1、Designation: E 2696 09Standard Practice forLife and Reliability Testing Based on the ExponentialDistribution1This standard is issued under the fixed designation E 2696; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of la
2、st revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice presents standard sampling procedures andtables for life and reliability testing in procurement, supply, a
3、ndmaintenance quality control operations as well as in researchand development activities.1.2 This practice describes general procedures and defini-tions of terms used in life test sampling and describes specificprocedures and applications of the life test sampling plans fordetermining conformance t
4、o established reliability require-ments.1.3 This practice is an adaptation of the Quality Control andReliability Handbook H-108, “Sampling Procedures andTables for Life and Reliability Testing (Based on ExponentialDistribution),” U.S. Government Printing Office, April 29,1960.1.4 A system of units i
5、s not specified in this practice.1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitat
6、ions prior to use.2. Referenced Documents2.1 ASTM Standards:2E 456 Terminology Relating to Quality and StatisticsE 2234 Practice for Sampling a Stream of Product byAttributes Indexed by AQLE 2555 Practice for Factors and Procedures for Applyingthe MIL-STD-105 Plans in Life and Reliability Inspection
7、3. Terminology3.1 DefinitionsSee Terminology E 456 for a more exten-sive listing of terms in ASTM Committee E11 standards.3.1.1 consumers risk, b, nprobability that a lot havingspecified rejectable quality level will be accepted under adefined sampling plan. E 25553.1.1.1 DiscussionIn this practice,
8、 the consumers risk isthe probability of accepting lots with mean time to failure u1.3.1.1.2 DiscussionFor the procedures of 9.7 and 9.8, theconsumers risk may also be defined as the probability ofaccepting lots with unacceptable proportion of lot failingbefore specified time, p1.3.1.2 life test, np
9、rocess of placing one or more units ofproduct under a specified set of test conditions and measuringthe time until failure for each unit.3.1.3 mean time to failure, u, nin life testing, the averagelength of life of items in a lot.3.1.3.1 DiscussionAlso known as mean life.3.1.4 number of failures, nn
10、umber of failures that haveoccurred at the time the decision as to lot acceptability isreached.3.1.4.1 DiscussionThe expected number of failures re-quired for decision is the average of the number of failuresrequired for decision when life tests are conducted on a largenumber of samples drawn at ran
11、dom from the same exponen-tial distribution.3.1.5 producers risk, a, nprobability that a lot havingspecified acceptable quality level will be rejected under adefined sampling plan.3.1.5.1 DiscussionIn this practice, the producers risk isthe probability of rejecting lots with mean time to failure u0.
12、3.1.5.2 DiscussionFor the procedures of 9.7 and 9.8, theproducers risk may also be defined as the probability ofrejecting lots with acceptable proportion of lot failing beforespecified time, p0.3.1.6 sequential life test, nlife test sampling plan wherebyneither the number of failures nor the time re
13、quired to reach adecision are fixed in advance but instead decisions depend onthe accumulated results of the life test.3.1.7 unit of product, nthat which is inspected to deter-mine its classification as defective or nondefective or to countthe number of defects. E 22341This practice is under the jur
14、isdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on StatisticalQuality Control.Current edition approved May 1, 2009. Published June 2009.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Servi
15、ce at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.1.8 waiting time, nin life testing, the tim
16、e elapsed fromthe start of testing until a decision is reached as to lotacceptability.3.1.8.1 DiscussionThe expected waiting time required fordecision is the average of the waiting times required fordecision when life tests are conducted on a large number ofsamples drawn at random from the same expo
17、nential distribu-tion.4. Significance and Use4.1 This practice was prepared to meet a growing need forthe use of standard sampling procedures and tables for life andreliability testing in government procurement, supply, andmaintenance quality control (QC) operations as well as inresearch and develop
18、ment activities where applicable.4.2 A characteristic feature of most life tests is that theobservations are ordered in time to failure. If, for example, 20radio tubes are placed on life test, and tidenotes the time whenthe ith tube fails, the data occur in such a way that t1#t2#.# tn. The same kind
19、 of ordered observations will occurwhether the problem under consideration deals with the life ofelectric bulbs, the life of electronic components, the life of ballbearings, or the length of life of human beings after they aretreated for a disease. The examples just given all involveordering in time
20、.4.3 In destructive testing involving such situations as thecurrent needed to blow a fuse, the voltage needed to breakdown a condenser, or the force needed to rupture a physicalmaterial, the test can often be arranged in such a way that everyitem in the sample is subjected to precisely the same stim
21、ulus(current, voltage, or stress). If this is done, then clearly theweakest item will be observed to fail first, the second weakestnext, and so forth. While the random variable consideredmostly in this guide is time to failure, it should be emphasized,however, that the methodology provided herein ca
22、n be adaptedto the testing situations mentioned above when the randomvariable is current, voltage, stress, and so forth.4.4 Sections 6 and 7 describe general procedures anddefinitions of terms used in life test sampling. Sections 8, 9,and 10 describe specific procedures and applications of the lifet
23、est sampling plans for determining conformance to establishedreliability requirements.4.5 Whenever the methodology or choice of procedures inthe practice requires clarification, the user is advised to consulta qualified mathematical statistician, and reference should bemade to appropriate technical
24、reports and other publications inthe field.5. Introduction5.1 The theory underlying the development of the life testsampling plans of this section, including the operating charac-teristic curves, assumes that the measurements of the length oflife are drawn from an exponential distribution. Statistic
25、al testprocedures for determining the validity of the exponentialdistribution assumption have appeared in the technical statis-tical journals. Professor Benjamin Epstein published a com-prehensive paper (in two parts) on this subject in the Februaryand May 1960 issues of Technometrics.3Part I of the
26、 papercontains descriptions of the mathematical and graphical pro-cedures as well as an extensive bibliography for referencepurposes. Numerical examples illustrating the statistical pro-cedures are included in Part II of the paper.5.2 It is important to note that the life test sampling plans ofthis
27、practice are not to be used indiscriminately simply becauseit is possible to obtain life test data. Only after the exponentialassumption is deemed reasonable should the sampling plans beused.5.3 Sections 6 and 7 describe general procedures anddescription of life test sampling plans. Section 8 descri
28、besspecific procedures and applications of sampling plans whenlife tests are terminated upon the occurrence of a preassignednumber of failures, and Section 9 provides sampling planswhen life tests are terminated at a preassigned time. Section 10describes sequential life test sampling plans. Section
29、8 covers:(1) acceptance procedures; (2) expected duration of life testsand cost considerations in selection of sample sizes; and (3) lifetest plans for certain specified values of a, b, and u1/u0. Section9 covers: (1) acceptance procedures; (2) life test plans forcertain specified values of a, b, u1
30、/u0, and T/u0; and (3) life testplans based on proportion of lot failing before specified time.Section 10 covers: (1) acceptance procedures; (2) graphicalacceptance procedures; and (3) expected number and waitingtime required for decision.5.4 Operating characteristic (OC) curves for the life testsam
31、pling plans of 8.1-8.5, 9.1-9.5, and Section 10 are shown inFig. A1.1 for the corresponding sampling plans in thesesections were matched with respect to their OC curves. The OCcurves in Fig. A1.1 have been computed for the life testsampling plans of 8.1-8.5 but are equally applicable for thesampling
32、 plans of 9.1-9.5 and Section 10.5.5 The procedures of this section are based on the premisethat the life tests are monitored continuously. If the tests aremonitored only periodically, the values obtained from thetables and curves are only approximations.6. General Definitions of Life and Reliabilit
33、y Test Terms6.1 Discussion of Terms and Procedures:6.1.1 PurposeThis section provides definitions of termsrequired for the life test sampling plans and procedures ofSections 7 through 10.6.1.2 Life TestLife test is the process of placing the “unitof product” under a specified set of test conditions
34、andmeasuring the time it takes until failure.6.1.3 Unit of ProductThe unit of product is the entity ofproduct that may be placed on life test.6.1.4 Specifying FailureThe state that constitutes a failureshall be specified in advance of the life test.6.1.5 Life Test Sampling PlanA life test sampling p
35、lan isa procedure that specifies the number of units of product froma lot that are to be tested and the criterion for determiningacceptability of the lot.3Epstein, B., “Tests for the Validity of the Assumption that the UnderlyingDistribution of Life is Exponential,” Technometrics, Vol 2, Feb 1960 an
36、d May 1960,pp. 83-101 and 167-183.E26960926.1.6 Life Test Terminated upon Occurrence of PreassignedNumber of FailuresLife test sampling plans whereby testingis terminated when a preassigned termination number offailures, r, occur are given in Section 8 of this practice.6.1.7 Life Test Terminated at
37、Preassigned TimeLife testsampling plans whereby testing is terminated when a preas-signed termination time, T, is reached are given in Section 9 ofthis practice.6.1.8 Sequential Life TestSequential life test is a life testsampling plan whereby neither the number of failures nor thetime required to r
38、each a decision are fixed in advance but,instead, decisions depend on the accumulated results of the lifetest. Information on the observed time to failure are accumu-lated over time and the results at any time determine the choiceof one among three possible decisions: (1) the lot meets theacceptabil
39、ity criterion, (2) the lot does not meet the acceptabil-ity criterion, or (3) the evidence is insufficient for eitherdecision (1) or (2) and the test must continue. Sequential lifetest sampling plans are given in Section 10 of this practice andhave the advantage over the life test sampling plans men
40、tionedin 6.1.6 and 6.1.7 in that, for the same OC curve, the expectedwaiting time and the expected number of failures required toreach a decision as to lot acceptability are less for thesequential life tests.6.1.9 Expected Number of FailuresThe number of fail-ures required for decision is the number
41、 of failures that haveoccurred at the time the decision as to lot acceptability isreached. For the life test sampling plans mentioned in 6.1.6,this number of failures is known in advance of the life test; but,for the sampling plans mentioned in 6.1.7 and 6.1.8, thisnumber cannot be predetermined. Th
42、e expected number offailures required for decision is the average of the number offailures required for decision when life tests are conducted ona large number of samples drawn at random from the sameexponential distribution. The expected number of failures canbe predetermined for the sampling plans
43、 mentioned in 6.1.6-6.1.8.6.1.10 Expected Waiting TimeThe waiting time requiredfor decision is the time elapsed from the start of the life test tothe time decision is reached as to lot acceptability. The waitingtime required for decision cannot be predetermined for any ofthe sampling plans mentioned
44、 in 6.1.6-6.1.8. The expectedwaiting time required for decision is the average of the waitingtimes required for decision when life tests are conducted on alarge number of samples drawn at random from the sameexponential distribution. The expected waiting time can bepredetermined for the sampling pla
45、ns mentioned in 6.1.6-6.1.8.6.2 Length of Life:6.2.1 Length of LifeThe terms “length of life” and “timeto failure” may be used interchangeably and shall denote thelength of time it takes for a unit of product to fail after beingplaced on life test. The length of time may be expressed in anyconvenien
46、t time scale such as seconds, hours, days, and soforth.6.2.2 Mean Time to FailureThe terms “mean time tofailure” and “mean life” may be used interchangeably and shalldenote the mean (or equivalently, the average) length of life ofitems in the lot. Mean life is denoted by u.6.2.3 Acceptable Mean Life
47、The acceptable mean life, u0,is the minimum mean time to failure that is consideredsatisfactory.6.2.4 Unacceptable Mean LifeThe unacceptable meanlife, u1(u1p0), is the minimum fraction of the lotthat may fail before time, T, and results in the lot beingconsidered unsatisfactory. The interval between
48、 p0and p1is azone of indifference in which there is a progressively greaterdegree of dissatisfaction as the fraction of the lot failing beforetime, T, increases from p0to p1.6.3.6 Acceptable Failure Rate During Period of TimeThe“acceptable failure rate during period of time,” G0,isthemaximum failure
49、 rate during the period of time that can beconsidered satisfactory.6.3.7 Unacceptable Failure Rate During Period of TimeThe “unacceptable failure rate during period of time,” G1,(G1 G0), is the minimum failure rate during the period oftime that results in the lot being considered unsatisfactory. Theinterval between G0and G1is a zone of indifference in whichthere is a progressively greater degree of dissatisfaction as thefailure rate increases from G0to G1.6.3.8 Life Test Sampling Plans Based on Failure RatesLife test sampling plans that are based
