost Arbitrary Large Networks Elart von Collani 213 15.1 Introduction, 213 15.2 Notations, 214 15.3 The Approximation, 215 15.3.1 Simple network, 215 15.4 Algorithms, 217 15.4.1 Compound system, 221 15.5 Accuracy, 223 15.5.1 Example 1: Network ARTI, 223 15.5.2 Example 2: Network K6, 224 15.5.3 Example 3: Network K7, 225 15.5.4 Example 4: Network ALG, 226 15.5.5 Example 5: LGR, 227 15.5.6 Example 6: Network EVA, 228 15.5.7 Example 7: Network DGN, 229 15.5.8 Example 8: FNW, 230 15.5.9 Example 9: Network TECL, 231 15.5.10 Example 10: Network RCG, 232 15.6 Conclusions, 232 References, 233
X 16. Reliability Analysis of Flow Networks Roland Jentsch 235 16.1 Introduction, 235 16.2 List of Used Symbols, 236 16.3 Definitions, 236 16.4 Flow Probability, 237 16.5 Computation of the Flow Probability, 237 16.5.1 The decomposition algorithm, 238 16.5.2 Special values of the demanded flow, 238 16.5.3 Special structures, 239 16.5.4 Computation by a generating function, 242 References, 244 17. Generalized Gram-Charlier Series A and C Approximation for Nonlinear Mechanical Systems Carten Sobiechowski 247 17.1 Introduction, 247 17.2 Formulation of the Problem, 248 17.3 Generalized Gram-Charlier-Series A Approximation, 249 17.4 Generalized Gram-Charlier-Series C Approximation, 251 17.5 Examples, 254 17.6 Conclusions, 259 References, 259 18. A Unified Approach to the Reliability of Recurrent Structures Valeri Gorlov and Peter Tittmann 261 18.1 Introduction, 261 18.2 The Decomposition of a Graph, 263 18.3 The All-Terminal Reliability of Recurrent Structures, 264 18.4 Generalizations and Open Problems, 270 References, 272 PART IV: PROCESS CONTROL 19. Testing for the Existence of a Change-Point in a Specified Time Interval Dietmar Ferger 277 19.1 Introduction, 277 19.2 A Family of Tests, 278 19.3 The Bootstrap-Test Family, 283
XI 19.4 The Proofs, 285 References, 289 20. On the Integration of Statistical Process Control and Engineering Process Control in Discrete Manufacturing Processes Rainer Göb 291 20.1 Introduction - Two Simple Examples, 291 20.1.1 An example from Statistical process control, 291 20.1.2 An example from engineering process control, 292 20.2 Comparison of SPC and EPC, 292 20.2.1 History and ränge of application of SPC and EPC, 293 20.2.2 Quality criteria in the parts and process industries, 293 20.2.3 Technical properties of production processes in parts and process industries, 293 20.2.4 Statistical tools of SPC and EPC, 293 20.2.5 Process changes in SPC and EPC, 294 20.2.6 Process monitoring in SPC and EPC, 294 20.2.7 Actions on the production process in SPC and EPC, 295 20.2.8 The structure of process modeis in SPC and EPC, 295 20.3 Models of Process Changes in SPC and EPC, 296 20.3.1 Process changes in SPC modeis, 296 20.3.2 Process changes in EPC modeis, 297 20.4 Process Control in SPC and EPC, 297 20.4.1 Process control as process monitoring in SPC, 298 20.4.2 Process control as process adjustment in EPC, 298 20.5 Problems of the Integration of SPC and EPC, 299 20.5.1 History of SPC/EPC Integration, 299 20.5.2 Models proposed in the literature for SPC/EPC Integration, 300 20.6 A General Model for the Integration of SPC and EPC, 301 20.7 Special Models for the Integration of SPC and EPC, 302 20.7.1 Additive disturbance and shift in drift parameter in the deterministic trend model, 302 20.7.2 Additive disturbance and shift in drift parameter in the random walk with drift model, 303 20.8 Discussion of SPC in the Presence of EPC, 304 20.8.1 Effect of simple shifts on EPC controlled processes, 304 20.8.2 Shifts occurring during production time, 304
Xll Contents 20.8.3 Effect of a biased drift parameter estimate, 305 20.8.4 Effect of constraints in the compensatory variable, 305 20.8.5 Effect of using a wrong model, 307 20.9 Conclusion, 308 References, 309 21. Controlling a Process with Three Different States Gundrun Kiesmueller 311 21.1 Introduction, 311 21.2 Process Model, 312 21.3 Control Model, 315 21.4 Long Run Profit Per Item, 316 21.5 Renewal-Strategy, 317 21.6 Inspection-Strategy, 318 21.7 Adjustment-Strategy, 320 21.8 Numerical Examples, 321 References, 322 22. CUSUM Schemes and Erlang Distributions Sven Knoth 323 22.1 Introduction, 323 22.2 Transition Kernel and Integral Equations, 327 22.3 Solution of the Integral Equations, 330 22.4 Numerical Example, 336 22.5 Conclusions, 336 References, 337 23. On the Average Delay of Control Schemes H. G. Kramer and W. Schmid 341 23.1 Introduction, 341 23.2 On the Average Delay of a Generalized Shewhart Chart, 343 23.2.1 Bounds for the average delay, 344 23.2.2 The average delay for exchangeable variables, 350 23.3 A Comparison of Several Control Charts, 352 23.4 Conclusions, 358 References, 358 24. Tolerance Bounds and C p k Confidence Bounds Under Batch Effects Fritz Scholz and Mark Vangel 361 24.1 Introduction and Overview, 362
xm 24.2 Effective Sample Size and its Estimation, 363 24.3 Tolerance Bounds, 364 24.3.1 No between batch Variation, 365 24.3.2 No within batch Variation, 365 24.3.3 The interpolation step, 366 24.4 Confidence Bounds for OL, Cy and C p k, 366 24.4.1 No between batch Variation, 367 24.4.2 No within batch Variation, 368 24.4.3 The interpolation step, 368 24.5 Validation, 373 24.6 Sample Calculation, 374 24.7 Concluding Remarks, 377 Appendix, 378 References, 378 Subject Index 381