IEC 61710:2013 specifies procedures to estimate the parameters of the power law model, to provide confidence intervals for the failure intensity, to provide prediction intervals for the times to future failures, and to test the goodness-of-fit of the power law model to data from repairable items. It is assumed that the time to failure data have been collected from an item, or some identical items operating under the same conditions (e.g. environment and load). This second edition cancels and replaces the first edition, published in 2000, and constitutes a technical revision. The main changes with respect to the previous edition are listed below: the inclusion of an additional Annex C on Bayesian estimation for the power law model. Keywords: power law model, Bayesian estimation, reliability of repairable items<\/p>\n
| PDF Pages<\/th>\n | PDF Title<\/th>\n<\/tr>\n | ||||||
|---|---|---|---|---|---|---|---|
| 6<\/td>\n | English CONTENTS <\/td>\n<\/tr>\n | ||||||
| 9<\/td>\n | INTRODUCTION <\/td>\n<\/tr>\n | ||||||
| 10<\/td>\n | 1 Scope 2 Normative references 3 Terms and definitions 4 Symbols and abbreviations <\/td>\n<\/tr>\n | ||||||
| 11<\/td>\n | 5 Power law model <\/td>\n<\/tr>\n | ||||||
| 12<\/td>\n | 6 Data requirements 6.1 General 6.1.1 Case 1 \u2013 Time data for every relevant failure for one or more copies from the same population 6.1.2 Case 1a) \u2013 One repairable item Figures Figure 1 \u2013 One repairable item <\/td>\n<\/tr>\n | ||||||
| 13<\/td>\n | 6.1.3 Case 1b) \u2013 Multiple items of the same kind of repairable item observed for the same length of time 6.1.4 Case 1c) \u2013 Multiple repairable items of the same kind observed for different lengths of time Figure 2 \u2013 Multiple items of the same kind of repairable item observed for same length of time <\/td>\n<\/tr>\n | ||||||
| 14<\/td>\n | 6.2 Case 2 \u2013 Time data for groups of relevant failures for one or more repairable items from the same population 6.3 Case 3 \u2013 Time data for every relevant failure for more than one repairable item from different populations Figure 3 \u2013 Multiple repairable items of the same kind observedfor different lengths of time <\/td>\n<\/tr>\n | ||||||
| 15<\/td>\n | 7 Statistical estimation and test procedures 7.1 Overview 7.2 Point estimation 7.2.1 Case 1a) and 1b) \u2013 Time data for every relevant failure <\/td>\n<\/tr>\n | ||||||
| 16<\/td>\n | 7.2.2 Case 1c) \u2013 Time data for every relevant failure <\/td>\n<\/tr>\n | ||||||
| 17<\/td>\n | 7.2.3 Case 2 \u2013 Time data for groups of relevant failures <\/td>\n<\/tr>\n | ||||||
| 18<\/td>\n | 7.3 Goodness-of-fit tests 7.3.1 Case 1 \u2013 Time data for every relevant failure <\/td>\n<\/tr>\n | ||||||
| 19<\/td>\n | 7.3.2 Case 2 \u2013 Time data for groups of relevant failures <\/td>\n<\/tr>\n | ||||||
| 20<\/td>\n | 7.4 Confidence intervals for the shape parameter 7.4.1 Case 1 \u2013 Time data for every relevant failure <\/td>\n<\/tr>\n | ||||||
| 21<\/td>\n | 7.4.2 Case 2 \u2013 Time data for groups of relevant failures <\/td>\n<\/tr>\n | ||||||
| 22<\/td>\n | 7.5 Confidence intervals for the failure intensity 7.5.1 Case 1 \u2013 Time data for every relevant failure 7.5.2 Case 2 \u2013 Time data for groups of relevant failures <\/td>\n<\/tr>\n | ||||||
| 23<\/td>\n | 7.6 Prediction intervals for the length of time to future failures of a single item 7.6.1 Prediction interval for length of time to next failure for case 1 \u2013 Time data for every relevant failure <\/td>\n<\/tr>\n | ||||||
| 24<\/td>\n | 7.6.2 Prediction interval for length of time to Rth future failure for case 1 \u2013 Time data for every relevant failure <\/td>\n<\/tr>\n | ||||||
| 25<\/td>\n | 7.7 Test for the equality of the shape parameters \u03b21,\u03b2 2, …, \u03b2 k 7.7.1 Case 3 \u2013 Time data for every relevant failure for two items from different populations <\/td>\n<\/tr>\n | ||||||
| 26<\/td>\n | 7.7.2 Case 3 \u2013 Time data for every relevant failure for three or more items from different populations <\/td>\n<\/tr>\n | ||||||
| 27<\/td>\n | Tables Table 1 \u2013 Critical values for Cramer-von-Mises goodness-of-fit testat 10\u00a0% level of significance <\/td>\n<\/tr>\n | ||||||
| 28<\/td>\n | Table 2 \u2013 Fractiles of the Chi-square distribution <\/td>\n<\/tr>\n | ||||||
| 29<\/td>\n | Table 3 \u2013 Multipliers for two-sided 90 % confidence intervals for intensity function for time terminated data <\/td>\n<\/tr>\n | ||||||
| 30<\/td>\n | Table 4 \u2013 Multipliers for two-sided 90 % confidence intervals for intensity function for failure terminated data <\/td>\n<\/tr>\n | ||||||
| 31<\/td>\n | Table 5 \u2013 0,95 fractiles of the F distribution <\/td>\n<\/tr>\n | ||||||
| 32<\/td>\n | Annex A (informative) The power law model \u2013 Background information <\/td>\n<\/tr>\n | ||||||
| 33<\/td>\n | Annex B (informative) Numerical examples Table B.1 \u2013 All relevant failures and accumulated times for software system <\/td>\n<\/tr>\n | ||||||
| 34<\/td>\n | Figure B.1 \u2013 Accumulated number of failures against accumulated timefor software system Figure B.2 \u2013 Expected against observed accumulated times to failurefor software system <\/td>\n<\/tr>\n | ||||||
| 35<\/td>\n | Table B.2 \u2013 Calculation of expected accumulated times to failure for Figure B.2 <\/td>\n<\/tr>\n | ||||||
| 36<\/td>\n | Table B.3 \u2013 Accumulated times for all relevant failuresfor five copies of a system (labelled A, B, C, D, E) Table B.4 \u2013 Combined accumulated times for multiple items of the same kind of a system <\/td>\n<\/tr>\n | ||||||
| 37<\/td>\n | Figure B.3 \u2013 Accumulated number of failures against accumulated timefor five copies of a system <\/td>\n<\/tr>\n | ||||||
| 38<\/td>\n | Table B.5 \u2013 Accumulated operating hours to failure for OEM product from vendors A and B <\/td>\n<\/tr>\n | ||||||
| 39<\/td>\n | Figure B.4 \u2013 Accumulated number of failures against accumulated time for an OEM product from vendors A and B <\/td>\n<\/tr>\n | ||||||
| 40<\/td>\n | Figure B.5 \u2013 Accumulated number of failures against time for generators Table B.6 \u2013 Grouped failure data for generators <\/td>\n<\/tr>\n | ||||||
| 41<\/td>\n | Figure B.6 \u2013 Expected against observed accumulated number of failures for generators <\/td>\n<\/tr>\n | ||||||
| 42<\/td>\n | Table B.7 \u2013 Calculation of expected numbers of failures for Figure B.6 <\/td>\n<\/tr>\n | ||||||
| 43<\/td>\n | Annex C (informative) Bayesian estimation for the power law model <\/td>\n<\/tr>\n | ||||||
| 44<\/td>\n | Table C.1 \u2013 Strengths and weakness of classical and Bayesian estimation <\/td>\n<\/tr>\n | ||||||
| 48<\/td>\n | Table C.2 \u2013 Grid for eliciting subjective distribution for shape parameter \u03b2 Table C.3 \u2013 Grid for eliciting subjective distribution for expected number of failures parameter \u03b7 <\/td>\n<\/tr>\n | ||||||
| 49<\/td>\n | Figure C.1 \u2013 Plot of fitted Gamma prior (6,7956, 0,0448) for the shape parameter of the power law model Figure C.2 \u2013 Plot of fitted Gamma prior (17,756 6, 1447,408) for the expected number of failures parameter of the power law model <\/td>\n<\/tr>\n | ||||||
| 50<\/td>\n | Table C.4 \u2013 Comparison of fitted Gamma and subjective distributionfor shape parameter \u03b2 Table C.5 \u2013 Comparison of fitted Gamma and subjective distribution for expected number of failures by time T = 20 000 h parameter \u03b7 <\/td>\n<\/tr>\n | ||||||
| 51<\/td>\n | Table C.6 \u2013 Times to failure data collected on system test <\/td>\n<\/tr>\n | ||||||
| 52<\/td>\n | Table C.7 \u2013 Summary of estimates of power law model parameters <\/td>\n<\/tr>\n | ||||||
| 53<\/td>\n | Figure C.3 \u2013 Subjective distribution of number of failures <\/td>\n<\/tr>\n | ||||||
| 55<\/td>\n | Table C.8 \u2013 Time to failure data for operational system <\/td>\n<\/tr>\n | ||||||
| 56<\/td>\n | Figure C.4 \u2013 Plot of the posterior probability distribution for the number of future failures, M <\/td>\n<\/tr>\n | ||||||
| 57<\/td>\n | Figure C.5 \u2013 Plot of the posterior cumulative distribution for the number of future failures, M <\/td>\n<\/tr>\n | ||||||
| 58<\/td>\n | Bibliography <\/td>\n<\/tr>\n<\/table>\n","protected":false},"excerpt":{"rendered":" Power law model. Goodness-of-fit tests and estimation methods<\/b><\/p>\n |