Advances in Stochastic Models for Reliability, Quality and Safety

Transcription

1 Advances in Stochastic Models for Reliability, Quality and Safety Waltraud Kahle Elart von Collani Jürgen Franz Uwe Jensen Editors Birkhäuser Boston Basel Berlin

2 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 Introductory Example and Preliminaries, Assembling Discrete Linnik and Discrete Stable Distributions, Calculation of Probabilities, Characterization via Survival Distributions, Asymptotic Behaviour, 15 References, Acceptance Regions and Their Application in Lifetime Estimation Klaus Dräger Confidence Bounds Based on Acceptance Regions, Basic notations, Confidence bound and system of acceptance regions, The algorithm "System of lower /^-acceptance regions", Quality of lower confidence bounds, Optimality of the algorithm, Quick determination vs. good quality, 27

3 2.2 Confidence Bound for the Expectation of a Weibull Distribution, The model, Applying the algorithm "System of lower /3-acceptance regions", Quality of lower confidence bounds for the expectation, 34 References, 35 On Statistics in Failure-Repair Models Under Censoring Jürgen Franz 3.1 Introduction, Survival Data Analysis Under Censoring, Nonparametric Estimators for Rr(t), The Failure-Repair Model Under Censoring, The general model, Model under Koziol-Green assumption, 46 References, 50 Parameter Estimation in Renewal Processes with Imperfect Repair Sofiane Gasmi and Waltraud Kahle 4.1 Introduction, A General Model, Specifications, Parameter Estimation in the General Model, Estimation of the parameters of failure intensity, 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, Limit Theorem, 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

4 vn 6.2 Least Squares and Minimum Distance Methods, General, Least Squares and minimum distance estimators for the three-parameter Weibull model, Modelling of River Drain Data, The data, General, Analysis of river Danube data, 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 Introduction, Sequential fc-out-of-n Systems With Unknown Model Parameters, Estimation in Specific Distributions, Sequential A;-out-of-n Systems With Known Model Parameters and Underlying One-Parameter Exponential Family, Example: Sequential 2-out-of-4 System, 109 References, Stochastic Models for the Return of Used Devices Berthold Heiligers and Jürgen Ruf Introduction, Additive Models for Returns, Model Fit, 119 References, Some Remarks on Dependent Censoring in Complex Systems Tina Herberts and Uwe Jensen Introduction and Summary, Dependence of the Components Within a Parallel System, Estimation of F by means of Fjk, Estimation of F by means of the Kaplan-Meier estimators of Fj, Estimation of F by means of multivariate Kaplan-Keier estimators, 133

5 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 Introduction, The Likelihood Function if Both Damage Increments and Failure Time are Observed, An Example, Appendix: Proofof Lemma , 151 References, 152 Boundary Crossing Probabilities of Poisson Counting Processes with General Boundaries Axel Lehmann Introduction, The Homogeneous Poisson Process, Upper boundary case, Lower boundary case, Two boundary case, The Nonhomogeneous Poisson Process, A Special Mixed Poisson Process, 164 References, 165 Optimal Sequential Estimation for Markov-Additive Processes Ryszard Magiera Introduction, The Model and Sampling Times, Efficient Sequential Procedures, Minimax Sequential Procedures, 174 References, 180 Some Models Describing Damage Processes and Resulting First Passage Times Heide Wendt Introduction, Basic Definitions, 184

6 ix 13.3 System Failure Time in the Case of Independent Marking, ML-Estimates for parameters in the distribution of the System failure time, 190 References, Absorption Probabilities of a Brownian Motion in a Triangulär Domain Erik Zierke Introduction, A Random Walk Result and Some Used Limit Theorems, The Case of Equal Drifts, The Case of Opposite Drifts, 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 Introduction, Notations, The Approximation, Simple network, Algorithms, Compound system, Accuracy, Example 1: Network ARTI, Example 2: Network K6, Example 3: Network K7, Example 4: Network ALG, Example 5: LGR, Example 6: Network EVA, Example 7: Network DGN, Example 8: FNW, Example 9: Network TECL, Example 10: Network RCG, Conclusions, 232 References, 233

7 X 16. Reliability Analysis of Flow Networks Roland Jentsch Introduction, List of Used Symbols, Definitions, Flow Probability, Computation of the Flow Probability, The decomposition algorithm, Special values of the demanded flow, Special structures, Computation by a generating function, 242 References, Generalized Gram-Charlier Series A and C Approximation for Nonlinear Mechanical Systems Carten Sobiechowski Introduction, Formulation of the Problem, Generalized Gram-Charlier-Series A Approximation, Generalized Gram-Charlier-Series C Approximation, Examples, Conclusions, 259 References, A Unified Approach to the Reliability of Recurrent Structures Valeri Gorlov and Peter Tittmann Introduction, The Decomposition of a Graph, The All-Terminal Reliability of Recurrent Structures, 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 Introduction, A Family of Tests, The Bootstrap-Test Family, 283

8 XI 19.4 The Proofs, 285 References, On the Integration of Statistical Process Control and Engineering Process Control in Discrete Manufacturing Processes Rainer Göb Introduction - Two Simple Examples, An example from Statistical process control, An example from engineering process control, Comparison of SPC and EPC, History and ränge of application of SPC and EPC, Quality criteria in the parts and process industries, Technical properties of production processes in parts and process industries, Statistical tools of SPC and EPC, Process changes in SPC and EPC, Process monitoring in SPC and EPC, Actions on the production process in SPC and EPC, The structure of process modeis in SPC and EPC, Models of Process Changes in SPC and EPC, Process changes in SPC modeis, Process changes in EPC modeis, Process Control in SPC and EPC, Process control as process monitoring in SPC, Process control as process adjustment in EPC, Problems of the Integration of SPC and EPC, History of SPC/EPC Integration, Models proposed in the literature for SPC/EPC Integration, A General Model for the Integration of SPC and EPC, Special Models for the Integration of SPC and EPC, Additive disturbance and shift in drift parameter in the deterministic trend model, Additive disturbance and shift in drift parameter in the random walk with drift model, Discussion of SPC in the Presence of EPC, Effect of simple shifts on EPC controlled processes, Shifts occurring during production time, 304

9 Xll Contents Effect of a biased drift parameter estimate, Effect of constraints in the compensatory variable, Effect of using a wrong model, Conclusion, 308 References, Controlling a Process with Three Different States Gundrun Kiesmueller Introduction, Process Model, Control Model, Long Run Profit Per Item, Renewal-Strategy, Inspection-Strategy, Adjustment-Strategy, Numerical Examples, 321 References, CUSUM Schemes and Erlang Distributions Sven Knoth Introduction, Transition Kernel and Integral Equations, Solution of the Integral Equations, Numerical Example, Conclusions, 336 References, On the Average Delay of Control Schemes H. G. Kramer and W. Schmid Introduction, On the Average Delay of a Generalized Shewhart Chart, Bounds for the average delay, The average delay for exchangeable variables, A Comparison of Several Control Charts, Conclusions, 358 References, Tolerance Bounds and C p k Confidence Bounds Under Batch Effects Fritz Scholz and Mark Vangel Introduction and Overview, 362

10 xm 24.2 Effective Sample Size and its Estimation, Tolerance Bounds, No between batch Variation, No within batch Variation, The interpolation step, Confidence Bounds for OL, Cy and C p k, No between batch Variation, No within batch Variation, The interpolation step, Validation, Sample Calculation, Concluding Remarks, 377 Appendix, 378 References, 378 Subject Index 381