AD-Ai42 385 DETERMINING THE TOTAL SHIP RELIABILITYSDISTRIBUTION(U) /ASSOCIATION OF SCIENTISTS AND ENGINEERS OF THE NAVAL

SEA SYSTEMS COMMAND WASHINGTON DC P HARTMAN ET AL.

UNCLASSIFIED MAR 84 F/G 13/i8 NL

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AD-A142 385ST

ANNUALTECHNICALM

U 1984

-"TH ENGINEERING CHALLENGE OF A GROWING NAVY

gDETERMINING THE TOTAL SHIP RELIABILITY DISTRIBUTION12 Dr. Patrick Hartman

* ... JPalmer LuetienMorton Buckberg .

C-3

ASSOCIATION OF SCIENTISTS AND ENGINEERS Of THE NAVAL SEA SYSTEMS COMMAND * DEPARTMENT OF THE NAVY * WASHINGTON. D.C 203

Determining The Total Ship Reliability Distribution

Patrick HartmanMechanical Engineer

Palmer LuetienOperations Research Analyst

and

Morton Buck bergAssistant for Reliability and Maintainability

Mater ials and Assurance Engineering OfficeNaval Sea Systems Command

March 1984

% Approved for Public ReleaseDistribution Unlimited

The views expressed herein are the personal opinions of theauthors and are not necessarily the official views of theDepartment of Defense or of the Department of the Navy

V "

4

ABSTRACT .

"Acquistion cost and reliable equipment performance are twoprimary considerations in procurement decisions for ships and shipsystems. An important question in this decision process is, howdoes the variation in equipment failure rate (performance quality)effect the satisfactory operation of the total system. Untilrecently the effect of equipment performance quality has beendifficult to predict. 0

A new method of analysis has been developed which modelsvariation in equipment failure rate 44dTBF))with the gammadistribution, calculates the system's response to this MTBFvariation, and replicates the performance of the parentdistribution with the cumulative beta distribution. Oncedetermined, the cumulative beta distribution enables the engineeror naval architect to easily calculate the upper and lower limitsof performance for the parent reliability distribution.

The distribution of MTBFs for 115 "identical" commercialcomputers is shown as an illustration of equipment performancequality variation discussed. Example cases using the gamma andbeta distributions are shown to demonstrate ease of use

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Table of Contents

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"I. Introduction . . . . . . . . . . . . . . . . .. 1

II. Reliability Impact of Spares Quality ........... 1III. Distribution of Spares MTBFs .................... 2IV. Modeling System Sensitivity with the

Cumulative Beta Distribution ................... 3 k"V. Calculating the Reliability

Distributions For Very Large Systems ........... 5VI. Conclusions .................................... 5

VII. References ..................................... 7

List Of Fiauresl

Figure 1 - TIGER Computer Program FlowFigure 2 - NAVSEA TIGER R&M AnalysesFigure 3 - The Increased Maintenance Burden

Caused by Lowered Equipment MTBFsFigure 4 - MTBF Data for Commercial Computers Fit

with the Gamma Cumulative Distribution FunctionFigure 5 - Typical Shape of the Gamma Probability

Distribution Function (pdf)Figure 6 - Series and Standby Reliability Block Diagrams (RBDs)Figure 7 - Example Cases Showing How Various MTBFs Can Occur

for any Given EquipmentFigure 8 - Probability Density Functions (pdf) of the Series

System for Two Different Mission DurationsFigure 9 - Reliability Performance Data and Cumulative Beta

Distribution for the Series SystemFigure 10 - Cumulative Beta Distribution for the Standby SystemFigure 11 - Range of TIGER Applications

List Of Tables

Table I - Varied Reliability Performance for the Example SystemsGiven Gamma Distributed Equipment MTBFs

Appendix A - Reliability and the Gamma and Beta DistributionFunctions

Figure A-i Reliability of an Equipmentwith and without a Standby Versus Time

Figure A-2 Weibull Plot of Commercial ComputerMTBF Distribution

ii

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I. IntroductionIncreasingly sophisticated computer simulations have been

developed by HAVSKA 05MR to predict the reliability,maintainability, and availability (RHA) of total ships based onthe failure and repair characteristics of the hundreds ofessential equipment which make up the ship.

The most sophisticated of these simulations, TIGER,(References 1,2) has been used to analyze each Navy ship classconstructed during the past decade as exemplified in Figures 1 and2. This computer program can be used to analyze the sensitivity oftotal ships and systems to variations in equipment performancequality. The distribution of mean time between failures (MTBFs)of 115 midentical" commercial computers (Reference 3) is shown asan illustration of the "quality" variation discussed.

II. Reliability Impact of Spares Quality

Consider the complexity associated with supplying Navycombatant ships with spare parts for hull mechanical andelectrical (HM&E) machinery. For example, failure of the mainpropulsion system would seriously degrade a ship's ability toperform a primary warfare mission. Spares are carried at sea forroutine maintenance. The supply system allocates spares for eachmaintenance level. The three levels of maintenance are:

o organizational (ships crews)o intermediate (tender or Navy

specialized activity)o depot (Navy or commercial shipyard

or industrial activity)

In this paper we will concentrate on parts support for theorganizational level of maintenance.

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HM&E equipment is designed integral to the ships hull and is .--

intended to have a life equal to the service life of the ship. Itis not cyclic; it does not become obsolete every ten years, but itdoes require substantial maintenance and overhaul and in manycases survives longer than its original manufacturer.

Essentially, preventive maintenance tasks consist of cleaning,purification, lubrication, packing and sealing, and conditionmonitoring. Navy ships because of their global commitments maytake on a variety of contaminated substances. The most criticalto the ship's performance and equipment life is main propulsionfuel oil where the contaminants include sulphur, dissolved metals,sludge, seawater, etc. If the preventive maintenance tasks arenot completed properly on a rigorous schedule, failures are

'* induced prematurely and wear-out begins. Overhaul or replacementis required taking the combatant out-of-service. What happens thento a fifteen year old main propulsion diesel engine, overhauledtwice, showing an increased failure rate, and a decreased timebetween overhauls. Where do the spare parts come from? Who makesthem? How reliable are the repair parts/components, etc.

If the Navy owns the manufacturing drawings, competitiveprocurement is required by law and the lowest bidder "qualifiednreceives the award. Does the drawing contain sufficientinformation to enable replication of the original part? Mostoften the answer is no. Proprietary manufacturing processes,tolerances and material-treating vary considerably amongsuppliers. The resulting parts are look-alikes, but will theirMTBFs live up to the original parts. For example, an attachedlube oil pump manufactured by other than the original equipmentsuppliers does not have a preproduction unit tested by mounting onthe engine and subjected to the environmental stresses of theoriginal pump. How long it will last is not determined untilplaced in service. There is not sufficient incentive for themanufacturer to produce more reliable, better quality part and theNavy does not normally test part reliability and quality prior toFleet use. The results can be catastrophic to ship availabilityand performance - an unacceptable operational limitation. Wemust be able to predict parts criticality; predetermine the costpenalties for poor reliability and quality, and standardize highreliability and quality into the parts procurement process. Figure3 shows the drastically increased maintenance burden incurred whenthe MTBF of replacement parts does not meet the value to which thesystem was designed.

III. Distribution of Snares MTBFs

There is considerable scatter in the failure data used toobtain the *point estimate" of MTBF. Each unit is unique andconstruction or use can produce variation in the distribution ofMTBFs. Good quality control in manufacturing and rigorousoperating philosophies can reduce the variance and result in a

2

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sharply peaked probability density function (pdf) with a samplepopulation concentrated around the mean of the distribution. Morescattered MTBF distributions have been observed in actualpractice. Reference 3 presented a histogram of MTBF's of 115"identical" computers in which the range of MTBF's extended from200 to 7200 hours, mean 1740 hours, standard deviation 1450 hours.This histogram data has been reduced and fit with the gammadistribution which is defined for MTBFs ranging from zero toinfinity. (References 4, 5, & 6, see Appendix A). The cumulativedistribution function (CDF) of this data is plotted in Figure 4.Figure 5 depicts a case in which there is considerable scatter inthe MTBF values, with the majority of the MTBFs falling below 2000

A.. hours and a few beyond 4000 hours. In a distribution, like this,skewed to the right, (Figure 5) the median and mode both occur atvalues less than the distribution mean.

The mean and unbiased standard deviation of the histogramdata were used to determine whether or not the two parameter,integer gamma distribution fits the observed MTBF distribution.Figure 4 shows that the gamma distribution with an integer shapefactor of b=2.0 fits the general distribution of the data quitewell. (See Appendix A and Reference 7 for analysis used.)

If there is very little deviation in MTBF and the distributionapproximates a spike, the point estimate yields a very good

'-.. prediction of operation in the field, but the variation shown inFigure 4 significantly effects the reliability of the system. Aspointed out previously, an MTBF distribution skewed to the right(gamma) has significantly more than half of its MTBF values belowthe mean. Therefore system reliability predictions from the pointestimate will be higher than actually observed in the field morethan half the time.

IV. Modeling System Sensitivity with theCumulative Beta Distribution

Overall, the gamma distribution closely represents "realworld" variation in these MTBF values and allows direct use inconjunction with existing total ship reliability simulationsthrough the use of a compact subroutine developed in References 8and 9 for high speed computer generation of gamma distributedvariates. The inputs for this subroutine are the mean andstandard deviation of the underlying distribution.

Test cases have been prepared to illustrate the methodologyrequired to determine total system sensitivity to variation inequipment MTBF. System . of Figure 6 consists of two differenttypes of equipment (X and Y) each of which exhibit exponentially(References 10 & 11) distributed failure probabilities with MTBFsof 150 and 200 hours respectively.

Ixample cases are calculated using two reliability block

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TK7diagrams (REDs) of Figure 6 for equipment X and Y; a simple seriessystem (System 1) and a series system in which each block has astandby to be operated upon failure of the first (System 2).

Table 1 shows significant variation in reliability at 25 and100 hours (mission time) for this system when its t4TBFs follow agamma distribution as might happen when spares are procurred atdifferent times from various manufacturers (Figure 7). ByEquation l,of Appendix A, equipment X has a point estimate

*reliability of 0.85 at 25 hours and equipment Y yields 0.68. Thereliability of System 1 is the product of these two reliabilities

* or 0.75, but not many of the units have met the reliabilitypredicted from the point estimate.

For the 100 hour example, less than 50%i of the samples have areliability of 0.2 or better. Table 1 illustrates thedistribution of 25 and 100 hour reliabilities obtained with thegamma distributed bITBFs presented earlier. To determine the

* . distribution of System 1 reliabilities, the data was analyzed andfit with the beta distribution which is ideally defined for thezero to one range of the reliability parameter. (References 4, 5,

* & 6, see appendix A).

By calculating the mean and unbiased estimates for thestandard deviation and variance of the reliability data, the two

N parameters of the beta distribution can be obtained. (Reference12). Figure 8 shows the probability density functions obtainedfor the time data in Table 1 using the beta distribution. Figure9 shows the cumulative beta distribution (incomplete betafunction) for 25 and 100 hour System 1 reliability data. Thedistribution (solid line) passes through the actual data pointstaken from Table 1, median ranked as in Reference 13 to determineeach point's representative portion of the total population CDF.

The cumulative beta distribution accurately estimates thesystem's reliability distribution and provides for calculation of

V the percentile of this reliability distribution. As shown inFigure 9, 90% of the population (10th percentile) has areliability of 0.50 or greater after running 25 hours.

These examples indicate that the cumulative beta distributionestimates the percentiles for a system reliability distribution.The procedure for determining the reliability distribution isstraight-forward and computationally efficient. Only ten

predictions of system reliability were required for the test case *4. shown.

ofFigure 10 shows the enhanced reliability increase performanceofthe System and quantifies the increased reliability gained by

* installing a standby unit as a back up. The reliability increaseobtained by adding a standby unit is evident from the shift in 25and 100 hour reliabilities to higher levels than those achieved

IN 4

-- -]

with the simple series system in Figure 9. Note that thecumulative beta distribution is capable of fitting the reliabilityperformance distributions of high and low reliability systems.This is a feature of the right and left hand symmetry(coefficients A and B in Equation 4 References 5 and 10; seeAppendix A) that exists with the beta distribution.

V. Calculatina The ReliabilityDistributions For Very Large Systems

Simple hand calculations are not sufficient to predict thereliability of these complex systems References 14 and 15.Simulations such as NAVSKA'S TIGER computer program (Reference

1) calculate the failure and repair characteristics of systemscontaining several hundred to several thousand different types ofequipment (see Figures 2 & 13). Note that in these systems,complex operating rules are required to efficiently perform theanalysis of the time dependent reliability. The range of TIGERanalyses covers the transistor level, ship level, and multi-shiptask force R&M assessments.

The quality of ship board spares and their MTBF variation doesmake a difference in the distribution of total system reliabilityperformance. Significantly different MTBFs can be obtained with"look-alikes" from different vendors. This investigation showedthat the real world data is skewed and that the MTBFs of somehighly complex equipment were gamma distributed. Reliabilitycalculations based on the mean of the MTBF distribution can besignificantly higher than those determined with an MTBFdistribution skewed to the right as with the gamma. This pointsout a probleml since spares may not live up to the MTBFs of theoriginal equipment, the in service reliability may besignificantly lower than predicted unless the variance of theMTBFs is symmetrically distributed.

The NAVSEA TIGER R&M computer program used with gammadistributed MTBFs and the cumulative beta distribution enables theengineer and architect to rapidly determine the upper and lowerlimits of total system reliability performance.

Very few calculations were required for the example casesshown here. The only data required to calculate the betadistribution parameters were the mean and unbiased estimate of thestandard deviation of the system's reliability values as obtainedfrom the standard NAVSKA TIGER R&M computer runs. The twoparameter oumulative beta distribution fits the reliability datawell and provides an easy method of calculating the percentiles ofthe parent distribution for various forms of distributed equipment

1..data.

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VII. References

1. TIGER MANUAL, NAVSEA Report No. TE660-AC-MMD-010t p* Washington, DC, 20362, October 1980.

Z. Hartman, P.3., Luetjen, P., and Mandel D., "ReliabilityAnalysis of Total Systems Using Component Failure Data,' ASMEJournal of Enaineerina Resources Technoloay, Vol. 105, No. 2,June 1993, pp. 217-221.

3. Goldberg, H., Extendina the Limits of Reliability Theory,Wiley, New York, 1981, pg.2.

4. Abramowitz, M. and Stegun,, I. A., ed., Handbook ofMathematical Functions, National Bureau of Standards AppliedMathematics Series 55, U.S. Government Printing Office.Washington, DC 20402, May 1968, pp. 257, 258,263, 944, 945.

a.. 5. Ireson, W. G., ed., Reliability Handbook, McGraw-Hill, NewYork, 1966, pp. 4-37 to 4-50.

6. Hogg, R. V. and Craig, A. T., Introduction to Mathematical.S.atistics, MacMillan, New York, 1978, pp. 138-139.

7. Neibull, N., *A Statistical Distribution Function of WideApplicability,* ASME Journal of Applied Mechanics Vol. 19,September 1951, pp. 293-297.

8. Naylor, T. H. at.al., Computer Simulation Technioues,Wiley, New York, 1966, pp. 87-89.

9. Scharge, L., "A More Portable Fortran Random NumberGenerator," ACM Tranactions on Mathematical Software, Vol.5, No. 2, June 1979, pp. 132-138.

10. VonAlven, N.H., ed., Reliability Engineerina, PrenticeHall, Englewood Cliffs, New Jersey, 1964, pp. 234-267.

11. Anstadter, B.L., Reliability Mathematics, McGraw-Hill, New

York, 1971, pp. 121-148, 265-257.

12. Hartman, P. J., "Noise Induced Hearing Loss Resulting fromCumulative Fatigue Damage of Cochlear Hair Cells," Ph.D.

Dissertation, University of Rhode Island, Kingston, RhodeIsland, June 1976, University Microfilms International 37/05,P2390-B Order No 76-25,521, pp. 136-137.

13. Johnson, L. G., "The Median Ranks of Sample Values in TheirPopulation With an Application to Certain Fatigue Studies,"Industrial Mathematics, Vol. 2, 1951, pp. 1-9.

6

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".,.,."14. REX Users Manual, NAVSEA Report NO. TE6S0-AC-MMD-010,' " Washington, DC, 20362, October 1990.15. Luetien, P. and Hartuan, P. J., "Simulation with the

Restricted Erlang Distribution," Proc. Ann. Reliability &• " .Maintainability Symp., 1992 January, pp. 233-237.

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Reliability and the

S4 Gamma and Beta Distribution Functions

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..-""' NOMENCLATURE: -

a gamma distribution, scale factor

b gamma distribution, shape parameter .

A beta distribution, first parameter

B beta distribution, second parameter

C mean of the beta distribution 0,

D variance of the beta distribution

" 2.7182... base of natural system oflogarithms

f(t) probability density function (pdf)of the failure distribution at time t

F(t) cumulative distribution function(CDF) of the f(t)being examined, the probability that an equipment hasfailed at or before time t

r( ) gamma function

X(t) instantaneous failure rate for exponential systems

m arithmetic mean of the data set

MTBF Mean Time Between Failures

MTTF Mean Time to (First) Failure

N number of failures or number of units failed

4

pdf probability density function •

R reliability 1 1-Pr(fail) -t

. - f(t) dt - 1-F(t)f0unbiased standard deviation of the data using N-1 samplefactor

t time in hours

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RELIABILITY CHARACTERISTICS:

For the exponental case, the reliability of any singleequipment is (References 10 & 11)*:

(- x t)R e (1)

How long an indiviaual equipment performs depends on its meantime between failures (MTBF) for repairable units and means timeto failure (MTTF) for nonrepairablo cases (1). MTBF is defined

:as

MTBF - It/number of failures (2)

The MTBF calculation is often history dependent and exhibitsan increasing hazard rate since the success paths in most simplesystems having redundancy decrease with time (10). In the case ofthe exponental equipment, the hazard rate is constant and thusMTBF equals WTTF where:

MTTF - It/number of units tested (3)

System performance can be enhanced by increasing the MTBF orMTTF of the individual units through better design, less severeoperating environments, and by adding standby units to take overupon failure. Figure A-1 shows that standby redundancy iseffective, but at substantial cost; e.g., another unit,installation cost, supporting structure, and maintenance burden.

R +e tote (4)

The added reliability obtained by adding a standby unit is evident

in Equation (4).

* References for appendix are the same as shown in the body ofV the paper.

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GAMMA DISTRIBUTED MTBFs:

The "real world" differs from Equation (2) in that there isconsiderable scatter in the life data used to obtain the "pointestimate" of MTBF. This fact was graphically presented in ahistogram of MTBFs for 115 "identical" computers in Reference 3.The data from this histogram has been extracted and presented inTable A-1.

-: -: The case in which there is considerable scatter in the MTBFvalues, with the majority of the failures taking place quite earlyand a few units lasting much longer. In a distribution skewedthis way (right), the median and mode both occur at values lessthan the distribution point estimate (the mean, Equation (2)).

This type of MTBF distribution has been observed in actualpractice. The mean and unbiased standard deviation of the dataare used to determine the two parameter, integar gammadistribution fit to the observed MTBF distribution. The gamma

a, distribution (References 4, 5, and 6) equals:

b (b-i) (-ar), <..,(5)

Cl r (Wi

where; 0 t (CD

a ', 0 andb 0.

for"Some of the deviation from the actual data can be accountedfrby the use of an integer shape factor b-2.0 versus the 1.4

calculated from the mean and standard deviation of the data set.Additional divergence appears to be caused by the complex(bimodal) distribution of the data set itself which is evident astwo distinct, straight lines when plotted on Weibull paper asshown in Reference 7.

The bimodal attributes of the computer MTBF data ischaracterized by two distinct straight lines. Note that the righthand curve (1000 to 10000 hour MTBF) has a Weibull shape slightlygreater than 1.0 indicating an exponential distribution whereasthe left curve is definitely non-exponental with a shape of 2.0.(Figure A-2)

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BETA DISTRIBUTION

Reliability values obtained from the parent distribution arefit with the beta distribution (References 4, 5, 6);where the beta distribution is defined as:

r' (A+5) (A-)) (B-1)f(y:AB) - .. .. (y (I-y) ) (6)

r (A) r (B)

where; 0

y is the variable of interest and J(A) is the gama gunction withA as its argument.

To provide a replicate of the parent. The mean of the betadistribution is:

AC

,M

A + B (7)

and the variance is:A B 72

D-s (8)2

[(Ai+) (A&+ I)]

Solving these Equations (7 and 8) for the two unknowns (A and B)as a reference (II)s

(C2 - CD)

2 A

D (9)

and

2 3(C 2C * C + CD D()

By calculating the mean and unbiased estimates for the standard

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deviation and variance of the reliability data, the parameters Aand B of the beta distribution can be obtained.

The cumulative beta distribution accurately quantifies thesystem's reliability distribution and provides for the calculationof the lower confidence limits of system performance.

The cumulative beta distribution is:

F(R;AB) f(y;A,B)dy (11)

0

where:

0 s R 1 1,A t 0, and B ' 0.

The replicate distribution of the parent distribution is aspecimen of the parent. The replicate distribution will have 0number of samples with each sample consisting of one observation(sample size N-1). For a continuous parent distribution samplingmay be with or without replacement. The replicate asymptoticallyaproaches the parent as the number of samples ,Q, increases.

The beta fit method, for prediction, is used in lieu of thecoverage calculations, for demonstration, using order statisticsRef. 4. The beta percentile is used as the estimator for thefractile of the parent distribution.

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Table A-1 MTBF Samples From 115 "Identical" Computers

For One Year's Operation

CumulativeMTBF Frequency Distribution

Class Mark** of occurence Function(Hours) (fi)

n

50 0 0.0150 2 0.02250 3 0.04350 7 0.10450 5 0.15550 7 0.21650 5 0.25750 9 0.33950 2 0.35950 2 0.37

1050 6 0.421150 5 0.461250 8 0.531350 4 0.571450 2 0.58

1550 3 0.611750 1 0.621850 5 0.661950 1 0.672050 4 0.702150 6 0.762250 2 0.772450 1 0.782550 3 0.812650 1 0.823150 5 0.863350 1 0.873550 1 0.883650 1 0.893750 1 0.90

") 4050 2 0.914450 2 0.934550 3 0.964650 1 0.975450 1 0.975650 1 0.987050 1 0.997150 1 1.0

** 100 Hour Class Interval Referenoc 3.

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CUMULATIVE DISTRIBUTION OFCOMPUTER MTBFs S

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