Power law model - Goodness-of-fit tests and estimation methods

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

Potenzgesetz-Modell - Anpassungstests und Schätzverfahren

Modèle de loi en puissance - Essais d'adéquation et méthodes d'estimation des paramètres

La CEI 61710:2013 spécifie les procédures pour l'estimation des paramètres du modèle de loi en puissance en fournissant les intervalles de confiance pour l'intensité de défaillance, les intervalles de prédiction pour les défaillances futures et pour déterminer l'adéquation du modèle de loi en puissance avec les données relatives aux entités réparables. L'hypothèse prise est que les données de durées avant défaillance ont été collectées à partir d'une ou de plusieurs entités identiques fonctionnant dans les mêmes conditions (par exemple d'environnement et de charge). Cette deuxième édition annule et remplace la première édition parue en 2000. Cette édition constitue une révision technique. Les modifications principales par rapport à l'édition précédente sont les suivantes: Introduction d'une Annexe supplémentaire C traitant de l'estimation bayésienne pour le modèle de loi en puissance. Mots clés: modèle de loi en puissance, l'estimation bayésienne, fiabilité des entités réparables

Model eksponentnega pravila - Preskusi ujemanja in metode vrednotenja (IEC 61710:2013)

Ta mednarodni standard določa postopke za vrednotenje parametrov modela eksponentnega pravila, ugotavljanje intervalov zaupanja za intenzivnost odpovedi, ugotavljanje napovednih intervalov za čas delovanja do prihodnje odpovedi in preskušanje ujemanja modela eksponentnega pravila s podatki na podlagi popravljivih elementov. Predvideva se, da so bili podatki o času delovanja do odpovedi zbrani na podlagi dela ali drugih identičnih delov, ki delujejo v istih pogojih (npr. okolje in obremenitev).

General Information

Status
Published
Publication Date
05-Sep-2013
Technical Committee
Drafting Committee
Current Stage
6060 - Document made available
Due Date
06-Sep-2013
Completion Date
06-Sep-2013

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2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.Model eksponentnega pravila - Preskusi ujemanja in metode vrednotenja (IEC 61710:2013)Modèle de loi en puissance - Essai d'adéquation et méthodes d'estimation des paramètresPower law model - Goodness-of-fit tests and estimation methods29.020Elektrotehnika na splošnoElectrical engineering in general03.120.30Application of statistical methodsICS:Ta slovenski standard je istoveten z:EN 61710:2013SIST EN 61710:2014en01-maj-2014SIST EN 61710:2014SLOVENSKI

STANDARD
EUROPEAN STANDARD EN 61710 NORME EUROPÉENNE
EUROPÄISCHE NORM September 2013

CENELEC European Committee for Electrotechnical Standardization Comité Européen de Normalisation Electrotechnique Europäisches Komitee für Elektrotechnische Normung

CEN-CENELEC Management Centre: Avenue Marnix 17, B - 1000 Brussels
© 2013 CENELEC -

All rights of exploitation in any form and by any means reserved worldwide for CENELEC members.

Ref. No. EN 61710:2013 E
ICS 03.120.01; 03.120.30
English version
Power law model -
Goodness-of-fit tests and estimation methods (IEC 61710:2013)
Modèle de loi en puissance -
Essais d'adéquation et méthodes d'estimation des paramètres (CEI 61710:2013)
Potenzgesetz-Modell -
Anpassungstests und Schätzverfahren (IEC 61710:2013)

This European Standard was approved by CENELEC on 2013-06-26. CENELEC members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration.

Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CENELEC member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CENELEC member into its own language and notified to the CEN-CENELEC Management Centre has the same status as the official versions.

CENELEC members are the national electrotechnical committees of Austria, Belgium, Bulgaria, Croatia, Cyprus, the Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the United Kingdom.

SIST EN 61710:2014
EN 61710:2013 - 2 -

Foreword The text of document 56/1500/FDIS, future edition 2 of IEC 61710, prepared by IEC/TC 56 "Dependability" was submitted to the IEC-CENELEC parallel vote and approved by CENELEC as EN 61710:2013.

The following dates are fixed: • latest date by which the document has to be implemented at national level by publication of an identical national standard or by endorsement (dop) 2014-03-26 • latest date by which the national standards conflicting with the document have to be withdrawn (dow) 2016-06-26

Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. CENELEC [and/or CEN] shall not be held responsible for identifying any or all such patent rights.

Endorsement notice The text of the International Standard IEC 61710:2013 was approved by CENELEC as a European Standard without any modification. In the official version, for Bibliography, the following notes have to be added for the standards indicated:

IEC 61703 NOTE Harmonised as EN 61703. IEC 61164:2004 NOTE Harmonised as EN 61164:2004 (not modified). SIST EN 61710:2014

- 3 - EN 61710:2013
Annex ZA
(normative)

Normative references to international publications with their corresponding European publications

The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.

NOTE

When an international publication has been modified by common modifications, indicated by (mod), the relevant EN/HD applies.

Publication Year Title EN/HD Year
IEC 60050-191 1990 International Electrotechnical Vocabulary (IEV) -
Chapter 191: Dependability and quality of service - -
SIST EN 61710:2014
SIST EN 61710:2014

IEC 61710 Edition 2.0 2013-05 INTERNATIONAL STANDARD NORME INTERNATIONALE Power law model – Goodness-of-fit tests and estimation methods

Modèle de loi en puissance – Essais d'adéquation et méthodes d'estimation
des paramètres

INTERNATIONAL ELECTROTECHNICAL COMMISSION COMMISSION ELECTROTECHNIQUE INTERNATIONALE XA ICS 03.120.01; 03.120.30 PRICE CODE CODE PRIX ISBN 978-2-83220-797-0

® Registered trademark of the International Electrotechnical Commission
Marque déposée de la Commission Electrotechnique Internationale ®

Warning! Make sure that you obtained this publication from an authorized distributor.

Attention! Veuillez vous assurer que vous avez obtenu cette publication via un distributeur agréé. SIST EN 61710:2014 colourinside

– 2 – 61710 © IEC:2013

CONTENTS FOREWORD ........................................................................................................................... 5 INTRODUCTION ..................................................................................................................... 7 1 Scope ............................................................................................................................... 8 2 Normative references ....................................................................................................... 8 3 Terms and definitions ....................................................................................................... 8 4 Symbols and abbreviations ............................................................................................... 8 5 Power law model .............................................................................................................. 9 6 Data requirements .......................................................................................................... 10 6.1 General ................................................................................................................. 10 6.1.1 Case 1 – Time data for every relevant failure for one or more copies

from the same population .......................................................................... 10 6.1.2 Case 1a) – One repairable item ................................................................. 10 6.1.3 Case 1b) – Multiple items of the same kind of repairable item observed for the same length of time ......................................................... 11 6.1.4 Case 1c) – Multiple repairable items of the same kind observed for different lengths of time ............................................................................. 11 6.2 Case 2 – Time data for groups of relevant failures for one or more repairable items from the same population ............................................................................. 12 6.3 Case 3 – Time data for every relevant failure for more than one repairable item from different populations .............................................................................. 12 7 Statistical estimation and test procedures ....................................................................... 13 7.1 Overview ............................................................................................................... 13 7.2 Point estimation .................................................................................................... 13 7.2.1 Case 1a) and 1b) – Time data for every relevant failure ............................. 13 7.2.2 Case 1c) – Time data for every relevant failure .......................................... 14 7.2.3 Case 2 – Time data for groups of relevant failures ..................................... 15 7.3 Goodness-of-fit tests ............................................................................................. 16 7.3.1 Case 1 – Time data for every relevant failure............................................. 16 7.3.2 Case 2 – Time data for groups of relevant failures ..................................... 17 7.4 Confidence intervals for the shape parameter ........................................................ 18 7.4.1 Case 1 – Time data for every relevant failure............................................. 18 7.4.2 Case 2 – Time data for groups of relevant failures ..................................... 19 7.5 Confidence intervals for the failure intensity .......................................................... 20 7.5.1 Case 1 – Time data for every relevant failure............................................. 20 7.5.2 Case 2 – Time data for groups of relevant failures ..................................... 20 7.6 Prediction intervals for the length of time to future failures of a single item ............ 21 7.6.1 Prediction interval for length of time to next failure for case 1 –

Time data for every relevant failure .................................................................... 21 7.6.2 Prediction interval for length of time to Rth future failure for case 1 –

Time data for every relevant failure ........................................................... 22 7.7 Test for the equality of the shape parameters kβββ ..., ,,21 .................................... 23 7.7.1 Case 3 – Time data for every relevant failure for two items from different populations .................................................................................. 23 7.7.2 Case 3 – Time data for every relevant failure for three or more items from different populations .......................................................................... 24 Annex A (informative)

The power law model – Background information ................................ 30 Annex B (informative)

Numerical examples .......................................................................... 31 SIST EN 61710:2014

61710 © IEC:2013 – 3 –
Annex C (informative)

Bayesian estimation for the power law model .................................... 41 Bibliography .......................................................................................................................... 56

Figure 1 – One repairable item ............................................................................................. 10 Figure 2 – Multiple items of the same kind of repairable item

observed for same length of time .................................................................................................................................. 11 Figure 3 – Multiple repairable items of the same kind observed for different lengths of time ...................................................................................................................................... 12 Figure B.1 – Accumulated number of failures against accumulated time for software system .................................................................................................................................. 32 Figure B.2 – Expected against observed accumulated times to failure for software system .................................................................................................................................. 32 Figure B.3 – Accumulated number of failures against accumulated time for five copies of a system ........................................................................................................................... 35 Figure B.4 – Accumulated number of failures against accumulated time

for an OEM product from vendors A and B ............................................................................................... 37 Figure B.5 – Accumulated number of failures against time for generators ............................. 38 Figure B.6 – Expected against observed accumulated

number of failures for generators ............................................................................................................................ 39 Figure C.1 – Plot of fitted Gamma prior (6,7956, 0,0448) ...................................................... 47 for the shape parameter of the power law model ................................................................... 47 Figure C.2 – Plot of fitted Gamma prior (17,756 6, 1447,408)

for the expected number of failures parameter of the power law model ........................................................................ 47 Figure C.3 – Subjective distribution of number of failures...................................................... 51 Figure C.4 – Plot of the posterior probability distribution

for the number of future failures, M ............................................................................................................................. 54 Figure C.5 – Plot of the posterior cumulative distribution

for the number of future failures, M ............................................................................................................................. 55

Table 1 – Critical values for Cramer-von-Mises goodness-of-fit test at 10 % level of significance........................................................................................................................... 25 Table 2 – Fractiles of the Chi-square distribution .................................................................. 26 Table 3 – Multipliers for two-sided 90 % confidence intervals for intensity function for time terminated data ............................................................................................................. 27 Table 4 – Multipliers for two-sided 90 % confidence intervals for intensity function for failure terminated data .......................................................................................................... 28 Table 5 – 0,95 fractiles of the F distribution .......................................................................... 29 Table B.1 – All relevant failures and accumulated times for software system ........................ 31 Table B.2 – Calculation of expected accumulated times to failure for Figure B.2 ................... 33 Table B.3 – Accumulated times for all relevant failures for five copies of a system (labelled A, B, C, D, E) ......................................................................................................... 34 Table B.4 – Combined accumulated times

for multiple items of the same kind of a system .................................................................................................................................. 34 Table B.5 – Accumulated operating hours to failure for OEM product from vendors A and B .................................................................................................................................... 36 Table B.6 – Grouped failure data for generators ................................................................... 38 Table B.7 – Calculation of expected numbers of failures for Figure B.6 ................................. 40 Table C.1 – Strengths and weakness of classical and Bayesian estimation ........................... 42 SIST EN 61710:2014

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Table C.2 – Grid for eliciting subjective distribution for shape parameter β ........................... 46 Table C.3 – Grid for eliciting subjective distribution

for expected number of failures parameter η .......................................................................................................................... 46 Table C.4 – Comparison of fitted Gamma and subjective distribution for shape parameterβ .......................................................................................................................... 48 Table C.5 – Comparison of fitted Gamma and subjective distribution for expected number of failures by time 20 000 hT= parameterη .......................................................... 48 Table C.6 – Times to failure data collected on system test .................................................... 49 Table C.7 – Summary of estimates of power law model parameters ...................................... 50 Table C.8 – Time to failure data for operational system ........................................................ 53

SIST EN 61710:2014
61710 © IEC:2013 – 5 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION ____________
POWER LAW MODEL –
GOODNESS-OF-FIT TESTS
AND ESTIMATION METHODS

FOREWORD 1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising all national electrotechnical committees (IEC National Committees). The object of IEC is to promote international co-operation on all questions concerning standardization in the electrical and electronic fields. To this end and in addition to other activities, IEC publishes International Standards, Technical Specifications, Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “IEC Publication(s)”). Their preparation is entrusted to technical committees; any IEC National Committee interested in the subject dealt with may participate in this preparatory work. International, governmental and non-governmental organizations liaising with the IEC also participate in this preparation. IEC collaborates closely with the International Organization for Standardization (ISO) in accordance with conditions determined by agreement between the two organizations. 2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international consensus of opinion on the relevant subjects since each technical committee has representation from all interested IEC National Committees.

3) IEC Publications have the form of recommendations for international use and are accepted by IEC National Committees in that sense. While all reasonable efforts are made to ensure that the technical content of IEC Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any misinterpretation by any end user. 4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications transparently to the maximum extent possible in their national and regional publications. Any divergence between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in the latter. 5) IEC itself does not provide any attestation of conformity. Independent certification bodies provide conformity assessment services and, in some areas, access to IEC marks of conformity. IEC is not responsible for any services carried out by independent certification bodies. 6) All users should ensure that they have the latest edition of this publication. 7) No liability shall attach to IEC or its directors, employees, servants or agents including individual experts and members of its technical committees and IEC National Committees for any personal injury, property damage or other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC Publications.

8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is indispensable for the correct application of this publication. 9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of patent rights. IEC shall not be held responsible for identifying any or all such patent rights. International Standard IEC 61710 has been prepared by IEC technical committee 56: Dependability. 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. The text of this standard is based on the following documents: FDIS Report on voting 56/1500/FDIS 56/1508/RVD

Full information on the voting for the approval of this standard can be found in the report on voting indicated in the above table. SIST EN 61710:2014

– 6 – 61710 © IEC:2013

This publication has been drafted in accordance with the ISO/IEC Directives, Part 2. The committee has decided that the contents of this publication will remain unchanged until the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data related to the specific publication. At this date, the publication will be

• reconfirmed, • withdrawn, • replaced by a revised edition, or • amended.

IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates that it contains colours which are considered to be useful for the correct understanding of its contents. Users should therefore print this document using a colour printer.

SIST EN 61710:2014
61710 © IEC:2013 – 7 –

INTRODUCTION This International Standard describes the power law model and gives step-by-step directions for its use. There are various models for describing the reliability of repairable items, the power law model being one of the most widely used. This standard provides procedures to estimate the parameters of the power law model and to test the goodness-of-fit of the power law model to data, to provide confidence intervals for the failure intensity and prediction intervals for the length of time to future failures. An input is required consisting of a data set of times at which relevant failures occurred, or were observed, for a repairable item or a set of copies of the same item, and the time at which observation of the item was terminated, if different from the time of final failure. All output results correspond to the item type under consideration. Some of the procedures can require computer programs, but these are not unduly complex. This standard presents algorithms from which computer programs should be easy to construct. SIST EN 61710:2014

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POWER LAW MODEL –
GOODNESS-OF-FIT TESTS
AND ESTIMATION METHODS

1 Scope This International Standard 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). 2 Normative references The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. IEC 60050-191:1990, International Electrotechnical Vocabulary (IEV) – Chapter 191: Dependability and quality of service 3 Terms and definitions For the purposes of this document, the terms and definitions of IEC 60050-191 apply. 4 Symbols and abbreviations The following symbols and abbreviations apply: β shape parameter of the power law model βˆ estimated shape parameter of the power law model UBLBββ, lower, upper confidence limits for β 2C Cramer-von-Mises goodness-of-fit test statistic ()MC21γ− critical value for the Cramer-von-Mises goodness-of-fit test statistic at γ level of significance 2χ Chi-square goodness-of-fit test statistic ()υχγ2 γth fractile of the 2χ distribution with υ degrees of freedom d number of intervals for groups of failures ()[]tNE expected accumulated number of failures up to time t []jtE expected accumulated time to jth failure

SIST EN 61710:2014
61710 © IEC:2013 – 9 –

()[][]itNEˆ estimated expected accumulated number of failures up to ()it []jtEˆ estimated expected accumulated time to jth failure ()21,ννγF γth fractile for the F distribution with ()21,νν degrees of freedom i general purpose indicator j general purpose indicator k number of items L, U multipliers used in calculation of confidence intervals for failure intensity λ scale parameter of the power law model λ estimated scale parameter of the power law model M parameter for Cramer-von-Mises statistical test N number of relevant failures jN number of failures for jth item ()tN accumulated number of failures up to time t ()[]itN accumulated number of failures up to time ()it R difference between the order number of future (predicted) failure and order number of last (observed) failure T accumulated relevant time *T total accumulated relevant time for time terminated test jT total accumulated relevant time for jth item RURLTT, lower, upper prediction limits for the length of time to the Rth future failure TN+1 estimated median time to (N+1)th failure it accumulated relevant time to the ith failure jit ith failure time for jth item tN total accumulated relevant time for failure terminated test jNt total accumulated relevant time to Nth failure of jth item ()()itit,1− endpoints of ith interval of time for grouped failures ()tz failure intensity at time t ()tzˆ estimated failure intensity at time t UBLBzz, lower, upper confidence limits for failure intensity 5 Power law model The statistical procedures for the power law model use the relevant failure and time data from the test or field studies. The basic equations for the power law model are given in this clause. Background information on the model is given in Annex A and examples of its application are given in Annex B. The expected accumulated number of failures up to test time t is given by: ()[]βλttNE= with 0,0,0[[[tβλ SIST EN 61710:2014

– 10 – 61710 © IEC:2013
where λ

is the scale parameter; β is the shape parameter (10<<β corresponds to a decreasing failure intensity; 1=β corresponds to a constant failure intensity; 1[β corresponds to an increasing failure intensity). The failure intensity at time t is given by: ()[]1)(−==βλβttNEdtdtz with 0[t Thus the parameters λ and β both affect the failure intensity in a given time. Methods are given in 7.2 for maximum likelihood estimation of the parameters of λ and β. Subclause 7.3 gives goodness-of-fit tests for the model and 7.4 and 7.5 give confidence interval procedures. Subclause 7.6 gives prediction interval procedures and 7.7 gives tests for the equality of the shape parameters. The model is simple to evaluate. However when 1<β, theoretically ()∞=0z (i.e. )(tz tends to infinity as t tends to zero) and 0)(=∞z (i.e. )(tz tends to zero as t tends to infinity); but this theoretical limitation does not generally affect its practical use. 6 Data requirements 6.1 General 6.1.1 Case 1 – Time data for every relevant failure for one or more copies

from the same population The normal evaluation methods assume the observed times to be exact times of failure of a single repairable item or a set of copies of the same repairable item. The figures below illustrate how the failure times are calculated for three general cases. 6.1.2 Case 1a) – One repairable item For one repairable item observed from time 0 to time T, the relevant failure time, it, is the elapsed operating time (that is, excluding repair and other down times) until the occurrence of the i-th failure as shown in Figure 1.

Key A operating time, B down time Figure 1 – One repairable item Time terminated data are observed to *T, which is not a failure time, and failure terminated data are observed to Nt, which is the time of the Nth failure. Time terminated and failure terminated data use slightly different formulae. B A 0 1 2 3 T B IEC

996/13 SIST EN 61710:2014
61710 © IEC:2013 – 11 –

6.1.3 Case 1b) – Multiple items of the same kind of repairable item observed for the same length of time It is assumed there are k items, which all represent the same population. That is, they are nominally identical items operating under the same conditions (e.g. environment and load). When all items are observed to time *T, which is not a failure time (i.e. time terminated data), then the failure time data are combined by superimposing failure times ()Niti...,2,1,= for all k items on the same time line as shown in Figure 2.

Key A item 1 B item 2 C item k D superimposed process Figure 2 – Multiple items of the same kind of repairable item

observed for same length of time 6.1.4 Case 1c) – Multiple repairable items of the same kind observed for different lengths of time When all items do not operate for the same period of time, then the time at which observation of the jth item is terminated ()kjTj,...,2,1=, where kTTT<<<...21,is noted. The failure data are combined by superimposing all the failure times for all k items on the same time line as shown in Figure 3. The times to failure are Niti,...,2,1,=, where N = the total number of failures observed accumulated over the k items. D T* 0 C B A t 1 t 2 t 3 t N-1 t N IEC

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Key A item 1 B item 2 C item 3 D item k t time Figure 3 – Multiple repairable items of the same kind observed for different lengths of time If each item is a software system then the repair action should be done to the other systems which did not fail at that time. 6.2 Case 2 – Time data for groups of relevant failures for one or more repairable items from the same population This alternative method is used when there is at least one copy of an item and the data consist of known time intervals, each containing a known number of failures. The observation period is over the interval ),0(T and is partitioned into d intervals at times )(...)2()1(0dttt<<<<. The ith interval is the time period between )1(−it and )(it, where ()1,2,...,,00 idt==and ()tdT=. It is important to note

...

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