IS 15038:2001 IEC 61164 (1995)
( Reaffirmed 2005 )
WR%?Wi'm . Yih-mw$ fadvl m ~ W%Pi q-d
Indian Standard
RELIABILITY GROWTH -- STATISTICAL AND ESTIMATION METHODS TEST
ICS 03.120.01; 03.120.30; 21.020
Q BIS 2001
BUREAU
MANAK
OF
BHAVAN,
INDIAN
STANDARDS
ZAFAR MARG
9 BAHADUR SHAH NEW DELHI 110002
September 2001
Price Group 10
Reliability of Electronic and Electrical Components and Equipment Sectional Committee, 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), 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. In the adopted standard, certain conventions are, however, not identical to those used in Indian Standards. Attention is particularly drawn to the following: a) b) Wherever the words `International Standard' appear referring to this standard, they should be read as `Indian Standard'. Comma (,) has been used as a decimal marker while in Indian Standards, the current practice is to use a point (.) as the decimal marker.
Only English language text has been retained while adopting this International Standard. CROSS REFERENCES In this adopted standard, reference appears to the following International Standard for which Indian Standard also exists. 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 Corresponding Indian Standard Degree of Equivalence
IEC 50(19 I) :1990 International Electrotechnical Vocabulary (IEV) -- Chapter 191: Dependability and quality of service
IS 1885 (Part 39):1999 Electrotechnical vocabulary: Part 39 Reliability of electronic and electrical items (second
revision)
Not Equivalent
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 IEC 605-4:1986 IEC 605-6:1986 IEC 1014:1989 Equipment reliability testing -- Part 1: General requirements Equipment reliability testing -- Part 4: Procedures for determining point estimates and confidence limits from equipment reliability determination tests Equipment reliability testing -- Part 6: Tests for the validation of a constant failure rate assumption Programmed for reliability growth
1S 15038:2001
IEC 61164(1995)
Indian Standard
RELIABILITY GROWTH -- STATISTICAL AND ESTIMATION METHODS
1 Scope
TEST
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. These procedures deal with growth, estimation, confidence intervals for system reliability and goodness-of-fit tests. 2 Normative
references
The following normative documents contain provisions which, through reference in this text, constitute provisions of this International Standard. At the time of publication, the editions indicated were valid. All normative documents are subject to revision, 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. Members of IEC and 1S0 maintain registers of currently valid International Standards. IEC 50(191): 1990, International
quality of service Equipment reliability reliability testing testing reliability testing - Part 1: General requirements for determining point estimates Electrotechnical Vocabulary (IEV) - Chapter 191: Dependability and
IEC 605-1:1978,
IEC 605-4: 1986, Equipment
and confidence
- Part 4: Procedures
limits from equipment reliability
determination
tests of a constant failure rate
IEC 605-6: 1986, Equipment
assumption
- Part 6: Tests for the validity
IEC 1014:1989, 3 Definitions
Programmed
for reliability
growth
For the purposes of this standard the terms and definitions together with the following additional terms and definitions:
delayed 3.1 end of a test. modification: A corrective
of IEC 50(191)
and IEC 1014 &ply,
modification
which is incorporated
into the system at the
NOTE - A delayed
modification is not incorporated during the test.
.
3,2 improvement effectiveness factor: The fraction by which the intensity of a systematic faiIure is reduced by means of corrective modification.
IS 15038:2001 IEC 61164(1995)
type I test: A test which is terminated at a predetermined 3.3 through a time which does not correspond to a failure.
NOTE - Type I test is sometimes called time terminated test.
time or test with data available
type II test: A reliability growth test which is terminated upon the accumulation of a specified 3.4 number of failures, or test with data available through a time which corresponds to a failure.
NOTE Type 11test is sometimes called failure terminated test.
4
Symbols
For the purpose of this international
standard, 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; KB = ~ Ki
i=l
M N Ni
N(T) E[N(T)]
parameter of the Crarm%-vonMises test (statistical) number of relevant failures number of relevant failures in i-th interval accumulated number of failures up to test time T expected accumulated number of failures up to test time T endpoints of i-th interval of test time for grouped data analysis current accumulated relevant test time accumulated relevant test time at the i-th failure total accumulated relevant test times for type II test total accumulated relevant test times for type I test y fractile of the %2distribution with v degrees of freedom general symbol for failure intensity y fractile of the standard normal distribution projected failure intensity current failure intensity at time T current instantaneous mean time between failures projected mean time between failures
f(il); f(i)
T ~ TN F
x;(v)
z
`Y `P :(T) E)(T) (3P
2
IS 15038:2001 IEC 61164(1995)
5
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. Except in the projection technique (see 7.6), the model is applied to the complete set of relevant failures (as in IEC 1014, figure 2, characteristic (3)) without subdivision into categories. The basic equations for the power law model are given in this clause. Background model is given in annex B. The expected accumulated number of failures up to test time T is given by:
E[N(T)]=kTP, with L>O, ~>0, T>O
information
on the
where k is the scale parameter
~ is the shape parameter (a function of the general effectiveness of the improvements; Oc ~ e 1, corresponds to reliability growth; ~ = 1 corresponds to no reliability growth; P >1 corresponds to negative reliability growth). The current failure intensity after T h of testing is given by:
Z(T) =+
E[N(T)] = l~T&*,
with T> O
Thus, parameters A and ~ both affect the failure intensity achieved in a given time. The equation represents in effect the slope of a tangent to the N(T) vs. T characteristic at time T as shown in IEC 1014, figure 6. The current mean time between failures after T h of testing is given by:
@T).~
Z(T)
Methods are given in 7.1 and 7.2 for maximum likelihood estimation of the parameters L and-#. Subclause 7.3 gives goodness-of-fit tests for the model, and 7.4 and 7.5 discuss confidence interval procedures. An extension of the model for reliability growth projections is given in 7.6. The model has the following characteristic it is simple to evaluate; 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, and IEC 1014, clause 6); it gives the unrealistic indications that z(T) = w at T = O and"that growth can be unending, that is z(T) tends to zero as T tends to infinity; practical use; however, these limitations do not generally affect its features:
it is relatively slow and insensitive in indicating growth immediately after a corrective modification, and so may give a low (that is, pessimistic) estimate of the final e(T), unless projection is used (see 7.6);
3
IS 15038:2001 IEC 61164(1995) the normal evaluation method assumes the observed times to be exact times of failure, but an alternative approach is possible for groups of failures within a known time period (see 7.2.2).
6
Use of the model in planning
reliability
improvement
programmed
As inputs to the procedure described in 6.3 of IEC 1014, two quantities have to be predicted by means of reliability growth models: the accumulated programme; 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 accumulated relevant test time is then converted to calendar time from the planned test time per week or month, making allowance for the predicted total downtime (see below) and other contingencies, and the number of relevant failures is increased by judgment to include non-relevant failures and used to predict total downtime. The inputs to the model for these calculations will be the assumed parameters for the model, as already estimated from one or more previous programmed, and judged to be valid for the future application by similarity of the test items, test environment, management procedures and other significant influences.
7 7.1
Statistical Overview
test and estimation
procedures
The procedures in 7.2 utilize system failure data during a test programme to estimate the progress of reliability growth, and to estimate, in particular, the final system reliability at the end of the test. The reliability growth which is assessed is the result of corrective modifications incorporated into the system during test. The procedures discussed in 7.2.1 assume that the accumulated test time to each relevant failure is known. Subclause 7.2.2 addresses the situation where achual failure times are not known and failures are grouped in intervals of test time. Type I tests, which are concluded at F, which is not a failure time, and type II tests, which are concluded at failure time TN. use slightly different formulae, as indicated in 7.2.1 . An appropriate goodness-of-fit procedures of 7.2.1 and 7.2.2. test, as described in 7.3, shall be performed after the growth test
Subclause 7.6 addresses the situation where the corrective modifications are incorporated into the system at the end of the test as delayed modifications. The projection technique estimates the system reliability resulting from these corrective modifications. 7.2
7.2.1 Growth tests and parameter estimation
Case 1 - Time data for every relevant failure
This method applies only where the time of failure has been logged for every failure.
4
IS 15038:2001 IEC 61164(1995) Step 1: exclude non-relevant documentation. failures by reference to 7.1 of IEC 1014, and/or other appropriate
Step 2:
assemble into a data set the accumulated relevant test times (as defined in 9.5 of IEC 605-1) at which each relevant failure occurred. For type I tests, note also the time of termination of the test. Calculate the test statistic
Step 3:
or y~-(N-1)$
2 u = `=' N-1 TN y r row
m
where. N P'
TN Ti
is the total number of relevant failures; is the total accumulated relevant test times for type I test; is the total accumulated relevant test times for type 11test;
is the accumulated
relevant test time at the i-th failure.
Under the hypothesis of zero growth, that is, the failure times follow a homogeneous Poisson process, the statistic U is approximately distributed as a standard normal random variable with mean O and standard deviation 1. The statistic U can be used to test if there is evidence of reliability growth, positive or negative, independent of the reliability growth model. A two-sided growth at the a significance `level has critical values U1+12 and - U1_a12 , where U,+,2 is the (1-a/2). 100-th fractile of the standard normal distribution.
u<
test for positive
or negative
If then there is evidence continued with Step 4. If, however, of positive
-uI_@2 Ou u> UI-0J2 growth, respectively, and the analysis is
or negative reliability
-%cz/2
< 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. In this case, the hypothesis of exponential times. between successive failures (or a homogeneous Poisson process) is accepted at the a significance level. The critical values `1-a12 and - u1_a12 correspond to a one-sided test for positive or negative growth, respectively, at the cx/2 significance level. At the 0,20 significance level, the critical values for a two-sided test are 1,28 and -1,28. The critical value 1,28 corresponds to a one-sided test for positive growth at the 10 % significance level. For other levels of significance, choose the appr~priate critical values from a table of fractiles for the standard normal distribution. 5
IS 15038:2001 IEC 61164(1995) Step 4: calculate the summation: ~type I] or Sl=~ln(T''/~) 1=1 Step 5: [type II]
calculate the (unbiased) estimate of the parameter ~ from the formula: [type I] or
[type
111
Step 6:
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, from the formulae:
6(T) = l/2(T)
NOTES
.
1 ~(T) and @T) are estimates of the "current" failure intensity and MTBF at time T >0, for T over the range represented by the data. "Extrapolated" estimates for a future time T during the test programme, or at its expected termination time, may be obtained similarly, but used with the usual caution associated with extrapolation. Extrapolated estimates should not extend past the expected termination time.
.
If the test programme is completed, then 6(T), for T = T* or T = TN (as appropriate), the system configuration on test at the end of the test programme. 2
estimates the MTBF of
7.2.2
Case 2 Time data for groups
of relevant failures
This alternative method is for the case where the data set consists of known time intervals, each containing a known number of failures. It is important to note that the interval lengths and the number of failures per interval need not be constant. The test period is over the interval (O; T) and is partitioned into d intervals at times, O c t(1)< t(2) c ... < t(d). The i-th interval is the time period between t(i- 1) and t(i)v i = 1, 2, .... d, ?(0) = O, t(d) = T. The partition times t(i) may assume any values between O and T. Step 1: exclude non-relevant documentation. failures by reference 6 to 7.1 of IEC 1014 and/or other appropriate
IS 15038:2001 IEC 61164 (1 S95) Step 2: assemble
[r(i-l);t(i)],
info a data set the number of relevant failures
i = 1,...,d.
Ni recorded
d
in the i-th interval
Thetotal
number ofrelevant
failures is N=~Ni. 1=1 adjacent intervals should be
For each interval, piN shall not be less than 5, (if necessary, combined before this test) where:
t(i) t(i - 1)
Pi =
t(d)
Step 3:
for the d intervals (after combination the statistic
if necessary) and corresponding
failures
Ni, calculate
d (Ni -piN)2 X2=X i=] pi N
Under the hypothesis of zero growth, that is, the failure times follow a homogeneous process, the statistic X2 is approximately distributed as a X2 random variable
Poisson with d-1
degrees of freedom. The statistic X 2can be used to test if there is evidence of reliability growth, positive or negative, independent of the reliability growth model. A two-sided test for positive or negative growth at the a significance level has critical value CV=X;+; If X22CV (d-1)
then there is evidence of positive or negative reliability growth and the analysis is continued with Step 4. If
X20, from the formulae: i(T) =ip T~l
6(T)=l/z(T)
NOTES
.
1 Z(~) and @T) are estimates of the "current" failure intensity and MTBF at time T >0, for T over the range represented by the data. "Extrapolated" estimates for a future time T during the test phase, or at its expected termination time, may be obtained similarly, but used with the usual caution associated with extrapolation. Extrapolated estimates should not extend past the expected termination time. . 2 If the test program is completed, then @T) for T = t(d), estimates the MTBF of the system configuration on test at the end of the test phase.
7.3
Goodness-of-fit
tests
If individual failure times are available, use case 1, otherwise, use case 2. 7.3.1
Case 1 Time data for every relevant failure
The estimation method included in 7.2.1 shall first be used to estimate the shape parameter Cramer-von Mises statistic is then given by the following expression:
~.
The
c2(A'f)=--
where
M=N
L+![(WJ
1-
and T = T* for type I tests and T = TN for type II tests
M= N-l
~ O B for the relationship between in ~ and 6 and
For ~ <1, this line is decreasing. The visual agreement of these points with this line is a subjective measure of the applicability of the model. 7.4
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. If 0< ~ <1, there is positive reliability growth, if ~ = 1, there is no reliability growth, and if ~ >1, there is negative reliability growth. For a two-sided confidence interval on ~ when individual failure times are available, grouped failure times, use case 2. 7.4.1
Case 1 Time data for every relevant failure
use case 1. For
Step 1: Step 2:
calculate ~ from step 5 in 7.2.1
type I test
For a two-sided 90 % confidence interval on ~, calculate
DL =
X:,05 ;(2N)
2(N-1) X:,95; (2N)
Du =
2(N-1) The fractiles IEC 605-6. can be found in tables of the X2 distribution, for example in IEC 605-4 and
The lower confidence limit on ~ is
~LB=DLji
The upper confidence limit on ~ is
~uB=Duj
10
IS 15038:2001 IEC 61164(1995) One-sided 95 % lower and upper limits on ~ are pm and j3n, respectively.
Type II test
For a two-sided 90 % confidence interval on ~, calculate N"x;,~5; (2( N-1)) 2( N-l)(N-2) N"X:,95; (2( N-1))
DU =
DL =
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,
7.4.2 Case 2 Time data for groups of relevant failures
respectively.
These confidence 7.2.2. Step 1: Step 2:
interval procedures are' suitable when ~ has been estimated from grouped data as in
calculate ~ as in 7.2.2, step 4. calculate
t(i) P(i) =--, t(d)
with i=l,
2,...,d
Step 3:
calculate the expression
[P(i)t. A=~ ISI
lnP(i)P - P(i-l)p.
P(i)p - P(i-l)@
lnP(i-l)F~
Step 4:
calculate
c=+
Step 5: for an approximate two-sided 90 % confidence interval on ~, calculate ~=(l,64)C fi where N is the total number of failures.
11
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, respectively. 7.5
Confidence intervals on current MTBF
From 7.2.1, step 7, &T) estimates the current MTBF, (3(T). For confidence intervals individual failure times are available, use case 1. For grouped failure times, use case 2. 7.5.1
Case 1Time data for every relevant failure
on 6(T) when
Step 1: Step 2:
calculate $T) from 7.2.1, step 7. for a two-sided 90 % confidence interval, refer to table 2, type I, or table 3, type II, and locate the values L and U for the appropriate sample size N. the lower confidence limit on 6(T) is en =L.6(T) the upper confidence limit on @T) is
euB=u. aT)
Step 3:
One-sided 95 % lower and upper limits on @T) are eLB and (lm, respectively. 7.5.2
Case 2Time data for groups of relevant failures
These confidence 7.2.2. Step 1: Step 2:
interval procedures are suitable when ~ has been estimated from grouped data as in
calculate ~ as in 7.2.2, and calculate 6(T) as in 7.2.1, step 7 calculate
T(i) P(i) =--, T(d)
avec i=l,2,...,
d
12
K 15038:2001 IEC 61164 (1995) Step 3: calculate the expression
d A=~ j=l
(
P(i)b.
In P(i) P- P(i-l)P.
P(i)$ - F'(i-l)p
h~(i-l)~
Y
Step 4:
calculate
D=
Step 5:
for an approximate
two-sided 90 % confidence interval on @T), calculate S=(1,64). fi D
r
A
1 --+1
where N is the total number of failures. Step 6: the lower confidence limit on (3(T) is eLB=ii(T) 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, respectively. 7.6
Projection technique
(l-s)
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. The objective is to estimate the system reliability resulting from these corrective modifications. Step 1: Step 2: separate the category A and category B failures (see IEC 1014, definitions 3.10 and 3.11). identify the time of first occurrence of each distinct type of failure in category separate data set. Let 1 be the number of these distinct types. B, as a
Step 3:
perform steps 1 to 5 of 7.2.1 upon this data set, in order to estimate ~, using N = I and T* or TN as applicable to the complete set of data. assign to each of the I distinct types of category B failures in the data set of step 2 an improvement effectiveness factor, Ei, i = 1,...1. For each of the I distinct types of category B failures,
Ei, O S Ei S 1, is an engineering
Step 4:
assessment of the expected decrease in failure (see definition 3.1).
intensity resulting from an identified corrective modification
From these assigned values, calculate the average ~, or if preferred, postulate an average assigning the improvement effectiveness factor (e. g., 0,7) instead of individually Ei, i=l ,...1, as described above.
13
IS 15038:2001 IEC 61164(1995) Step 5: estimate the projected failure intensity and MTBF:
where
KA Ki is the number of category A failures;
is the number of observed failures for the i-th type of category B failures;
T = T* or TN, as used in step 3 above.
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. In this case the projected failure intensity is:
The projected MTBF is ep =lIZP
14
IS 15038:2001 IEC 61164(1995)
Table 1Critical values for Cram6r-von Mises goodness-of-fit test at 1070 level of significance
M 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 z 60 Critical value of statistic 0,154 0,155 0,160 0,162 0,165 0,165 0,167 0,167 0,169 0,169 0,169 0,169 0,169 0,171 0,171 0,171 0.171 0,172 0,172 0,173
NOTE - For type 1tests, M = N, for type
11 tests, M = N-1
15
IS 15038:2001 IEC 61164(1995)
Table 2 Two-sided 90 YOconfidence intervals for MTBF from type I testing
N L
u
6,490 4,460 3,613 3,136 2,826 2,608 2,444 2,317 2,214 2,130 2,060 1,999 1,947 1,902 1,861 1,825 I I 1,793 1,765 I
N
L
u
1,738 1,714 1,692 1,672 1,653 1,635 1,619 1,604 1,590 1,576 1,520 1,477 1,443 1,414 1,369 1,336 I 1,311
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0.175 0,234 0,281 0,320 0,353 0,381 0,406 0,428 0,447 0,464 0,480 0,494 0,508 0,521 0,531 0,543 I 0,552 0,56 I
21 22 23 24 25 26 27 28 29 30 35 40 45 50 60 70 80 100 I I
0,570 0,578 0,586 0,593 0,603 0,606 0,612 0,618 0,623 0,629 0,652 0,672 0,689 0,703 0,726 0,745 0,759 0,783
I
19 20
I
I
1,273
I
NOTE -For N> 100
L=++.05+,,+~ "=+,...,5+,,2/3
where
I
I
U. ~ +Y12 is the 100 (0,5 + y / 2) -th fmctile of the standard normal distribution.
IS 15038: IEC 61164
Table 3- Two-sided 9070 confidence intervals for MTBF from type II testing
IN 13 4 5 6 7 18 19 10 I ILIU I 0.1712 0,2587 0,3174 0,3614 0,3962 0.4251 0.4495 0,4706 t I I 4,746 3,825 3,254 2,892 2,644 2.463 2,324 2,216
2001 (1995)
N 21 22 23
24 25 26 27 28 29 30 35 40 45 50 60 70 80 I ! I t I
L
v
1,701 1,680 1,659
I I I 1,790 1,623 1,608
0,6018 0,6091 0,6160
0,6225 0.6286 0,6344
I
11 12 13
0,6400 I
0,6452 0,6503 0,6551 0.6763 0,6937 0,7085 0,7212 0,7422 0,7587 0.7723 0,7938 I I I I I I I
1,592
1,578 1,566 1,553 1.501 1,461 1,428 1,401 1,360 1,327 1.303 1,267
0,4891 0,5055 0,5203
I 0,5337 0,5459 I
2,127 2,053 1,991
1,937 1,891
I
14 15
16 17 18
0,5571 0,5674 0,5769
I 0,5857 I
1,876 1,814 1,781
1,752
I
19
100
I
,=++uo,5+,,2~
.1
"=+.o,$+,,$j
where
U0,5 +y/2
is the 100
(0,5+ y 12)-th fractileof the standardnormaldistribution.
17
IS 15038:2001 IEC 61164(1995)
Annex A
(informative)
Numerical examples
A.1
Introduction
The following numerical examples show the use of the procedures discussed in clause 7. Table A. 1 is a complete data set used to illustrate the reliability growth methods when the relevant failure times are known, and table A.2 shows these data combined within intervals suitable for the grouped data analysis. Tables A.3 and A.4 provide data for the projection technique when corrective modifications are delayed to the end of test. Goodness-of-fit tests, as described in 7.3, are applied when applicable. These examples may be used to validate computer programs designed to implement the methods given in clause 7.
A.2 Current reliability assessments
The data set in table A. 1 corresponds to a test finishing at 1000 h. These data are used in the examples of A.2. 1 and A.2.2 for type I and type II tests, respectively, and combined in table A.2 for the example of A.2.3 for grouped failures. A.2. 1 Example
1: Type I test - Case 1Time data for every relevant failure
This case is covered in 7.2.1. Data from table A.1 are used with test finishing at 1000 h. a) Test for growth
u=
At the 0,20 significance level, the critical values for a two-sided test are 1,28 and 1,28. Since U<- 1,28, there is evidence of positive reliability growth and the analysis is continued.
-3,713. estimation
b) Parameter
The estimated parameters of the power law model are: ~= 1,0694 ~= 0,5623 c) Current A4TBF The estimated current MTBF at 1000 h is 34,2 h. d) Goodness-of-fit
C2(M) = 0,038 with M = 52. At the 0,10 significance
Since C2(M) <0,173, e) Confidence
interval
level, the critical value from table 1 is 0,173. the power law model is accepted (see 7.3 and figure A. 1).
on P
A two-sided 90 % confidence interval on ~ is (0,4491; 0,7101). f) Confidence
interval on current MTBF
A two-sided 90 % confidence interval on the current MTBF at 1000 h is (24,2 h; 48,1 h).
18
IS 15038:2001 IEC 61164(1995) A.2.2
Example 2: Type II test - Case 1 - Time data for every relevant failure
This case is covered in 7.2.1. Data from table A. 1 are used with test finishing at 975 h. a) Test for growth At the 0,20 significance level, the critical values for a two-sided test are 1,28 and -1,28. Since U C 1,28, there is evidence of positive reliability growth and the analysis is continued.
U = -3,764.
b) Parameter
estimation
The estimated parameters of the power law model are: ~= 1,1067 fi = 0,5594 c) Current MTBF The estimated current MTBF at 975 h is 33,5 h. d) Goodness-of-fit C2(M) = 0,041 with M = 51. At the 0,10 significance level, the critical value from table 1 is 0,173. Since C2(M) <0,173, the power law model is accepted (see 7.3 and figure Al).
e) Confidence interval on ~
A two-sided 90 % confidence interval on ~ is (0,4646; 0,7347). f) Confidence
interval on current MTBF
A two-sided 90 % confidence interval on the current MTBF at 975 h is (24,3 h; 46,7 h). A.2.3
Example 3Case 2Time data for group relevant failures
This case is covered in 7.2.2. Data from table A. 1 are used. The failures have been grouped over intervaIs of 200 h to give the data set in table A.2. The analysis of this data set gives the results described below. a) Test for growth X2 = 595 with four degrees of freedom. At the 0,20 significance level, the critical value is 6,0. Since X2 >6,0, there is evidence of positive or negative reliability growth and the analysis is continued. b)
Parameter estimation
The estimated parameters of the power law model are: ~ =0,9615 ~ = 0,5777 c)
Current MTBF
The estimated current MTBF at 1000 h is 33,3 h. d)
Goodness-of-fit
X2 = 2,175
with three degrees of freedom. At the 0,10 significance level, the critical value is 6,25. Since X2< 6,25, the power law model is accepted (see 7.3 and figure A.2).
Confidence interval on ~
e)
A two-sided 90 % confidence interval on ~ is (0,3202; 0,8351).
19
1S 15038:2001 IEC 61164(1995)
f) ,Conj7dence interval occurrent MTBF
A two-sided 90 % confidence interval on the current MTBF at 1000 h is ( 16,6 h; 49,9 h).
A.3
Projected
reliability
estimates
This example illustrates the calculation of a projected reliability estimate (see 7.6) when the corrective modifications have been incorporated into the system at the end of test. A.3. 1 Example 4 The basic data used in this example are given in table A.3. There are a total of N = 45 relevant failures with KA = 13 category A failures which received no corrective modification. At the end. of the 4000 h test, 1 = 16 distinct corrective modifications were incorporated into the system to address the KB = 32 category B failures. The category for each relevant failure is given in table A.3. Each category B failure type is distinguished by a number. Table A.4 provides additional information used for the projection.
Steps in the procedure
Step 1: identify category A and B failures. Times of occurrence and the category A and B failures are identified in table A.3. The failure times for the 16 distinct category B types are indicated in table A.4, column 2. Step 2: identify first occurrence of distinct category B types. The times of first occurrence of the 16 distinct category B types are given in table A.4, column 3. Step 3: analyze first occurrence data. The data set of table A.4, column 3, is analyzed in accordance follow below:
Parameter estimation
with steps 4-8 of 7.2.1. The results
The estimated parameters of the power law model are: ~= 0,0326 ~= 0,7472
First occurrence failure intensity estimation
The estimated current failure intensity for first occurrence 0;0030 h-l. Goodness-of-fit C*(M) = 0,085 with M = 16. At the 0,10 significance
of distinct category B types at 4000 h is
level, the critical value from table 1 is O,171. of distinct
Since C*(M) eO,171, the power law model is accepted for the times of first occurrence category B types.
20
IS 15038:2001 IEC 61164(1995) Step 4: assign effectiveness factors
An example of assigned individual effectiveness factors for each corrective modification is given in table A.4, column 5. The average of these 16 effectiveness factors is 0,72. An average in the range of 0,65 to 0,75 is typical, based on historical experience. Step 5: estimate projected failure intensity. To calculate the projected failure intensity, the following values are needed:
T KA = =
4000h 13 16
1 b E
Ki Ei
=
= 0,7472 = 0,72
table A.4, column 4 table A4, column 5
The estimated projected failure intensity at T = 4000 h (the end of test) is 0,0074 h-l. Step 6: estimate projected MTBF. The projected MTBF is 135,1 h. NOTE- With no reliability growthduring the 4000 h test, tlie MTBFover this period is estimated by (4 000/45) =
88,9 h. The projected MTBF is the estimated increase in MTBF due to the 16 corrective modifications and the corresponding effectiveness factors. The sensitivity of the projected MTBF to the assigned effectiveness factors is often of interest. If only an average effectiveness factor of 0,60 were assigned, the projected MTBF would equal 121,3 h. An average effectiveness factor of 0,80 would give a projected MTBF of 138,1 h.
21
Is 15038:2001 IEC 61164(1995)
Table A.1 - Complete data - all relevant failures and accumulated test times for type I test; F=lOOOh, N=52
2 41 I20 307 556 792
4 43 196 329 571 803
10 45 217 357 621 805
15 47 219 372 628 832
18 66 257 374 642 836
19 88 260 393 684 873
20 97 281 403 732 975
25 104 283 466 735
39 105 289 521 754
Table A.2 - Grouped data for example 3, derived from table A.1
I
I
Group numb
I
I
Number of failures
I
I
Accumulated relevant test time at end of group interval
I
I
1
2 3 4 5
20
13 5 8 6
400 600 800 1000
22
IS 15038:2001 IEC 61164(1995)
Table A.3 - Complete data for projected estimates in example 4all relevant failures and accumulated test times; P=4000h, N=45, KA=13, KB=32, Z=16
Accumrdated relevant test times, Ti Classification per category A/B, including disdnct catego~ B types 150 BI 253 B2 475 B3 540 B4 564 B5 722 B5 871 A
Ti
636 A
996 B6
Category
Ti
1003 B7
1025 A
1120 B8
1209 B2
1255 B9
1334
1647
1774
1927
category
B1O
2508 A
B9
2601 B1
B1O
2635 B8
Bll
2731 A
Ti
2130 A
2214 A
2293 A
2448 A
2490 B12
Category
Ti
2747 B6
2850 B13
3040 B9
3154 B4
3171 A
3206 A
3245 B12
3249
B1O
3420
B5
Catego~
Ti
3502
3646 B1O
3649 A
3663 B2
3730 B8
3794 B14
3890 B15
3949 A
3952 B16
Category
B3
Table A.4 Dktinct types of category B failures, from table A.3, with failure times, time of first occurrence, number observed and effectiveness factors
I
1
Column no
I
3 I 4 5 I
I
2 Failure times
I
I
13
Type
I
I I
h
I The I
at fmt occurrence
150 253 475 540 564 996 1003 t 1120 1255 1334 1927 2490 2850 3794 3890 I 3952 I i
Number observed
2 3
Assigned I effectiveness
I I 0,7 0,7 0.8 I I
I
L-1-;
2 4 5 6 7
253; 1 209; 3663 475; 3502 540; 3154 564: 722; 3420 996; 2747 1003
2601
I
I I
I I 121
I
2 3 2 1 3 3 4 1 2 1 1 1 1 I t
0,8 0,9 0,9 0.5 0.8 0,9 0,7 0,7 0.6 0,6 0,7 0.7 0,5 i
18
9 10 11 12 13 14 15
I
1120:2635:3730 1 255; 1 647; 3040 i 334; 1 774; 3 249; 3646 1927 2 490; 3245 2850 3794 3890 3952
I
16
23
IS 15038:2001 IEC 61164(1995) Expected test time at failure
lOOOh
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) Failures Test time 0,20
0,10 0,09 0,08 0,07 0,06
0,05
0,04 200 h 300 h 400h 500h 700h 900h 2000h Accumulated test time
Figure A.2 - Observed
and estimated accumulated
failureslaccumulated
test time
based on data of table A.2 with power law model
25
IS 15038:2001 IEC 61164(1995)
Annex B
(informative)
The power law reliability growth model - Background information
B.1
The Duane postulate
The most commonly accepted pattern for reliability growth was reported in a paper by J.T. Duane in 1964. In this paper, Duane discussed his observations on failure data for a number of systems during development testing. He observed that the accumulated number of failures N(T), divided by the accumulated test time, T, was decreasing and fell close to a straight line when plotted against Ton in-in scale. That is, approximately, ln(N(T)/T)=5-aln T, withd>O, a>O number of failures is approximated
Duane interpreted these plots and concluded that the accumulated by the power law function,
fV(T)=ATp, withA>O,
Based on this observation,
(3=1-a
Duane expressed the current instantaneous
failure intensity at time T as
~N(Z')=A~T&l,
with T>O
which gives the instantaneous
MTBF
The exponent cx=1- ~ is sometimes called the "growth rate". 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.
B.2
The power
law model
L.H. Crow, in 1974, considered the power law reliability growth pattern and formulated the underlying probabilistic model for failures as a non-homogeneous Poisson process (NHPP), {N(T), T > O}, with mean value function
E[N(T)] = LTP
and intensity function
26
IS 15038:2001 IEC 61164(1995) The Crow NHPP power law model has exactly the same reliability growth pattern as the Duane
postulate, for example, they both have the same expression ATP for the expected number of failures by time T. However, the NHPP model gives the Poisson probability that N(T) will assume a particular value, that is, . n AT$ e () = ~1
kT"
Pr[N(T)=n]
,withn =0,1,2,..,
Also, under this model
~[~Tjp]=~, with~=
where Tj is the accumulated time to the j-th failure. This gives the useful first order approximation
1,2,...
+]& ;
for the expected time to the j-th failure.
("1
ID
,avec j=l,2,...
When ~ = 1, then z(T)s k, and the times between successive failures follow an exponential distribution with mean l/L (homogeneous Poisson process), indicating no reliability growth. The intensity function z(T) is decreasing for ~ c 1 (positive growth), and increasing for ~ >1 (negative growth). 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. These methods include maximum likelihood estimation of the model parameters and system reliability, confidence interval procedures and objective goodness-of-fit tests. The NHPP power law model was extended by Crow in 1983 for reliability growth projections. B.3 1
Reference documents and
Reliability Crow, L. H., 1974, "Reliability Analysis for Complex Repairable Systems". Biometry, ed. F. Proschan and R.J. Serfling, pp. 379-410. Philadelphia, PA: SIAM.
2
Annual Reliability
Crow, L. H., 1983, "Reliability Growth Projection From Delayed Fixes". Proceedings and Maintainability Symposium, pp. 84-89, Orlando, FL. Duane, J. T., 1964, "Learning Curve Approach to Reliability Monitoring". 2: pp. 563-566.
of the 1983
3
IEEE Transactions
on
Aerospace
27
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of Indian
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