Statistical and Probabilistic Models in Reliability

by Ionescu, D. C.; Limnios, N.

• ISBN13: 9780817640682
• ISBN10: 0817640681
• Format: Hardcover
• Copyright: 1999-03-01
• Publisher: Springer Verlag

Summary

Presents twenty-four carefully edited chapters providing an up-to-date survey of new models and methods for reliability analysis and applications in science, engineering, and technology. DLC: Reliability (engineering)--Statistical methods.

Table of Contents

Foreword	xiii	(2)
Preface	xv	(4)
Contributors	xix	(6)
List of Tables	xxv	(2)
List of Figures	xxvii	(4)
Glossary of Terms	xxxi
PART I: STATISTICAL METHODS	3	(124)
1 Statistical Modeling and Analysis of Repairable Systems	3	(24)
Bo Henry Lindqvist
1.1 Introduction	3	(1)
1.2 "Major Events" in the History of Repairable Systems Reliability	4	(2)
1.3 Notation and Basic Definitions	6	(3)
1.4 Classification of Repair Actions	9	(2)
1.5 The Trend-Renewal Process	11	(2)
1.6 Statistical Inference in Trend-Renewal Processes	13	(3)
1.7 Trend Testing	16	(2)
1.8 Monte Carlo Trend Tests	18	(3)
1.9 Concluding Remarks and Topics for Further Study	21	(1)
References	22	(5)
2 CPIT Goodness-of-Fit Tests for Reliability Growth Models	27	(12)
Olivier Gaudoin
2.1 Introduction	27	(2)
2.2 The Conditional Probabilty Integral Transformation	29	(1)
2.3 CPIT GOF Tests for the Homogeneous Poisson Process	29	(1)
2.4 CPIT GOF Tests for the Jelinski-Moranda and Goel-Okumoto Models	30	(1)
2.5 CPIT GOF Tests for the Power-Law Process	31	(1)
2.6 Experimental Results	32	(3)
2.7 Conclusion	35	(1)
References	36	(3)
3 On the Use of Minimally Informative Copulae in Competing Risk Problems	39	(12)
Tim Bedford
3.1 Competing Risk	39	(3)
3.2 Bounds Without Assumptions on a Dependence Structure	42	(1)
3.2.1 Peterson bounds	42	(1)
3.2.2 Crowder-Bedford-Meilijson bounds	42	(1)
3.3 Estimators Using Dependence Assumptions	43	(3)
3.3.1 The copula-graphic estimator	45	(1)
3.4 Minimally Informative Copulae	46	(2)
3.5 Examples	48	(1)
3.5.1 Example 1	48	(1)
3.5.2 Example 2	48	(1)
3.6 Conclusions	49	(1)
References	49	(2)
4 Model Building in Accelerated Experiments	51	(24)
V. Bagdonavicius
M.S. Nikulin
4.1 Introduction	51	(1)
4.2 Additive Accumulation of Damages Model and Its Submodels	52	(6)
4.3 Generalized Multiplicative Models	58	(7)
4.4 Generalized Additive and Additive-Multiplicative Models	65	(1)
4.5 Models Describing the Influence of Stresses to the Shape and Scale of Distribution	66	(1)
4.6 The Model of Sedyakin and Its Generalizations	67	(2)
4.7 The Heredity Hypothesis	69	(1)
References	70	(5)
5 On Semiparametric Estimation of Reliability From Accelerated Life Data	75	(16)
V. Bagdonavicius
M.S. Nikulin
5.1 Introduction	75	(2)
5.2 Estimation in the AAD Model	77	(3)
5.3 Properties of Estimators	80	(5)
5.4 Estimation, When Stresses Change the Shape of Distribution	85	(1)
5.5 Estimation in AFT Model, When G is Completely Unknown and r is Parametrized	86	(2)
References	88	(3)
6 Analysis of Reliability Characteristics Estimators in Accelerated Life Testing	91	(10)
Leo Gerville-Reache
Valentina Nikoulina
6.1 Introduction	91	(2)
6.2 Parametric Estimation	93	(4)
6.3 Nonparametric Estimation	97	(2)
6.4 Conclusion	99	(1)
References	99	(2)
7 Chi-Squared Goodness of Fit Test for Doubly Censored Data With Applications in Survival Analysis and Reliability	101	(12)
M. S. Nikulin
V. N. Solev
7.1 Introduction	101	(2)
7.2 Weak Convergence of the Process U(n)(t)	103	(4)
7.3 The Weak Convergence of the Process U^*(n)(t)	107	(2)
7.4 The Test Statistics	109	(1)
References	110	(3)
8 Estimation of Kernel, Availability and Reliability of Semi-Markov Systems	113	(14)
B. Ouhbi
N. Limnios
8.1 Introduction	113	(2)
8.2 Estimator of the Semi-Markov Kernel	115	(2)
8.3 Estimation of the Markov Renewal Matrix and Its Asymptotic Properties	117	(1)
8.4 Estimation of the Semi-Markov Transition Matrix and Its Properties	118	(2)
8.5 Reliability and Availability Estimation	120	(2)
8.5.1 Availability	120	(1)
8.5.2 Reliability	121	(1)
8.5.3 Asymptotic properties of the estimators	121	(1)
8.6 Application	122	(1)
References	123	(4)
PART II: PROBABILISTIC METHODS	127	(124)
9 Stochastical Models of Systems in Reliability Problems	127	(16)
Vladimir S. Korolyuk
9.1 Introduction	127	(1)
9.2 Reliability Problem for a Redundant System	128	(3)
9.2.1 Repairable duplicated system	128	(1)
9.2.2 Sojourn time in a subset of states	129	(2)
9.3 Problems of Singular Perturbation	131	(2)
9.4 Analysis of Stochastic Systems	133	(4)
9.4.1 Phase merging scheme	133	(2)
9.4.2 Heuristic principles of phase merging	135	(2)
9.5 Diffusion Approximation Scheme	137	(4)
References	141	(2)
10 Markovian Repairman Problems. Classification and Approximation	143	(10)
Vladimir S. Korolyuk
Nicolas A. Derzko
Vladimir V. Korolyuk
10.1 Introduction	143	(2)
10.2 Classification of Repairman Models	145	(2)
10.3 Asymptotical Analysis of Queueing Process	147	(3)
References	150	(3)
11 On Limit Reliability Functions of Large Systems. Part I	153	(32)
Krysztof Kotoworcki
11.1 Introduction	154	(5)
11.2 Limit Reliability Functions of Homogeneous Systems	159	(5)
11.3 Limit Reliability Functions of Nonhomogeneous Systems	164	(8)
11.4 Remarks on Limit Reliability Functions of Multi-State Systems	172	(10)
11.5 Summary	182	(1)
References	183	(2)
12 On Limit Reliability Functions of Large Systems. Part II	185	(14)
Adam Cichocki
Dorota Kurowicka
Beata Milczek
12.1 Domains of Attraction of Limit Reliability Functions	185	(4)
12.2 Asymptotic Reliability Functions of a Regular Homogeneous Series-"k out of n" System	189	(3)
12.3 Limit Reliability Functions of Homogeneous Regular Series-Parallel Systems of Higher Order	192	(5)
References	197	(2)
13 Error Bounds for a Stiff Markov Chain Approximation Technique and an Application	199	(14)
Olivier Pourret
Jerome Collet
Jean-Louis Bon
13.1 Introduction	199	(1)
13.2 Notations	200	(1)
13.3 Approximation Techniques	200	(3)
13.3.1 A path-based technique	200	(2)
13.3.2 Bobbio and Trivedi's algorithm	202	(1)
13.4 Main Results	203	(2)
13.4.1 Equivalence	203	(2)
13.4.2 A non-conservative case	203	(1)
13.4.3 Error bounds	204	(1)
13.5 Numerical Example	205	(2)
13.5.1 Model used	205	(1)
13.5.2 Results	205	(2)
13.6 Conclusion	207	(1)
Appendix	208	(2)
A.1 Proof of Proposition 13.3.1	208	(1)
A.2 Proof of Proposition 13.4.1	208	(1)
A.3 Proof of Theorem 13.4.1	208	(2)
References	210	(3)
14 On the Failure Rate of Components Subjected to a Diffuse Stress Environment	213	(12)
A. Le Breton
J.-L. Soler
14.1 Introduction	213	(1)
14.2 The Mathematical Model	214	(1)
14.3 General Results	215	(3)
14.3.1 The case of a stress starting from a fixed level	216	(1)
14.3.2 The case of a stationary stress process	217	(1)
14.4 Particular Case of Interest	218	(2)
14.4.1 Instantaneous action of the stress	218	(2)
14.4.2 Cumulative action of the stress	220	(1)
14.5 A Shot-Noise Model With Diffuse Stress	220	(2)
14.6 Conclusion	222	(1)
Appendix (Proof of Lemma 14.3.1)	222	(2)
References	224	(1)
15 Modelling the Reliability of a Complex System Under Stress Environment	225	(10)
Christina Zahalca
Mohamed Chardi
15.1 Introduction	225	(1)
15.2 Modelling the Stress	226	(1)
15.3 System of n Identical Components Subjected to an Homogeneous Poisson Stress Process	227	(2)
15.4 Some Particular Configurations of the n Identical Component System	229	(1)
15.5 Architecture and Stress Influence	230	(2)
15.6 Example -- System of Two Identical Components Subjected to a Common, Homogeneous Poisson Stress Process	232	(1)
15.7 Conclusions	233	(1)
References	234	(1)
16 On the Failure Rate	235	(8)
Gheorghe Oprisan
16.1 Introduction	235	(1)
16.2 Failure Process	236	(2)
16.3 Semi-Markov Process	238	(4)
References	242	(1)
17 Asymptotic Results for the Failure Time of Consecutive k-out-of-n Systems	243	(8)
Brahim Ksir
17.1 Introduction	243	(1)
17.2 Strong Laws for the Failure Time of the System	244	(3)
References	247	(4)
PART III: SPECIAL TECHNIQUES AND APPLICATIONS	251	(98)
18 Two-State Start-Up Demonstration Testing	251	(14)
N. Balakrishnan
P.S. Chan
18.1 Introduction	252	(1)
18.2 Probability Generating Function	253	(3)
18.3 Probabilities and Recurrence Relations	256	(6)
References	262	(3)
19 Optimal Prophylaxis Policy for Systems With Partly Observable Parameters	265	(14)
B. P. Harlamov
19.1 Introduction	265	(1)
19.2 One-Server System	266	(2)
19.2.1 Mathematical model	266	(1)
19.2.2 Coefficient of readiness	267	(1)
19.3 Two-Server System	268	(4)
19.3.1 Mathematical model	268	(3)
19.3.2 Coefficient of readiness	271	(1)
19.4 Optimization	272	(5)
19.4.1 Functional equation	272	(1)
19.4.2 Continuous semi-Markov process	273	(1)
19.4.3 Evaluation of functionals	273	(3)
19.4.4 Process of maximal values	276	(1)
19.4.5 Inversed Gamma-process	276	(1)
References	277	(2)
20 Exact Methods to Compute Network Reliability	279	(16)
Corinne Lucet
Jean-Francois Manouvrier
20.1 Introduction	279	(1)
20.2 Definitions and Notation	280	(2)
20.3 Enumeration	282	(3)
20.3.1 State enumeration	282	(1)
20.3.2 Path enumeration-Cut enumeration	283	(2)
20.4 Reduction With Factoring	285	(2)
20.5 Decomposition	287	(4)
20.5.1 The principle	287	(2)
20.5.2 Algorithm implementation	289	(1)
20.5.3 Complexity	290	(1)
20.5.4 Adaptation for other relability problems	291	(1)
20.6 Conclusion	291	(1)
References	292	(3)
21 On Matroid Base Families and the Reliability Computation of Totally Amenable Systems	295	(12)
A. Behr
L. Camarinopoulos
21.1 Preliminaries	295	(2)
21.2 Algorithmic Complexity of Reliability Computation and Domination Theory	297	(2)
21.3 Matroid Base Families	299	(2)
21.4 On the Complexity of Computing the Reliability of Matroid Base Family Systems	301	(3)
21.5 Conclusions	304	(1)
References	304	(3)
22 The Computer-Assisted Analysis of the Semi-Markovian Stochastic Petri Nets and an Application	307	(14)
Anatoli Paul Ulmeanu
Dumitru Cezar Ionescu
22.1 Introduction	307	(2)
22.2 Background Material in the Stochastic Behavior of Petri Nets	309	(2)
22.3 Computer-Assisted Analysis of the Semi-Markovian Petri Nets	311	(4)
22.4 Application	315	(4)
22.5 Conclusions	319	(1)
References	320	(1)
23 Incremental Approach for Building Stochastic Petri Nets for Dependability Modeling	321	(16)
Nicolae Fota
Mohamed Kaaniche
Karama Kanoun
23.1 Introduction	321	(1)
23.2 Presentaiton of the Incremental Approach	322	(1)
23.3 Guidelines for Modular Construction of GSPN Models	323	(6)
23.4 Example: Duplex System	329	(5)
23.5 Conclusions	334	(1)
References	334	(3)
24 Lifetime of High Temperature Working Pipes	337	(12)
E. Alamoreanu
R. Iatan
R. Chirita
R. Ceausu
24.1 Introduction	337	(1)
24.2 Failure Risk	338	(1)
24.3 Defining Reliability	339	(1)
24.4 Mathematical Model for Lifetime Estimations	339	(2)
24.5 Simulating Reliability	341	(1)
24.6 Algorithm of Simulation	341	(2)
24.7 Simulating Reliability for Components	343	(1)
24.8 Simulating System Reliability	343	(4)
References	347	(2)
Index	349