7 AD-AlGA 031 ARMY MAIERIEL SYSTEMS ANALYSIS ACTIVITY ABERDEEN PROV-EYC F/B 12/1CONFIDENCE INTERVALS FOR THE RELIABILITY OF A FUTURE STEM CONIETCIU)SEP 81 6 BMILLER

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9 t. fr*1UtIWW.lj2. GOVT ACCESSION NO. 3. RECIPI NT'S CATALOG NUM R

Technical Repart, . 343 __ _ _ _ _ __, -

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S. TYPE OF REPORT & PERIOD CC- -ED

Confidence Intervals for the Reliability of aFuture System Configurationv .1 .. .- . . *..... .. ...... . ... "6.... . . . PERFORMING ORG. REPORT NL -

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p Grady W.Ailler!

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U S Army Materiel Systems Analysis ActivityAberdeen Proving Ground, MD 21005

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cj~ Sep ar 1981',/ '16

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Weibull ProcessReliability GrowthReliability ProjectionConfidence IntervalRisk Assessment

20. ABSrRACT (Va e rover" M f noeeemy amd Identify by block numbev)

-An i-nferential procedure is presented which provides confidence intervalsfor a future reliability parameter when reliability growth testing is onlypartially completed. Hypothesis tests based on this method are uniformly mostpowerful unbiased. These results are applicable if (1) the system failure ratecan be modeled as the intensity function of a Weibull process and (2) effortsto improve reliability are assumed to continue at a steady rate throughout theintervening period of testing. The usefulness of this methodology is

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illustrated by evaluating the risk of not reaching some future reliabilitymilestone. If such risk is unacceptably high, program management may have timeto identify problem areas and take corrective action before testing has ended.As a consequence, a more reliable system may be developed without incurringoverruns in the scheduling or cost of the development program.

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TABLE OF CONTrENTS

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1. INTRODUCTION .. .. ..... ....... ....... .......

2. SPECIFICATION OF MODEL............ . ... . .. .. .. .. .. .. .

3. DERIVATION OF RESULTS. .. ..... ....... ........ .. 8

3.1 Point Estimators. .. ..... ........ ....... .. 83.2 Reduction of the Parameter Space. .. ..... ..........83.3 Inferential Procedures. .. ..... ........ ...... 10

4. EXAMPLE .. .. ........ ....... ........ ..... 10

REFERENCES .. .. .... ........ ....... .........13

DISTRIBUTION LIST .. ... ....... ....... .........15

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CONFIDENCE INTERVALS FQR THE RELIABILITY OF A FUTURE SYSTEM CONFIG['RATP N

1. INTRODUCTION

Reliability growth management is a critical function in the develop:programs of major defense systems.1 It consists of planning, monitoring, andcontrolling the growth of reliability parameters throughout system developmentin order to achieve the reliability milestones for each test phase and for theoverall program. A key factor in this process is the ability to assess therisk of not meeting a reliability requirement and to make such assessment atan early stage in the current test phase. If this risk is unacceptably high,the program manager may then have an opportunity to take remedial actionbefore test time or other program resources are exhausted. The risks of fail-ing to achieve program goals or contractual requirements can therefore beminimized. Instead of having to react to program shortcomings after the fact,management can exert positive control over the growth process to accomplishreliability objectives.

Reference 1 (pp. 10, 23, 28, 64-66, 75-78) discusses the use of reliabilitygrowth models to project reliability estimates beyond the present test time tosome future time, such as the end of the current test phase. These projectionsare valid only if test conditions remain relatively constant and the develop-ment effort continues at its previous level. The projected reliabilityestimates are compared with future milestones in order to assess whether thereliability enhancement program is likely to reach a successful conclusion.

One of the problems with assessing a program by this method is how toevaluate the accuracy of the reliability projections. Such projections areonly point estimates and do not reflect the uncertainties that accompanyrandom sampling from a probabilistic model. In this paper we show how toquantify these uncertainties when the Weibull process is used to model andforecast reliability growth. The result is an objective appraisal of currentprogram risks, and this appraisal can be factored into those managementdecisions which may impact on future reliability parameters.

The Weibull process model has been successfully applied to the reliabilitytest results of many complex defense systems. It is introduced in Section 2in a parametric form that is especially suited to the problem of forecasting.The basic features of this model are described in Appendix C of Reference 1,which includes confidence interval procedures for the reliability of thecurrent system configuration. (See also References 2 and 3.) The theorydeveloped in Section 3 extends these latter results to provide inferential

IDepartment of Defense, Reliability Growth Management, Military Handbook 189,Naval Publications and Forms Center, Philadelphia, PA, February 1981.

2Bain, L. J. and M. Engelhardt, "Inferences on the Parameters and CurrentSystem Reliability for a Time Truncated Weibull Process," Technometrics,Vol. 22, pp. 421-426, August 1980.

Xrow, L. H., Confidence Interval Procedures for Reliability Growth Analysis,Technical Report No. 197, U S Army Materiel Systems Analysis Activity,Aberdeen Proving Ground, MD, June 1977.

5

procedures for future reliability levels. These procedures are illustratedin Section 4, where confidence intervals are obtained for the reliabiif tobe achieved at future points in a test phase which is still in progress. "cobtained by an equivalent technique is the risk of not achieving a certainreliability level at the end of the test phase.

2. SPECIFICATION OF MODEL

Consider a reliability growth test phase which has been underway for Tunits of testing. We shall hereinafter regard these test units as time,although they could equally well represent other units such as distance.Suppose that the test phase began at time 0, but is planned to continue foran additional S units of testing till test time T+S, at which point the systemconfiguration will have failure rate R. Our objective is to make inferencesabout the parameter R.

A Weibull process is a nonhomogeneous Poisson process with an intensityfunction that can be expressed as a multiple of some power of the test time.For the particular test phase described above, an intensity function of theappropriate parametric form is

r(t) = R[t/(T+S)]O-1 , (1)

where R>O, 6>O, and O<t<T+S. As shown in Figure 1, the function r(t) modelsthe failure rate of the system configuration as it changes over a reliabilitygrowth test phase of length T+S, and the failure rate at the end of the (asyet uncompleted) test phase is given by r(T+S) = R.

The failure rate model in Figure 1 shows a decreasing trend during futuretesting from time T to time T+S. This trend reflects our previously statedintention to continue reliability improvements throughout this period. Thecase in which reliability is constant from T to T+S is treated in Reference 4.

According to the scenario of this paper, test results are available forthe test period from time 0 to the (current) time T, but the system testingfrom time T to time T+S has not yet been accomplished. Let N be the numberof failures that occur before time T and Tl ... , TN the observed failure

times (O.T I < ... T N<T). Then the Poisson process with intensity function

r(t) has a sample function density given by

f NT , ...,I T N(n, tl , ...,9 tn )

"Miller, G., "Efficient Methods for Assessing Reliability," Proceedings of theNineteenth Annual U S Army Operations Research Symposium, Part Ill, pp. 33-42,October 1980.

6

exp[_(RT/)QI-B] if N = 0, (2.1)

n Rn [(T+S)/t] , exp[-(RT/)Q if N = n>O, ( )i=l

where Q = (T+S)/T; n = 0, 1, ... ; and O<t I <... <tn<T. (See e.g., Reference 5.)

Z'r(t) =R[t/(T+S)]" -

LU.

-j

---------------------------...................

II

0OBSERVED TEST TIME T FUTURE TEST TIME 1+5

f 4 .. RELIABILITY GROWTH TEST PHASE

FIGURE 1. Intensity Function for the Case p<1.

5 Snyder, D. L., Random Point Processes, John Wiley and Sons, New York, NY,1975.

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3. DERIVATION OF RESULTS

?.1 Point Estimators

The Weibull process model is used in applications where Pr(NjUille s!,)all, and therefore the likelihood expression in Equation (2.2' -aimaxiolized to obtain point estimators for 6 and R as follows:

NN/ ln(T/T.), (3)ii l

R N /QIT. (4)

As would be expected, the expression in (3) is identical to the estimator forin Reference 2 (Equation (4)). The projected mean time between failures

(MTBF) for the system configuration at the end of the test phase (time T+S) is

estimated by R

The point estimators in Equations (3) and (4) are convenient becauseof their simplicity, but were obtained without conditioning formally on theevent N-O. As a practical matter, inferences on the two-parameter WeibullDrocess are possible only when N>O, and we shall condition on this-event inthe sequel without further mention.

.2 Reduction of the Parameter Space

NLet V Y 7 ln[(T+S)/Ti], and observe from Equation (2) that V is a

i=l

sufficient statistic for S. It follows from Reference 6 (pp. 134-140) thatuniformly most powerful unbiased (UMPU) hypothesis tests on the future failurerale R can be constructed by utilizing t,,e conditional distribution of N given,_'v. 7o obtain this distribution, we begin by determining the conditionaldistribution of V given N=n.

Given N=n, the random variables T1 .... Tn are distributed as the

order statistics from n independent distributions with cumulative distributionfunction

t TF(t) f r(x)dx/f r(x)dx

0 0

(t/T) , (5)

,I-ehmann, E. L., TestinqgStatistical Hypotheses, John Wiley and Sons, New York,.Y, 1959.

8

where O<t<T. Let X be a random variable with distribution function .-straightforward calculation shows that the random variable ln[(T+S)/X]distributed over the interval (InQ,-) according to

Pr{ln[(T+S)/X] < y} = l-exp[-(y-lnQ)B],

where InQ<y<w. This latter function is a two-parameter exponential distr-bution function on the interval (InQ,-). The conditional distribution of Vgiven N=n is therefore the sum of n such distributions, all independent, andconsequently is a three-parameter gamma distribution with density function

f V vN(Vn)

n (v-nlnQ)n- exp[- (v-nlnQ)]/(n-l)!, (7)

where nlnQ<v<-.

The random variable N is Poisson distributed with mean value

o-(RT/a)Q - , so that (conditional on N>O)

Pr(N=n) = [l-exp(-a)]- enexp(-O)/n!, (8)

n = 1, 2 ...... Thus the joint density function of V and N is

fV,N(v,n) = fVIN(vln) Pr(N=n)

exp(-e-sv) (RTQ)n (v-nlnQ )n -l

= l-exp(-) n.)n-i)! (9)

where n = 1, 2, ... and nlnQ<v<.

In the case Sz0 (forecasting zero time into the future), we see thatInQ=O and that the results in this paper generalize certain results in [3]and [2] on inferences for current system reliability. In the case S>O, theabove inequality nlnQ<v< implies that N has finite support, given V=v:

Pr(0<N<v/lnQIV=v) = 1. (10)

Given V=v, let G(v,S) be the greatest integer less than v/InQ if S>O andG(v,S) if S = 0.

We can now write down the conditional distribution of N qiven V=v as

p(n;R) £ Pr(N=nlV=v, N>0)

(RTQ)n (v-nlnQ)n- /n!(n-l) (ll)G(v,S)I (RTQ)k (v-klnQ)k ' /k !(k- l )!

k=1

9

where n = 1, 2, .... G(v,S). This expression for p(n;R) can be readii 1-uated at minimal cost with an electronic computer.

3.3 Inferential Procedures

A conservative I-q confidence interval for R can be constructed by

G(vS) nobtaining values R1 and R2 which satisfy I (k;R 1 ) = LI and X p(k;R 2) 2

k=n k=1

where cl + a2 = a. The corresponding confidence bounds for R-I (the MTBF at

test time T+S) are R2 and R1 -I Because N is a discrete random variable,

construction of exact confidence intervals would require ranOLmization. AUMPU test of H0 :RR 0 versus HI:R>R 0 at significance level a calls for rejection

G(v,S)of H0 if p(k;R 0 )5a. Other UMPU hypothesis tests can be constructed in

k=n-l

a similar manner. If R0 is the MTBF goal for the end of the test phase

(time T+S), then the risk of not achieving this goal may be evaluated asn

p(k; R0 ).k~l

4. EXAMPLE

Suppose that a reliability growth test phase has been in progress forT=200 hours and is scheduled to continue for another S=200 hours. From thetest data up to time 200, we wish to obtain an 80 percent confidence intervalfor the MTBF at time T+S = 400. The following failure times ti were recorded

(n=21): 2.2, 3.3, 4.5, 5.3, 5.8, 20.3, 27.4, 34.1, 55.2, 58.4, 61.4, 62.,78.3, 78.4, 91.9, 97.7, 112.4, 116.9, 142.4, 176.8, 181.5.

Equations (3) and (4) yield 6 = .591 and - 21.4, and it is also ofinterest to observe that v/lnQ = 72.3. Thus G(v,S) = 72, so that the set ofpositive integers less than or equal to 72 is a support of the conditionaldistribution of N given V=v.

With Equation (11) we obtain by iteration the values R2- 12.7 and72 21

R1 = 38.6 such that p(k;R = .10 and 2 p(k;R 2 = .10. The intervalk=21 k=l

(12.7, 38,6) is therefore an 80 percent confidence interval for the MTBF attime 400.

By successively taking S=O and S=100, we can obtain in a similar manner80 percent confidence intervals (10.7, 26.0) and (11.9, 32.7) for the !ITBF attimes 200 and 300, respectively. All three confidence intervals are shown inFigure 2 for comparison purposes.

10

40

L

20- -

z

UjW

0

0 200 400

TEST HOURS

FIGURE 2. Eighty Percent Confidence Intervals for Currentand Future MTBF Parameters.

Suppose further that an MTBF goal of 15.0 has been set as a milestone forthe end of the current reliability growth test phase (T+S n 400). Based onthe data up to time T = 200, the risk of not achieving this qoal is

1 p(k;15 - ) - .20. In view of such a result, the program manager should feelk=1optimistic about this aspect of the development proqram, but will probably wantto avoid any actions which might adversely affect the overall reliabilityenhancement effort.

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REFERENCES

I. Department of Defense, Reliability Growth 'Management, Military Hanu,189, Naval Publications and Forms Center, Philadelphia, PA, February

2. Bain, L. J. and M. Engelhardt, "Inferences on the Parameters and CurrentSystem Reliability for a Time Truncated Weibull Process, Technometri- ,Vol. 22, pp. 421-426, August 1980.

3. Crow, L. H., Confidence Interval Procedures for Reliability GrowthAnalysis, Technical Report No. 197, U S Army Materiel Systems AnalysisActivity, Aberdeen Proving Ground, Maryland, June 1977.

4. Miller, G., "Efficient Methods for Assessing Reliability," Proceedings ofthe Nineteenth Annual U S Army Operations Research Symposium, Part III,pp. 33-42, October 1980.

5. Snyder, D. L., Random Point Processes, John Wiley and Sons, New York,NY, 1975.

6. Lehmann, E. L., Testing Statistical Hypotheses, John Wiley and Sons,New York, NY, 1959.

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13

DISTRIBUTION LIST

No. 0tConies Oga n i zati on

1? CoimmanderDefense Technical Information CenterATTN: DDC-TCCameron StationAlexandria, VA 22314

CommanderU S Army Materiel Developmentand Readiness Command

ATTN: DRCPA-S5001 Eisenhower AvenueAlexandria, VA 22333

CommanderU S Army Electronics Research

and Development CommandATTN: DRDEL-AP-QA2800 Powder Mill RoadAdelphi, MD 20783

DirectorU S Army TRADOC Systems Analysis ActivityATTN: ATAA-SLWhite Sands Missile Range, NM 88002

DirectorU S Army TRADOC Systems Analysis ActivityATTN: ATAA-TWhite Sands Missile Range, NM 88002

2 ChiefDefense Logistics Studies Information ExchangeU S Army Logistics Management CenterATTN: DRXMC-DFort Lee, VA 23801

CommanderU S Army Concepts Analysis Agency8120 Woodmont AvenueBethesda, MD 20014

Reliability Analysis CenterATTN: Mr. I. L. KrulacGriffiss AFB, NY 13441

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I Director, BRL, Bldg 328

1 Director, BRLATTN: DRDAR-TSB-S (STINFO Branch)Bldg 305

1 Director, HEL, Bldg 520

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