Statistical Reliability Engineering

by Gnedenko, Boris; Pavlov, Igor V.; Ushakov, Igor A.; Chakravarty, Sumantra

• ISBN13: 9780471123569
• ISBN10: 0471123560
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-05-03
• Publisher: Wiley-Interscience

Summary

Proven statistical reliability analysis methods-available for the first time to engineers in the West While probabilistic methods of system reliability analysis have reached an unparalleled degree of refinement, Russian engineers have concentrated on developing more advanced statistical methods. Over the past several decades, their efforts have yielded highly evolved statistical models that have proven to be especially valuable in the estimation of reliability based upon tests of individual units of systems. Now Statistical Reliability Engineering affords engineers a unique opportunity to learn both the theory behind and applications of those statistical methods. Written by three leading innovators in the field, Statistical Reliability Engineering: * Covers all mathematical models for statistical reliability analysis, including Bayesian estimation, accelerated testing, and Monte Carlo simulation * Focuses on the estimation of various measures of system reliability based on the testing of individual units * Contains new theoretical results available for the first time in print * Features numerous examples demonstrating practical applications of the theory presented Statistical Reliability Engineering is an important professional resource for reliability and design engineers, especially those in the telecommunications and electronics industries. It is also an excellent course text for advanced courses in reliability engineering.

Author Biography

BORIS GNEDENKO (deceased) was the chairman of the Probability Theory Department of Moscow State University in Russia. IGOR PAVLOV is Professor of Mathematics at Moscow Technical University in Russia. IGOR USHAKOV is Principal Engineer at QUALCOMM, Inc. in San Diego, California.

Table of Contents

Preface	xv
PART I UNIT RELIABILITY ESTIMATION
Main Knowledge of Statistics	3	(57)
Introduction	3	(8)
Reliability Indices	3	(2)
Main Tasks of mathematical Statistics	5	(2)
Sample	7	(4)
Main Distributions	11	(8)
Continuous Distributions	11	(3)
Discrete Distributions	14	(3)
Special Distributions	17	(2)
Point Estimation	19	(13)
Introduction	19	(1)
Properties of Estimators	19	(6)
Methods of Estimation	25	(5)
Sufficient Statistics	30	(2)
Confidence Intervals	32	(12)
Introduction	32	(2)
Construction of Confidence Intervals	34	(4)
Confidence Estimation of Exponential distribution	38	(1)
Normal Distribution, Known Variance ***2	39	(2)
Approximation Based on Levy-Lindeberg Theorem	41	(1)
Clopper-Pearson Confidence Intervals	41	(1)
Approximation for Binomial Distribution	42	(1)
Confidence Interval for Parameter of Poisson Distribution	43	(1)
Test of Hypothesis	44	(16)
Introduction	44	(1)
Two Simple Hypotheses	45	(1)
Neymann-Pearson Criterion	46	(5)
Sample Size	51	(2)
Composit Parametric Hypotheses	53	(4)
Problems	57	(3)
Plans of Tests with a single Censorship	60	(40)
Introduction	60	(3)
Test Without Replacement	60	(2)
Test Plans with Renewable Units	62	(1)
Exponential Distribution	63	(10)
Test Plan [N U r]	63	(3)
Test Plan [N U T]	66	(2)
Test Plan [N U (r, T)]	68	(1)
Test Plan [N R r]	69	(1)
Test Plan [N R T]	70	(2)
Plan [N R (r, T)]	72	(1)
Exponential Methods for IFR Distributions	73	(9)
Plan [N U r]	73	(6)
Test Plan [N R T]	79	(3)
Estimation of Unit Reliability Indices for General Parametric Case	82	(10)
Point Estimate of Reliability Indices	83	(2)
Confident Limits for Reliability Indices	85	(7)
Nonparametric Confidence Limits for Distribution Function	92	(1)
Bootstrap Method of Approximate Confidence Limits	93	(7)
Problems	98	(2)
Censored Samples	100	(45)
lndroduction	100	(1)
Independent Random Censorship	100	(3)
Markov Model of Censored Testing Without Renewal	103	(1)
Nonparametric Estimates of Reliability Function	104	(12)
Kaplan-Meier Estimate	104	(5)
Discrete Scheme of Testing	109	(3)
Unbiased Estimator	112	(1)
Nonparametric Estimator of Resource Function	113	(3)
Asymptotic Confidence Limits	116	(2)
Accurate Confidence Limits	118	(4)
More General Models of Censored Tests: Markov Model with Renewal	122	(3)
Appendices	125	(20)
Markov Moments of Time	125	(1)
Markov Censored Test Model Without Renewal	126	(1)
Unbiasedness of Estimate (3.12)	127	(4)
Proof of Formula (3.18)	131	(1)
Nonparametric Confidence Limits for Test Markov Model [MMU]	132	(7)
Consistency of Confidence Limits (3.24) and (3.25)	139	(3)
Markov Test Model [MMR] with Censorship and Renewal	142	(1)
Problems	143	(2)
Bayes Methods of Reliability Estimation	145	(12)
Introduction	145	(1)
Point Estimates and Confidence Limits	146	(2)
Bayes Point Estimates	146	(1)
Sufficient Statistics	147	(1)
Bayes Confidence Limits	148	(1)
Binomial Model	148	(3)
Exponential Model	151	(3)
General Parametric Model	154	(3)
Problems	156	(1)
Accelerated Testing	157	(22)
Introduction	157	(1)
Basic Notions and Probabilistic Model	158	(8)
ITF Distributions and Accelerated-Life Reliability Model	158	(1)
Stress Severity in Terms of TTF Distribution	158	(1)
Time Transformation Function for the Case of Constant Stress	159	(1)
Accelerated-Life Model (Linear Time Transformation Function)	159	(1)
Cumulative-Damage Models and Accelerated-Life Model	160	(1)
Proportional-Hazards Model	161	(1)
Some Popular AL (Reliability) Models for Percentiles	162	(1)
Accelerated-Life Model for Time-Dependent Stress	163	(1)
AL Reliability Model for Time-Dependent Stress and Miner's Rule	164	(2)
Accelerated-Life Test Data Analysis	166	(13)
Exploratory Data Analysis (Criteria of Linearity of Time Transformation Function for Constant Stress)	166	(2)
Statistical Methods of Reliability Prediction on the Basis of AL Tests with Constant stress	168	(8)
Exploratory Data Analysis for Time-Dependent Stress	176	(1)
Statistical Estimation of AL Reliability Models on the BASIS of AL Test with Time-Dependent Stress	176	(3)
PART II SYSTEM RELIABILITY ESTIMATION
Testing with no Failures	179	(29)
Introduction	181	(1)
Series System	181	(3)
General Expression for Best Lower Confidence Limit	184	(2)
Systems with Monotone Structure	184	(2)
Best Lower Confidence Limit for No-Failure Test	186	(1)
Structures with Convex Cumulative Hazard Function	186	(4)
Series Connection of Groups of Identical Units	189	(1)
Series-Parallel System with Identical Redundant Units	189	(1)
Series Connection of K our of N subsystems	190	(1)
Series-Parallel Structure with Different Redundant Units	190	(6)
Parallel Connection	192	(2)
Parallel-Series System	194	(2)
Systems with Complex Structure (Identical Tests)	196	(5)
Computation of Confidence Limit for System bases on a Known Limit for Another System	199	(2)
Appendix	201	(7)
Confidence Clopper-Pearson Limits for Parameter of Binomial Distribution	201	(1)
Best Lower Confidence Limit for PFFO in Case of No-Failure Test	202	(1)
Maximum of Convex Function on Convex Set	203	(2)
Proof of Theorem 6.1	205	(1)
Proof of Theorem 6.2	205	(1)
Problems	205	(3)
System Confidence Limits Based on Unit Test Results	208	(74)
Introduction	208	(2)
Practical Applications	208	(1)
Formulation of the Problem	209	(1)
Calculation by Direct Substitution	210	(12)
Point Estimates for Units	210	(1)
Confidence Limits	211	(1)
General Method	212	(4)
Arbitrary Distribution of Statistic S	216	(6)
Series Structures	222	(9)
Binomial Model	222	(1)
Lidstrem-Madden Method	223	(8)
Parallel Structures	231	(6)
System of Different Units	231	(3)
System with Replicated Units	234	(3)
Series-Parallel System	237	(12)
Superreliable System	237	(3)
Method of Hyperplane	240	(3)
Method of Hypercube	243	(1)
Method of Truncated Hypercube	244	(2)
Modified Hyperplane Method	246	(3)
Approximate Methods	249	(15)
Method of Equivalent Tests	249	(5)
Method of Reduction	254	(1)
Method of Reduction for Complex Systems	255	(2)
Method of Fiducial Probabilities	257	(4)
Bootstrap Method	261	(1)
Normal Approximation	261	(3)
Methods of the Use of basic Structures	264	(11)
Basic Ensemble of Trivial Algorithms	265	(3)
Constructing New Algorithms from the Basis Collection of Trivial Algorithms	268	(3)
Constructing the Optimal Algorithm	271	(4)
Bayes Method	275	(3)
Appendix	278	(4)
Derivation of Formulas (7.26) and (7.27) for Confidence Limits in the Arbitrary Distribution of Test Results x and Statistic S(x)	278	(2)
Computation of Confidence Limits for Binomial Testing Plan	280	(1)
Problems	281	(1)
Confidence Limits for Systems Consisting of Units with Exponential Distribution of Time to Failure	282	(46)
Introduction	282	(1)
Method of Replacement	283	(5)
Description of method	283	(1)
Conditions of Method Application	284	(1)
Systems with Series--Parallel Structure	285	(3)
Systems with Complex Structure	288	(6)
``Recurrent'' Structures	288	(4)
Monotone Structures	292	(2)
Systems with Repairable Units	294	(6)
System with a Series Structure	294	(2)
Series--Parallel System with Loaded Redundancy	296	(1)
General Model of Series Connection of Renewal Subsystems	297	(3)
Method of Fiducial Probabilities	300	(10)
Introduction and Method Description	300	(4)
Conditions of Method Correctness	304	(1)
Systems with Series--Parallel Structure	304	(2)
Systems with Complex Structure	306	(1)
Systems with Renewal	307	(3)
Method of Tangent Functions	310	(4)
Appendix	314	(14)
Confidence Limits for Quasi-Convex and Quasi-Concave Functions	314	(7)
Construction of Confident Limits by Substitution Method for Function of Type (8.17)	321	(2)
Proof of Quasi-Convexity of Reliability Index for Renewable Systems	323	(1)
Conditions of Correctness of Method of Fiducial Probabilities for Calculation of Confidence Limits for System PFFO	324	(4)
Sequential Criteria of Hypothesis Test and Confidence Limits for Reliability Indices	328	(76)
Introduction	328	(8)
Sequential Confidence Limits and Hypothesis Test Criteria	328	(3)
Sequential Independent and Identical Test	331	(5)
Sequential Wald Criterion	336	(20)
Introduction	336	(8)
Standard Sequential Wald's Plan of Reliability Index Control	344	(11)
Truncated Sequential Wald's Plans	355	(1)
Sequential Test of Composite Hypotheses for Multidimensional Parameters	356	(5)
Sequential Rules of Composite Hypothesis Exclusion	357	(2)
Sequential Criterion for Composite Hypothesis Test	359	(2)
Sequential Confience Limits	361	(6)
Construction of Sequential Confidence Limits	361	(4)
Test Reliability Hypotheses with Sequential Confidence Limits	365	(1)
Scheme of Dependent Tests	366	(1)
Sequential Confidence Limits and Tests of Hypotheses for Systems on the Base of Unit Test	367	(20)
Binomial Model	367	(7)
General Parametric Model	374	(2)
Markov Model of Tests with Censorship and Unit Renewal	376	(5)
Sequential Confidence Limits for Availability Coefficient of Renewable Unit	381	(6)
Appendix	387	(17)
Computation of Accurate Values of Characteristics of Sequential Test Plans	387	(4)
Wald's Equivalency	391	(1)
Sequential Optimal Test Criterion for Composite Hypotheses	392	(9)
Scheme of Dependent Tests	401	(3)
Monte Carlo Simulation	404	(21)
Nature and Purpose	404	(1)
General Description of a Discrete Simulation Model	405	(1)
Detailed Example of Algorithmic Description	406	(4)
Random Numbers	410	(2)
Sample Computer Simulation	412	(1)
Modeling Network Reliability	413	(12)
Simulating with Fixed Failed States	418	(2)
Modeling Link Failures Until System Failure	420	(2)
Accelerated Simulation	422	(3)
Monte Carlo Simulation for Optimal Redundancy	425	(28)
Introductory Remarks	425	(1)
Formulation of Optimal Redundancy Problems	426	(1)
Algorithm for Trajectory Generation	427	(2)
Description of the idea of the Solution	429	(2)
Solution of Direct Optimization Problem	431	(7)
Inverse Optimization Problem	438	(5)
Appendix	443	(10)
Consistency and Asymptotic Unbiasedness of solution to Direct Problem	443	(5)
Consistency of Inverse Problem of Optimal Redundancy	448	(2)
Restrictions on System MTBF	450	(3)
Appendix	453	(6)
Bibliography	459	(10)
Solutions to Problems	469	(28)
Index	497