RELIABILITY DATA ANALYSIS MODELThere are a number of reasons why the constant failure rate assumption is so often made. First, the constant failure rate is often assumed primarily

/4 P 7<$ ++/ RIA-80-U289

ACS Technical Library

5 0712 01013404 6

R-WV-T-6-22-73 AD

TECHNICAL LIBRARY

RELIABILITY DATA ANALYSIS MODEL

R.RACICOT

BENET WEAPONS LABORATORY WATERVLIET ARSENAL

WATERVLIET, N.Y. 12189

JULY 1973

TECHNICAL REPORT

AMCMS No. 611102.11.85300.01

DA Project No. 1T061102B14A

Pron No. B9-2-67162-01-M2-M7

APPROVED FOR PUBLIC RELEASE DISTRIBUTION UNLIMITED

DISPOSITION

Destroy this report when it is no longer needed. Do not return it

to the originator.

DISCLAIMER

The findings in this report are not to be construed as an official

Department of the Army position unless so designated by other authorized

documents.

Unclassified Security Classification

DOCUMENT CONTROL DATA -R&D (Security clsasification ol title, body of abstract and indexing annotation mumt be entered whan the overall report la clmaelfied SL

I. ORIGINATING AC Tl vi TV (Corporate sulhor)

Watervliet Arsenal Watervliet, N.Y. 12189

24. RlfO»T ItCUBITY CLASSIFICATION

2b. GROUP Unclassified

3. REPORT TITLE


4. DESCRIPTIVE NO T E s (Type ol report and Inclutt v» dates)

Technical Report 3 AUTHORIS) (Fire! name, middle Initial, Immlnmmm)

R. Racicot

« REPORT OA TE

July 1973 7a. TOTAL NO OF PASES

_8JL

lb. NO. OF RIFI

-2JL »a. CONTRACT OR GRANT NO

AMCMS No. 611102.11.85300.01 b. PROJEC T NO.


Pron No. B9-2-67162-01-M2-M7 d.

9a. ORIGINATOR'S REPORT NUMBERIS)

R-WV-T-6-22-73

96. OTHER REPORT NOW (/Inv olh« numtara that may be aeeljned thla report)

10. DISTRIBUTION STATEMENT

Approved for public release; distribution unlimited.

II- SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

U. S. Army Weapons Command

13. ABSTRACT

A computer model has been prepared to assist in the analysis of reliability test data. Essentially, the model computes point estimates and confidence limits for mission reliability of components, subsystems, and systems from component failure data. The main features of the model are: 1. Performs goodness-of-fit tests to determine the best fit probability distribution of component failure times, 2. Computes maximum likelihood estimates of distribution parameters, 3. Computes point estimates of reliability for the renewal nonconstant failure rate case, and 4. Computes lower confidence limits for component, subsystem, and system reliability for the constant failure rate case.

DD FORM I MOV •• 1473 RF.-LACE4 DO FORM 1471. I JAN 44. WHICH IS

oasOLKTS FOR ARMY USB.

^cSrU^ftjA,

Unclassified Security Classification

KEY WO*DS

Reliability

Statistics

Probability

Distribution Selection

Computer Model

Failure Rate

Bayesian Statistics

Hnrlassifif-d Security Classification

R.WV-T.6-22-73


R.RACICOT

BENET WEAPONS LABORATORY WATERVLIET ARSENAL

WATERVLIET, N.Y 12189

JULY 1973

TECHNICAL REPORT

AMCMS No. 611102.11.85300.01


Pron No. B9-2-67162-01-M2-M7

APPROVED FOR PUBLIC RELEASE DISTRIBUTION UNLIMITED

RELIABILITY 'MTA ANALYSIS ''OPEL

ABSTRACT Cross-Reference

Pat a

A computer model has been Prepared to assist in the analysis o* reliabilitv test data. Essentially, the model confutes noint estimates and confidence limits for mission reliability of comnonents, subsystems, and systems from component failure data. The main features of the model arc: 1. Performs ôodness-of-fit test-s to determine the best fit probability distribution of component failure times, 2. Commutes maximum likelihood estimates of distribution parameters, 3. Comnutes noint estimates cf reliabilitv for the renewal non- constant failure rate case, and 4. Confutes lower confidence limits for comnonent, subsystem, and system reliability for the constant failure rate case.

peliability

Statistics

">robabilitv

Distribution Selection

Computer Model

Failure Rate

Bavesian Statistics

FOREWORD

This project was supported in part by the Project Manager's

Office for Close Support Weapon Systems and by USASASA for the Future

Rifle System.

The author would like to thank R. Hagjjerty ^d R^ Scanlon of the

Benet Weapons Laboratory for preparing a number of the comnuter sub-

routines used in the model.

TABLE OF CONTENTS

Introduction 5

Purpose of Reliability Corn-nut er Model fi

Tyne of Data Considered for Analysis r,

Type of Information Generated 7

Usual Assumptions for Data Analysis and Their Limitations ^

Main Features and Status of the Present Model \/\

Analytical Methods 15

Computation of Component Mission Reliability 1C

(a) Component Reliability Based on Age 17

(b) Mission Reliability of a Component Within a System jo

(c) Numerical Solution for the Renewal Rate 9f>

(d) Numerical Solution for Mission Reliability ?R

(e) Average Mission Reliability ->0

Computation of System Mission Reliability 30.

Probability Distribution Selection 37

(a) Candidate Distributions 32

(b) Least Squares Fit of Data 35

(c) Computation of the Standard Error 3A

(d) Kolmogorov-Smimov Goodness of Fit 30

Maximum Likelihood Estimates of Parameters 4]

Bayesian and Classical Confidence Intervals for Component Reliability 45

Bayesian Confidence Intervals for System Reliability 5]

(a) Constant Failure Rate Components in Series r^j

(b) Non-Constant Failure Rate Components in Series 53

Computer Program 53

References 55

Appendix I-Input Format for Programs DISTSEL and RELIAB 57

(a) Overall System and Subsystem Data Cards 57

(b) Element or Component Data Cards 57

(c) Sample Problem 53

Appendix II-Output Data for Sample Problem 60

DD Form 1473

figures

1. Component Failure Rate vs Component Age 18

2. Component Failure Rate vs System Age for the Case of Ideal Repair 21

3. Component Renewal Rate vs System Age 25

4. Component Reliability vs System Age 25

Tables

I. Average System Reliability for Equal Components Having a Weibull Distribution of failure Times \n


INTRODUCTION

Current Army regulations 111 reouire the snecification of ouanti-

fied reliability goals in the develonment of new weanon systens.

rhese reliability goals must also be verified bv tests nrior to final

acceptance and fielding of the systen. Reliability, bv definition,

is a nrobabilistic quantitv. In addition, the amount of testing that

can be conducted is limited by cost and time considerations and conse-

quently reliability assessment must be conducted within a statistical

framework. To assist in this assessment effort a statistical data

analysis computer model has been nrenared and is described in this

report. Essentially, the model computes noint estimates and confi-

dence limits for reliability of components, subsystems and system

from component test data.

The snecific reliability indnx considered is the mission relia-

bility defined as the probability that a given system will success-

fully perform its intended function without failure for a specified

period of time under a given set of conditions. Time can be given

in terms of clock time, rounds, cycles, miles, etc. The svstem mav be

required to nerform a number of missions over its exnected life.

Army Regulation AR70S-50, "Army Materiel "el i abil itv and ''attain- ability," Headquaters Dent, of the Army, Washington, D.C., 8 January 1968.

PURPOSE OF THE RELIABILITY COMPUTER MODEL

The reliability model is intended to serve a number of useful

purposes. First, reliability analyses for the realistic situations

encountered in weapon system testing and for all but the simplest

assumptions made for computation entail rather complicated mathemati-

cal formulation and tedious computation. The computer model conse-

quently provides a systematic means of incorporating more complex

analyses with less restrictive assumptions into a routine data analy-

sis procedure. Second, the model provides greater speed in obtaining

results of data analyses. Third, the mathematical procedures and for-

mulations incorporated into the model provide a means of standardizing

reliability assessment among design agencies, test agencies and the

user. Finally, the model provides a basis for determining the type

and amount of test data required to perform a particular kind of

analysis. This provides insight into planning of tests and data

collection procedures required.

TYPE OF DATA CONSIDERED FOR ANALYSIS

The type of data considered for analysis is failure and suspen-

sion data on a component or a number of components making up a sub-

system or system. Definition of failure depends on the particular

requirements and the mission profile of a givep system and can

include part breakage, out-of-tolerance performance, incipient

failure, safety hazard, etc. Suspended data is obtained when a compo-

nent is removed from the test without failure, the test is terminated

without failure or when the failure of a component is attributable to

the failure of another component or cause.

Since failure rates and resulting mission reliabilities are

generally transient, all data is assumed collected as a function of

system age. Snecifically, the age given in terms such as rounds, time

or miles on individual romnororts within a systor at failure or sus-

pension is required. In pen^ral if is not enough to record onlv the

total test time and number of failures except in the case of constant

failure rate.

TYPE OF INFORMATION nr. NEGATED

The tvpe of information generated bv the model includes results

of analyses for each individual component within a subsvstem, for

each subsystem making UP a given svstem and for the entire system.

Component information includes results of distribution selection

procedures, Kolmogorov-Smimov goodness-of-fit tests, maximum likeli-

hood estimates of distribution parameters, noint estimates of the

mission reliability as a function of svstem age ror the non-constant

failure rate case and confidence limits on average mission reliahilitv

over the system life for the constant failure rate case. Subsystem

and system information generated includes no^nt estimates and confi-

dence intervals on mission reliability for the constant failure rate

case.

A more complete description of required input data for the model

and the resulting output information will be presented in subseouent

sections dealing with the computer program.

USUAL ASSUMPTIONS FOR DATA ANALYSIS AND THEIR LIMITATIONS

It is worthwhile at this point to review the usual assumptions

made in performing a reliability assessment for a given system and

the desirability of relaxing these assumptions for some of the realis-

tic cases encountered in weapon system testing. Undoubtedly, the most

common assumption made in reliability data analysis is the assumption

of constant failure rate for components and systems. The most signi-

ficant implication of this assumption is that the probability of

failure of a component or system is independent of its age. Components

are assumed to fail at completely random points in time. This, of

course, does not permit the full treatment of the cases of early

failures and wear-out failures common to mechanical components.

There are a number of reasons why the constant failure rate

assumption is so often made. First, the constant failure rate is

often assumed primarily to simplify analysis and computation. Second,

straightforward techniques for determining confidence intervals on

component reliability from test data are readilv available for con-

stant failure rate. Third, testing procedures are greatly simplified

since only the total number of failures and total test time are re-

quired for data analysis. Also, the amount of data required for each

component is not as critical as in those cases where theoretical

8

distribution selection must be considered. Finally, for system relia-

bility verification tests, individual component data is not required

witb all failures being treated as system failures. This is true

onlv if all components have enua] test times.

Assuming a constant failure rate can lead to three tynes of

errors; the first is in the computation of reliability for fixed '1TRr

(mean time between failures), the second is in the computation of

confidence limits, and the third is in the estimation of 'TB^ when

susnension or censored data has been generated. The magnitude of the

error in computing reliability for fixeH ^(TRT7 cannot be determined ;n

general. For illustrative purposes, however, consider ^ hypothetical

case in which a system is made up of coual components in series.

Assume further that each component has a Weibull distribution and an

*1TBF (mean time between failure) enual to twice the expected system

life. The number of missions over system life is assumed to be ISO.

The magnitude of the error in assuming constant failure rate can now

be determined for this particular case. Table I lists the results of

average system reliability for different numbers of components in the

system and for various values of the Weibull shape narameter R. FOT

a shape parameter 8 equal to 1.0 the Weibull distribution reduces to

the exponential which describes the distribution for constant failure

rate. For values of R greater than 1.0 the failure rate is an increas-

ing function of time characteristic of wear-out phenomena. The greater

the value of R the more peaked the distribution is about the mean.

As can be seen from Table I, large differences can be obtained when

It--aHtîH«oocNN « N CO N C t«HNH OCOHrHVNM(N\COO

i-< m t~- oo c. c. c. c c c.

cococcoocc

u o z o l-l

00 p—I

c: p oo

HCII/INCHSVCIT, i/;

vC'S-^TfNvCKSvCOOC c

r- oo c. c ci c ci c- c ci

cccceecrcc <r

s

2-

00

IX!

oa <

ai Z C a.

c: U -J <r

o- u:

OL C IJ-

V F- >—i

•J w 03 < >—i

_•

uu a:

• IX! co H IX! to 5 >- i—i

OO H

u_- IX! (.' s <r rD a -J UJ i-<

> < < IX

c c C LO c

• • to i—I C •—I

hO

o c

II II II H

oo Z c t-H

IX. OO ao 00

t 1—1 IX! UJ T T U- i—i

| (-1

O H

IX) Z z 5 cc o £ IX! UJ 1—4

to C2 oo » B 00 o >- 1—1 u 00 z

< i—i ~3 IX! or

oo OO

1X3

oo >• 00

IS < UJ > <

oo

IX!

a.

6 u

1X3 CO

6 z

£ oo r^ r~ i-( c. <J M 00 M N tfC H\CM

00 C". c c c-

C t^ oo r: I>J c r»> c r~ r-i N to r vc oo \C 00 c c c. C. C. C Cr C. a c c. c e

cccccccccc

\CHH0OKlflWHTfM

(NNMNvCMyfioOCC S S f M» f . c-, Cl c a vC 00 Cs C C. C". C. C. C C ff. Ci ffl C C1. O: ffi ff, O C. • »•»••»••• coeceecccc

r-r^foc\ivcM\cooc. c. ve s * s « a o c c. f,

•-H| \C 00 Cl Cl Cl C o a, c-. c. Cl C7> Cl Cl Ci C. C. Cl C. Cl Ol O) O^ C) ff Cl C. C. C Ol • •»•••*••• ooccccoocc

a CLOQ rt X >-. 00 a>

•-> r-H 4) 1—1

3 1 Xi fc< •w rt a> a. S

cccoccooco HNIO^l/HDMOOlC

in

assuming exponential or constant failure rate for Weihul1 components

that in actuality have 3 values greater than 1.0, ôr example, a svs-

tem of SO equal components, each component having a Weihul1 share para-

meter equal to 3.0 would have an averarc svsten reliability of 0.072.

Assuming a constant failure rate op the other hand would yield a con-

nuted reliability of 0.846 which is significantly different than the

true value for this case.

Reliability for the Weihul 1 distribution withf,>l was computed

using renewal theory where, in this rase, a component is replaced or

renewed upon failure. The mathematical formulation for the renewal

case is described in the Analytical Methods section of this report.

The second source of error in assuming constant failure rate is in

the computation of confidence limits. Consider as an examnle nroblem

a test sample o^ failure times given as 2000, 2200, 1800, 1000 and

2100. For this nroblem the '1TBF is 2000 and the standard deviation is

158. Lower confidence limits on the true '1TBF can readily be computed

assuming different underlying distributions of failure times. ror

examnle, the lower 00° confidenced moans for the exponential (constant

failure rate)and the normal (increasing failure rate) distributions

are 1250 and 1758 respective^'. It is common in reliability verifica-

tion tests to rcnuire that the lower confidenced "ITBF (or reliability)

exceed a given fixed value for accentance. If the requirement in the

above problem were 1500, the assumption of the exponential distribution

would lead to rejection and the assumption of the normal distribution

would lead to acceptance. Although the above is an oversimplification,

11

it does indicate the possible error in assuming the wrong distribution

in determining confidence limits.

The third source of error in assuming constant failure rate lies

in the unrealistic handling of suspension data in estimating population

parameters. Consider as an example problem a test sample of failure

times given as 3800, 3900, 4100 and 4200 and a samnle of suspension

times given as 3500, 4000, 4000 and 4500. In this example, eight dif-

ferent components were tested but onlv four failed. The point esti-

mates of MTBF assuming exponential and Weibull distributions are 8000

and 4130 respectively. As can be seen the '1TBF for constant failure

rate is nearly twice the value for the nonconstant failure rate Weibull

distribution.

In the first two sources of error discussed above the computed

reliability is generally conservative for components and systems that

wear out. Reliability and lower confidence limits are conservative in

that lower than true values are computed. In the third source of error

the resulting reliability is generally nonconservative. The unrealis-

tic handling of suspension data seems to be the most significant

source of error in assuming constant failure rate for testing of

mechanical systems.

A second assumption or requirement which is often made in relia-

bility analysis is that the data sample be complete. Consequently

only the failure times in a given test are considered for analysis.

The main reason for this assumption is that statistical methods that

treat suspended data along with the failure data are in many cases

12

restrictive, complicated or not available. It is clear, however, that

susnended data, particularly data involving suspension times that are

greater than some or all of the failure times should contribute signifi-

cant information toward determininn the unknown population parameters.

This was observed in the above discussion of the constant failure rate

assumption. Suspepded data can result in manv ways. In weanon svstem

testing, for example, it cap arise from the removal of a comnonent from

test without failure, comnletiop of a system test nrior to failure or

through failure resulting directly from the failure o*" another compo-

nent or cause such as accident.

A third assurmtiop which is often made in determining confiHence

limits on system reliability from svstem tests is that the svstem can

be treated as if it were a single comnonent with no differentiation

being made as to which component within the svstem actually fails.

All failures are treated as svstem failures. A necessary underlving

assumption for this approach is that either all components have con-

stant failure rate or all components have failed a number of times so

that steady state conditions prevail. In this instance confidence

intervals on system reliability are readilv derived usin^ the theory

applicable to the constant failure rate case. A limitation o*" this

assumption is that results of individual component or subsvstem tests

cannot be included with the system test results. Mso, if components

are redesigned during the course of a test, which is often the case in

large weapon system testing, total test time on the redesigned compo-

nents are not the same as the total system test time. The usual

13

methods of analyzing system test data consequently do not annly in

this case.

Finally, large sample theory is often assumed in computing confi-

dence intervals since methods to handle small samples may not exist

or are too difficult to use.

MAIN FEATURES AND STATUS OP THE PRESENT MODEL

The reliability data analysis model presented in this renort con-

tains a number of general solutions to overcome many of the limitations

of the usual assumptions discussed in the previous section. The main

features of the model are summarized as follows:

a. Performs goodness-of-fit test to determine the best fit

probability distribution of component failure times. The theoretical

distributions considered are the exponential, normal, lognormal,

Weibull and gamma.

b. Computes maximum likelihood estimates of population para-

meters for general theoretical distribution of failure times.

c. Can handle suspended data which results when a component is

tested without failure.

d. Computes point estimates of reliability for the renewal case;

that is, for the non-constant failure rate case.

e. Computes lower confidence limits on component, subsystem

and system reliability for the constant failure rate case.

14

The present version of the reliability data analysis computer

model does not contain all o^ the features ultimately planned for in

the final version. Mnst significant o*" the limitations is the assumn-

tion of constant failure rate in determining confidence intervals

although point estimtes n* reliability are determined fnr the general

non-constant failure rate case. In addition components are assumed to

he in series for system reliabi]ity computation and no provision is

presently made for nreventive maintenance narts replacement of comno

nents. The analytical methods section o* this report descrihes the

techniques to be employed in removing these limitations with work on

future versions of the model presently being undertaken.

The remainder of this report contains a discussion of the analyti-

cal methods used for commutation followed by a presentation of the

computer program with example input and outout information.

ANALYTICAL METHODS

The purpose of this section is to briefly outline the mathematical

and statistical methods used or planned in the reliahility data

analysis model. A complete discussion of probability and statistical

theory will not be presented in this report and the reader is referred

to the references cited for more complete discussions. Much of the

theory presented is straightforward and readily found in the litera-

ture. However, some of the statistical and computational methods used

are the result of research efforts at the Watervliet Arsenal and will

IS

be the subject of forthcoming reports and publications.

There are a number of good texts on the general subiect of reli-

ability. The texts found particularly useful to this writer are those

by Lloyd and Linow [2], Barlow and Proschan [3], ^nedenko, Belyayev

and Solowev [4] and Pieruschka [5].

Computation of Component Mission Reliability.

Although one of the computational goals in th<* model is system

reliability it is necessary to develon the analytical methods for

computing system reliability by considering the individual components

or elements making up the svstem. This is narticularlv true in the

case of non-constant failure rate components.

At the present time there is considerable confusion among design

engineers on the definition of component reliability when the compo-

nent is part of a system. The confusion lies primarily on the time

reference used in computing reliability. Many texts on reliability

consider time in terms of component age. However, for system reliabi-

lity the system age is the important time reference. Since components

that fail within a system are replaced or renewed at random noints in

2 Lloyd, D.K., and Linow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1962).

3 Barlow, R.E., and Proschan, F., "Mathematical Theory of Reliability," John Wiley ft Sons, New York (1965).

4 Cnedenko, B.V., Belyayev, Yu.K., and Solovyev, A.D., "Mathematical Methods of Reliability Theory," Academic Press, New York (1969).

5Pieruschka, E., "Principles of Reliability," Prentice-Hall, Englewood Cliffs, New Jersey (1963).

16

time, comnonent apes are generally not known a ~riori as a function of

system age. In this instance renewal theorv must be emnloved t-o

determine the transient mission reliability as a function of system

age. This asnect of the nroblem will be considered in the following

sections.

(a) Component Reliability Baser! on Component Ape.

Consider first a single con-<-nen* '/here the time reference

is component age. Let

f(t) = Probability densitv distribution o*" coT->on°nt failure times or tine between Fiilur^q. t

F(t) = / f(t)dt = cumulative distribution r>mofion. 0

• Probabilitv that the component will *ai 1 in the time interval (0,t).

R(t) • Reliability defined as the probability that the comnonent will not fail in the time interval fO,t) where the comnonent is assumed new at tine o.

= 1 - P(t)

Aft) = Conditional failure rate where X(t)dt is the nrohabilitv of failure in the time interval (t,t+dt) "iven that the comnonent has survived to time t.

The conditional failure rate > ft") defined above is a useful and

descrintive quantity in reliability theory. It can be determined fron

the probability distribution of failure tines as follows:

Aft) = fft)/fl-^ft))

= fft)/Rft) m

Figure 1 shows descrintivelv the failuro rate for three tvmcal cases

in reliability; decreasing, constant and increasing failure rate.

17

Component Failure Rate

1 MTBF"

Increasing railure Rate flVear Out âil ires) /

Constnnt \ (Independent of A.rre')

necrensin" fRarlv railures )

0 Component Ar;e

Figure 1. Component failure Rate vs Component Ape

The decreasing failure rate implies that the longer a component survives

the lower its probability of failure. This is typical of early failures

in which components containing manufacturing or material flaws tend to

fail early. The increasing failure rate is tynical of components that

wear out where the longer a component survives the higher the probabi-

lity of failure. Mechanical components in which early failures have

been eliminated generally have increasing failure rates caused by such

phenomena as fatigue, corrosion, erosion and abrasion. The constant

failure rate is typical of components which have a probability of

failure independent of its age. This can result, for example, from

a situation in which excessive loads can cause failure where load

fluctuations are nurely random in time such as wind loads.

18

Examples:

1. Exponential Distribution

For this case

f(t) = Xe"Xt

F(t) = l-e'Xt

Rft) = e"Xt

A(t) = > = constant f2)

The exponential distribution describes the constant failure rate cas**.

2. Weibull Distribution

p R-l B f(t) = — - e

F(t) - 1 - e-(t/n)B

R(t) = e"(t/n^

B t6-1 A(t) = w ^ (3)

in which B and n, are distribution or population parameters. Note that

for the Weibull distribution B values less than, equal to and greater

than unity yield failure rates which are decreasing, constant and

increasing respectively. This characteristic makes the Weibull distri.

bution a useful one in reliability analysis.

(b) Mission Reliability of a Component Within a System [3,

Consider next the situation of a component within a system.

The auantity of interest here is the reliability of the component for

10

3 Barlow, R. E., and Proschan, F., "Mathematical Theory of Reliability," John Wiley F, Sons, New York (1965).

4 flnedenko, B. V,, Belyayev, Yu.K., and Solovyev, A. D., "Mathematical Methods of Reliability Theory," Academic Press, New York (1069).

Pieruschka, E., "Princinles of Reliability," Prentice-Hall, Ent»lewood Cliffs, New Jersey (1963).

20

a mission time interval of (t, t • T) wh^re t is the system aj>e and T

is the mission length. Prior to time t the component could have

failed and been replaced one or more times. Figure 2 denicts the

conditional failure rate Aft) of a component within a svstem showing

failure noints for an increasing failure rate.

Component failure Rate

nailures with Subsequent Penewals

System Are

Figure 2. Component Failure Rate vs Svstem Age For The Case Of Ideal Repair

In general the failure times of components within a system are not

known in advance as depicted in Figure 2 and consenuently must be

treated probabilistically. This is accomplished by defining another

quantity h(t) called the renewal rate or unconditional failure rate

which is an ensemble average of the failure rate over the population

of all systems. The renewal rate is defined such that

h(t)dt = unconditional probability of component failure in system time interval (t,t+dt).

21

The renewal rate for a given component is a function of the

underlying failure distribution given by the following equation:

h(t) = f(t) + fZ h(x)f(t-x)dx. 0

Derivation of this equation is presented in references ]?.], [3] and

elsewhere. Logically, equation (4) can be derived from the theorems

of total and conditional probabilities [6] alonp with the definitions

of the terms in equation (4). Multiplying both sides of equation (4)

by dt and considering the integral as a sum we have

f(t)dt • Probability that the original component fails

in the time interval (t, t + dt).

h(x)dx • Probability that a failure and subsequent renewal

occurred in time interval (x, x • dx).

f(t-x)dt • Conditional probability that a component which

was renewed at time x fails in the time interval

(t, t + dt).

h(x)f(t-x)dxdt • Unconditional probability of failure in time

interval (t, t • dt) for a component which could

fail at time x.

f(t)dt +

/ h(x)f(t-x)dxdt • Total probability of failure which is the sum of 0

all possible conditions which could lead to

failure in the time interval (t, t + dt),

= h(t)dt by definition.

The mission reliability can now be determined from the renewal

22

2 Lloyd, D.K., and Linow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1D62).

3 Barlow, R. E., and Proschan, F., "Mathematical Theory of Reliability," John Wiley 5 Sons, New York (1965).

Papoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (1965).

23

rate h(t) using the relation from reference f4]

R(t,T) = 1-F(t+T) + / [l-r(t+T-x)lhCx)dx (5) 0

in which

R(t,r) = Mission reliability at svstem time t for a

mission length T,

= Probability of no failure in (t, t+x).

l-F(t+r) = Probability that the original component in the

system has not failed at time t+T.

l-F(t+T-x) • Probability that a component which failed at

time x has not failed at time t+T.

h(x)dx • Probability of failure at time x.

Typical examples of the renewal rate and mission reliability of an

increasing failure rate component are deoicted in Figures 3 and 4. As

can be seen the renewal rate increases with svstem age until about a

system time enual to the MTBF (mean time between failure) of the

component. The renewal rate then approaches a constant value enual to

the reciprocal of the MTBF. The reliability OP the other hand de-

creases from a value of 1.0 at t=0 and approaches a constant value.

The asymptotic values of h(t) and R(t,T) can be derived directlv from

equations (4) and (5) by passing t to the limit infinity. These

values are given by the relations

h(») = l/'TTBF

1 R(«yr) - jjjrr/ [1-F(x)|dx.

T (6)

4Hnedenko, B.V., Belyayev, Yu.K., and Solovyev, A.D., "Mathematical Methods of Reliability Theory," Academic Press, New York (1969).

24

Component Renewal Rate

<1TB^

0 ' Svsten Are

Figure 3. Component Renewal Rate vs System Ape.

1.0

Comnonent Reliability

0 Svstexii Aê

Figure 4. Component Reliability vs System Age.

25

For the exponential distribution (constant failure rate) the

renewal rate and reliability are both constant throughout system life

-AT and are equal to X and e respectively.

It should be noted at this point that for a high reliability sys-

tem (say .85 or greater) the MTBF of the comnonents comnrising the

system should be of the same order of magnitude as the expected or

required system life. Consequently, the components within a system

are essentially exercised within the transient nortion of their lives

thus indicating the general importance of considering renewal theorv

in determining component reliability for high reliabilitv systems,

(c) Numerical Solution for the Renewal Rate.

A number of numerical techniques were investigated to

solve the integral equation (4) for the renewal rate h(t). The most

computationally efficient solution investigated thus far was through

the use of finite difference methods [7], Finite difference yields a

direct solution for the renewal rate. Other more general technioues

investigated which give comnlete solutions to the renewal problem

(e.g. probability distribution of total number of failures or time to

nth failure) involved the use of orthogonal expansions of Lanlace

transforms. Two sets of orthogonal functions considered were trigono-

metric and Laguerre nolynomials. The finite difference solution,

although limited to solution of the renewal rate, generally required

less computer time for given accuracy. The solution is also quite

general in that it can be used to determine renewal rate for most

theoretical distributions of failure time applicable to reliability

7 McKelvey, R. W., "An Analysis of Approximate Methods for Predholm's Integral Equation of the First Kind," December 1956, ADr>5f)530.

26

theory.

In solving equation (4) using finite differences, the density

f(t) and the renewal rate h(t) are discretized over a fixed time inter-

val generally the system life. In this case f(t) and h(t) are written

as

f(t) = f(kAt) = fk

h(t) = h(kAt) = hk

t = kAt where k=n,l,...,n.

The integral equation (4) can then he written as

t A k_1 / f(t-x)h(x)dx • _ I [f(kAt-iAx)h(jAx) + 0 2 i=n

•f(kAt-(i*l)Ax)hC(i*l)Ax)]

Ax v

j = l [f. .h. • f. . .h. .] J k-i j k-i-1 i + lJ

for k = 1,... ,n

0 for k = o.

(7)

Substituting into equation (4) then gives

j=0

for k • l,...,n (8)

= f for k = o.

Equation (8) renresents n+1 equations with n+1 unknowns which can be

readily solved using Gaussian elimination.

27

A difficulty is encountered in some situations where f(0) • «

such as in the case of the Weibull distribution with shape narameter

8<1. In this instance f(0) is fixed at a finite value such that the

actual area under f(t) in the interval (0, At) (i.e. F(At)) is made

equal to the finite difference area (f(0) + f(At))At/2.

(d) Numerical Solution for Mission Reliability.

Once the renewal rate is determined the transient

mission reliability can then be computed from equation (5):

R(t,T) » l-F(t+x) + fZ p-F(t+T-x)lh(x)dx. n

Numerically, this equation is difficult to solve since the integral

must be evaluated over the whole interval (0,t) for each different

value of t. By making use of the renewal equation an equivalent

equation for R(t,r) can be derived which is commitationally more

efficient. Integrating the renewal equation (4) from 0 to T gives

H(T) » F(T) + / H(T-x)f(x)dx (9) 0 j

where H(T) = / h(t)dt. 0

Making the substitution T=t+x in equation (9) gives the following

equation:

t+T H(t+T) » F(t+T) + / H(t+T-x)f(x)dx. (10)

0

Integrating by parts and using the fact that H(0) = n and F(0) = n

then gives

2«

t+T H(t+T) = F(t+T) + / P(x)h(t*T-x)dx

0 t+T

= F(t+T) + / T7(t+T-x)hfx)dx. fll) n

From equation (11) the following equation can he readilv derived:

t t + T

/ fl-F(t+T-x)]h(x)dx = r(t*x] - J ri-^ft*T-x)]hfx)dx n t

(12) t

where the definition H(t) = / h(x)dx was used. o

Substituting equation (12) into equation (S) for mission reliability

finally yields

t+T R(t,T) = 1 - / ri-T7(t+x-x)lh(x)dx (13)

t

Note that in this equation the integral is evaluated onlv over the

interval (t, t+T) and that the value of F(v) is reouired onlv over

the interval (0,x). This simplifies the numerical solution for mission

reliability.

(e) Average Mission Reliability.

Interval reliability as discussed in the previous

section is a transient function of system age. Weapon svstem renuire-

ments, however, generally specify one value of mission reliability for

the system. This snecified value could represent the lowest reliability

to be experienced by the system or an average value over system life.

In the computer model average system reliability is comnuted using the

relation

1 ? R(T) - - I R.(T) (14)

n i.i i

in which

R-CT) = Reliability for the ith mission

n • Expected number of missions over system life.

Computation of System Mission Reliability.

In general, system reliability can be computed directly from

the component reliabilities using an appropriate reliability model

[2,8], For example, for the series reliability model, system reliabili-

ty Rs(t) is determined from

n

Rs(t) « TTRi(t) (is) i-1

in which

R^(t) • Reliability of the ith component

n = Total number of components.

A series model is applicable whenever the failure of a single

component within a system results in a failure of the system. This

model has been initially chosen for the data analvsis computer model.

In the more general case where redundancy and load sharing are

inherent in the system, reliability models are more complicated but

csn be derived in most given situations [2,81.

30

2 Lloyd, D. K., and Linow, M., "Reliability: Manajremrnt, Methods, and Mathematics," Prentice Hall, flnj>lewood Cliffs, New Jersey (lOri?).

g Bazovsky, I., "Reliability Theory and Practice," Prentice-Flail, Englewood Cliffs, New Jersey (10^2).

31

Probability Distribution Selection.

One of the more difficult and questionable aspects of data

analysis is the selection of a theoretical distribution applicable to

a given set of data. This is particularly true for small data sarnies

where information is ultimately required at the tails of the distri-

bution. In the case of component failure data, theoretical considera-

tions, prior history and experience play a large part in distribution

selection. For example, for particular failure modes such as fatigue

the lognormal and Weibull distributions have been found to yield Rood

characterizations of the data [9, 10],

The data analysis computer model includes a distribution fitting

program to assist in the selection of a best-fit theoretical distri-

bution for use in reliability computation. Essentially, this program

computes the standard error for a number of different candidate

distributions. The standard error is a measure of the deviation of

the data from the theoretical cumulative distributions.

The Kolmogorov-Smimov statistical goodness-of-fit test is also

used to determine which of the theoretical distributions can be

rejected for given confidence level. Final selection of the distri-

bution can then be made based on the computer results, theoretical

considerations and/or personal experience and judgment.

(a) Candidate Distributions.

Following are the candidate distributions considered

for characterizing component failure times. In order to standardize

terminology used to describe the distribution parameters, the terms

32

Dolan, T. J., "Basic Concents of Fatirnj* Damage in Metals," 'fetal Fatif»ue, McGraw-Hill, New York (1950), Chanter 3, np. 3 -67.

' Freudenthal, A. M., and Shinozuka, "'., "Structural Safety Under Conditions of Ultimate Load Failure and âtipue," Columbia Univcrsitv, Mew York, N.Y., October 1P61, WADD Technical êôrt fil-177.

S3

scale, shape and location parameters are used for all distributions

and are defined below.

(1) Exponential

f(t) = ~e't/n (16)

where n = MTBF = scale parameter

= 1/X

X = constant failure rate.

(2) Normal 1 t-U 2

f(t) - e 2 ° (17) /2if a

where u = *fi*BF • scale parameter

o • Standard deviation • shane parameter.

(3) Weibull

6 t-y e-i -(^)B

f(t) = (ft rS-J e n (18)

where u • >1TBF = y +nr(l+l/8) 2 2 2

a = Variance = rf [r(l*2/B)-r (1*1/6)]

H = Scale parameter

3 • Shape parameter

y • Location parameter.

(4) Lognormal

(t-y)/2T 8

1 2 1 - rsT(*n(t>Y)-n)

f(t) * e 2P (19)

34

where V = '1TBF » y+e 2 ->

a = Variance = e (e -1)

p • Average {£n t} = Scale parameter

B = Standard Deviation f?,n tl • Shape parameter

Y = Location parameter.

(5) r.amma

f(t) . Ii^!ll e"^ C20) T(B)p3

where u = lTBP = Y*BP

2 2 a = Variance « Bp

p = Scale parameter

B = Shane narameter

Y = Location parameter.

More detailed discussions of these distributions can be found

in the literature [6,11],

(b) Least Squares Fit of Data [12].

For each distribution other than Hamma a least

squares fit of the data to the distribution is made. Henerally, the

theoretical distribution is linearized as far as nossible to simnlify

computation. In the case of the Camma distribution maximum likelihood

estimates of oarameters are used rather than solving the more difficult

nonlinear nroblem.

The cumulative distribution function £(t) is used in fittintj the

35

Papoulis, A., "Probability, Random Variables, and Stochastic Pro- cesses," McGraw-Hill, New York (1965).

nireson, W. G., Editor, "Reliability Handbook" McGraw-Hill, New York (1966), Chapter 4.

12 Draper, N. R., and Smith, H., "Annlied Regression Analysis," John Wiley 5 Sons, New York (1966).

.36

data with the median ranks being used as the true values of the cumula-

tive distribution associated with each ordered data point [13]. The

median ranks for a complete data sample are computed using the relation

i-0.3 MR. = . (21)

.1 n-0.4

where n = Data sample size

i = l,...,n

= Order numbers for the data where the data is

sorted in increasing order.

When suspended data has been generated along with failure data,

the median ranks are determined for the failure data onlv with the

suspension times being used to modifv the median ranks ri3]. In this

instance the failure and suspension data are sorted in increasing

order. The sample size n in equation (21) is now the total number of

failure and suspension data items. The order number i associated with

each failure item is determined as follows; initiallv i is set equal

to zero. An increment is then computed which is to be added to i to

give the order number for the first failure item using the general

relation

n • 1 - (Previous Order Number) New Increment =

1+(Number of Items rollowing the Present Suspension Set).

C22)

If there are no suspension data prior to the first failure, then the

increment is 1.0 and the order number for the first failure item is

1.0. This procedure is repeated for each subsequent failure item

1 ^ Johnson, L.G.t "Theory and Technique of Variation Research," Hlsevier, New York (1964), Chapter 8. _

using equation (22) in each case. The median ranks are then computed

using equation (21).

The median ranks are considered to be the true values of ^(t)

associated with the ordered failure times t.. The distribution para-

meters for a given theoretical form of F(t) are thon determined by

minimizing the sum of the squares of the error between the theoretical

distribution evaluated at the failure times and the median ranks or

between the transformed distribution and median ranks.

Consider as an example the Weibull distribution. In this case

t-Y B

F(t) = 1 - e "HH . (23)

Rearranging equation (23) and then taking double logs gives

£n£n(l/(l-F(t))) » B(An(t-y)-inn). (24)

For fixed Y, this equation is linear in £n(t-Y). The error between

the transformed theoretical distribution and the data is then given by

e. » 6(P-n(ti-Y) - £nn) - lnta(l/(l-»1R )) (25)

where t. • ith failure time l

MR. = ith median rank. l

The oarameters 8, o. and Y are then found which minimize the total

2 r 2 square error c • 2,ej* A similar approach is used for other distri-

butions.

(c) Computation of the Standard Error [12],

In general, the standard error in the fitting of

12Draper, N.R., and Smith, H., "Applied Regression Analysis" John Wiley g Sons, New York (1966).

38

a theoretical distribution to data is not the same as the error dis-

cussed in the previous section. The standard error is a measure of

the difference between the untransformed theoretical distribution and

the median ranks. The standard error is defined bv the following

equation:

( I Crct.)-MR.)2)1/2 i = l * T

s.e. = Standard Error = —— (26) n - n - 1

where F(t.) = Theoretical distribution evaluated at the failure 1 tines tj.

»|R. a Median ranks

n = Number o^ failure noints

p - Number of nonulation parameters estimated in determining p(t).

The smaller the s.e. the closer the data fits the theoretical distri-

bution.

(d) Kolmogorov-Smimov Goodness of Fit N4"|.

A non-parametric distribution has been derived bv

Kolmogorov [15] and Smirnov f16] for a particular statistic d which

is a measure of the fluctuation of sample data about the theoretical

distribution from which the sample is drawn. The statistic d is de-

fined as the maximum absolute deviation between the theoretical and

observed cumulative nrobability distributions. The theoretical distri

bution is fixed by specifying both the functional form and the para-

meters^

Siegel, S., "Nonparametric Statistics for the Behavioral Sciences," McGraw-Hill, New York (1956), Chapter 4.

30

Kolmogorov, A., "Confidence Limits for an Unknown Distribution Function," Annals Mathematical Statistics, Vol. 12, (1941), nn. 461- 463.

16Smimov, N. V., "Table for Estimating the Hoodness of Pit of Empirical Distributions," Annals Mathematical Statistics, Vol. 19, (1948), pv. 279-281.

40

In application of the K-S statistic, a Riven set of data is

hypothesized to have heen drawn from a Riven theoretical distribution.

The statistic d is then determined from the data and compared to the

K-S distribution of d. That is, for given significance level a or

confidence level (1-ct) the theoretical value d is determined from

K-S tables and compared to the computed d. If d>d , the hypothesis

is rejected. If d<d then there is no sufficient reason to reject the

given theoretical distribution and the hypothesis may be accepted.

The statistic d is determined for the candidate distributions

in the computer model to provide bases for reiecting distributions and

to help indicate the best-fit distribution.

Maximum Likelihood Estimates of Parameters [17],

Maximum likelihood estimates of population parameters are

determined for each component within a svstem for which failure data

has been generated. There are a number of reasons for generating

maximum likelihood estimators as part of the reliability model. First,

maximum likelihood yields estimates of parameters which have a number

of desirable attributes. For example, if an efficient estimator for

small samples exists (i.e. one with minimum variance), then maximum

likelihood provides such an estimator. In general then minimum con-

fidence intervals can be derived in this case. Maximum likelihood

estimators are also consistent in that thev anoroach the true para-

meter values as sample size increases. In addition, maximum likeli-

hood estimators are asymptotically normal as sample size increases

Mood, A.M., and Craybill, r.A., "Introduction to the Theory of Statistics," McGraw-Hill, New York (1063), Chapter 8.

41

for most cases. This simplifies determination of confidence intervals

for large samples.

A second reason for determining maximum likelihood estimates is

that they can be computed for relatively general underlying probability

distributions through the use of routine mathematical procedures.

Finally, the numerical procedures planned to be used in the computer

model to determine confidence intervals require the maximum likelihood

estimates of parameters to improve computational efficiency.

The basic idea behind maximum likelihood estimation is relatively

straightforward. One assumes first that a sample of nf failures and

n suspensions have been generated from a given theoretical distribu-

tion with parameters c^ * (a. ,a2,... ,ct ). The failure and suspension

data are designated as t^. (i=l,... ,nf) and t .(i=l,...,n 1 respectively.

The joint probability distribution of the random sample of failure and

suspension times can be written as

L(t;o/) » gt(tfl,..*,tfn »tsl»---»tsn ;i) (2?)

where L(t;a) » Defined as the likelihood function

g. « Joint distribution of the sample outcome

t = Vector of the sample data values

a • Vector of parameters for given underlying distributions of failure times.

It is next assumed that each failure and suspension time is a

statistically independent outcome. Equation(27) can then be written

in terms of the failure density f(t) and the cumulative distribution

42

F(t) as follows [6]

ns

L(t;o) - C | | f(tf.;op [1-F(t .;o)] i«l i=»l

(28)

where C • Normalizing constant such that the area under L(t;a) is unity

f(t;a) • Theoretical density distribution of failure times.

F(t;o) • Cumulative distribution function

= J* f(x;o)dx n

[1-F(t;ct)] = Probability that the failure time is greater than t. This term is used to represent the nrobabilitv of obtaining the suspension times.

As can be seen, the likelihood function L is defined as the

multivariate probability distribution of the random failure and sus-

pension times. Solving for the parameters c^ such that L is maximized

consequently gives the highest probability density for the given sample

outcome t. The parameter values which maximize L are denoted as a

and are called the maximum likelihood estimates of cu

In practice in L given by equation (29) is maximized rather than

L to simplify computation. This can be done since the maximum of

any positive function and its log are equivalent.

Jin L(t_;a) = In C * I In f (t_. ;a) i-1 "

"s • I An[l-F(t :o)] (29)

j-1 *">

Panoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (1965).

43

Consider the Weibull distribution as an example. For this case

F(t;ci) = 1 - e n

where a • (n,B,Y).

The likelihood function can then be written as

nf

An L * An C + £ r£nB-£nn.+ (B-l)(£n(t -Y) i=l fl

*£/*" "s V"7* - Ann) - (—••—D ] ~ f (—~~) (30)

n iî *

Solving for n,B and y which maximize An L yields the maximum likeli-

hood estimators for the Weibull parameters.

Likelihood functions such as given by equation (30) are maximized

in the computer model using Rosenbrock's algorithm ("18] which is a

general solution of the unconstrained minimization problem for non-

linear functions. Minimizing the negative of An L maximizes the

function.

Once point estimates of the failure distribution parameters are

generated for a given component, the mission reliability can be com-

puted using the methods previously presented.

1 8 Rosenbrock, H.H., "Automatic Method for Findinp the Greatest or Least Value of a Function," Computer .Journal, Vol. 3, (1061), pj). 175-184.

44

Bayesian and Classical Confidence Intervals for Component

Reliability.

Classical methods of confidencing component mission reli-

ability for small samples are generally available onlv for the constant

? failure rate case. In this instance the x distribution represents

the inferencing distribution for the "fTBF from which confidenced reli-

ability can be derived [2]. Bayesian methods will consequently be used in

the computer model for the non-constant failure rate components since

this method provides a svstematic means o^ determining confidence inter-

vals for more general problems. A complete discussion of Bayesian

statistics will not be presented in this report and the reader is

referred to the books by Lindley [19] for a more complete discussion

of this topic.

Bayesian inferencing is based, as the name implies, on Bayes"

probability theorem which is essentially a conditional probability

statement:

Bayes' Theorem:

f(X.x) = f(y;x)f(x) = f(x;y)f(y) (SI)

in which

f(y»x) • Joint probability density of random variables x and y.

f(y;x) • Conditional probability of y given x

f(x;y) = Conditional probability of x given y

2 Lloyd, D. K., and Lipow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1962).

19 Lindley, D.V., "Introduction to Probability and Statistics from a Bayesian Viewpoint," Part 1: Probability and Part 2: Inference, Cambridge University Press, Cambridge (1965).

46

f(x)» f(y) • Marginal distribution of x and y respectively.

From equation (.31)

f(y) f(y;x) = f(x;y) *— " (37)

f(x)

In the Bayesian aonroach thn unknown mrameters of the distrihu-

tion of failure times are considered to be random variables. The

sample outcome t from a given test are also random variables which

are dependent on the distribution narameters. Bayes' theorem, eouation

(32), can be written in this instance as

f(a) f(a;x) = f(x;a) (33)

f(x)

in which

f (a;x) = Density distribution of narameters given the test sample x.

f (x_;a) = Density distribution of the samnle outcome given the narameters m.

f(a),f(x) = Marginal densities of rx_ and x_ respectively.

In equation (33) the distribution f(x;ot) is iust the likelihood

function L(x_;ot) by definition. The distribution f(oi) is called the

prior of a and f (rt;x) is called the posteriori distribution of r^

since it is determined after the samnle x_ is determined and is con-

sequently conditioned by the test results.

The prior distribution f(a) generally reflects nrior knowledge

47

or prior data on the population parameters a before the test results

x have been generated. For the case of weapon system components there

is at the present time little or no prior information and consequently

f(a) must reflect in some way our prior ignorance of the values of a.

Choosing the prior distribution to reflect decrees of ignorance is one

of the more difficult and questionable aspects of Bayesian statistics.

Much research work is presently being conducted on the question of

prior distribution selection. Priors which represent maximum ignorance

for some of the simpler problems have been found [19,70]. It has also

been found, however, that a uniform nrior on the parameters yields

confidence intervals that are generally close to exact in a classical

frequency sense for a number of reliability indices. The uniform

distribution assigns equal probability to all values of a random

variable over a given range. The robustness of uniform priors is

discussed somewhat in reference [19]. Uniform priors have consequently

been used in our reliability statistics work and have also been found

to be generally robust.

The distribution f(x) in eauation (33) is considered to be a

constant and is determined so that the area under f(o;x_) is unity.

Equation (33) can now be rewritten in the following form:

f(a;x) - C L(xjo) (34)

where f (a;x) • Posteriori distribution of ot_ given x_.

C = Constant such that area under f (nyx) is unity. The constant C contains the constant terms f(x) and f(ct).

4*

19 Lindley, D. V., "Introduction to Probability and Statistics rrom a Bayesian Viewpoint," Part 1: Probability and Part 2: Inference, Cambridge University Press, Cambridge fl9f,S).

20 Jaynes, E. T., "Prior Probabilities," IEEE Transactions on Svste^s Science and Cybernetics, Vol. SSC-4, No. 3, Scot. 1^8, no. 227-241

49

L(x;a) • Likelihood function.

In computing confidenced reTiability our interest is not on the

•parameters themselves but rather on a function of the parameters.

From probability theory [6] the cumulative distribution of any function

z of the random variables a is determined by the relation

F_(z;x) = /// f(a;x)dn f35)

in which

F (z;xj = Posteriori distribution of Z

D « Domain of Z_ such that Z(o)<z

f(a;x) • Posteriori distribution of a i»iven by equation (34).

Once the posteriori distribution ^(z;x) is known confidence

intervals can be constructed on z for any given confidence level CL

by determining the apnronriate Percentage points directs from F(z;50.

For example, a lower confidence limit is determined by solving the

following equation for z. :

Fz(zL;x) « 1 - CL f.36)

in which

Zt * Lower confidence limit.

CL = Confidence level.

Since component mission reliability is a direct function of the

failure distribution and hence of the parameters, confidence intervals

Panoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York ("1965).

SO

on mission reliability can be determined using the above formulation.

Solution of equations (34), (35) and (36) for confidence limits can

be accomplished numerically using computer routines.

Bayesian Confidence Intervals for Svstem Reliability.

(a) Constant Failure Rate Comnonents In Series.

Confidence intervals on component reliability for con-

2 stant failure rate can be determined using the x distribution f2].

For a system of components in series, however, classical confidence

intervals can be readily determined only if all components have eoual

test times. In this instance the svstem can be treated as if it were

a component with all failures being counted as system failures regard-

2 less of which components fail. The x distribution can again be used

to determine confidence limits.

For series components with unequal test times, however, classical

methods do not generallv apply. For this case Bayesian statistics, as

discussed in the previous section, can be employed to determine confi-

dence intervals.

Consider the series svstem

R = | R. (37) i=l 1

in which R = Svstem reliability, s

n = Number of components, c

R. = Reliability of the ith component, l

2 Lloyd, D. K., and Lioow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1062),

For constant failure rate, R. = e i and the system reliability

becomes

nc -A.T -A T

R = ||e = e S (38) 5 i=l

where n c x = y x.

i=l x

The system failure rate A is enunl to the sum of the individual s

component failure rates. The posteriori probability density of the

individual component failure rates can be determined usin.c? equation

(34) by letting ot = A . Since A is equal to the sum of independent — i s

random variables, the distribution for A is equal to the convolution s

of the individual component distributions [6]. Performing the required

convolution is generally tedious and it is at this point that a simpli-

fying assumption is made. Using the Central Limit Theorem for sums of

random variables [6] it is assumed that the oosteriori distribution for

As is Gaussian. Consequently, all that is required is the mean and

variance of A to define the oosteriori distribution. These are deter- s

mined using the relations

nc E(A J = I E(A.) (30)

s i=l

nc Var(As) = I VarCXj) M")

i=l

Where E(*) • Expected value

Var(*) = Variance

Papoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (T>65).

52

Confidence intervals can now be constructed for X and hence for system

reliability using equations (39) and (40).

The accuracy of making the Gaussian assumption was checked and it

was found that a system with about ten components and one to three fail-

ures experienced ner comnonent yielded relatively accurate confidence

intervals in comparison to true values. In the example problem consi-

dered test times for each component were assumed equal which permitted

the comoutation of classical confidence limits.

(b) Non-constant Failure Rate Components In Series.

For the series svstem eouation (37) applies:

"c

R - | j R s i-1 1

Taking logs of both sides of this eouation yields

nc In R = 7 InR.

i-1

Equation (41) represents a sum of independent random variables

and consequently the posteriori distribution of InR can be determined

using the Gaussian assumption as discussed in the previous section.

Confidence intervals can then he constructed from this distribution.

COMPUTER PROGRAM

Two computer programs have been prepared to perform the compu-

tations required for reliability data analysis. One program called

53

DISTSEL performs goodness-of-fit computations and is used to assist in

distribution selection given component failure and suspension data.

Included in this program are computations of the standard error and

Kolmogorov-Smirnov statistic. At the nresent time, the theoretical

distributions that can be handled are the exponential, two and three

parameter Weibull, two parameter lognormal, two parameter famma and

the normal.

The second computer program called RRLIAB Performs corn-nutations

required to determine reliability and confidence limits for components,

subsystems and system given component data. The component data in-

cludes failure and suspension times and the associated theoretical

distributions to be assumed.

The fortran listings of the computer programs are lengthv;

DISTSEL and RELIAB contain 17 and 25 subroutines respectively. These

listings are consequently not contained in this report but can be

made available upon request. The innut data format for both programs

is presented in Appendix I and output data for a sample problem are

given in Appendix II.

As a final note, it should be emphasized that the computer models

are flexible. Input and output data formats can readily be changed

for particular situations, perhaps to render the programs compatible

with a given computer data file system. Improved versions of the

programs will also be generated in the future.

54

REFERENCES

1. Army Regulation AR705-5D, "Army ''ateriel Reliability and Maintain- ability," Headquaters Dent, of the Army, Washington, D. C.f 8 January 1968.

2. LLOYD, D. K., and LIPOW, M., "Reliability: Management, Methods, and Mathematics," Prentice-Mall, Englewood Cliffs, New Jersey (1962).

3. BARLOW, R. E., and PROSCHAN, F., "Mathematical Theorv of Reli- ability," John Wiley F, Sons, New York (1965).

4. GNEDENKO, B. V., BELYAYEV, Yu.K., and SOLOVYEV, A. n., "Mathema- tical Methods of Reliabilitv Theory," Academic Press, New York (1969).

5. PIERUSCHKA, E., "Principles of Reliability," Prentice-Hall, Engle- wood Cliffs, New Jersev (1963).

6. PAPOULIS, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (1965).

7. McKELVEY, R. W., "An Analysis of Approximate Methods for Fredholm's Integral Equation of the First Kind," December 1956, AD650530.

8. BAZOVSKY, I., "Reliability Theory and Practice," Prentice-Hall, Englewood Cliffs, New Jersey (1067).

9. DOLAN, T. J., "Basic Concents of fatigue Damage in Metals," Metal Fatigue, McGraw-Hill, New York (1959), Chanter 3, np. 39-67.

10. FREUDENTHAL, A. M., and SHINOZUKA, M., "Structural Safety Under Conditions of Ultimate Load Failure and Fatigue," Columbia Univer- sity, New York, N.Y., October 1961, WADD Technical Report 61-177.

11. IRESON, W. G., Editor, "Reliability Handbook," McGraw-Hill, New York (1966), Chapter 4.

12. DRAPER, N. R., and SMITH, H., "Applied Regression Analysis," John Wiley T, Sons, New York (1966).

13. JOHNSON, L. C, "Theory and Technique of Variation Research," Elsevier, New York (1964), Chapter 8.

55

14. SIEGEL, S., "Nonparametric Statistics for the Behavioral Sciences," McGraw-Hill, New York (1956), Chanter 4.

15. KOLMOGOROV, A., "Confidence Limits for an Unknown Distribution Function," Annals Mathematical Statistics, Vol. 12, (1941), op. 461-463.

16. SMIRNOV, N. V., "Table for Estimating the Goodness of Fit of Empirical Distributions," Annals Mathematical Statistics, Vol. 19, (1948), pp. 279-281.

17. MOOD, A. M., and GRAYBILL, F. A., "Introduction to the Theory of Statistics," McGraw-Hill, New York (1963), Chanter 8.

18. ROSENBROCK, H. H., "Automatic Method for Finding the Greatest or Least Value of a Function," Comnuter Journal, Vol. 3, (l°r,n) f pp. 175-184.

19. LINDLEY, D. V., "Introduction to Probabilitv and Statistics from a Bayesian Viewnoint," Part 1: Probability and Part 2: Inference, Cambridge University Press, Cambridge (1965).

20. JAYNES, E. T., "Prior Probabilities," IEEE Transactions on Svstems Science and Cybernetics, Vol. SSC-4, No. 3, Sent. 1968, nn. 227-241,

56

APPENDIX I

INPUT FORMAT FOR PROGRAMS DISTSEL AND RELIAB

The following listings give the card by card description of the

innut data and format required by the reliability data analysis

programs.

(a) Overall System And Subsystem Data Cards

Card No.

Column No. Description Format

1

2

4 to 4 +NSIJB

1-80

1-10 11-20 21-30

1-40

1-80

Overall system identification.

Number of missions over svstem life. Specified system life. Desired confidence level for reliability.

Number of components per subsystem for which data is supplied. UP to four subsystems can be handled.

Alphanumeric

110 FlO.o F10.0

4110

Subsystem identification. NSUB is the number of subsystems where one card is used to describe each subsystem. Alphanumeric

(b) Element or Component Data Cards

The following data cards are required for each element

starting with the elements of the first subsystem down to the elements

of the last subsystem:

57

Card No.

Column No.

Description format

1 2-10

11-80

2 to Tlj

2-11

12-80

Theoretical distribution code number*, Element identification code number. Description of element.

Code number for determining if data on card is a failure time, a suspension time or last card ôr current element**. n.r is the number of failures and n is the number of suspensions. Failure nr susnensiop time. If this is the last card then the remaining columns are blank. Description o*- the failure or suspension.

II 19

Alohanumeri c

II

^10.0

Alnhanuneri r

Distribution Code Number: 1 - Exponential 2 - Two parameter Weibul1 3 - Three parameter Weibul1 4 - Two parameter lognormal 5 - Three narameter lo^nonnal 6 - Two parameter gamma 7 - Three parameter gamma 8 - Normal

Failure or Suspension Code Number: 1 - Suspension 2 - Failure 3 - End of data

(c) Sample Problem

The following is a listing of data innut for a hypothetical reli

ability data analysis problem:

58

SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS 10 2000.0 0.70 3 1 1 0

SAMPLE PROBLEM SUBSYSTEM A SAMPLE PROBLEM SUBSYSTEM B SAMPLE PROBLEM SUBSYSTEM C 1100000001 SAMPLE PROBLEM ELEMENT M 2 1200.0 SPACE FOR rMLURE DESCRIPTION 2 000.0 SPACE FOR FAILURE DESCRIPTION 2 1500.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1350.0 SPACE FOR FMLURE DESCRIPTION 2 1000.0 SPACE FOR FAILUF-H DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1100.0 SPACE FOR FAILURE DESCRIPTION 2 850.0 3 1100000002

SPACE F0" PAIUPE DESCRIPTION

SAMPLE PROBLEM ELEMENT A? 2 1500.0 SPACE ÔR FAILURE DESCRIPTION 1 8000.0 3 1100000003

SPACE FOR SUSPENSION DESCRIPTION

SAMPLE PROBLEM ELEMENT A3 1 0000.0 3 2200000001

SPACE FOR SUSPENSION DESCRIPTION

SAMPLE PROBLEM ELEl*ENT Bl 2 1200.0 SPACE ÔR FAILURE DESCRIPTION 2 000.0 SPACE FOP. FAILURE DESCRIPTION 2 1500.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1350.0 SPACE FOR FAILURE DESCRIPTION 2 1000.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1100.0 SPACE FOR FAILURE DESCRIPTION 2 850.0 3 4300000001

SPACE FOR FAILURE DESCRIPTION

SAMPLE PROBLEM ELEMENT Cl 2 1200.0 SPACE FOR FAILURE DESCRIPTION 2 000.0 SPACE FOR FAILURE DESCRIPTION 2 1500.0 SPACE ÔR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1350.0 SPACE FOR FAILURE DESCRIPTION 2 1000.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE "OR FAILURE DESCRIPTION 2 1100.0 SPACE ÔR FAILURE DESCRIPTION 2 850.0 3

SPACE FOR FAILURE DESCRIPTION

'•ODEL

5P

APPENDIX II

OUTPUT DATA TT)R SAMPLE PROBLEM

The following pages contain output data renerated bv p-rorrams

DISTSEL and RELIAB for the sample innut data presented in Appendix I

60

RESULTS FROM PROGRAM DISTSEL

FAILED SPECIMENS AND MEDIAN RANKS...

850.0 0.07447 SPACE FOR FAILURE DESCRIPTION 900.0 0.18085 SPACE norc FAILURE DESCRIPTION 1000.0 0.28723 SPACE FOR FAILURE DESCRIPTION 1100.0 0.39362 SPACE noi> FAILURE DESCRIPTION 1200.0 0.50000 SPACE FOR FAILURE DESCRIPTION 1300.0 0.60638 SPACE TOR FAILURE DESCRIPTION 1300.0 0.71277 SPACE "OR FAILURE DESCRIPTION 1350.0 0.81915 SPACE FOR FAILURE DESCRIPTION 1500.0 0.92553 SPACE FOR FAILURE DESCRIPTION

61

STANDARD ERROR 0^ ESTIMATE...

DISTRIBUTION

EXPONENTIAL NORMAL LOC-NORMAL WEIBULL-2 PARM. WEIBULL-3 PARM. GAMMA-2 PARM.

S.E.

0.280808E on 0.566338E-01 0.657507E-01 0.530806E-01 0.642205E-01 0.83200OE-O1

DISTRIBUTION SELECTED... WEIBULL-2 PARAMETER

K-S STATISTIC ( 0.141) IS LESS THAN TABLE VALUE ( 0.388) THEREFORE MAY ACCEPT HYPOTHETICAL DISTRIBUTION AT 10 PERCENT LEVEL

62

)DflTA

1500 J_

SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS MODEL MARCH 1973

100000000 SAMPLE PROBLEM ELEMENT Al

EXPONENTIAL DISTRIBUTION...LEAST SQUARES ESTIMATE MEAN LIFE = HUB. 173

1339 __

1195 __

X X

1067 __

952 __

850 __

-+- •+• -t- -t- -+- .001 01 .05

f>7,

.10 .20 .HO .60 .80 ."90

CUMULATIVE DISTRIBUTION


DATA

isoo _i_

lOOOOOOOO SAMPLE PROBLEM ELEMENT Al

NORMAL DISTRIBUTION...LEAST SQUARES ESTIMATES MEAN LIFE

STANDARD DEVIATION =»2H5.0730

1166.665

137Q __

1240 __

1110 __

980 __

650 __

.001 .20

M

HO .60 .80 .90 .95


SAMPLE PROBLEM TO DEMONSTRATE RELIRBILITT DATA ANALTSIS MODEL MARCH 1973

DATA

1SQO _L

lOOOOOOOO SAMPLE PROBLEM ELEMENT HI

LOG NORMAL DISTRIBUTION...LEAST SQUARES ESTIMATES

STANDARD DEVIATION =2S9.UU38

MEAN LIFE = 1175.337

1195 __

.001

65

.40 .60 .BO .30 .95

CUMULATIVE DISTRIBUTION 99


DRTfl

1500

1UU0O00OU SAMPLE PROBLEM ELEMENT Al

2-PARM. HEIBULL...LEAST SQUARES ESTIMATES SHAPE = 5.587

SCALE = 1259.OD

1339 __

1195 __

1067 __

9SE _.

850 __

.001 .01 1— .05

66

•+- -+- •+• -i—

.10 .20

CUMULATIVE .40 .60 .80 .90

DISTRIBUTION


.DflTfl

903

100000000 SAMPLE PROBLEM ELEMENT Al

3-PARM. HEIBULL...LEAST SQUARES ESTIMATES SHAPE = 2.494

SCALE = 648.866

LOCATION = 597.253

700 __

543 __

421 --

326 __

253 __

H .001

•+• -+- -+- •+• •+ •+• 01 .05

67

.10 .20 .40 .60 .80 .90



lOOOOOOOO SAMPLE PROBLEM ELEMENT Al

GAMMA DISTRIBUTION...2-PARAMETER MAXIMUM LIKELIHOOD ESTIMATES ALPHA = 30.71U

BETA = 37.98U

Q Q

ao CD

un

o i—iCl . to

CO

az I—

•—,LD

DQ

LU >

l—CO

u un

en

a

o o

^0.00 90.00 100.00 110.00 120.00 DATA *10'

68

130.00 140.00 150.00

M N c rH

•—

D e <

00 F- >—1

ca •• < < f-H f- M -J < -J j Q UJ w u: c c

g U z

«* di UJ < c 7" a oo •—< UJ c »—i (J J c 00 »Tr UJ c cr > c c O.

< cj --

C c c

LO f. rH >H

c 5 Pi U-

c c II II II

o c II (1 <r c 00 c c 5r* •r— "Z

f- 1 UJ c cr c UJ UJ UJ oo < •-i c c UJ H H H f~ C o. H • c i—< 00 00 oo •J z <-< c c j >• >- >• 13 > < _) c c a. 00 oo 00 to H •—< c cr tr a. CO ca ca UJ i—i 00 cc r: •—i c 77 5 J J3 c UJ < c oo < 00 ca 00 U CO

CO < -J II ii • 00 y 00 ^~ 00 r co < M UJ UJ c l-H UJ l-H UJ »-H OU HH >-< M a' Ex. F- a: H "

CO £ 1-3 f- UJ >—i c < 00 H 00 H UJ oo £ u. J c H >- >• >> c UJ c-< c < oo Ci 00 B CO Ci

r -J ^~ c c ce o ca C ca c

UJ H UJ c [3 S J2 U- r: u. H Z UJ ?r

H e II T" 00 CO co < HH J UJ oo e U oo oo CO K c UJ H >- c J >—i jj £ T. H £ H E- a- 00 oo cr UJ M UI UJ Z uj ?: 00 p >- c > J UJ 2 | J UJ

cc T z oo j2" co cr • UJ C3 •»- ca Q w <2 UJ c -J II or O UJ o UJ o w

z c ^r; e C OS -J C- 2 c: J UJ I—1 V UJ c N UJ oo u D_ UJ Cu UJ D, UJ a ;r~ UJ 1—1 U 5"

c H u 00 s^ UJ oo UJ u UJ u UJ u c UJ OO 1—1 z UJ H

% -J c -J e -J c

H H > u o II c 00 a. p. a. IU oo UJ >—< i—( >> 1 y e; y cr r o;

y C i23 & oo f u oo < UJ < UJ < UJ a; ~~ oo oo ?: ca UJ OO ca W ca CO 03 •J •j 00 •-H IS O ^ J » § •

*j :£ OQ < a 2£ 71 CJ 00 UJ » • C a-. * o u3 » Z • s __• z K is 2"~ u ~J a [X. u ^~ 7~ •- a. o UJ Q c UJ o c UJ UJ UJ

pt H UJ r: >-< f- H fe UJ a. oo f- C£ o u. ct en 00 00

-J >- b UJ >—i t-H UJ UJ >< >" >- Q. 00 oo UJ PQ OO u ca CO OO 00 OO r I—c Q.

b 00 UJ jr* i 02 ca C3

< JJJ a X t—1 CL 5 3 B r^ 00 H O UJ Z 7~ 00 7-. 2: oo t/5 w

69

J

c c c c c

c c in

O ©

zzzzzzzzz ocooooooo HHHHHHHHH

CLDuCLCLCL.Ci.fr.CLi MHHHHHMMM

uuuuuuuuu nwwwwnwww lUHIUUUUlUUU 0.0.000030.0

WUJUPWWWWK cc ctix or: oc c£ o: or c = 3 = 3=13332

HHHHHHHHH <<<;<<<<<:< t (L U B- li. tl. U. U U-

OOOOOOOOO U.U.LLU.O.B-UUU.

UUUUUUUUU uuuuuuuuu to<<<<<<<<< UO.CLCLQ.CLCLCLCLCL JWWWWWWWWW

UJ ce 3 • • • » JOOOOOOOOC t-tLOOCCOOOLOC <»mCHNKlMMifl

z c o.

rs «- o- c c U r—(

^r CO !•* H-l t^. 55 • H Ol .—t c IN C

u Is- vC Q

vC i vC II CO a vC to to

• CO < r vC u c vc y to •J •—' (-1 gj r -i H

••H uj w H z ct •—i r> 3 UJ fc to vC

II J & UJ vC i-t 3 c h r^ vC

to < J a < vC vC u u. - .

t-H CQ T~ vC vC < 3 t-i vC vC

•H LU U- to (3 vC vC f- O vC vC tt. jr m vC l~~ \C u z O o vC vC • a: o o: Q vC vC VC 3 K z U C vC vC vC •J H o o t VC l-H M < (-H to T vC vC —i < HH i H »—I vC vC U- > J -J .—1 vC

u CQ = UJ «-H vC U- Q »-l to S*T, VC

o C u •H • o S~ cc -J vO II

u «: to vc 3 < i—i Q

1—1

•H Z

2§ u H •tf p—1 X UJ < < CO U

i—i UJ 1 i > u u f- ^ 5" u •H l-H UJ I—c u. <5^ Q H H S -J o 0^ N

gg o UJ r >- <

CL Q

UJ c;

SS 3 H j| i UJ

tr < < a

1—1 HH UJ x *J < K Z u <

B: o; JJ-* i u > H < < H CO CO < to

70

U-. LO C C TT "* rj n

00 00 • » c c

1,". u-; c c vT vr "<» ^r ro r^ TT *t oc or; » c c

rH

C >• C -J c c fc o UJ cr LO c m 1/5 UJ .-i

LO e

c <£

Osl

OC • c

C

CM •«• OC

c

S a to l—l

>- >-l

UJ

<r

cr c

c c c c C —' c c- c m c <r

or VC as J cr. C oo UJ ^r o >-c c • cc r-j i—i 02 UJ c c <» to < cr. r-~

00 to I—I ^~~ • cr. • 1—1 -J -J a to Q 7~ 10 LO UJ I—I

D c c c <r d <r ii >—i r—< j IT) cr c II >• vC vC u < UJ \C • c CL" rj- •c 7. >- £ Cr vC c c >• UJ rj r-; O H H f~

H i *JT e c (- > »» •cr >—< UJ z • • c n C >-i UJ oc oc H <^ •J J T e i—i vC c c i—i -1 • r I—I lo UJ l-H UJ v£ c to <—i en, c e ^~ > CQ -3 UJ UJ r-H • J co o I—I BE UJ < u: -3

UJ 7: t-i

r—1 i—l < < I—i

u -3 o O

-J l-H -3

"»- £ II II c

r\i cr. UJ

II -J UJ t UJ

U L" UJ UJ U DC

Qj O cc 1 to cr i—i 2 BS 2 _i nc UJ II n UJ 7. hj UJ I—I w a CC eo c H i—( c to •J I—I LO LT. C S £T. U3 C rrr o D' K l—l H < CO c c I—I >-i U cr" LO u UJ UJ II UJ to -3 > < vC vC U. to u, z C- E- H £ r to 3 cr; l—l n rr z LO 2 UJ ^~ ?; UJ Eg

< H-t l—l to UJ J N Osl c < o a

UJ o o 7~ < H y UJ 1? UJ TJ- <» u U HH -3 a: >-( < CL. l—l Cr" ec 00 00 a u. o- U- fc or cr H U- U- »—i • •

SB UJ a 2

y < < ?: C c Q -J c C g UJ o < CO CQ a. Cu o

r UJ UJ 7Z r U- J ec UJ UJ H u UJ < IX c or. i-i cc 3 H -3 CL < >—i CQ cc g, or UJ a. UJ y^ co to < """ o < S =3 O > s UJ G. O c Q to to -J •P-. ?: CJ < 1—1 u a W >J

71

UJ

p to

UJ 2:

c c c c c c C

c D CO

c c • c c

I—I LO c

II

to

c CO *H KM II <* 2 3 UJ W -J (X T~ HH CO »-l < 3 H U. CO

u- u. to oow oc a:

CQ OQ <

55o 2 Z H

2 O

u to UJ c 2 c P-H

UJ

to

8

o c c c c c c > e c

to

UJ 5 5 I—I

< CL

U. o UJ

a.

to

c o LO

a. c to o 3 C CO 00

U- o

UJ >

UJ UJ J X a. H £ >-i

3 PJ CO <

c u a UJ 5f 5 to to < to

UJ II Ci

CO -J tU l-H r < l-H U. H

U- (U o a: cr

8 UJ n 2 u -J C

to < l-H i—i UJ u g a.

to H U- l-H to c Di

2 O UJ fe i—i tr l-H to < a

-J -" M X J s UJ r B t-H U_ J o

fc >* 2 Pa

X £ 2 to

c c c

c c c c c c

o c c © o o o o c c c o o ©

• C LO C C CD c

c c o e C II

e LO c.

Of UJ

5 < a. UJ

< to

2 o l-H

< l-H > UJ Q

q cc <

UJ <

< CO

77

n C C C C C

LT. LT. I~- t~-

C. C.

0". c • • c o

v. ir.

»-H t-H c c c-. c • » cr c

c UJ c oo FH

LT.

I/)

c.

c. cr

UO

r».

i—i c r- c-. • c

;JJ cc a.

OO >-H

>- H

UJ

•—1

c C T* C *j c cr. c c

C [^ t-H <: C TT UJ \C 7. _) c c cr. CO .—I e t-H c • £ 01 t-H CQ UJ c c

r>«. OO < c: r^ ?:

c a • oo

t-H t-H rr •

cr a i—i

e C c

c r ur UJ t-H < II

CM •J C e C II >- ^c vT u

Z" UJ e • C DC #—i i—' 2 >- £ c c C fc

UJ cr. C. O s II H uj H £ c c cr > f-~ F HH — Z. 1 • C II c t-H UJ c c H <? J J UJ UJ t-H c cr C I—1 J • • l-H !- UJ t-H —i T c cr oo l-H Ci o o JJT oo > M UJ UJ UJ U0 • .J no c i-H T: (JJ <

_J Z PM c-, c < l-H u -3

fc it © CC II -J >- UJ UJ U r II N UJ u H U L' U ex Eu o CC i oo K l-H LT. Ul Z T: z -3 cr UJ II M UJ •z. -J r»> h» UJ t-H w c CC 02 C F l-H c oo J t-H v£ vC a I- C UJ O => e a CC l-H H < CC ^* r—( l-H t-H U a W Q UJ UJ II r

OO -J > < co Ol U- oo U Z a H H < 00 gg DC l-H r- r» ^: oo 2 UJ r 2 UJ UJ UJ t-H t-H 00 UJ -5 cr. cr. C < C Q UJ o O g ^~ < 3E H - . UJ F UJ • • U U fH -J a: i—i I <£ CL 1—( cr 2 £ o c Q u. a. EL t

fV H U. a. t-H

t UJ Q Z ~~ < <: 2 o O Q -3 z UJ C < 00 02 0- 0- O *-• UJ UJ < UJ t-H <-i u 00 H t-H >—i rjp cr cc i is > s T7* (L

-J CE UJ UJ f- ~r UJ UJ s cc o or l-l DC rr f- -J a. < t-H 00 aa a. UJ g. UJ U UJ oo 00 < o UJ a. o cc a oo oo -J 2 z u < t-H u c oo J

73

c c c CJ

c to c •-1

c c F •—1

Z o

cr

e Q UJ

5 UJ - u c

l-l

to to <

to UJ

<

2 ec o. y to J i—i to UJ *r-

fe a c H UJ UJ •-I

z H z t-H P &—. c UJ H

52 1—1 Cr- ^~T i—i UJ to D C o

a 2: J K S. UJ H-I CQ a. «e rr UJ

UJ O c to ti. to F

t-H Ci • to u ^~ 2 O M cr O c >—< u. «e C DC <T. U-

|H O c z u 2

££ cr. u. c >—i

to •H to

2£ e f UJ y fc to UJ

f- UJ II t-H < 03 1 i F < J E- CU i—c to < few to to DC UJ U

ti z 2 Es s •-H

§=§ c o c to l-( i—i >—i a 2

O J UJ to M to a 8 •-(

U OQ cr: 2 2 O J3 UJ UJ UJ 1 •J •-r- <

u- a: ~J a. S a. c < M H O o. •-i to >—i to c u J <

< —« H XS c •-< UJ D co w [X to to Cl H j ^ H J H UJ < l—l 1 -Jt O. [L u to u S M -J r; r O O UJ o o £ 5 to 3 F UJ 0 _J UJ CO a ex: UJ 5 UJ | i—i a: UJ UJ —1 •J F 5 <

CQ CQ < a. c ••H u. g I |

•*— <

UJ S < o 2 2 H to F UJ 2

74

c c. c <r o e cr c c c c e c c

C c cr c c c c c c c c c c

• c 1 i—i ,—1

M cr >« c -:

o z" cr UJ c to C c cr UJ r~i

c c

c c c c o c

• f—1

cr c C C c c

• •—i

c. a.

to l-H

UJ

c

cr oo

c cr c c C" lO cr to

CZ. c M < C 1^ (U c Z ] cr c cr. ca c c l-H c • i c l-H CC UJ r^ cr

c to < K Z * to l-H

l-H 3 c Q * cr T" c c UJ •—( y—i

tO -J * •X c

II >«

cr c c cr

a •

c * • c l-H

UJ c C o ̂— II f- UJ t t *

* c c cr c II c UJ

cr cr

cr C H r^ u

r i—i * • cr i-H ~J • i l-H H UJ l-H

J * c cr tfl M cr t—I t-H *" to > M UJ IU (U * » J co c l-H z UJ <

NJ Z »—i * c < —i * cr > (—1 CJ -J

H * II c c II *J >• UJ UJ UJ T~ * ' n * rj UJ UJ f- u L" u a; UJ c cr F to C" l-H z z z -J C- (U ll II CU Z Z J UJ l-H U Q cc CC a H »-i c to —1 l-H cr c c 6j a UJ o J3 c cr Dc w l-H F- «* CC cr cr l-H •-H U c: 00 u UJ tu n

r to —1 ^> < cr O b to u z

C_ F H < to D CrT HH C o * . to Z UJ »- z w UJ cr. 83 HI t—1 to UJ -J c c O < o a UJ c o 7" 7~ < f~ 7~ UJ F UJ cr cr CJ u •-- J ex: l-H < 2 0- (—1 s z cr e c a u (X a

fc or H EL u. HH • • UJ a z

T~ < < z e o Q •J !-H r—1 z LU c < tO ca a a. O ^~ UJ UJ < UJ HH t-i U to H i—i •H 1 CT PS t c > Z 7" U-

_! « UJ UJ H UJ UJ < pr. o a HH or D H -J a. < I-H £ CO a. cr UJ Cu UJ U W to UJ

to < £ y o X < g 5 6 UJ > £ b UJ

UJ a o. o

2 o to to 2 7~ z z u < l-H u a 10 -3

75

CU

C

r-^ r c c c u c c c <—i CO c Tt M

c i^ p c c

to • E-

M

vj

c. 1—1

r

r C U

c UJ

Cr- •7. ?: r^ ?: •7. y. ? 5?! 7- \C n CO c (U O o c o c o O c O <. CO CO 03 l-H t-l 1—1 i—i l-H t—i i—i l-H >—i » CO < J

1 H S- H E- E- s- E- H H vc cu CU cu a. a. c. a. a. C Cu cv vC' J co 1

1—c t-i i—< »—* i—i 1—c t—t t—i M 1—1 (-H UJ fcC cu c: K a BC C£ a c f—i E- IT-

D u y U U L; u U u c 1—1 cu M CO

UJ CO UJ

CO OJ

CO UJ

CO UJ

CO UJ

CO cu

a. cu

CO cu

cu f- t—1 c

f—. C: p c H c C c c C ~ u: E CO TT z II -J a cu --H

UJ UJ cu UJ a; tu UJ cu cu cu »—I [ • 5 E~ r- LT. •^r - f^ Or: c. c: a c tr c. c CO < ZJ c. < 00 oo C CU ^ [— — rj 7~ P rz: rj ' u. u. •—( CJ

*• to r> to

-3 -J J J -j -J -J -J _: •J <• p l-H c c o: UJ >—i 1—1 i—i >—i i—i ^H i—* I—1 1—1 ^ CL u CO £- LO cc W,

cr < < < < < < < <r <• F- O CO r: to P>- K c u. u u CL u u u u f- u 7~" CU to a fM c «—i • UJ ^~" c c CN i—' • u cc cr cr cc a; Ct. cr c" c c: CI Cr c c c C OJ to tv cr.

C c C c O c C C C C C ] ! »—i Z ti u; S to lO a. C co £ LO a u u. u. CL u u u. U i_3 H O -^. • • cr. r: SK 0\ o t—i <r i—i CO UJ -•- to \£ Tf e

r r-H UJ UJ UJ UJ UJ w UJ u_ UJ < i—i fe H l-H l-H in cr

M u u u U CJ u CJ CJ CJ U •— J p J M r-~ E- UJ II co < < < < <r < < < < CU C2 —> u CU •—i o: <r •J

r a. a. n. Cu a. a. a. a. C- U C l—t CO < S h«

| UJ oo co co co CO CO CO co co co c Cf J cu l-H OJ ll ^ i-< e F J or: 2 J i—i II

o_ 7~ c f- cu c CO •"; cu p

y UJ to >—I t: < 1—< C3 a cu 7" oc 2-. c -J UJ CO II UJ < Cl e "—1 o S3 p: 11 II >r C u 23 CJ •7-. c: cr 2- UJ Q 1—1 l-H

c p cu UJ p P * • • • • • • • CU < — 2 CO KH C a: r-H H Cc K •J 0. T J c a c o c c c c c *~.. E- < i-H < X cu CU <: c Cu l-H co HH l-H CO o c c c c c LO c < to u a J p < H f- l-H

—i cu CU 7~ CU BJ > CO CU EL co u U U E- F- Ui cu y Tj CU H J H >—i >—i cu CJ l-H U u < -i- a J a. u. U- CO u H H a y ^J 2 c a tf II a ^~ c o UJ O < < c <r: cu < < J^ co < F- M— cu c; *• e >- CL a. CU DC UJ CO CrT cc cu Eij cu =: <r. i o- &' CJ < 2 UJ UJ -3 J •—. f- a. cu < CU cu < C.

as 03 < a. P H l-H > J O- K z T~ ^T* E- r l-H •-H cu c ^< z; <r j! CU H

^ T: H CO < ^ « r. c *r -«-r to c r-. <M «c- C C H c c c c c c c c c * • * » c c c- c

00 VC 00 >C ri to "-f or. r-- •-<

oo cr • »

c c

c c c c c- c c c cr c c c- c c <r c c c e c • • • • c- c cr c c c c c to oc to oo

i- M N X O f) N ir, PH i-- e e- C «-H C vC c ^r oo CN C C rH C c c c c cr c cr c • • • • c c c c

cr f-i ci to to rr- n r i \C c to vc c c- • • e cr

c c c c c c c tN

cr o c c c c c c c c c c c cr cr c cr c c cr • • • • c- cr c c cr cr o e (N MM h

C i-i 00 C CM CM tO vC C M MK; c c to I-- C «N C> »H C C i-i c cr C c Cr e e c e • • • • c c c c

00

to c CM o-. \C 00 "3- tO CM M Ci r- C. 00 • • Cr C

e- l-H

tu •* aa oc 2 fc, sC C

7 C c c C c LO rH

oo CO

• l-H to to c v£ sC C C Cr e cr or r~j 1*. c 1—1 Cl oo

>-c LT r-> to

cr C

cr C

C c

c c

cr c

t»> or io

oc IN 2" r-3- SC

CM LO

2- oa -: CN Cfi o c- C o c e c c c ^c ro II >- "-J- Cr tU t\i i—' > cr CM • • • * c~ c 1—1 C Of oi i-r

tu a r- to c c c cr c c c c C O -=- UJ C, IO

Eu S t= • L0

* c c II c

cr *—>

cr rH

c c c • c • cr

• i—i > tu • •

tu r >—i to tO C ^H i—i r—i C c c c -J c c

LO e co l-H ec tu EU IN • -n 02 c 3 2 CM )—i c < < tt tu •—i c >• l-i

t II o CM tu

II _5 t tu c n II Cf P to K 1-1 J BE: [U tu se 2 cr cr c O r-j to IO cr J 02 aa Q H i—i C cr C cr cr ri rj ««t L.0 to -) 1—1 1-^ -<J C 3 C c Cr; (£! II (—1 c c O c C-J C kC L0 b- < as CM i-H a tO u tU tu - w to c o c: C to oc cn •ra- J > < CT) io a H r- < ^~ to e cr o C C c »—i LO se — et: l-H IT. to

7~ *7- tu tu or r p—i l-l • • » I » tu c r—i c cr t/3 tu -J v. to tu O d * f < f- T o C c C O F- c c c o tu i tu ci r-- j or M < £ o. t—i c cr C Cr < cr o o c c & a: <7i \o a. u H Cr- H u U- u-) c LO C K • • * • • i—* • • ^~ 3 < < 2 O c •—I i—( (N C c err c cr en NJ o c < CO ec C 0- o jr j tu tu < to f- t—i •—i

r B! oc < 1 c >

•J CrT tu tu H LU LU a 8S c-r D E

M

NJ a, < i—i CQ CQ w tu O- tu 1/3 li-

< < y o X

- i cU i—i i &

tu > t

ce: a to to _• s s H a -1 i—^

77

M <

OQ UJ < or. -3 UJ or.

o

4) 3 C

«-> c o u

o CL

-<

UJ UJ u a:

UJ u UJ

O u a UJ

U-

U-

HI 0C U UJ UJ £ a. o to .J

78

tu i 00

c c c c c c c

s?

c c » er

c c 10 c

g

zzzzzzzzz ocooooooc HHHHHHHMH

cacx£a.a.CLa.o. HMMMHMMMM

UcJUUUCJUUU WWWtOCOWlOlOtO wuuamuiuwu; COQCQOCCO

unuituuiuHu o: c K a or K e e:

HHHHHHHHH <<<<:<<<<< u ii.u-t-uu.uutL

aBrftrBrB-orKDrc ooooooooo uu. u-uuuu-u u wwaiuiufKuu! uuuuuuuuu

waca.Q.&.aa.t.cL rwiflwtflwwwwB

tu 3 . . . . •JCOCOCOCCC t-iuicoccocmo <OOClOHN(OIOMin Li. _| ^H _| _ ,_« _| ,_,

tu ?: c a. •

r*. c c 0 c >—1 to *» M N *— w P 01 Cr: f-H C r U

t^ \C c

vC II to Q sC to to

« GO < •*-

a. to ~J

•-I KH tu 7~ FH H

tu tU f~ s £ H-1 p? [U g to r:

ii -J Be tu C »-H | i c p K>

00 < -J S r oo ^c 00 (U Li i-< CQ *» vC r» ?: < rr i—i t^ * c t-( u u to fe

i-r, 0i \C f- O oo <NI M

u •r LU KJ I—1 t~- tu 2- c o LO 01 «M Ct O S e CL LO CNI • 3 M s u 1 Q *r 00 r- io J H o Q o —-1 lO rH HH < l-l to w • 1 00 CM < l-l t h> i-< >-i l> c tr. U- > •J X -J 0

tu S ~J J3 u tu CM U Q i—i to < ^ ^H

O OL tu 1—1 00 o F a: K

tu -J 00 II

tu or to o c E < l-l a tu el «. i—i

• z

tu 3 a -3 S 1 03

to

it

a:

II t^ o \C I—1 l-H H

> 8- 2 i—i $ X IU (U H <

< to u oc -3 < •M

P— P 1—1

u u l-l H

U] p 3 tu r | > LU

HH l-l Si l—l -3

u U- o 3 3 a

l ** O

U3 § — . 8 >- < a.

< a. a tu a:

££ 8 < £X £ 9 UJ

QC LU UJ

c <

UJ 3 •H h-l 59 p

Q l-H X c? £ < a.

CO < < P ? d b to

u CO < oo

7n

c c c c e c c c c c c c c c c c c c o c

» • • a c c c c c c c c *r c T c.

C r-- r-l r~- C 1C00 M C H r 1ft cc c n c c cc 10 C t-i c c c c c c cr e c c • • • • cr c c cr

"C o 9 r -^ •- c c

IT •—

C v£ r. r- r- r^ oc c c. c |v o

c c

c cr cr c c c c c c c c c c c o c c c e c • • • • c c c c c c c c M K M CC

C tO rr OC C i-< lO TT c c c c O oc to c C M to to C C •-( c c c c c cr c c c • • • • c c c c

CM c vc r

C i-i vC or

c c • • c c

c c c c c c cr

c. c c cr c oc v£ M c c c cr c ^c v cr. e c cr cr cr e •c «* cr c cr c c OC vC W c c c cr e C r> tM » • • • c c iH cr cr c c c cr c c c c c C c c c c c r; Is n i*» • • • •

o c c cr

TT r-H LT. rr LO LT; h- oc c. to c c Cr OC

• • c c .-

C

PS vC cr.

ffi e rr oa r- o 2 £

OC »-H

C1 C C c cr C c sC TT r: oo • to

to r^ tN C c cr c c I"*. c m C l-H C LT;

Q to •—i c C e cr c VC cr LT. • C CN HH U"> Ol C c cr c c o cr c ro cr. ll cr 10

H _J lO n • (NJ e cr cr cr cr cr cr. CN >- cr c.

6" UJ Tf oc c c • » • • c c r-H e a c -a- K cr •—i c c cr cr c C cr c c >- UJ e to

Sfe fe » » c cr c O cr c cr c cr H > cr t-.

IS c c || c l—l vO i-H VC • • • • >-H UJ • •

UJ u t—i c l-H ^H l-H c <r c c J •-H e •J ? cr to l-H c. UJ U4 UJ cr -J Q2 O

>j 2 >» • <c *^ U. UJ i—i

_r II

cr cr II

i—i >-

T~ II cr UJ u: (- Eu o II n a CM E oo cs l-H

-J S UJ ai S' c C cr e rr 1— l-H c -J ca ca a H »-i c c c cr c l-H to e ri to ' -J l—c c i^ O rr c a: a: UJ •—< C o C c- cr r~j vC r~- H < ca cr to E* CO u UJ

11 II

— to c c o c cr -s- c rxi J > < cr c

a. i to c cr <r cr cr cr v£ LT. cc ec l-H cr —i .-- 2

•i

i—i l-H » • » • 9 UJ cr l-H c c CO UJ J cr cr UJ o o y H -" cr cr cr cr c H cr cr c c US

it u cr <N

J K l-H < < Qu l—l C o cr o < cr C cr cr c cc cr cr VD CL. U- 1 f- u u LO c w cr OL- • * • • • l-H • • y < < 2 o o. l-H r-H CM c C C c cr cr -J ^H C < co ca Bu Q. C -J UJ UJ < to H M >—i K c < | t: > -J K (U UJ f- UJ UJ S < Cr^

a P J a. < >—1 CQ oa UJ UJ c o. UJ 00 co < £8 X 5 § r 2 rr LU H UJ •-H y 5

if. I-I IU O > 2 £ O CO to -J 2 2 H ce L^ <r l-H

sn

5

T3 0)

u to UJ c a CO

(-1 CQ UJ < K h-i 3 •J J tt! HH a: <

U- Z c H 2

>-i f- T to i-« 2 J O

l_> EU U C z z UJ H-I

e *- ©I c c

c c C i- c c C LO o c O 00 c • c cr

ii f-

Z o u

UJ z o a. I o

CO < a UJ Z i—i

r UJ i a

-J J UJ H-I > CO UJ < •J hH

UJ UJ

z UJ o a UJ HH U a z Z UJ o a u H«

a z UJ o l-H U U- l-H K C_3 UJ UJ 3: a o to -J

81

M < f- U.

B

• • * • • vo c * oo r- \C c « »o * M l/i * H H

* + *

0) a r

§

§ UJ c c u

i— I-I i— r rr

UJ

a to u z

UJ o o l-H

u. K to c UJ z

03 UJ >-

S 10 z s w

u O

UJ

to UJ

< a

c c c c c » » • • » c c c c c c c c c c lO IT. C LO LO c c. c-. c c

O I-I —i c c

C". —i c o c

o: i—1 CN tO t—1 t—1 f- UJ Z «

c o o c c c c c o c £§ o o c c o

e o c o c- UJ z c c o o o J c c o o c w o o e e o c

1—1 H H H N M

a Sfc (

• Z l-H K UJ J i i—* K! oc •—i IH C HH tt c rr c c C l-H 03 i—i rr c r— r—1 u <

| c c Ifl C C r—1 <N tO

Z l-H L^ c- M u~. ir: < <r <t C J < c TT r* C C u UJ

to oc

t # oc • 00 • £ fefe K 2 e C c C cr UJ UJ UJ UJ » O 7" 3E tj? U UJ UJ UJ o > *_^ cr - -J J 3 < c

C - < UJ UJ

c UJ UJ c •r. >J J NJ C UJ cc 03 03 t» H c O O • to c c: c^

UJ H c

to a. a. a.

E-r M D2 (U UJ UJ < J II [ • —) •J —i T" *-> LO LT. c oc c w c^ a. ex l-H 03 er N c 1—1 VC *• T r E- < vC vC c «9 » ~2 s < < < tO l-H <tf i—i c oc c UJ UJ to 00 00 UJ -J rs C c vT r- > J

UJ *» |v c oc cc UJ IT, cc

z"* CC • c- • c • 00

a 00

1 3 • %

C

l-H • o c 1—1 C c UJ c UJ c c tr U c ^~ <—1 (N tO a. > z r\: UJ UJ c c c

< UJ Q l-H U z O u

II

UJ

-3 UJ

u. o z c

-J a.

< 00 •

• •

c c c c C c r-l

c c o c c o c c c c l-H l-H

cc t—< r\i 1-0 i—l r—I a H l-H s E- UJ c c c C e UJ H UJ z oa c o e e c l—l 2 C_ H

c c o c C tt. o HH to c e o c c l-H l—l C£ >•

tu ^: c c c c e u to ^J w *J o c o c c UJ to to CO tu o c cr Q c c IX •-I UJ 3

t-H f—4 •—< i—I CM M U-} ^— a to

8?

fc ~ w J 3 tu cu

BQ E u r U w

(U DO s Z3 PS co a: co a

fc6" fca

ea w S3 =3 -J to g. CO D.

£u to W CO J J OQ 00 o o tt K a. ft.

t-< •—1

w o u o -J c •J c ft. o ft. © r c •H c < o < c CO o CO © • c • ©

I—V • <N • M T3 • • 4) 3 i C •H CO i *J >- >- c o § g o 3 3 V—' co co

83

c ^c KI Tf r> rr r- w M r c r- c vC r r- c. rt c oc vC *—i rH c r—«

1 LO •- C vC c CO oc IH c ^-<

C oe vC C c c *- C oc ir, r- L" r r-- OC c or c OC >- r^ • >- oc C" >- =1

M • -i • C J • • -J •

s c £ c c

c fe

c c

c c >

c c c

M u c UJ c ft c f- o UJ c II c: UJ c o UJ c < II a f- c n a H c II II c p C II t ~_

<r o . ' < c •—. -r cr u K <r >- u c c U c< c

Q-

C

>- co UJ

c c £ E-

i—i to UJ c c t t W uo

c >- l-H Hi fc. t*» M -3 I—I B t~^ l-H l-H l-H c r- i-H

u -J • -J I-H • J -J ~ • _j l-H >~ •J c M CC t i-3 c l-H >—1 t -J C H-

>- cc H t—1 CO < •—i ca oc Ki co f- < I-H < < 1—1 I—I < < < l-H < < i—i »—l (—1 -*

[X »—i fN •J. 1-J (1 KH Kl l-H -J U t—1 -5 KJ t-. II -J UJ l-H II J -J M II -J I-H C UJ cr.

% UJ C e." 03

^ UJ Dr UJ CJ H UJ

OC tu Q < e UJ «( a: UJ c <r z B: < co 1-1 < J 02 l-H 3 —1 C= l-H < J t—. 1 z ij E- £ s e. z ^J H UJ 7:

» - — r -J lo ta z

«J O tu to O c UJ to ^.. O c UJ > c UJ 7. i—i C Z UJ t—i z 1—1 Cr- z UJ I—I 7- •—I c z m HH c 00 o -J to tfl o NJ to to c J to "~ to Q O to 5* to cr u to J" LO cr u m 5 UJ l-H tL) UJ »-i UJ M UJ UJ >-H UJ >-H UJ UJ l-H Eu

& ^~ U CJT u 5" F 7~ U c y y h- 5 U tr u r F 7". Z 7. to 7. 2" z W Z z 2

w >• u UJ i—i UJ a >- [X UJ l—l UJ Q >- ti UJ HH UJ C >* CO C C § C UJ to c c. c- e UJ CO cr C L c uj CO cc M t—i u C£ l-H 1—1 u CO l—c I-I u co —1 u (1 CO ti z J2 UJ ft to B 7. I"*1 u u to u- T: " to H z: to Z UJ to E- z to Z UJ to H 7^ to Z UJ to < o <r o a < c <f r C < c < o c

a: r G a M K u u l-H K 5 u C_3 i-H ft c •-I c (1 C •-i C u. c^ l-H c u. o EL E- 5* UJ c z ft f- y UJ c 7: u H *" UJ c z

w UJ Z UJ c to UJ Z UJ o CO ft! s*. UJ c to P- UJ f- i—i 1—1 u >- UJ H i—i l-H u >- UJ H l-H >— tj H E- tO *- tv E- to >»- u. E- to H- U- -J i—i £ >- S »—1 c* >—i

SE >- or l-H e: •-H

fc >- a »H CJ — ~J CO UJ U g -J CO UJ U UJ o to UJ u ty

CO 1—1 i—i aa F UJ i—i l-H CO P UJ * •-H l-H CO F UJ S UJ CO c £2 UJ a. c CC o " tu o_ o C3 o ~~~ UJ c o Cr <

>—i

-3 [U e*

C: tu U 2 UJ Q i—i U-

u

a. W a to 2 < l-H -3

s C UJ r- 7. UJ Q i—i tv

s

ft. to a cr. •J < l-H -J UJ c CH tu

UJ - l-H ft § c_-

£_ to c to J

8^1

to 7. C

UJ >* /-^ u I- • 5r »-H ec UJ J » \C (M CJ D •-* U IflC c

00 c c __ lO LT.

-J < Is- c c UJ {-. t-~ oc cc c w • • •

cc z c c c UJ • c * c u c > »~ -J -c

(I

c D

< f- CO »-• fe; c

| c u

co < J

or

UJ

10 •< to CO 5 to

u c >- <

to

UJ H f- «-> < J ?" •"' t-H CC

to l-i tu -J

UJ H c

c e: Q. >

<

CMC er. >-i vt c «a- e c cc c "C VC t^ JN cc cc cc oo oc

• • • c c- c

< ea u

c to

or; BE W Hi go T

£3 2 UJ

LU H o; to w >. CO to 5 eo £ S Z to

.-< rvj to

E,g CO to to >- >> >-

c to to to c CO co cc c cr to to to c c *• •- «-

" *•.

l>> UJ UJ UJ • J -J -J c CO

c CO CO o o oc ct

II

a;

o. a.

UJ UJ J -J •J -J Ui to a. CL a.

1 2" UJ < ' 5 < <

-J c H to to to • to • • • Oi cr >- • » • u c %2 i • • z (M i—< rsi to UJ ' •

Q to c. cc K >-i II UJ UJ Ui u. u CO CO CO z c (U

o £-

o l-l

z o 2 z z

o H (-1 * . 2- r u s- Ui Ul UJ >—1 Z a. H H f~ a c >-H to to to (—1 1—1 at: fc >- >- u co u to

CO CO UJ to to CO a. KH Oi z: => p to - Q to to to

85

to r» c •—i

rr u a to <£ i—i to •--

c to c »—1

c OC •J OC >- rr u: vC J to C c

c cc Sj IN

^ Tl (U lO \C 0 »

to • e c >-> C T" c to 0 tu c >• f- C II ~ II co <£ c < M cr cr

5 t-4 El Sp

C •-! < kJ M E C -J E~ »—1 •J ij h- t-i < C3 H (-H • S3 c <r OS < C <

•—i - c C CO > o < CO >—i tu 2 tu *-> t-i CO u c <3 co •J J co 2 U CO IU < »—i tu tU l-H K F- p c c u r

BE u l-H Z 2

tu c P c 2 r c tu < oc C 3 •-H CJ o: o tu U CO B- 2

fe* p CO r < 2 tu C Q

gfc 7"' tu CJ t-i M H C U

S M i~ CO tu C 2 tu -J co >- 2 tu O C HH (U CO i-( HH U

03 *~ U- C < H J or: t-i Cr" f- l-l 7- <f tU CJ tu

•J M H H tu 3= I" tu O o tu a. o tu a: •J CQ O

s. H CZ CO i-J

c m ce CJ a. 2

U tu a •J ii a. u~ r 2 < c co u

86

Documents

RELIABILITY DATA ANALYSIS MODELThere are a number of reasons why the constant failure rate assumption is so often made. First, the constant failure rate is often assumed primarily