is.15038.2011 (iec 61164)

इंटरनेटमानक Disclosure to Promote the Right To Information Whereas the Parliament of India has set out to provide a practical regime of right to information for citizens to secure access to information under the control of public authorities, in order to promote transparency and accountability in the working of every public authority, and whereas the attached publication of the Bureau of Indian Standards is of particular interest to the public, particularly disadvantaged communities and those engaged in the pursuit of education and knowledge, the attached public safety standard is made available to promote the timely dissemination of this information in an accurate manner to the public. “जान1 का अ+धकार, जी1 का अ+धकार” “प0रा1 को छोड न' 5 तरफ” “The Right to Information, The Right to Live” “Step Out From the Old to the New” Mazdoor Kisan Shakti Sangathan Jawaharlal Nehru IS 15038 (2011): Reliability Growth – Statistical Test and Estimation Methods [LITD 2: Reliability of Electronic and Electrical Components and Equipment] “!ान $ एक न' भारत का +नम-ण” Satyanarayan Gangaram Pitroda “Invent a New India Using Knowledge” “!ान एक ऐसा खजाना > जो कभी च0राया नहB जा सकता ह” है” ह Bhartṛhari—Nītiśatakam “Knowledge is such a treasure which cannot be stolen” ‘/ L___ IS 15038:2001 IEC 61164 (1995) ( Reaffirmed 2005 ) WR%?Wi’m .01.120.30. Yih-mw$ fadvl ~ W%Pi q-d m Indian Standard RELIABILITY GROWTH — STATISTICAL AND ESTIMATION METHODS TEST ICS 03. I I ‘1 .120. INDIAN STANDARDS 9 BAHADUR SHAH NEW DELHI 110002 ZAFAR MARG Price Group 10 September 2001 I i 1.020 /-”- Q BIS 2001 BUREAU MANAK OF BHAVAN. 21. 03. ) as the decimal marker. Attention is particularly drawn to the following: a) Wherever the words ‘International Standard’ appear referring to this standard.__— Reliability of Electronic and Electrical Components and Equipment Sectional Committee. certain conventions are. however. In the adopted standard. not identical to those used in Indian Standards. Only English language text has been retained while adopting this International Standard. reference appears to the following International Standard for which Indian Standard also exists.) has been used as a decimal marker while in Indian Standards. was adopted by the Bureau of Indian Standards on the recommendation of Reliability of Electronic and Electrical Components and Equipment Sectional Committee and approval of the Electronics and Telecommunication Division Council. they should be read as ‘Indian Standard’. LTD 03 NATIONAL FOREWORD This Indian Standard which is identical with IEC 61164 (1995) ‘Reliability growth — Statistical test and estimation methods’ issued by the International Electrotechnical Commission (IEC). the current practice is to use a point (. b) Comma (. CROSS REFERENCES In this adopted standard. The corresponding Indian Standard which is to be substituted in its place is listed below along with its degree of equivalence for the edition indicated: International Standard IEC 50(19 I) :1990 International Electrotechnical Vocabulary (IEV) — Chapter 191: Dependability and quality of service Corresponding Indian Standard IS 1885 (Part 39):1999 Electrotechnical vocabulary: Part 39 Reliability of electronic and electrical items (second Degree of Equivalence Not Equivalent revision) The technical committee responsible for the preparation of this standard has reviewed the provisions of the following International Standards and has decided that they are acceptable for use in conjunction with this standard : / IEC 605-1:1978 Equipment reliability testing — Part 1: General requirements IEC 605-4:1986 Equipment reliability testing — Part 4: Procedures for determining point estimates and confidence limits from equipment reliability determination tests IEC 605-6:1986 Equipment reliability testing — Part 6: Tests for the validation of a constant failure rate assumption IEC 1014:1989 Programmed for reliability growth l’! . Equipment and confidence reliability reliability limits from equipment IEC 605-6: 1986. .. At the time of publication. . Equipment IEC 605-4: 1986. and IEC 1014 &ply. Equipment reliability testing . . International quality Electrotechnical Vocabulary (IEV) .Part 6: Tests for the validity of a constant failure rate assumption IEC 1014:1989. the editions indicated were valid. 3.Part 4: Procedures determination for determining point estimates tests . constitute provisions of this International Standard.—.1 end of a test. Programmed for reliability growth 3 Definitions For the purposes of this standard the terms and definitions together with the following additional terms and definitions: delayed 3. into the system at the . estimation. All normative documents are subject to revision.A delayed A corrective modification of IEC 50(191) which is incorporated modification is not incorporated during the test.. These procedures deal with growth.- . through reference in this text.Part 1: General testing reliability testing requirements . 2 Normative references The following normative documents contain provisions which. IEC 50(191): 1990.Chapter 191: Dependability and of service IEC 605-1:1978. confidence intervals for system reliability and goodness-of-fit tests. modification: NOTE .2 improvement effectiveness factor: The fraction by which the intensity of a systematic faiIure is reduced by means of corrective modification. Members of IEC and 1S0 maintain registers of currently valid International Standards. —- 1S 15038:2001 IEC 61164(1995) . -’- Indian Standard RELIABILITY GROWTH — STATISTICAL AND ESTIMATION METHODS 1 TEST Scope This International Standard gives models and numerical methods for reliability growth assessments based on failure data from a single system which were generated in a reliability improvement programme. and parties to agreements based on this International Standard are encouraged to investigate the possibility of applying the most recent editions of the normative documents listed below. time or test with data available NOTE . NOTE – Type 11test is sometimes called failure terminated test.3 through a time which does not correspond to a failure.4 number of failures. . the following symbols apply: scale and shape parameters for the power law model critical value for hypothesis test number of intervals for grouped data analysis mean and individual improvement effectiveness factors number of distinct types of category B failures observed general purpose indices number of category A failures number of category B failures Ki number of i-th type category B failures observed. 1. _ IS 15038:2001 IEC 61164(1995) type I test: A test which is terminated at a predetermined 3. 4 Symbols For the purpose of this international standard. projected failure intensity :(T) current failure intensity at time T E)(T) current instantaneous mean time between failures (3P projected mean time between failures 2 / i .._. or test with data available through a time which corresponds to a failure. f(i) endpoints of i-th interval of test time for grouped data analysis T current accumulated relevant test time ~ accumulated relevant test time at the i-th failure TN total accumulated relevant test times for type II test F total accumulated relevant test times for type I test x.(v) y fractile of the %2distribution with v degrees of freedom z general symbol for failure intensity ‘Y y fractile of the standard normal distribution i=l ‘P . type II test: A reliability growth test which is terminated upon the accumulation of a specified 3.- 1.Type I test is sometimes called time terminated test. KB = ~ Ki M parameter of the Crarm%-vonMises test (statistical) N number of relevant failures Ni number of relevant failures in i-th interval N(T) accumulated number of failures up to test time T E[N(T)] expected accumulated number of failures up to test time T f(i–l).. . 1 and 7. The equation represents in effect the slope of a tangent to the N(T) vs. Background model is given in annex B. The basic equations for the power law model are given in this clause. and IEC 1014. T>O where k is the scale parameter ~ is the shape parameter (a function of the general effectiveness of the improvements. — The power law model The statistical procedures for the power law reliability growth model use the original relevant failure and time data from the test. Subclause 7. pessimistic) estimate of the final e(T). T characteristic at time T as shown in IEC 1014. however. P >1 corresponds to negative reliability growth). Oc ~ e 1. + IS 15038:2001 IEC 61164(1995) 5 . with T> O Thus. The current failure intensity after T h of testing is given by: Z(T) =+ E[N(T)] = l~T&*.4 and 7.-. The current mean time between failures after T h of testing is given by: k @T). parameters A and ~ both affect the failure intensity achieved in a given time. with L>O. The model has the following characteristic features: it is simple to evaluate. and so may give a low (that is. An extension of the model for reliability growth projections is given in 7.5 discuss confidence interval procedures. ~ = 1 corresponds to no reliability growth. information on the The expected accumulated number of failures up to test time T is given by: E[N(T)]=kTP.6). figure 2.6. the model is applied to the complete set of relevant failures (as in IEC 1014. and 7. practical use. characteristic (3)) without subdivision into categories.2 for maximum likelihood estimation of the parameters L and-#.3 gives goodness-of-fit tests for the model. when the parameters have been estimated from past programmed it is a convenient tool for planning future programmed employing similar conditions of testing and equal improvement effectiveness (see clause 5.~ Z(T) Methods are given in 7. unless projection is used (see 7. ~>0. 3 / -- .6). that is z(T) tends to zero as T tends to infinity. Except in the projection technique (see 7.. it gives the unrealistic indications that z(T) = w at T = O and”that growth can be unending. these limitations do not generally affect its it is relatively slow and insensitive in indicating growth immediately after a corrective modification. clause 6). corresponds to reliability growth. figure 6. 2.6 addresses the situation where the corrective modifications are incorporated into the system at the end of the test as delayed modifications.2.2. which are concluded at F. and the number of relevant failures is increased by judgment to include non-relevant failures and used to predict total downtime. Subclause 7. 7 Statistical 7. in particular. relevant test time in hours expected to be necessary to meet the aims of the the number of relevant failures expected to occur during this time period. The reliability growth which is assessed is the result of corrective modifications incorporated into the system during test. and type II tests. as indicated in 7.3 of IEC 1014.2). management procedures and other significant influences. 7.Time data for every relevant failure This method applies only where the time of failure has been logged for every failure. test. but an alternative approach is possible for groups of failures within a known time period (see 7.2. which are concluded at failure time TN. IS 15038:2001 IEC 61164(1995) —. The inputs to the model for these calculations will be the assumed parameters for the model.1 assume that the accumulated test time to each relevant failure is known.1 Growth tests and parameter estimation Case 1 .2 addresses the situation where achual failure times are not known and failures are grouped in intervals of test time.2. The procedures discussed in 7.2.2 7. the final system reliability at the end of the test. and judged to be valid for the future application by similarity of the test items. making allowance for the predicted total downtime (see below) and other contingencies.3. test environment. The projection technique estimates the system reliability resulting from these corrective modifications. 4 1 . — the normal evaluation method assumes the observed times to be exact times of failure. Type I tests. as already estimated from one or more previous programmed. and to estimate. use slightly different formulae. An appropriate goodness-of-fit procedures of 7.2. as described in 7. two quantities have to be predicted by means of reliability growth models: the accumulated programme.1 and 7.. which is not a failure time. shall be performed after the growth test Subclause 7. 6 Use of the model in planning reliability improvement programmed As inputs to the procedure described in 6.2 utilize system failure data during a test programme to estimate the progress of reliability growth.1 test and estimation procedures Overview The procedures in 7.2.1 . The accumulated relevant test time is then converted to calendar time from the planned test time per week or month. U1_a12 . A two-sided test for positive or negative growth at the a significance ‘level has critical values U1+12 and . independent of the reliability growth model. failures by reference to 7. the hypothesis of exponential times.28 and -1..1 of IEC 1014. the critical values for a two-sided test are 1. For type I tests.20 significance level. At the 0. —. the failure times follow a homogeneous Poisson process. at the cx/2 significance level. . 5 . between successive failures (or a homogeneous Poisson process) is accepted at the a significance level.+.u1_a12 correspond to a one-sided test for positive or negative growth.. The critical value 1. u< of positive -uI_@2 Ou u> UI-0J2 or negative reliability -%cz/2 growth.2 is the (1-a/2). In this case.28. note also the time of termination of the test. positive or negative./ . where U. If then there is evidence continued with Step 4. The critical values ‘1-a12 and . and the analysis is < fJ < %cl/2 then there is not evidence of positive or negative reliability growth at the a significance level and the growth analysis is terminated. that is. TN is the total accumulated relevant test times for type 11test. 100-th fractile of the standard normal distribution. respectively. Under the hypothesis of zero growth.5 of IEC 605-1) at which each relevant failure occurred. For other levels of significance. and/or other appropriate Step 2: assemble into a data set the accumulated relevant test times (as defined in 9. If. choose the appr~priate critical values from a table of fractiles for the standard normal distribution. Ti is the accumulated relevant test time at the i-th failure.+ Step 1: exclude non-relevant documentation. N is the total number of relevant failures. The statistic U can be used to test if there is evidence of reliability growth. however. respectively.— IS 15038:2001 IEC 61164(1995) . Step 3: Calculate the test statistic or y~-(N-1)$ 2 u = ‘=’ row N-1 TN y r m where.28 corresponds to a one-sided test for positive growth at the 10 % significance level. the statistic U is approximately distributed as a standard normal random variable with mean O and standard deviation 1. P’ is the total accumulated relevant test times for type I test. 2 Case 2 – Time data for groups estimates the MTBF of of relevant failures This alternative method is for the case where the data set consists of known time intervals.. for T = T* or T = TN (as appropriate). 7 ~type I] .1$ or $ Sl=~ln(T’’/~) 1=1 Step 5: [type II] calculate the (unbiased) estimate of the parameter ~ from the formula: [type I] or [type Step 6: 111 calculate the estimate of the parameter 1 from the formula: ~= N/(T*)p [type I] or [type 111 Step 7: calculate the estimated failure intensity 2(T) and mean time between failures 1$(T). for any test time T >0.. .. 1 ~(T) and @T) are estimates of the “current” failure intensity and MTBF at time T >0. failures by reference 6 to 7. < t(d). 2. It is important to note that the interval lengths and the number of failures per interval need not be constant. the system configuration on test at the end of the test programme. . “Extrapolated” estimates for a future time T during the test programme.. each containing a known number of failures.1) and t(i)v i = 1. If the test programme is completed.. The i-th interval is the time period between t(i.. Step 1: exclude non-relevant documentation. The partition times t(i) may assume any values between O and T. . then 6(T). T) and is partitioned into d intervals at times. but used with the usual caution associated with extrapolation.1 of IEC 1014 and/or other appropriate .. may be obtained similarly.— IS 15038:2001 IEC 61164(1995) Step 4: calculate the summation: . ?(0) = O.. for T over the range represented by the data. or at its expected termination time. d. .. Extrapolated estimates should not extend past the expected termination time. The test period is over the interval (O. 2 7. O c t(1)< t(2) c . . from the formulae: 1’ 6(T)= l/2(T) NOTES . t(d) = T.2. Step 4: levels a and several degrees of freedom for example in IEC 605-4 and IEC 605-6. (if necessary. 1=1 For each interval..+. i = 1. which satisfies the following equation: .d. independent of the reliability growth model. the shape parameter & The maximum likelihood estimate of ~ is the value ~. The statistic X 2can be used to test if there is evidence of reliability growth. calculate d (Ni -piN)2 X2=X i=] pi N Under the hypothesis of zero growth. (d -1) for various significance d–1 can be found in tables of the X2 distribution. the hypothesis of exponential times and the growth analysis is terminated. (d-1) -. combined before this test) where: adjacent intervals should be t(i)– t(i .. for the original data set assembled in Step 2. between successive failures (or a homogeneous Poisson process) is accepted at the a significance level. If X2<CV then there is not evidence of positive or negative reliability growth at the a significance level In this case.t(i)]. the failure times follow a homogeneous process. that is. the statistic X2 is approximately distributed as a X2 random variable Poisson with d-1 degrees of freedom. A two-sided test for positive or negative growth at the a significance level has critical value CV=X. critical-values X~_a. Thetotal number ofrelevant failures is N=~Ni.IS 15038:2001 IEC 61164 (1 S95) Step 2: assemble info a data set the number of relevant failures Ni recorded in the i-th interval d [r(i-l)... calculate the maximum likelihood estimate of . If X22CV then there is evidence of positive or negative reliability growth and the analysis is continued with Step 4. positive or negative. piN shall not be less than 5.1) Pi = Step 3: for the d intervals (after combination the statistic t(d) if necessary) and corresponding failures Ni. ) terms may be normalized with respect to t(d) and then the final term in (t(d)) disappears. but used with the usual caution associated with extrapolation. 2 If the test program is completed. may be obtained similarly. otherwise.t(i . If the statistic C2( M) exceeds the critical value corresponding to M in the table.3.- IS f15038 :2001 IEC 61164(1995) ! –—4 .3 Goodness-of-fit tests If individual failure times are available.< TM. J Note that r(0) = O and also t(0) ~in t(0) = O All t(. 1 Z(~) and @T) are estimates of the “current” failure intensity and MTBF at time T >0. from the formulae: Step 6 i(T) =ip T~l 6(T)=l/z(T) NOTES .1 in t(d) = O. --7.I)B 1 . Table 1 gives critical values of this statistic for 10 % significance level. estimates the MTBF of the system configuration on test at the end of the test phase. for any test time T >0..( 2’) and mean time between failures G(T).1 shall first be used to estimate the shape parameter Cramer-von Mises statistic is then given by the following expression: ~.2. the model shall be accepted. . Step 5: calculate the estimate of the parameter A from the formula i= N/t(d)B / calculate the estimated failure intensity . then @T) for T = t(d). “Extrapolated” estimates for a future time T during the test phase.. ..- ~Ni 1=1 h 1 t(i)pln t(i) – t(i-l)pln t(i– 1) t(i)~ . use case 2. The L+![(WJ c2(A’f)=— 1- where M=N and T = T* for type I tests M= N-l and T = TN for type II tests ~ <Tz <. An iterative method must be used to solve this equation for ~ .. Extrapolated estimates should not extend past the expected termination time. use case 1. 8 .. Otherwise.-. 7. for T over the range represented by the data.1 Case 1 – Time data for every relevant failure The estimation method included in 7. then the hypothesis that the power law model adequately fits the data shall be rejected. or at its expected termination time. . E[~].. adjacent intervals should be combined before the test.I)b] ei = ~[t(i)P For each interval. j=l The expected number of failures E[N(t(i))] is estimated by i[iV(t(i))] = L(i)b 9 k .2. For d intervals (after combination if necessary) and with Ni the same as in 7. Tj. J ~ . calculate the statistic: ..3. the graphical procedure described below may be used to obtain additional information about the correspondence between the model and the data.. d’. on identical ()L linear scales. The visual agreement of these points with a line at 45° through the origin is a subjective measure of the applicability of the model. IS 15038:2001 IEC 61164(1995) . E[Tj] may be estimated by: lf~ [’) ‘[Tj]= ~ . the number of observed failures from O to r(i) is N(t(i)) = ~ Nj .. From annex B. For the graphical procedure. for example in IEC 605-4 and IEC 605-6.’ .1). When the data set consists of known time intervals.. number of failures in the time interval [t(i . Tj. the graphical procedure described below may be used to obtain additional information about the correspondence between the model and the data. N 11~ The expected failure times. then the hypothesis that the power law model adequately fits the grouped data shall be rejected. ei shall not be less than 5. For each interval endpoint t(i). 1. each containing a known number of failures. q against the observed time to the j-th failure.2.-— When the failure times are known. and if necessary.“ :.. as in the example of figure A.t(i . an estimate of the expected time to the j-th failure.- x’=i(Ni-ei)2 ei i=l The critical values of this statistic for d-2 degrees of freedom can be found in tables of the X2 distribution.2 Case 2 – Time data for groups of relevant failures This test is suitable only when @ has been estimated using grouped data. If the critical walue at a 10 % level of significance is exceeded. as in 7.2. 7. is plotted ~s# . t(i)] is approximated The expected by: ... Witi j=l. are then plotted against the observed failure times.2. The lower confidence limit on ~ is ~LB=DLji The upper confidence limit on ~ is ~uB=Duj 10 for example in IEC 605-4 and -. . 7. —. and if ~ >1.95. If 0< ~ <1.05.1 use case 1. The visual agreement of these points with this line is a subjective measure of the applicability of the model.! t(i) ln~+(~-l)ln and also plotting the line as in the example (~-l) of figure A.2.1 Step 2: type I test For a two-sided 90 % confidence interval on ~.4. (2N) Du = 2(N-1) The fractiles IEC 605-6.4 Confidence intervals on the shape parameter The shape parameter ~ in the power law reliability growth model determines if the model reflects growth and to what degree. For ~ <1. . T>O See annex B for the relationship between in ~ and 6 and and -ct. if ~ = 1. T.. calculate DL = X:. i= ~ ~ .(2N) 2(N-1) X:.4 &[N(t(i))l = ~(i)k~ This gives t(i) The graphical procedure consists of plotting H .. can be found in tables of the X2 distribution.. .. there is positive reliability growth.n N(t(i)) . use case 2. this line is decreasing. 7._—— IS 15038:2001 IEC 61164(1995) . For Case 1 – Time data for every relevant failure Step 1: calculate ~ from step 5 in 7. For a two-sided confidence interval on ~ when individual failure times are available.2. there is no reliability growth. there is negative reliability growth. grouped failure times. ..P(i-l)p.2. — -–— L. .. with i=l.~5. (2( N-1)) DU = 2( N-l)(N-2) The lower confidence limit on ~ is fJLB=DL$ The upper confidence limit on ~ is ~UFj=DU”~ One-sided 95 % lower and upper limits on ~ are pm and ~n.64)C fi where N is the total number of failures. calculate ~=(l..2. respectively.. lnP(i)P ..2 Case 2 – Time data for groups These confidence 7. lnP(i-l)F~ A=~ ISI Step 4: P(i)p . -q Type II test For a two-sided 90 % confidence interval on ~.d calculate the expression [P(i)t.2. 2.2. Step 2: calculate . 7. 11 / .IS 15038:2001 IEC 61164(1995) One-sided 95 % lower and upper limits on ~ are pm and j3n. calculate DL = N“x.95. of relevant failures interval procedures are’ suitable when ~ has been estimated from grouped data as in Step 1: calculate ~ as in 7. (2( N-1)) 2( N-l)(N-2) N“X:.4.P(i-l)@ calculate c=+ Step 5: for an approximate two-sided 90 % confidence interval on ~.- t(i) P(i) =—. t(d) Step 3: respectively. step 4. &T) estimates the current MTBF.1 Case 1- on 6(T) when Time data for every relevant failure Step 1: calculate $T) from 7. aT) One-sided 95 % lower and upper limits on @T) are eLB and (lm. 7.2..2. step 7 Step 2: calculate T(i) P(i) =—..2 Case 2- Time data for groups These confidence 7.1.2. type I. of relevant failures interval procedures are suitable when ~ has been estimated from grouped data as in Step 1: calculate ~ as in 7. refer to table 2. Step 2: for a two-sided 90 % confidence interval. respectively.6(T) the upper confidence limit on @T) is euB=u. or table 3. step 7. For confidence intervals individual failure times are available. type II.IS 15038:2001 IEC 61164(1995) Step 6: the lower confidence limit on ~ is Pm=@(l-s) The upper confidence limit on ~ is One-sided 95 % lower and upper limits on ~ are pm and ~m.. step 7.5. use case 1. (3(T).2.. respectively.5.2.1. Step 3: the lower confidence limit on 6(T) is en =L. and calculate 6(T) as in 7. 12 d .5 Confidence intervals on current MTBF From 7. use case 2.2. 7. 7.2. and locate the values L and U for the appropriate sample size N.2. T(d) avec i=l.1. For grouped failure times. calculate S=(1.- technique ~ The following technique is appropriate when the cotr~ctive modifications have been incorporated into the system at the end of the test as delayed modificati~ns..1.. Let 1 be the number of these distinct types.. definitions 3.6 Projection .11). Step 4: assign to each of the I distinct types of category B failures in the data set of step 2 an improvement effectiveness factor. or if preferred. 13 / . i=l . Ei. D fi where N is the total number of failures. Step 1: separate the category A and category B failures (see IEC 1014. respectively. is an engineering B..1. The objective is to estimate the system reliability resulting from these corrective modifications. g. as described above. Step 2: identify the time of first occurrence of each distinct type of failure in category separate data set. postulate an average assigning the improvement effectiveness factor (e. in order to estimate ~.P(i-l)P.7) instead of individually Ei. using N = I and T* or TN as applicable to the complete set of data.1). Step 3: perform steps 1 to 5 of 7.. O S Ei S 1.d - — -— K 15038:2001 IEC 61164 (1995) Step 3: calculate the expression d A=~ ( P(i)b. j=l Step 4: h~(i-l)~ Y P(i)$ .F’(i-l)p calculate D= r 1 —+1 A Step 5: for an approximate two-sided 90 % confidence interval on @T)..2.1 upon this data set. i = 1. 7. In P(i) P. calculate the average ~. For each of the I distinct types of category B failures. 0. From these assigned values.64). Ei..10 and 3. as a assessment of the expected decrease in failure intensity resulting from an identified corrective modification (see definition 3. Step 6: the lower confidence limit on (3(T) is eLB=ii(T) (l-s) The upper confidence limit on @T) is e“B=6(T)(1+S) One-sided 95 % lower and upper limits on @T) are eLB and eUB. Ki is the number of observed failures for the i-th type of category B failures. estimate the projected failure intensity and MTBF: where KA is the number of category A failures. If the individual Ei values are not assigned and only the mean ~ is available. then the middle term in the square brackets becomes: KB(l-~) where KB is the number of category B failures. as used in step 3 above. In this case the projected failure intensity is: The projected MTBF is ep =lIZP . T = T* or TN.—- IS 15038:2001 IEC 61164(1995) 1 “’”+ Step 5: .--.. 14 . 167 10 0. !+ _.169 12 0.169 13 0. M = N.160 6 0.154 4 0.169 15 0.171 17 0.172 z 60 0.165 9 0.171 18 0. M = N-1 15 / ! ! I .167 11 0.172 30 0..162 7 0.171 20 0.— IS 15038:2001 IEC 61164(1995) Table 1- Critical values for Cram6r-von Mises goodness-of-fit test at 1070 level of significance M Critical value of statistic 3 0.169 16 0.169 14 0.155 5 0. for type 11tests.171 19 0.For type 1tests.173 NOTE .165 8 0. ..508 1.443 16 0.460 22 0.613 23 0.447 2.759 I 0.381 2.635 9 0.552 20 I I I / 0..726 1.353 2.531 1.406 2.692 6 0.902 50 0.619 10 0.520 14 0. — IS 15038:2001 IEC 61164(1995) .606 1.612 1.618 1..714 5 0.765 I 100 I I 0.738 4 0.672 7 0.414 17 0.273 . 1.2/3 where U.5 + y / 2) -th fmctile of the standard normal distribution.311 1.590 12 0.653 8 0.826 25 0.586 1.+~ “=+.428 2.490 21 0.745 1.281 3.703 1.593 1.570 1.5+.603 1.369 18 0. ~ +Y12 is the 100 (0.999 40 0..214 29 0.825 70 0.793 I 80 1.689 1.672 1.— - Table 2 – Two-sided 90 YOconfidence intervals for MTBF from type I testing I N L u N L u 3 0.947 45 0.629 1.543 1.494 1.234 4.480 2...608 26 0..136 24 0.477 15 0.464 2.56 I I I 1.521 1.861 60 0.05+.576 13 0.652 1.578 1.175 6.130 30 0..783 I NOTE -For N> 100 L=++.444 27 0.- .623 1.060 35 0.604 11 0.336 19 I 0.320 3.317 28 0. 327 1.4891 2..6225 I 1.5055 2.6344 I 1.623 18 I 0.790 25 0.501 1.592 I I 10 0.324 27 0.6503 I 1.216 28 0.303 I 0.428 16 0.5674 1.4251 t 2.578 11 0.7938 I 19 0.5+.6091 1.Two-sided 9070 confidence intervals for MTBF from type II testing IN ILIU 13 I 0.6937 1.701 4 0.5+ y 12)-th fractileof the standardnormaldistribution.3174 3.937 40 0.267 I .6763 I 1.5857 I 100 1.876 50 0.752 80 ! 0.$j where U0.825 22 0.IS 15038: IEC 61164 Table 3.644 I 0.2587 3.566 12 0.=++uo.461 I 14 0.991 35 t 0.5337 I 15 0.360 18 0.6400 I 1.659 6 0.7422 1.3962 2.892 24 7 0.463 26 0.814 60 0.608 19 I 0.o.6018 1.053 30 I 0.746 21 0. 17 i -.5571 1.3614 2.680 5 0.5 +y/2 is the 100 (0.6452 I 1.1712 I N L v 4.5459 1.7587 I 1.1 .891 45 0.— 2001 (1995) .7723 I 1.781 70 I 0.7212 1.4495 I 2.6551 I 1.$+.7085 1.254 23 0.4706 2.401 17 0.6286 I 1.5203 1.2~ “=+.553 13 0.6160 1..5769 1.127 29 0. Goodness-of-fit tests. --- a) Test for growth At the 0.173.0694 ~= 0.28.3 for grouped failures.2 for type I and type II tests. Since U<.2 h. the critical value from table 1 is 0. 0. the power law model is accepted (see 7.1 are used with test finishing at 1000 h.20 significance level.2. as described in 7.7101). 1).28 and -3. are applied when applicable. 18 .10 significance level. A.2.4491. 1. respectively. Tables A.713. e) Confidence interval on P A two-sided 90 % confidence interval on ~ is (0. 1 Example 1: Type I test . the critical values for a two-sided test are 1.1. there is evidence of positive reliability growth and the analysis is continued. These data are used in the examples of A.173.2 shows these data combined within intervals suitable for the grouped data analysis.2 Current reliability assessments The data set in table A.1 h).28. and combined in table A. At the 0. 48.2. A.5623 c) Current A4TBF The estimated current MTBF at 1000 h is 34. These examples may be used to validate computer programs designed to implement the methods given in clause 7. 1 is a complete data set used to illustrate the reliability growth methods when the relevant failure times are known.3. Data from table A. u= b) Parameter estimation The estimated parameters of the power law model are: ~= 1.2.Case 1- Time data for every relevant failure This case is covered in 7.2.1 Introduction The following numerical examples show the use of the procedures discussed in clause 7. Since C2(M) <0.IS 15038:2001 IEC 61164(1995) Annex A . f) Confidence interval on current MTBF A two-sided 90 % confidence interval on the current MTBF at 1000 h is (24. Table A.038 with M = 52.1.2 for the example of A.– .3 and A. 1 corresponds to a test finishing at 1000 h.2 h. d) Goodness-of-fit C2(M) = 0. 1 and A.- (informative) Numerical examples A.3 and figure A.4 provide data for the projection technique when corrective modifications are delayed to the end of test. and table A. 2.3 h.173.0.3 and figure A. b) Parameter estimation The estimated parameters of the power law model are: ~ =0. the critical value is 6.2. 1 are used with test finishing at 975 h.9615 ~ = 0. Since C2(M) <0. b) Parameter estimation The estimated parameters of the power law model are: ~= 1.20 significance level.28.10 significance level.28.2. 19 / - . 0.764. d) Goodness-of-fit C2(M) = 0. e) Confidence interval on ~ A two-sided 90 % confidence interval on ~ is (0.5594 c) Current MTBF The estimated current MTBF at 975 h is 33. the power law model is accepted (see 7.0. IS 15038:2001 IEC 61164(1995) A.28 and -1.5 h.8351).3202.25. The analysis of this data set gives the results described below.041 with M = 51. the critical values for a two-sided test are 1. a) Test for growth U = -3. At the 0. At the 0. A. At the 0.173. 0.10 significance level. At the 0.2.2 Example _-— . Since X2 >6.Case 1 . 46.q 2: Type II test . Since X2< 6.1067 fi = 0..175 e) Confidence interval on ~ A two-sided 90 % confidence interval on ~ is (0. Data from table A.7347). Data from table A. the power law model is accepted (see 7.20 significance level.3 Example 3- Case 2- Time data for group relevant failures This case is covered in 7.3 h.Time data for every relevant failure This case is covered in 7.5777 c) Current MTBF The estimated current MTBF at 1000 h is 33. a) Test for growth X2 = 595 with four degrees of freedom. the critical value from table 1 is 0. 1 are used.2.4646. there is evidence of positive reliability growth and the analysis is continued. X2 = 2. f) Confidence interval on current MTBF A two-sided 90 % confidence interval on the current MTBF at 975 h is (24. the critical value is 6. d) Goodness-of-fit with three degrees of freedom.1.25. there is evidence of positive or negative reliability growth and the analysis is continued.2.7 h).2). Since U C– 1. The failures have been grouped over intervaIs of 200 h to give the data set in table A.3 and figure Al). 3.1S 15038:2001 IEC 61164(1995) f) .3.6 h. The results estimation The estimated parameters of the power law model are: ~= 0.2. of the 4000 h test. column 3. The data set of table A. There are a total of N = 45 relevant failures with KA = 13 category A failures which received no corrective modification.4. Step 3: analyze first occurrence data.7472 First occurrence failure intensity estimation The estimated current failure intensity for first occurrence 0.4. 1 = 16 distinct corrective modifications were incorporated into the system to address the KB = 32 category B failures. 1 Example 4 The basic data used in this example are given in table A. At the end.171. Step 2: identify first occurrence of distinct category B types.0030 h-l. Steps in the procedure Step 1: identify category A and B failures. the critical value from table 1 is O.3 Projected reliability estimates This example illustrates the calculation of a projected reliability estimate (see 7.3.085 with M = 16. 49.Conj7dence interval occurrent MTBF A two-sided 90 % confidence interval on the current MTBF at 1000 h is ( 16.1. Table A. Each category B failure type is distinguished by a number.6) when the corrective modifications have been incorporated into the system at the end of test.171. column 3.10 significance level. Since C*(M) eO. The failure times for the 16 distinct category B types are indicated in table A.0326 ~= 0.9 h). The category for each relevant failure is given in table A. of distinct category B types at 4000 h is Goodness-of-fit C*(M) = 0. -- The times of first occurrence of the 16 distinct category B types are given in table A. column 2. is analyzed in accordance follow below: Parameter with steps 4-8 of 7. At the 0. A. A. 20 / of distinct . the power law model is accepted for the times of first occurrence category B types.4. Times of occurrence and the category A and B failures are identified in table A.3.4 provides additional information used for the projection. If only an average effectiveness factor of 0.72 Ki - table A. i i ~ Step 5: estimate projected failure intensity.65 to 0. the projected MTBF would equal 121. tlie MTBFover this period is estimated by (4 000/45) = 88.With no reliability growthduring the 4000 h test.75 is typical. column 5 The estimated projected failure intensity at T = 4000 h (the end of test) is 0.. the following values are needed: T = 4000h KA = 13 1 = 16 b = 0. NOTE. The sensitivity of the projected MTBF to the assigned effectiveness factors is often of interest.1 h. To calculate the projected failure intensity.72.---- IS 15038:2001 IEC 61164(1995) Step 4: assign effectiveness factors t~ N { An example of assigned individual effectiveness factors for each corrective modification is given in table A. based on historical experience.60 were assigned. column 5. The average of these 16 effectiveness factors is 0.4.1 h. The projected MTBF is 135.7472 E = 0. Step 6: estimate projected MTBF.80 would give a projected MTBF of 138. column 4 Ei – table A4.0074 h-l. 21 k . An average in the range of 0.9 h. An average effectiveness factor of 0.4. ~-.3 h. The projected MTBF is the estimated increase in MTBF due to the 16 corrective modifications and the corresponding effectiveness factors. F=lOOOh..—. .> 2 4 10 15 18 19 20 25 39 41 43 45 47 66 88 97 104 105 I20 196 217 219 257 260 281 283 289 307 329 357 372 374 393 403 466 521 556 571 621 628 642 684 732 735 754 792 803 805 832 836 873 975 . derived from table A.1 I I Group numb 1 I Number of failures I I 20 I Accumulated relevant test time at end of group interval I 2 13 400 3 5 600 4 8 800 5 6 1000 ii 22 / I ) .Grouped data for example 3.1 .all relevant failures and accumulated test times for type I test.Complete data .. N=52 n 9’ .— Is 15038:2001 IEC 61164(1995) Table A. Table A.2 . 3420 564 3 0.9 7 1003 1003 1 0. Ti Classification per category A/B.8 121 540.7 11 1927 1927 1 0.5 I 9 1120:2635:3730 t 1120 i 3 t 0.9 10 i 334.6 13 2850 2850 1 0.6 14 3794 3794 1 0. 3502 3 The I I I at fmt occurrence I I I 150 253 475 I I 4 5 Number observed Assigned I 2 3 I effectiveness I I 0. with failure times. Z=16 Accumrdated relevant test times. --r all relevant failures and accumulated test times. N=45..7 4 18 I 2601 I 13 h I --- I I I i 0.7 16 3952 I 3952 I 1 I I. including disdnct catego~ B types Ti 150 253 475 540 564 636 722 871 996 Category BI B2 B3 B4 B5 A B5 A B6 Ti 1003 1025 1120 1209 1255 1334 1647 1774 1927 category B7 A B8 B2 B9 B1O B9 B1O Bll Ti 2130 2214 2293 2448 2490 2508 2601 2635 2731 Category A A A A B12 A B1 B8 A Ti 2747 2850 3040 3154 3171 3206 3245 3249 3420 Catego~ B6 B13 B9 B4 A A B12 B1O B5 Ti 3502 3646 3649 3663 3730 3794 3890 3949 3952 Category B3 B1O A B2 B8 B14 B15 A B16 Table A. 3154 540 2 0.7 12 2 490. time of first occurrence. 2747 996 2 0. KB=32. 1 209. 3040 1255 3 0.Complete data for projected estimates in example 4- -. 1 774.9 6 996.3 .3.5 23 ) I . number observed and effectiveness factors I I Column no 1 I 2 Type I Failure times L-1-. 3663 475. 2 I 253.8 1 255.8 5 564: 722. P=4000h.7 15 3890 3890 1 0. from table A.7 0.4 – Dktinct types of category B failures. 1 647. 3245 2490 2 0. KA=13. I I 0. 3646 1334 4 0. IS 15038:2001 IEC 61164(1995) 4’ ——— Table A.. 3 249. 800 h 600h 400 h 200 h ---- Oh Oh 200 h 400 h 600 h 800 h 1000h Observed test time at failure Figure A.1 – Scattergram of expected and observed test times at failure based on data of table A..1 with power law model 24 .— IS 15038:2001 IEC 61164(1995) Expected test time at failure lOOOh ... . 07 0.06 0.04 200 h 300 h 400h 500h 700h 900h 2000h Accumulated test time Figure A..10 0.05 0.20 0.08 0. I .Observed and estimated accumulated failureslaccumulated test time based on data of table A. IS 15038:2001 IEC 61164(1995) ‘-4 Failures 4.2 with power law model 25 I .09 0.‘/ —— . \ Test time 0..2 . T. MTBF The exponent cx=1. in 1974.-. .. fV(T)=ATp. approximately. 7 * The most commonly accepted pattern for reliability growth was reported in a paper by J. withA>O. The Duane postulate is deterministic in the sense that it gives the expected pattern for reliability growth but does not address the associated variability of the data.T. which gives the instantaneous number of failures is approximated failure intensity at time T as with T>O .. In this paper.~ is sometimes called the “growth rate”. (3=1-a Duane expressed the current instantaneous ~N(Z’)=A~T&l. divided by the accumulated test time. . T > O}. {N(T). with mean value function E[N(T)] and intensity function 26 -/ = LTP . —.. B.1 The Duane postulate t’ .Background information B. . Duane discussed his observations on failure data for a number of systems during development testing. a>O Duane interpreted these plots and concluded that the accumulated by the power law function. He observed that the accumulated number of failures N(T). considered the power law reliability growth pattern and formulated the underlying probabilistic model for failures as a non-homogeneous Poisson process (NHPP). was decreasing and fell close to a straight line when plotted against Ton in-in scale.- Annex B (informative) The power law reliability growth model .2 The power law model L. ln(N(T)/T)=5-aln T. Crow.IS 15038:2001 IEC 61164(1995) .H. Duane in 1964. withd>O... That is. Based on this observation. IS 15038:2001 IEC 61164(1995) The Crow NHPP power law model has exactly the same reliability growth pattern as the Duane postulate. L. with~= 1. J. 379-410. PA: SIAM.. 84-89. the NHPP model gives the Poisson probability that N(T) will assume a particular value. FL. F. When ~ = 1. The NHPP power law model was extended by Crow in 1983 for reliability growth projections. Pr[N(T)=n] n AT$ e () = ~1 kT” . Orlando. However.avec j=l. pp. indicating no reliability growth. ed.. These methods include maximum likelihood estimation of the model parameters and system reliability. and increasing for ~ >1 (negative growth).. “Reliability Analysis for Complex Repairable Systems”. 1964.. they both have the same expression ATP for the expected number of failures by time T. 1974. pp.J. that is. H.. T. Aerospace 27 IEEE Transactions on . Proschan and R.2... for example..3 1 2 Reference documents Reliability Crow. Also. This gives the useful first order approximation ID (“1 +]& . “Learning Curve Approach to Reliability Monitoring”. 1983. The intensity function z(T) is decreasing for ~ c 1 (positive growth). Proceedings and Maintainability Symposium. 563-566. Serfling. . and the times between successive failures follow an exponential distribution with mean l/L (homogeneous Poisson process).withn =0. 2: pp.2. H.2. confidence interval procedures and objective goodness-of-fit tests... “Reliability Growth Projection From Delayed Fixes”. L. . The NHPP power law reliability growth model is a probabilistic interpretation of the Duane postulate and therefore allows for the development and u’se of rigorous statistical procedures for reliability growth assessments. where Tj is the accumulated time to the j-th failure. under this model ~[~Tjp]=~.. then z(T)s k. B. Biometry.. Philadelphia. and of the 1983 Annual Reliability 3 Duane.1. for the expected time to the j-th failure. Crow. . GUWAHATI. Delhi .I. Campus. This Indian Standard has been developed from Dot: No.3239402 Telegrams: Manaksanstha (Common to all offices) Regional Offices: Telephone Central : Manak Bhavan.2541442 { 2542519. Marol.T. 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