mil-std-781c and confidence intervals on mtbf.pdf

7/27/2019 MIL-STD-781C and confidence intervals on MTBF.pdf

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MIL-STD-781C AND CONFIDENCE INTERVALSON MEAN TIME BETWEEN FAILURES* AES-7905

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MIL-STD-781AND .....

CONFIDENCE INTERVALS ON MEAN _TIME BETWEEN FAILURES ,

byJ0 f--e chmeeInstitute of A min on and ManagementUnion College

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UNION COLLEGE AND UNIVERSITYINSTITUTE OF ADMINISTRATION AND MANAGEMENT

Schenectady, New York 12308

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ACKNOWLEDGMENT

The author appreciates the substantial criticism andadministrative support of Professor L.A. Aroian, Union College,and Dr. Gerald J. Hahn, General Electric Corporate Researchand Development.

Accession For

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MIL-STD-781 and Confidence Intervals

SUMMARY

Various realistic examples illustrate how to obtainconfidence limits on the mean time between failures (MTBF) ofan exponential distribution from data obtained from one of thefixed-size or sequential test plans of MIL-STD-781C.

For fixed-length tests, the methods developed by B. Epsteinand the modifications of H.L. Harter are briefly discussed. Forthe sequential tests simple charts for newly developed methodsof Bryant and Schmee are given.


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INTRODUCTION

MIL-STD-781C "covers the requirements for reliabilityqualification tests and reliability acceptance tests forequipment that experiences a distribution of times-to-failure12that is exponential" 1 A set of standard test plans areprovided. They are either of the fixed length or the sequen-tial type. The performance requirement is specified in termsof mean-time-between-failure (MTBF). Sometimes dimensionsother than time are used, e.g. cycles. Then the performancerequirement is mean-cycles-between-failures. MIL-STD-781C isonly applicable when the times to failure follow the exponentialdistribution.

One of the major criticisms of a previous version of thestandard (MIL-STD-781B) was that equipment tested and acceptedby the statistical test plans often showed unacceptable timeto failure characteristics in the field. Such discrepanciesbetween a test method and the field may be due to statisticaland non-statistical reasons.

The test plans in MIL-STD-781B emphasized statisticalhypothesis testing of two distinct values of the MTBF, 0versus eI . Either 0o was accepted and e1 rejected, or viceversa. The accepted value was assumed to be the MTBF ofthe tested equipment. However, acceptance or rejection of astatistical hypothesis provides only limited insight intothe possible values of the MTBF. On the other hand, aconfidence interval calculated from the test data after


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acceptance or rejection of the equipment, provides a rangeof values of statistical hypotheses (or MTBFs) which couldnot be rejected on the basis of the test data. Thus a confi-dence interval is viewed as a collection of acceptable hypo-12theses. Confidence intervals are new in MIL-STD-781C

As a specific example, in a later section we calculatea confidence interval on the MTBF of some electronic equipmentfrom 80 hours to 241 hours. This means that an hypothesisthat the MTBF is between 80 hours and 241 hours would havebeen accepted, and not merely the 100 hours as stated in theaccepted hypothesis of that example. Rather than accepting(or rejecting) a single value for the. MTBF, with a confidenceinterval one can give a range of values for which a similardecision would have been reached. This is useful to know,because in MIL-STD-781 (and in other real world situations)the acceptance or rejection of the statistical hypothesisis frequently accompanied by a contractual acceptance or rejectionof equipment.

This paper presents an overview of classical methodsfor confidence intervals on the MTBF of an exponential distri-bution after completion of a life test of MIL-STD-781C. Themethods themselves are not limited to the standard, butapply(especially after fixed-length tests) after testing assumingan exponential distribution. The next section briefly reviewsthe test p~ans of MIL-STD-781C. This is followed by sectionson confidence intervals after fixed-length tests and afterthe sequential test plans.


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The following are limits to the subject treated inthis paper.

* Only the statistical aspects of the test plansare considered. Thus the important problem oflab versus field testing is not considered(see Yasuda 15).

* Only equipment with failure times that are eitherexponential or can be transformed to the exponentialcan be considered. Harter and Moore 10 looked atthe robustness of the test plans if the assumptionof exponentiality is not satisfied. In particular,they look at Weibull failure times.

e Only confidence intervals on the MTBF after astatistical test are discussed. This excludes thediscussion of prediction intervals or toleranceintervals. The various types of intervals are discus-sed in Hahn 6,7A prediction interval is an interval which containsa future outcome or outcomes with a specifiedprobability, for example,- the time to failure of a single equipment, or- the average time to failure of the equipment ina lot of size k, or- all the failure times of the equipment in a lotof size k.Prediction intervals are generally wider th3nconfidence intervals. Using a confidence inteivalwhen a prediction interval is required results in


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a wrong, overly optimistic answer. Toleranceintervals contain thekfailure times of a leasta specified proportion p (of the population) witf.a stated level of confidence. Tolerance intervalsare generally also wider than confidence intervals.Many times rather than confidence intervals,prediction intervals or tolerance intervals arethe answer. New methods have yet to be worked outfor these types of intervals.


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STATISTICAL TEST PLANS

The test plans of MIL-STD-781C serve two major purposes.In (preproduction) qualification tests they are used to ensurethat hardware reliability meets or exceeds the requirements.Also they are used to conduct (production) acceptance testseither through lot-by-lot sampling or for all equipment.

This section introduces the standard test plans. First,notation and definitions are given. Then fixed-length tests

and sequential tests are briefly described and compared.Notation:

f(t) (1/0) exp {-t/0) , t>O; the densityfunction of exponential failure times.

O = the true mean time between failures(MTBF) of the exponential distribution.

Qi = lower test MTBF is an unacceptable valueof the MTBF which the standard test plansreject with high probability.

o upper test MTBF is an acceptable value of0MTBF equal to the discrimination ratiotimes 01.

d = 0o/01, the discrimination ratio; d identifiesa test plan.

E = producer's risk; the probability ofrejecting equipment(s) with a true MTBFequal to 0


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consumer's risk, the probability ofaccepting equipment(s) with the true

equal to 01.tAi = standardized acceptance time; equipment

is accepted, if not more then i failuresoccur in t i01 hours.

tRi = standardized rejection time; equipmentis rejected, if at least i failuresoccur at or before tRi01 hours.

= demonstrated MTBF; as defined in thestandard it is the probable range of thetrue MTBF stated with a specified degreeof confidence. In this paper 0


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3. All equipment reliability test, number XVIITC(not covered in this paper).

Parameters of the Test Plans: The test plans in the abovefirst two groups are characterized by the way a test iseventually terminated (stopping rule, truncation), and, mostimportant, by the three parameters a, 8, and d. The decisionrisks a and 8 of the standard test plans are .1, .2, or .3;the discrimination ratio is either 1.5, 2.0, or 3.0.

For example, to test the statistical hypothesesH: 0 = 10 hours versus0 0HI: 01 = 5 hours

i.e. d = 2.0 with specified risks a= a= .1, one can eitherselect the fixed-length test XIIC or the sequential test IIIC.

Table C-I of MIL-STD-781C (12, p. 64), gives a summary ofthe parameters of each test plan.

The same test plan would be chosen for testingH : 0 = 30 hours versus0 0HI: 01 = 15 hours,

since the discrimination ratio d = 30/15 = 10/5 = 2 is thesame, assuming the same decision risks. However, the differenthypotheses make a difference, because the times to rejectionand times to acceptance are multiples of 01. Thus for acceptancein test plan XIIC, the second hypotheses requires three timesthe total ;est time of the first hypothesis, viz 15 x 18.8 hoursas opposed to 5 x 18.8 hours. In fixed-length tests the minimumtime to accept is always a multiple ofO 1 . The standard minimumtimes to accept are also given in Table C-I of MIL-STD-781C.


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In sequential test plans the standard acceptance timestAi and the standard rejection times tRi must be multiplied by01 to arrive at the actual acceptance and rejection times.For illustration, standard acceptance and rejection times fortest plan IIIC are given in Table 1, for the other sequentialtest plans they are in MIL-STD-781C (12, pp. 66-81). Forexample, for test plan IIIC tA0 = 4.40, tAl = 5.79 and so on.Thus, the first (second) hypothesis can be accepted, ifeither

- no failure occurs up to tAO01 = 4.40 x 5 hours(4.40 x 15 hours), or

- one failure occurs before tAO 1, and no failureoccurs between 4.40 x 5 hours (4.40 x 15 hours),andt = 5.79 x 5 hours (5.79 x 15 hours), andAlso on.

Nominal versus True Decision Risks: The nominal decisionrisks are used to identify comparable test plans. Becausefailures are measured by whole numbers, it is generally notpossible to construct a test with stated risks. The risksactually achieved are called true decision risks. They arevery close to the nominal risks.

For example, for test plan XIIC the nominal risks are= = 0.10, but the true risks are ax 0.096 and4

8 = 0.106. The true decision risks for the other test plansare given in Tables II-V of MIL-STD-781C (12, pp. 12-3).


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Selection of a Test Plan: One must choose between fixed-lenor sequential tests. The standard explains that a fixed-lentest must be chosen if

- the total test time is fixed in advance, or- an estimate of the true MTBF demonstrated is

required.Sequential tests are recommended when only an accept/rejectdecision is desired.

These preceding selection criteria seem rather arbitrbecause the maximum total test time (truncation time) of asequential test is hardly longer than the fixed-length minimacceptance time. For example, the truncation time for testplan IIIC is 20.6 01 hours, whereas the minimum acceptancetime for the equivalent fixed-length test plan XIIC is 18.8hours, at worst an increase of 1.8 O hours or 8.6 percent.However, sequential tests offer substantially earlier termintion times. Test plan IIIC terminates on the average after10.2 01 hours.

Bryant and Schmee 5 and the graphs of this paper provequivalent methods to those available for fixed-length testsfor estimation after a sequential test.Sample Size and Test Length: The standard also specifiesa minimum sample size for production reliability acceptanceof at least three equipments (unless otherwise specified), obetween 10% and 20% of the lot. The sample size for areliability qualification test is specified in the contract.Also, each equipment shall operate at least one-half theaverage operating time of all equipment on test.


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ESTIMATION AFTER A FIXED-LENGTH TEST

In estimation from life test data one must distinguishbetween time censored data, when the test is terminated aftersome predetermined time, and failure censored data, when thetest is terminated at the occurrence of a predetermined numberof failures. Each censoring mode requires different formulae.In life tests, such as those of MIL-STD-781, either censoringmode may occur: time censoring occurs, if the test is accepted;failure censoring, if the test is rejected. However, at thestart of the test one does not know, which of the two censoringmodes will occur, so that a set of formulae or tables fittingeach outcome must be specified.

MIL-STD-781C provides methods for estimation after afixed-length test (but not after a sequential test). In thissection two methods for estimation after a fixed-length test are

4presented. The first, due to Epstein , is the one currentlyincluded in MIL-STD-781C. It yields confidence intervals withhigher confidence levels than stated. The second method,proposed by Harter 9, seems to give narrower intervals at con-fidence levels closer to the stated ones than Epstein's method.Because of the form of the exponential distribution both methodsdo not require the actual failure times. Only the number offailures and the total test time are accumulated. The sameholds for the methods after se;uential tests described in thenext section.


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4Epstein's Method: Epstein proposes the following formulaefor two-sided (l-2y) 100% confidence intervals on the MTBFafter a fixed-length test:

After Acceptance:

2t 2t0= < E) 0 ifr > 0-- 2 2 -X (l-y, 2r+2) X (Y, 2r)

and

2tE)= -< 0 < if r =0- 2

X (*.-y, 2)After RNjection:

2t 2t0=


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8are girven in Harter2. For (l-y) 100% one-sided confidence intervals

one uses the same formulae as for (1-2y) 100%two-sided confidence intervals, For one-sidedlower intervals the left-hand side of the two-sidedformula is used (the upper limit is at infinity),and for one-sided upper intervals the right-handside of the two-sided formula is used (the lowerlimit is zero). Also note that there is noone-sided upper confidence interval with zerofailures (r=0).

!MIL-STD-781C oes not even give the formulafor r=0 for two-sided confidence intervals. Theobvious reason for this omission is that thisresults in an interval which is unbounded to theright.

3. The above formulae produce conservative confidenceintervals. This means that the true confidencelevel is usually higher than stated (see Epstein5and Fairbanks )

Example : In a fixed-length lifetest of electronic equipment it is desired to acceptthe equipment with probability 1-a =.9 when 0=0, = 100hours, and to reject it with probability 1-0 =.9when 0=0, = :v hours. Thus the discrimination ratiod = 2.0. Test Plan XIIC is selected for this test.


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From Table II of MIL-STD-781C (12, p. 12)wefind that this test plan results in acceptance,if not more than 13 failures occur in 18.80, =18.8 x 50 = 940 hours, and in a rejectionotherwise.Acceptance: Suppose that only r = 7 failuresoccur in 940 hours. So the test results inacceptance of the equipment. In this case thedata are time censored. Note that the seventhfailure occurred before 940 hours.

A two-sided 80% confidence interval on the MTBF is2 x 940 2 x 940E) < 0

mil-std-781c and confidence intervals on mtbf.pdf

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