DTIC_JUN ,I 51982

UNITED STATES AIR FORCE

AIR UNIVERSITY E

AIR FORCE INSTITUTE OF TECHNOLOGYWright-Patterson Air-force Base,Ohio

S 2 B 14Best Available Copy

CCON ?TDS'CE :IIITTERVAT-S- FOR SYST7,iREET BIL TmY ANTD AVA -A IL:: A OF"

.AINTAINED S7STEMS USINOI ,ONTE CARLO 7DBCHNIQIUES

THESIS

G;OR/.MA/81D-7 Mohamed S. 3. Hasaballait. Col. Exvztian Ar--Irw

Acprcvecý for public release; distribu't-on unlimi1ted

CONFIDENCE TINITERALS FOR SYSTEM RELIABIL TTY

AIND .'VAlLABlL=TY OFF MAI-N.TAINED SY'STEM"S

USITTG, MONTE CARLO TECHNIQUES

Presented t~the 7aculty of, the School c*-' Engin~eering

o,-: t're Air F:or-e -ns"it'.,,e of Technclogy

Air Uv-ýerz.tyj

in Partial '7ulfilIThent

Reýquirements fol- ,-hth D,ýýr.e c4"

MastLer of' OperaZ-ins .iesearc-

VeTA 3

MohamedO ''''Jo

SpD' Fas.abal-la

Lt. C ol. Egyplt.'.an Army

Gradua~'p ( ler ons Research

December 1981

Approcved fL~r pu>ý' rclea.se; distributiur unlAl. 1jt,ýd

Preface

With the steady increase in comp2exity of equipment,

n the stringency of operating conditions and in the positive

identif cation of system effectiveness recuirements, it is

becoming harder to satisfy the requirements; and more and

more emphasis is placed on preventive maintenance together

.th the speedy renn.por of pl at uii . L* as a means of

achievement. Test equipment is also becoming more and more

complex. Maintainability engineering is, therefore, an

attempt to achieve scme repair time objectives by specifyinga cor,.inat-or, o' desIn and human factors together wit, the

maintenance philosophy consistent with the other engineering

and cost constraints which exist.

The pi-pose of this study I. co present the confidence

intervals for system reliability and ava iability of main-

tained system using Mont Carlo techniques. The current lit-

erature indicates that Mont Carlo techniques offer the only

oractical method of analyzing large scale systems with general

failure and repair systems. it is hoped that this kind of

study will serve as a useful introduction to the var5.ous

types of problems which are of current interest.

I would like to thank my thesis Advisor, Professor

A. H. Moore, for his most valuable advice and guidance during

"this study, I would also like to thank Dr. joseph Cain,

my reader, for his help during the study. Also, i am

1i

grateful to Mrs. Barbara Folkeson for her help in typing

this thesis. I am also grateful to Mrs. Betty Marris and

Mrs. Mary Browning of the Air Force Institute of Technology

Library for their help in obtaining several references.

Finally I wish to reccgni%- the wonderful efforts of my wife,

Gawhra, who encouraged me to strive, to search, to study,

and, ultimately, to succeed.

Mohamed S. H. Hasaballa

li!i

TABLE OF CONTENTS

Page

Preface .................. .......................

List of Figures .............. ................. vi

List of Tables ........... .................. .. vii

Abstract viii

Chacrer

T. :ntroduction ............ .................

I'.r Survey of Technicues to Stu-ly Rieliability,.aintai.,abilty, and Availability of MaintainedSystems. .......... .................. .. L

?Reliability, Maintairab-ilit, and Ava211-abiit Def Ined .

The Basic Markovilan Fallure Models ....

Application of tuh Bas!M Yarkovlan Modelsto ReliabitLty .......... ............. 10

The Basic Markovian Repair Model. . .

Application of the Basic Markovian RepairModel to Malntainabillty. ........ 17

Specific Applications of Markov Processesto Reliability and Maintainability . ... 19

Redundant Systems ...........

Maintenance Systems .......... 38

Avallabilitv. ......... .............. .!

!I . Suxmmary of Point Estimates of Availability,l of- Mairtained Systems 51.....5

'omcu•-Iilon of Re!iability and Availabil!ity 5

. .. C - r•" e TA. . tF for Availab•..•ies ofMainta ied j stems, Exact Anai ytia! Methods 64

iv

Page

The Case of both Exponential Distributionzfor the Time-t-Fl and Time-to-Repair U.

Lowfer Confidence Limits Assuming Lowrncr-iallyDistributed Repair Times .... .......... .. 68

V. Yiont Cario Comparisons Confidence Limits forAvailability and Reliability .... .......... 73

Surmnary ............... ................... 73

Results . . . . . . . . . . . . . . . 75

Blioraph ............................ 78

Appendices ....................... 87

Appendix A - Major Reference Sources .. ....... .. 88

ApDendix B - General Reference Bibliogranhy ... 90

Acoendix C Supplemental Bibliography ... ...... 93

J t a . . . . . . . . . . . . . . . . . . . . . . . . Q9

7:t..............................................9

LIST OP FIGUFES

Figcure Page

1 Granh of the Basic Markovian Failure rModel

2 Graph of a Three-State Failure Model ......... 9

3 Graph of the Basic Markovian Repair Model . 16

4 n-th Extension of the Basic MarkovlanReoair Model .......... ................. ... 18

S Grap.. of Two-7 ..emet Non-Recalrable System 20

6 Graoh o' Two-Element RepalrabIe Sy.stem ... 2

Trans±iorzn Diagram for K-out-of-n System 28

8 Success Yodes for a Two-n1tt Standby Sysem 33

9 Failure State Diagram ....... ............ ... 35

0 ,va-. .labIty Graph for a Single ComponentSystem ................ .................... 42

ii Data Processor (#1) and Tape Units (*2) 46

12 Trar:siton-rat- Diazram for Computer System 47

13 Transition Diagrams for Two Component Systems

(a) Availability... ......... .............. 52

(b) Reliability ......... ............... 53

_ Availability Tran.sition Diagram Cor TwoComponent System with Two Repairmen (No Jo,'intEffort) ....................... ............. 56

15 Transition Diagrams for General System of TwoIdentical Components ...... ............. .. 57

1._ Mrkovan Rellabilty Model for Two ldentlcalParallel Elements and K Repairmen .. ...... 60

vI

LIST OF TAaLES

Table Page

3.1 System States fo- 2 Cnmporent -Ineparab• e

System . 51

3.2 Coefficients of / u' .w ... .......... 62

5.1 Results of the Double 'ont Carlo TechniqueExponental Failure and epair, "-Imes . . . 7 75

5.2 Results of the Double Mont Carlo TechniqueExponentia. Failure Time and LognormallyRepair T.me... ............ ................ 76

vl!~

ABSTRACT

This thesis presents the results of aSt extensive

literature search for finding the confidence intervals for

system reliability and availabIlity of mnIntained systems

using Mont Carlo techniques. The characteristics of system

reliabilIty and maintainability analysis are dIscussed. The

basic MIarkovlan failure ana repair models are developed. A

summary of the point estimates of ava'lab lity, relab-llt

of maintai-rd systems, and the exact analytical methods are

esent...d. 'Finally, the Bootstrap (Double Mont Carlo tech-

nique) is used to obtain the confidence ltmts for the avail-

ability of the maintained systems.

'.4i i

CONFIDENCE !NTERVALS FOR SYSTEM RELIABILITY

AND AVAILAEL•TYITY OF MAINTAINED SYSTEMS

USING MONT CARLO TECHNIQUES

I. introduction

The reliability and maintainability disciplines, as

the:.' exist today, has evolved primarily during the cast two

decades. 7.e impetus for these disciplines grew -arzely ouz

of military and space programs and was related to several

pr'ncpai considerations:

,a) the relatively high failure rates of equipment,

durinz the 1940's and early 1950's;

(b) the resultirn sharp increase in the cost of pro-

curement and maintenance;

(c' the continual increase in total oarts and func-

ticnal complexity; and

\d, the rco-gnized need to have an 'f't_'1-7,t _Fpproanh

to exclude or minimize the conditions that contributed to

fauity equipment.

Tn the evolution of the rellability and n--','..l

io'." dIsc1cZInes, many mathematical methodologies were devel-

oped to account for and explain the observations (failures

or -enars) generat•d by various equloments or systemts undzr

cons1.deratl.o, Sin e the coservations were (and cntinue

"to bel often random in nature, e. z , 3tocastic rather than1 • . . v ;.

II

deterninist-c, orobabllistic concepts were applied, to cor-

rectly analze the particular system. Since the Mont Carlo

technicues offer the best and valuable method of analyzing

larze scale s-:stems with general failure and re•air systems,

these technicues are used to calculate the confidence Inter-

vals for system reliability and availability of maintained

systems.

Rellabilty is a yardstick of the.. capabIl.ty of an

e-ulc-rr.ent tooperate without failures when put into service.

elIabili-.y predlcts mathematically the equipment's behavIor

under expecrt»; oce-ating conditions. More specifically, re-

il'biwiexcxresses in numbers the probability an equloment

will. onerate without failure for a given length of tIme in

an . . , r....r . •e ..... • it was * eA igne

Mi'intalnability is a more widely known term. Spe-

c.icallv, I7; is defined as the probabilIty that a failed

system can be made operable •n a specified interval of down-

time. Ma 4 ntenance actions can be classifiel into two cate-

gories. First, there is cff-schedule maIntenarnce necessi-

tated ty system in-service failure or maifunction. Its pur-

pose is to restore system operation as soon as possible by

replacing., repa.ring, or adjusting the component or compo-

nents which cause interruption of service. Second, there is

a scheduled maintenance at regular intervals; its purpose i.to keep the system In a condition consistent w•th Itsb

_n levels of cerformance, reliability, and, where applicable,

saey.

Availability is defined as the probability that a sys-

ter is operating satisfactorily at any poin.t in time. The

problem is to calculate the confidence intervals for systemriabiltv and availability of maIntained systems using

Mont Carlo techniques.

To achieve the study objectives, an extensive search

was made of the available literature. The results of this

search and the ensuing study are reported in the following

sectfons: Section Ii includes a survey of techniques to

study rela b ilIty , maintainability, and avallab 'ity of main-

tained systems; Section III contains a sumiary of point estI-

mates of availability and reliability of maintained systems;

Section IV contains the confidence limits for availabilities

of mainained systems - exact analytical methods; and final-

ly, Section V contains the Mont Carlo comparisons confidence

li-Its for avallabil ty and reliability. Appendix A contains

a list of major reference sources - that is, documents which

contained a majority of the references used in the study.

4pnendix 3 iq a reference bibliography containing literature

which is of a general nature, yet nertinent- to the second

objective of the thesis. This information has been included

as an aid to the reader for further study. Aopendix C Is a

sup'.emental bibliography containing two tyres of references:

those applicable• to the thesis but noc selected for detailed

analysis and those which appear to be applicable but were iot

readily obtainable.

Ii. Survey of Techniques to Study Reliability,

Maintainability, and Availability of

Maintained Systems

Reliability and maintainability engineering activi-

ties are performed for the purpose of determining and improv-

ing the reliability and maintainability of a particular sys-

tem. These activities often involve the analysis of physical

processes which have probabilistic transition events, in

which case the state of the orocess at any time t is a ran-

dom variable. Reliability and maintainability analyses, there-

fore, rely heavily upon the techniques of probability theory.

Reliability, Maintainability, and

Ayailability Defined

Reliability is the probability that a system will

perform its intended function satisfactorily for at least a

given time period under specified operating conditions. The

probabiIity of failure as a function of tLiae can be defined

by

p(t s t) = F(t), t ý 0 (2-1)

where t is a random variable denoting the failure time. Then

R(t) is the probability that the system will fail by time t.

if we define the reliability as the probability of success,

cr the probability that the system will perform its intended

function at a tIme t, we can write (Ref 43: 9)

1 I) P t >t 2 2

where R(t) is the reliability function. If the time to fail-

ure randor. variable t has a density function f(t), then

tR(t) = 1- F(t) = I- If(f)at = !f(T)aT (2-3)

0

Maintainability is a more widely known term. Spe-

cifically, maintainability is defined as the probability that

a failed system can be made operable in a specified interval

of downtime. Here the downtime includes the total time that

the system is out of service. Downtime Is a function of the

failure detection time, repair time, administrative time,

and the logistics time connected with the repair cycle.

Theoretically, for a product there exists a maintainability

function. The maintainability function describes probabilis-

tically how long a system remains in a failed state. Mathe-

matically

p(t 5 t) = m(t) (2-4)

where t is the total downtime.

Availability is defined as the probabilit, that a sys-

tem is operating satisf oily at any point in t and con-

siders only operating time and downtime, i.e., excluding the

idle time. Availability is a measure of the ratio of the

operating time of the system to the operating time plus down-

time. Thus it includes both reliability and maintainability.

,he relL•a_1ity and maintainability fields have

evolved as distinct disciplines since 1950. 3oth rely heav-

ily upon mathematical methodologies. Since they are concerned

*1

5e

with probabilistic concepts, many well-established mathemat-

ical concepts and techniques are applicable. One of" these

concepts is the theor:, of Markov processes. M~any of the

determinant-s of the reliability and maintainability of a sys-

tuem are random processes (sequence of' states), any given

state of which is dependent only upon the previous state in

the process. Such processes are, by definition, Markov pro-

cess; few reeecsthat present Markovr process theroy In-

clude applications of this technique to reliability and main-

tainability. Thus it is the purpose of this chapter to pro-

vide the Markov models for reliability and maintainability

analysis.

The Basic Markovian Failure Models

The basic Markovian failure model (Ref 30:18) consists

of two states and a single transition event. The system '-s

in state 0 if it is operating satisfactorily, and it is in

state I I~f it has failed. The transition event from state 0

to staite 1 is a system failure. Figure 1 shows the graph of

the basic Markovian model. At any point in time, if the sys-

tem is in state 0, it can either fail with pnrobability p.1

or not fail with probability p00 . System failure, state 1,

is considered to be the end of the process. Therefore, p,1

may either be ignored or set equal to I

The Markovian propertiet3 of the model result from the

dependence of the one-step transitfon probabilities on only

the inmmediatel', prrav~cns state. To amplify, the proballiit;

1711

Fig. 1

Graph of the Basic Markovian Failure Model ','rom Ref 13:18)

of the occurrence or nonoccurrence of failure in the time

interval, (t+At); it is assumed to depend only upon the sys-

tem being in the unfailed state at time t, regardless of the

past history of the system. The one-step transition proba-

bilities of the basic failure model depend upon and can be

derived from the failure characteristics of the system when

it is in state 0. The probability of a system failure in a

small time t is

I f(x)dxt

where f(x) is the probability density of the time to failure

cf the system. Since failure cannot occur unless the system

is operating, the conditional probability of failure in At,

given the system has not failed prior to t, is

t+At

I f(x)dxt (2-5)P O l cc

If(x)dxt

Equation (2-5) divided by At is the system failure rate (Ref

3:60)."

The hazard function is defined as the limit of the

failure rate as the interval approaches zero. Thus the haz-

ard function is the instantaneous failure rate:

h(t) =f(t)

also

f(t) = h(t) exp -h(-)a (2-7)0 i

Thus f(t), R(t), and h(t) are all related to each other and

any one implies the other two (Ref • 3:_).

Assuming stationary one-step transition probabilities

In the basic failure model, POI (qiAt), the failure rate

of the system is (qo0 .At)/At = Q,,, and h(t) = qOl. Letting

qCl X, if the time to failure is described by an exponen-

tial densit"y function, then

f(t) a e-t (2-8)

R(t) - !e-XTe Tt

-T

( t

i~e.,

and

f(t)R(t) (2-10)

Thus to achieve temporal homogeneity of the Markovian fail-

ure model, exponential failure times are usually assumed.

The basic Markovian model can be easily extended to

include system states other than total operability and total

failure. Figure 2 is the graph of such a 7jodel.

POO P 1 1 P 2 2

P02

Fig. 2

Graph of a Three-State Failure Model (frcm Ref 30:20)

State 0 and 2 are "system operating" and "system failed,"

respectively. State 1 is any state of system degradation

that changed the failure characteristics of the system but

does not cause total system failure. The transition event

from state 0 to state 1 is some form of system deterioration,

and the transition events 0 + 2 and 1 - 2 are the two differ-

ent modes of system failure that can occur. Because these

threeL tz- nsitiun events differ physically, it is usually the

9

F!

case that p0 1 ZP0 2 PI 2 Extensions of these basic failure

models to different system configurations require approximate

addition and redefinition of the possible system states and

their associated transition probabilities.

Application of the Basic Markovian

Models to Rell-ab--iity (-ef 30:21). System reliability,

R(t), is the probability that the system does not fail prior

to time t. in terms of the basic Markovian failure model,

Figure I, R(t) is the probability that the process does not

progress to state i prior t, time t for a contInucus-time

Markov failure model:

R(t) = 1- pl(t) :- Po(t) (2-l1)

Assuming stationary transition probabilities,

PO(t+•r) =p O(t) P po0 = PO(t)(I - qo 1 "AL)(2-12)

Po(t+At) = Po(t) - Po(t)qo].At

poct+tt) - o(t) -Pop(t)qo .0 At

t• 4- +o IAt = Po(t)qc~l

Using the definition of the derivatives of a function,

p'(t) = -qo! " PO(t) (2-13)

Laplace transforming of equation (2-13)

"•0'O -q0 PO 0S10 =•

Assunming p 0 (t 0 ) 1 and solving for p 0 (S)

PO (S) = (-1Ip0..C• - +q 0 1 (-a

and

r 1 *qlt (-5

L ![P 0 (s)] = p 0 (t) R(t) e (0-'5

for the corresponding discrete-time failure model

R(K) = p 0 (K) = p0 (p14 (2-16)

assuming stationary p,, and p0(O) = 1

R( = (1 - q 0 1 .At)K (2-17)

-he expression for the reliability of the three-state system

deascribed by the failure model in Figure 2 differs from that

cf the two-state model in that the number of different paths

resulting in system failure is increased. When the time

parameter of the model is continuous,

R(t) = p 0ot) + P1 (t) = 1 - P 2 (t) (2-18)

The probability P2(t) is determined as follows: from Figure

2 and the definition of Pij(At) (Ref 19)

(At)j II = ni (at) q ,2(At)

L 0 0 1

The n-

11

PO (t+At) p p0 (t)q 0 0 , At

-. o(t)[1 - (q0 .t + q~2 ~

o,(t+At) p0 (tY~')--t + r tqlL

-p0(t,)qo 1 .At + pD1 (t)( -

P (t+Lt) =pO(t)/q 'A U + P,(t)q2 'L + p(t)(2-19)

Taigderivatives,

po,'t) =-(qol + q0 2 )PO(t)

, t q0 p (t) -q 2 ,t

P 2 (t) qC*T2'0't) + cl2P 1(t) (-0

L~ap~lace trans~forming equation (2-20) and. assurn~ng p,(t.)=

and o It = 0, then

(s + c10 1 + q0 2\'00s) I

q0 1 p0 (s) - (s + q, 2 )F 1,(S\ =

q 0 2P0 (s) + q1 2PI(s) - sro.,(s) =0 (2-21)

By Cramer's Rule,

s+ c- ~02 0 1

-(s + q,.2 C

PS)q 0 2 q 1 2 0p2 (s s + q 1+q0 0 0

q1-(S + q 1 2 ) 0

q 0 2 q 12 -S

12

qoq`2,q02( +q1 (2-22)

P2 -5rs + q~Y + +ý

By partla2. ,,ractionls

B +

P2(s) s +s + q12 q2 +

q0q2 + q02 + )

A =- ( s + q,,)(ý5 'r qo q0 2 7

qCl 2+ q02 q12

B o - qo 1q 1 2 0 S + q2

+ q-', +q22

B s - -q12 +q12

+ ~ q 12'

,. obtain. the vaille of C we have

qo:Lql2 + '- 02 (s + q12)AB + 3c

......+ q 0 ( + 7 s $ + q 2 + q0

-- s-75-7 Bsos 02s++I + 'qc9 0

+ + (.9+ qs( j

A( 5 2 + sq0 1 + sq0 2 + sq.12

+q, 1 + a02 1 ) + 2

+ Bsqo1 + Bso0 2 + CS2 + Cs" 12

Taking the coefficients of s 2 in both sides

0 = As2 + 2 + 2

S A + B + C =0i

so C A + BN

q 0 1 + q2 -q 1 2

q-12 - q02 ] .

q0[ + q 0 2 - q!2

q1 2 - q 0 2

q1 2 - + q0 2 )

. 1, B - (q C -q! 2 + qO201+ -+12 q 002 - q 0 2'

(2-23'

pI (-q0) /q + q 0 2 - q 1 2 )2) s + q12

+ (-(q 1 2 t qO2)/ql2 - 02' • q 2S + a01 + q 0 2

[t 2 (s)] P2t) = 1 - B exp - q2t - C exp - (q 0 ! + q 0 9t

R(t) = 1 - p 2 (t)

R(t) - B exp - q 1 2 t + C exp - (q + q0 2 )t (2-214)

The reliability expression for the discrete-time model of

Figure 2 is

P0 (K) + p,(K) = - (K)

1P4

-~ -- == _ -- - _ _

where p 2 (K) is the sum of all possible paths that will cause

the ,stem to be in state 2 after the K-th step ocf the pro-

cess

p2 (K) a p 2 (O) + P (0')p 2 (K) + p 0 (0)P 0 2 (K) (2-25)

using the transition matrix method (Ref 19)

P2 (K)K) (2-2().

i.e., the third (last) element in the vector resulting from

t:,he multiplication of the initial-state probabilities vector

and the K-th step transition probability matrix. The relia-

bility expressions for more complex Markovian failure moaels

must be derived from the individaul model in terms of the

possible states and the failure characteristics of the sys-

tem being modeled. The general approacn, however, is the

same as shown above.

1he Basic Markovlan Reoair Model

The basic Markovian reoair model (Ref 3C:27) is shown

in Fure 3. -,The system states of this model are identical

to the states of the basic Markovlan failure model: state 0

if the system is operating and scate 1 if the system has

failed. The transition event of this model is any repair that

places a failed system in satisfactory operating condition.

From the failed state 1, the system can either be repaired

in t with crobabllity p., or not repaired in 6t with proba-

b ... y s,.... The repair process is completed when the system

Poo P!!

FIg. 3

Graph of the Basic Markovian Repair Model (fro- Ref 30:27)

is returned to operable condition. Thus, p0 0 = 1 the random

varlable, tine-to-repair of a particular system Is a funct-ion

of the repair characteristics (i.e., accessibillty, complex-

iby, 'u r .i i -urato. , etc ) of the s',, t. m and the -ch ........

tics of the repair activity (i.e., maintenance procedures,

non power, technical proficiency, etc.). The t-ansition prob-

ability, p0(t is derived from the probability density,

g't), of the system time-to-repair. When the repair-time

characteristics of the system remain unchanged regardless of

the past history of the system, the density, g(to, remains

unchanged and the repair model is Markovlan. The limit, as

t - 0, of the conditional probability that the system will

be reDaired in (t+6t) given that it is failed at t is the

instantaneous repair rate, analogous to the hazard rate of

the basic Markovian fallure model. To achieve stationary

p.,, exponential repair times are often assumed, in which

case, Pl, = w where q., Is a constant repair rate.

Because repair is the reverse process of failure,

except for the nature and direction of the transition events

as discussed above, the operation of the basic Markovian re-

pair model is similar to that of the basic Markovian model.

-- Application of the Basic Markovian

Repair Model to Maintainability (Ref 30:28). System

maintainability, M(t), is the probability that a failed sys-

is repaired within a specified time. In terms of the basic

Markovian repair model, Figure 3, M(t) is the probabillty

that the syste. transition from state 1 to state 0 within

the time interval (t 0 -t). For the continuous-time Markovian

repair model

M(t) = P 0 (t) = 1 - pl(t) (2-27)

Absuming p! 0 stationary and pl(tO) 1

N(t) = 1 - e(2-28)

The corresponding expression for M(t) of the basic discrete-

time repair model is

M(K) = 1 - pl(K) =J - (1 - ql 0 .t)• (2--29)

As it is used above, and consistent with the defini-

tion of maintainability, repair is corrective maintenance.

Each different mode of failure of the system requires a dif-

ferent repair action. Thus the basic Markovian repair model

can be extended to descrite the system at any desired level

of detaia. Figure t shows an n-th extension of the basic

17

'rL' ~ ~ ' __ ________ ____ ________.==

0 P0 Pl

?2)

n P22".nn

Fig. 4

n-th Extensicn of the Easic Markovian Repair Model

(from Ref 30:30)

Markovian. reair model. State 0 is the "system operating

satisfactorily" and states 1 through n represent system fall-

ure due to the possible modes -r malfunctions by which the

system can fail. It is reasonable t. assume that the time-

to-repair for each different mode of failure is different

and that P1. = P20= Pn0" For this model,,

nM(t) = D Pi(t0) p!)(t) (2-30)

Other extensions of the basic repair model depend upon sys-

tem configuration and system repair characteristics.

i8

Soecific Applications of Markov Processes

to Reliability and Maintainability

Numerous extensions and combinations of the basic Mar-

kovian failure and repair models were found during the liter-

ature search portion of this study. Representative models

showing the applicability of Markov process techniques to the

estimation of system reliabili'y and maintainability are pre-

sented in this section. Specific attention is given to Mar-

kov process applications in redundant systems and maintenance

systems. Examples of the combined failure and repair models

used to deter:mine the system availability are also discussed.

Redundant Systems. Shooman 'Ref 78) presents an exten-

sion of the basic Markovian failure model: a two element non-

repairable system. Figure 5 shows the graph of the system.

The author makes prov'ision in the model for failure hazards

which are a function of time, i.e., non-homogeneous. This

discussion, however, considers only the homogeneous model

with constant failure hazards, PJK= 2i'

The states of the system are defined as follows:

state 0 = both elements operating;

state 1 = element I failed, element 2 operating;

state 2 = element 2 failed, element 1 operating;

state 3 = both elements failed;

and the transitions may be described as follows:

= the failure rate associated with the •wo states

in question

19

~3a

Fig. 5

Graph of Two-Element Non-repairable System

XAt = the transition probability due to element fail-

ure in time interval Lt.

Using difference-differential equations procedures, the author

develos the following state probability expressions:

Po(t+At) -ri - (Al+ 2 )ItItPo(t) (2-D1)

P[lAt]pO(t) + [1- A 3 t] o(t) (2-32)

P 2 (t+At) = [3,2Lt]Po(t) + [1- YtItP 2 (t) (2-33)

't+At) = D .3•]i(t) + [4At]pp(t) + lp•(t) (2-34)

20

Th, solutions to these equations, with initial conditions

P(O) = ! and p,(O) py(O) = P3 (O) = 0, are

p0 (t) = e (2-35)

Pl(t) = 1 Fe-3 - e 1 (2-36)1 2 3 t

2 - e (2-37)P2+1 2 A1

p3(t) 2. - [p0 (t) + Pl(t) + p 2 (t)] (2-38)

For a two-element redundant (parallel) system, one failure

can be tolerated; and hence, there are three successful states:

Po(t), Pi(t), and p 2 (t). Since these states are mutually

exclusive, the expression for system reliability is

R(t) z po(t) + p!(t) + p2 (t)

or

R~ ) e-(X I + Xe2)t +' [e-X3 t I(k + X 2)t

R(t) =e + + kl X 1 [e+ 2 e3

+X 2 -X 4t - ( i + x2+ l+ f>- k4 e (2-39)

The author points out the complete generality of equa-

tion (2-39) and its usefullness in determining the reliabil-

ity of any two-element redundant system with constant hazard

elements (Ref 78:235). For example, if the hazard functions

are the same regardless o! tho stale of the system, i.e.,

21

I= x4 and X2 = 3) then

R(t) = e x + e e (X 2)t(2-40)

If the elements are identical and the failure rate with both

operating is Xb ard with a single element operating Ns, then

1 = x2 = xb and X3 = X4 & Xs which yields

-X t -2Xbt2•be b

R(t) - b s (2-41)2xb- s

Finally, i b X, then

R(t) = e-t (2 - e-t) (2.-42)

(Ref 43:217) shows that the expected time to system failure,

found by integrating equation (2-40) over the range of t, is

1 1 (2-43)Est = i-i+ - 2 -I+-

when all the units have the same failure rate k. Then equa-

tien (2-43) gives the nean time to failure for' a two compo-

nent system as

E (t) = 3/2X

s x 2

for a three component system, we have

Es(t) = !i/6x

and in genrai for n component •ystemr, the mean time to fail-

ure is

22

nE (t) z e/4. (2-44)s il

where 6 - Here it can be seen that the marginal gain

in the mean time to system failure decreases with each sub-

system added.

If equation (2-39) represents a two-element standby

system initially in state zero, then X, = x and X = 0A 2

since element 2 cannot fail prior to failure of element 1

because it is an unenergized state. Thus X3 = x B and X4 need

not be specified since if P0 (O) - 1 and X2 = 0, state 2 has

a probability which is zero. With these substitutions, equa-

tion (2-39) becomes

- Bt e•At

R(t) = (2-45)A- x B ýA- XB

in a similar application, Zorger (Ref 91) computes the relia-

bility of a "maintened" or repairable two-unit system. In

this model, the author assumes the failure rates are the same

regardless of the state of the system, e.g., X 1 - X and

2 = x 3" The repair rate of component i, ii, multiplied by

the time interval At gives the probability of transition due

to a repair. The graph of this system is shown in Figure 6.

Due to the fact that At is considered infinitesimal,

the probability of a double transition in at is understood

to be zero. System reliability is compcted on the basis of

the first time state 3 is reached; subsequent system repair

has no ef.ect on the reliability of the system. Hence,

23

,D 4

Fig. 6

Graph of Two-Element RF=pirable Syscem

p 3 2 (t) and p3 1 (t) are also considered to be zero. The author

shows that the reliability of the repairable system is

R(t) = 1 - L-p 0 (s) (2-46)

where

PO(s) - LSPo(S)/A(S)

3P(S) -- X! 2 A3 312s + xL + XID + 1j, + 12]

+ x2 A 22s2 + s(2X1 + X2 +

22

+ \+ XX, + +

2i

+ X AI[S' + s(% + 2> 2 + Ul) + X,21 C.

+ XIX2 + 1IX2 + i 1 Xl] + A0O[S3 +

+ s 2 (2X 1 + 2X2 + 1 +2 +s(

2 2 +X+ "2'2 + "1'1 1 2i+ + V2Xi + PlX2

+ 1 1 2) + AX 2X + X X2 +

+ Xl¾l'2 (2-47)

With A, representing the initial condition p (t 0 ) andii1

F= -s (s + X + 12)(s + X2 + V'N/(S + X + X2 )

U2 wX2 (s + X2 + Ul) - ll 1(s + XI + u2 )J (2-48)

letting AO , A = 1, A 0, 14 - 11 - u and X, =A0 0 A1 1 ~'22 '1 2 2.

X, the above expressions are greatly simplified

B(t)= .AeBt Be AtA- B (2-49)

where

A, B (-3X + v) ±--+ 6uV + v'T (2-50)

' 2

using equation (2-49), the author shcws that the maintained

system NTBF is (3X + u)/(2x 2 ).

Zorger makes no mention of system maintainability.

it can, however, be derived from the repairable system model

ty conslkierng only tne repair characteristics of the sys-

tem. Aszuming one repai.rman is available, the system is

25

repaired when either clement 1 or element 2 is operating.

Also, p 3 0 = 0 due to the infinitesimal probability of two

repairs in At. Hence,

M(t) = p 2 (t) + pl(t) - 1 - P 3 (t) (2-51)

By the methods used above,

P 3 (t+At) = P 3 (t)(i- .iat)(2 - i 2!t)

3 13 (ut + u2 At)P 3 (t) (2-52)

p'(t) = -(I. + uI 2 )p 3 (t) (2-53)

sP 3 (r) - A3 3 - -('l + u2)P3(s(2-54)

From the definition of maintainability, A3 3 = 1 and A2 2 =

Al • 0. Thus

sp 3 (S) - i = - + .i 2)p 3 (S)

i.e.,

P 3 (s)(s + I+ +2) 1

1

p 3 (s) = s + 1 + (2-55)3 S-+ Ij + u2)

p 3 (t) = e(Il + P t (2-56)

M(t) - 1 - e 1 2 (2-57)

Messinger (Ref 58) considers a "K-cut-of-h" redundant

ýt....... co-mosed uf identical eiements with constant

26

failure rates and no repair capability. The system will

operate when at least any K of its components are operating.

The states of the system are defined to be the number of

failed components; thus, the number of system states, n + 1,

increases linearly with the number of components, n, in the

structure. The component failure rate in the i-th state

(i.e., when i failures have occurred) is denoted by Xi. The

graph or transition diagram for this system is shown in

Figure 7. The reliability of the K-out-of-n structure is

computed by summing the probabilities of successful system

states 0,1.,2,...,n-k , i.e.,

n-KR(t) E p pi(t) (2-58)

i=0

The author points out that considerable computational effort

can be saved (especially if K is near n) by grouping all of

the system failed states into a single collective failed

state. (This is essentially equivalent to partitioning the

transition matrix into successful and unsuccessful states

as considered by Sandler (Ref 73:100).) This grouping is

permissible because the individual probabilities of being

in the failed states are Qf no interest in ca.culating

system reliability. Once the system enters a failed state,

it canno*, wIthcut repair, return to a good state. Thus,

the system can be modeled by n - K + 1 good states and one

collective falled state, a total of n - K + 2 states. using

a collective failed state, the transition diagram would

27

I 4-3

0

4.3)

* I1

r* cr

28I

appear as in Figure 7 with state (n-1) relabeled as (n-K)

and state (n) relabeled as (F) for the failed state. The

transition probabilities would also require appropriate

modification.

In the special case where the hazard rate is assumed

proportional to stress, i.e.,

S_ n •0 for i = 1,2,...,n-1 (2-59)i n - L "''

and the components are all assumed initially operational,

the author shows that the system reliability is

n-KR(t) = e t (2-60)

i=o

Reliability production formulas for standby redundant sys-

tems have been developed by numerous authors. Equation

(2-45) is an example of the reliability expression for a

two-element standby system. In general, the reliability of

a structure composed of N identical elements, of which N - 1

are in stanby (assuming exponential distributed failure times

and off-line failure rate equal to zero) is given by

R~) _Xt N (xt)JN Z (t (2-61)

J=0 jl

Benning (Ref 9) employs Markovian techniques to

derive the reliability expression for a structure involving

29

N identical components of which at least K (K < N) are re-

quired for system operation while N - K are in standby. The

author shows that

N- KR(t) e"K- "-K (Kt) (2-62) i

and that the mean life (ML) of this configuration is

ML (2-63)

For a similar K-out-of-N system 4n which all N components

a-e on-line, the mean life is

N-K ,ML= - (2-64)

i1 0 M 1

C...paring equation (2-63) and equaticn (2-64) term-by-term

shows that every term in equation (2-64) is less than or

equal to each term in equacion (2-63). This result, as the

author explains, shows that the K-out-of-N standby structure

is more reliable than the on-line structure providing switch-

ing is perfectly reliable.

Kapur (Ref 43:221) shows that for the perfect switch-

ing case Rn(t) - tne reliability function for a standby sys-s

tem that has n subsystems - is

Re- n (et) X /i. (2-65)i=0

wherc X is the failure rate and is identica! for eachn standb.y

30

unit. However, for larger k, the design of switcning cir-

cuitary would be quite complex if it were done automatically.

Hence, it appears that standby redundancy would be most use-

ful in those applications in which manual switching or re-

placement of components would be tolerated (Ref 13:137).

Enstein and Weinstock (Ref 28) consider a similar sys-

tem composed of N units of which m are on-line and N-m are

off-line (standby). Of the m on-line units, at least K

must be operating properly except for an "acceptable dcwn-

time" which is less than or equal to t. units of time. In

other words, even if less than K units are operational, the

system is still considered to be performing satisfactorily

as long as this condition persists for less than time to.

It is assumed that each element of the system has indepen-

dent exponential failure and repair times and that there are

N available repairmen. The authors consider the system to

be in either one of two states, the good state or the bad

state. If thp system has had at least one failure (less than

K units operational) for longer than to prior to the time

in question, it is considered to be in a bad state. "Con-

ditional availability" is defined by the authors as the prob-

ability of being in a particular state at time t conditioned

on the fact that the system is not in a bad state at time t.

Strinivason (Ref 8Q) applies a Markcv model to a stand-

by redundant system where the swltchover is not instantaneous.

Switcho.er time IS c ('-P tn be a random variable and

is accumulated from the Instant action is initiated to bring

31

the standby system to the active state, to the instant at

which the standby unit becomes operational. The author

assumes exponentially distributed failure times, an arbi-

trary repair distribution, and an arbitrary distribution for

the probability that switchover is completed in (0, t).

After deriving a lengthy equation for the probability dis-

tribution of first time to failure, the author then obtains

an explicit expression for the expected time to system fail-

ure. His results show that if cost factors are of no con-

sideration and if the objective is to have maximum MTBF,

"The best policy is to initiate switching action . " just

at the moment the subsystem becomes standby (Ref 80:176).

Kaour (Ref 43:221) considers the case of imperfect

sw.tchicg. . He first looked at a situation where the switch

simply fails to operate when called upon. The probability

that the switch performs when required is ps" For the two-

unit standby system, it was shown that

tR2(t)' = Rl(t) + ps ff(tl)R 2 (t - t )at 1 (2-66)

and for a three-unit system

R3(t = R 2 (t) + p 2r (2-67)

s s s'3

He considered that the switch is a complex piece of equip-

ment and has a constant failure rate of • Thus the re-s

liability function for the switch is

-t¾ (t) -A St s32

32

and the switch can fail before it is needed. When he con-

siders the two-unit standby system, the reliability at time

t is

Rs /U (t .5 t t s >t t 2 >= - t 1 )](2-68)

where tS is the random varlable representing time to switch

failure. Figure 8 shows the success models for a two-unit

standby system.

EI

• EE

o ( Eý

0

t t1 It

Time

Fig. 8

Success Modes for a Two-Unit Standby System (Ref 43:219)

Equation (2-68) becomes

t(t = R1 (t) + ifl(t )R (t )R( ( t - )t (2-69)

33

by substitution for R s(t )

2 / t -x tR R,(t) + ff,(t )e R2 (t - t 1 )•t 1 (2-70)

The author considers the special case where all subsystems

have a constant failure rate X, then equation (2-70) reduces

to

R2 (t) = e + - (l - e , t 2 0 (2-71)s Xs

He finally considers the case of three-unit standby systems

which have a constant failure rate X; he deduced

R(t) " = e EI + - - (1 - e s)]

-+ )[) -eXst

_xt - x te

+ e ( Xl Xs)2 e - ex

- s e, t a 0

(2-72)

Anderson (Ref 2) employs a Markov chain model to

analyze the reliability of a special class of redundant

systems. These systems include those which operate in a

standby mode for a long period of time in anticipation of

participation in a single mission. The system consists of

N modules of which q are active on-line spares. The system

is in operating condition when at least N - q modules are

operating; the system has failed when more than q modules

have failed. The following assumptions are made:

(1) The module rate (X) iS constant.

34~

(2) The module repair rate (p) is also constant and

independent of the number of failed modules (one module is

repaired at a time).

(3) Repairs may be made during the standby interval

but not during the mission interval.

(4) The system has been in the standby mode sufficient-

ly long to Insure its being in a steady-state condition,

with respect to reliability, when it enters the mission

interval.

The failure state diagram for this system is shown in Figure

9. The state number corresponds to the number of fdiled

modules. Transitions from state k to k + I occur at faii.-

ure rate XK and transition from state k to k - . occur atKL

repair rate uK.

•0 I 2 x k

U1 W2 ý13 Pq Uq-i UN

Threshold ofSystem failure

Fig. 9

Failure State Diagram (from Ref 2:22)

35

The general differential equation for this system is

the same as those developed by McGregor (Ref 57). Anderson

manipulates these general equations to determine the prob-

ability of' mission success, Pms' and the expected downtime

per year, DT/Yr, for the assumptions and constraints (nota-

bly the initial conditions) of the given system. Since

there is no recair during the mission interval, the proba-

bility of mission success is simply the sum of the individ-

ual probabilities of being in an unfailed state, i.e.,

qE PK(tm) (2-73)I K=o.P

The solution for pK(trm) involves the solution of the set

of a..multaneous differential equations which apply during

the mission interval. The author derives an explicit equa-

tion for p (tm) which is expressed in matrix form as

p 0 (tm) ý00 0 0 ... 0 0 (tm

Pl(tm) 010 .ll 0 0 IPl(tm r 0)p 2 (t-) 420 12 1 •22 "'" 0 P (tm = 0)

n(t i n0 nl n2 '"nn Pn m

where

36S

k-1 K -(n - r)xt mRKi = f(n- x1) z e (2-74)

x~i r~iT (r - y)

Applying these equations to a two-element redundant system

with module failure rate X, the author obtains

-2xtP0 (tm) e mp( 0 ) (2-75)

[r-Xt -2tm -Xtm(tp'(t ) 2 2e P0 (O) + e (2-76)

[ -Xt -2ýtm

P 2 (tm) - 2 e m + e P0(0)

+ - xtm] D( 0 ) + 1P2 (O) (2-77)

The probability of the system being available, PA'

(i.e., above the defined threshold of system failure) upon

reaching the steady state condition during the standby inter-

val, is given by

q A= ) (2-78)

K=O

The probability of being in state K at the end of the stand-

by interval is

PK(ts n for K = 0,1,...,n

(n K-

(n -J - (2-79)J 0

r,4ally, the expected system downtime per year is given by

37

0

DT/yr 8760(1 - PA) hours/year (2-80)

The author alc presents two sets of curves; one set

shows the probability of mission failure as a function of n,

MTBF/MTTR (mean time to repair) and t /MTEF fcr different

values of q. The second set of curves give the expected sys-

tem donwtime per year as a function of n, q, and MTBF/MTTR.

These curves are helpful in gaining insight regarding the

tradeoffs among the various system parameters. Several other

examples of Markov process applications to the reliability

analysis of redundant systems were found in the literature.

Ir most cases, these additional applications are extensions

or modifications of the examples already presented. The

interested reader may refer to Sandler (Ref 73) for an exten-

sive general discussion of the application of Markov proces-

ses to the reliability analysis of various system configura-

tions.

Maintenance Systems. Brosh (Ref 16) utilizes a Mar-

kovian model to analyze a multi-service maintenance system

with "first come, first served" repair (queue) discipline.

The system is inspected at fixed periods of time, classi-

fied into one of a finite number of decisions. The state

of the system is defined by the number of machines in the

system, i = 0,,.1..,N, where N is the maximum number of ma-

chines the system can contain. There is only one maintenance

crew. However, repairs may be performed in two different

ways. This results in two different probab'lities of

38

completing maintenance in a unit of time. A set of costs is

associated with being in each state and using each type of

service. An additional cost is incurred when a machine ar-

rives for service and the system is full. The author presents

a procedure for choosing the optimal policy for controlling

the system by deriving the steady-state transition probabil-

ities. Maintenance policies are considered by partitioning

the state zpace Into two sets - one containing those states

in which the unit is maintained with service type two. The

measure of effectiveness is the expected average cost per

observation period over an indefinite sequence of observa-

tions. Analysis of the maintenance policy space reveals the

interrelationship betwe,1 system variables and optiz.al main-

tenance policies; e.g., disconnected policies arp driinated

by connected policies except for one case (c2)1(c)11 (12/wl)

and (c 3 /c 1 ) =()/(I) for which any policy chosen will yield

the same value (cl, (X)/(ul)) for the cost function (Ref 16:78).

Natarajan (Ref 64) considers another important aspect

of the system maintenance problem - that o.i assigning repair

priority to a particular type of component. The author pre-

sents a Markovian characterization of an anti-aircraft system

consisting of redundant radars working in conjunction with

redundant computers. System failure occurs only when both

radars or both computers are in a failed condition. As soon

as any component fails, repairs are initiated. If, in the

interim, a component of the other parallel system fails, it

must either wait for repairs or it can preempt tne unit in

39

_________________________________ ____________________________________________________________

repair. The first case is known as first-in, first-out (FIPO)

discipline; the second case is known as preemptive di4scipline.

The author assumes exponential distributions for the compo-

nent failure rates (X and X,2 and repair rates (wI and v2).

He derives somewhat lengthy expressions for mean time to

systeim failure for both the FIFO repair policy and the pre-

emptive policy. When no repairs are permitted, the mean time

to system failure is

1X {5 2X1 - ^2 ,2-81)

E(T) = ( + ) -Ii + I 2 22( +2X + X

The author discusses the problem of optimal priority

assignments to which component ana what priority and compares

variatlons exhibited by the mean time to system failure with

a priority discipline imposed on the repair process of dif-

ferent components. Numerical results provide the author with

the following conclusions:

(i) FIFO discipline for repair of priority components

is more effective than a preemptive disciplIne when > 2

and v 1 >2 or when XI < X 2 and v < P2" Hence, the pre-

emptive disciplines should be used only under special circum-

stances governed by emerging or risk considerations.

(2) When a preemptive discipline is imposed on the

repair process, higher mean time to system failure will re-

sult If > X, and v < i2 (Raf 64:107).1 2

As a final example of the use of Markov processes in

a maintenance environment, Eckles (Ref 26) discusses a system

40

that deteriorates stocastically and is further complicated

by the fact that the state of the system is unobservable

unless an inspection is performed. The author develops a

model to determine the appropriate sequence of actions (re-,

placements, repair:, Inspections, etc.) that minimizes the

total cost of operation (including such factors as downtime,

inef.ficiency, and salvage value). Maintenance of the system

is characterized by a discrete-parameter, non-stationary

Markov process. Prior to each transition, the decision maker

selects one of a finite number of available actions. The

action selected and the system's age determine the subsequent

one-step transition probabilities and the conditional (on the

syst-m states) distribution of the next measurements. Cost

iz dependent on the action taken and on the system's state

assigned to each possible transition. The author shows that

the action that minimizes the discounted value of expected

immediate and future costs (assuming optimum future actio:.z)

is determined by the system's age and the posterior distri-

bution over the states (Ref 26:16). Optimum maintenance

policies can then be calculated using a dynamic programming

method.

Availability. The expression for system availability,

A(t), integrates the probabilities of reliability and main-

tainability. A(t) may be defined as the probability that

a predicted percentage of operations of time duration T will

not have any malfunctions which cannot, through maintainability,

41

I

be restored to service within the permissible repair time

constraint (Ref 5:8). Figure 10 shows the basic availabil-

ity graph for a single component system. The differential

equations for each state probability may be written directly

from the figure (Ref 58:33).

1!- Xat J.- ujAt

uAtj

Fig. 10

Availability Graph for a Single Component System

•Ot, -i po(t)l

IL (2-82)

Pi(t - Pl(t

A'suming the initial conditions p0 0) = 1 and pl(1) = 0, the

solution of equation (2-82) yields for state probabilities

PO +

-(+ + u)t.•(t) = 1- p 0 (t) = •k. - " eU-~t POt A (2-84)

42

The availability of the single component is the probability

that the system is in state zero, i.e., A(t) - p 0 (t). For

"large t, the availability approaches a steady state value

Ass = lim A) (2-85)t + A+

noting that X - !/(MTBF) and i (!)/(MTTR), As can be re-

written as

MTBF (2-86)Ass MTBF + MTTR

Kapur (Ref 43:227) attained the same results for the

availability as the above equations by stating that for the

exponential distribution and for some interval of time Lt,

he stated

posystem failure during At] = XAt (2-87)

and

p[repair during at/system failure] = w4 t (2-88)

He defines the availability function, A(t), by using equa-

tions (2-87) and (2-88) as follows

A(t+At) = A(t)(' - XAt) + [1 - A(t)]pAt

= A(t) - XA(t)At + IiAt - ijA(t)At

or

A(t+At) - A(t) = -(X + p)A(t) + vAt

43

Taking the limit as Lt 0

a A(t) -(x + v)A(t) +Dt

A'(t) = -(X + i,)A(t) + i.'

which is a d.fferential equation whose solution is

U + X e-( + •)A(t) - T- T V + U T X

A lim A(t) = Ut -) X +

Mean repair rateMean failure rate + Mean repair rate

or equivalently

Mean time to failure

A - - 7ean time to repair + Mean time to failure

(2-90)

He stated that, fortunately, it so happens that if the down-

time, p. d. f., is taken as something other than exponential,

the same steady state solution results. in practice, fre-

quently the log normal is used as the downtime, p. d. f.

Kapur defined the intrinsic availability in the sa-me way as

the availability. The only difference will ba that the mean

repair rate in the availability function will be replaced by

the mean active repair rate. Thus the steady state intrin-

sic availability is given by

Mean time to failureI ~Mean active repair time + Mean trime to failure

(2-91)

44

Dayon (Ref 23) utilizes the steady-state availability

concept to analyze a computer system consisting of a data

processor and tape units. The purpose of the analjsis is to

solve for the MTTR of the redundant system. The author

points out that defining the system states and formulating

the appropriate system steady-state availability transition

rate diagram is the step requiring the greatest a-gree of

ingenuity and expertise. By contrast, subsequent steps to

obtain a numeilcal solution for the system MITTR involves

only routine mathematical manipulations (Ref 23:153). The

system under discussion consists of a data processor (unit

#42 requiring two tape units (units #2) for data storage.

A third tape unit is on standby redundancy. A block diagram

of the system is shown in Figure 11. Implied in the model,

but not explicitly stated, are the assumptions of a Markovian

tape system with con3tant element failure and repair rates.

Other features of the system include:

(1) All three tape units are identical.

(2) Any tape unit can be removed off-line and repaired

while the system is energized.

(3) All units are de-energized when the system enters

a fall state.

(4) Repair is performed on a FIFO basis.

(5) The system is re-energized and operated as soon

as there are enough units repaired to have full system capa-

(6) There is one repairman.

45

* ~ # _ - -'.

Fig. U1

Data Processor (#1) and Tape Units (#2)

The assumed failure and repair rates for the respec-

tive units are:

X! = 0.01/hour 1i 1.0/hour

x 2 = 0.02/hour P2 - 2.0/hour

The states of the system are defined as:

State 0: all the units up

State 1: one tape unit down

State f 1: two tape units down

State f2: one tape and the data processor down

State f : data processor down

The steady state availability transition-rate diagram

for this system is shown in Figure 12. The loops showing

the uroba0ility of remaining in each state have been omitted

22

- 3l

Fig. 12

Transition-rate Diagram for Computer System

to enhance clarity. The repair transition from state f2, to

state f3 is indicative of the FIFO repair discipline. if

the policy had been to resume operation as soon as possible,

the repairman would have ceased work on the tape unit when

the system entered f2 and performed maintenance on the data

processor until the system was returned to state 1. The

author develops the availability matrix and substitutes the

appropriate failure and repair rates. Numerical solution

yields (after "normalizing" the state probabilities) a

value for MTTR equal to 1.46 hours.

47 .

11! 1 1 P

A-0.5 2.0 0 1.0 PI

0.04 -2.05 2.0 0 0 Pf (2-92)

0 0 0.04 -2.0 0 0 Pf 2l

0 0 0.01 0 -2.0 0 pf

Grace (Ref 34) discusses the evaluation of steady-state

system availability when there are a limited number of re-

pairable spares for the various types of on-line units. The

particular system considered consists of five on-line vnits

and two spares of a given type, with one maintenance man

(associated with a replacement rate 6) and one repairman

(associated with a repair rate u) assigned to the unit type.

Failure, repair, and replacement times are assumed to be

exponentially distributed. Hence, the availability of I units

of type a, Aail can be determined from a Markov model which

includes all possible states and transitions of the on-line

and spare units of type a. The author includes a complete

transition diagram (33 states) and discusses a procedure for

writing the resulting equations directly from the transition

diagram. For this system, the steady-state probability RiJ

of being in state iJ is obtained by solving the set of equa-

tions:

(nX + sa)l 0 0 J i0l + 6fi, -I

[nx + (s - 0)s]f0 1 = u 02 + 611, 0 (2-93)

etc.

48

Where n is the number of on-line positions, a is the off-line

(spare) failure rate, and s is the number of spares. The

normalizing condition

Z IT 1 (2-94)i ,j

is required for a unique solution. The steady-state proba-

bility that i on-line units are not operating is given by

,= ii (2-95)

if i units have failed, the probability that any 1 units are

good is

(I Z (2-96)

(n) (n)The author shows that the total probability that any 1 par-

ticular units are working is

Aa. I=p Et Z n (2-97)

Schick (Ref 76) employs a Markov process to determine

the availability or "operational readiness" of a system which

is subject to inspection and repair. Except for inspection

and repair periods, the system is kept in a normal mode from

which it is called if its operation is required. Failure of a

primary part causes immediate shutdown, inspection, and

4~9

repair. Failure during checkout is detectable only at peri-

odic inspections while failure of some equipment is liable

to two types of failure. Examples of such systems include

missile launch facilities and computer repair systems.

Grippo (Ref 37) also employs a Markov process to

evaluate the reliability and availability of large, complex

systems. The author's approach enables the analyst to intro-

duce system operation or maintenance constraints without

adding appreciably to the complexity of the solution. Al-

gorithms are derived for obtaining computer-aided transient

and steady-state solutions to both reliability and availa-

bility problems. The author also presents a unique method

for integrating the differential equations that can result

from a Markov process.

Sasaki (Ref 74) utilizes implied Markov process tech-

niques to develop a set of charts and nomographs which can

be used to evaluate trade-off characteristics between system

reliability and maintainability. The author presents a slide

methods (using the charts and nomographs) of achieving desired

values of system (mission) availability for duplex parallel

redundancy and duplex switch over redundancy.

Finally, Hevesh and Harrahy (Ref 38) discuss the

effect of failure on the availability or "readiness" of phased,

array radar systems under different structural and repair

caoacitv conditions. The author also developz . xpressions

for the availability of parallel operated equipment, whether

truly or quasi-redundant.

50

| !INMM

III. Summary of Point Estimates of

Availability, Reliability of

Maintained Systems

Computation of Reliability

and Availability

The techniques for obtaining the reliability and avail-

ability of a repairable structure with state dependent hazards

and repair rates is best illustrated by an example (Ref 58).

Consider the following two components parallel system with:

components xI and x

components hazards XI' X2

repair rates ul' 12

number of reoairmen: 1

In general, five states are needed as tabulated in Table 3.1.

Table 3.1

System States for 2 Component Repairable System

Good States Bad States

0 - x1, x2 3 - !, 2 (x1 failed before x

1 - XI X2 - ½2 1 (x 2 failed before xI)

2 - xl, x 2

The transition diagrams for calculating the system reliabil-

ity and availability are given in Figure 13. The distinction

between the availability and reliability transition diagrams

Is that the availabilitv transition diagram repairs are

51, • 6

I-

ca

4o ~0

"< • 0

" "• 0

.oN7 E-i .-

• " 52

Ix-41

0

Q.)

En0)

41)

ccli

V2

17

(-V'

N - CI X.

-53

allowed from all system states, while for the reliability

tradrsition diagram repairs are allowed only from the gocd sys-

tem states. Therefore, summing the good state probabilities

obtained from the availability diagram gives the probability

that the system is good at time t, while summing the good

state probabilities obtained from the reliability transition

diagram gives the probability that the system has never en-

tered a failed state. System states that distinguish the or-

der in whlch component failures occur are necessary for the

availability model since there is only one repairman, and

according to the repair policy, he will worn on the component

that fails first. If the system is in state number three (Ri'

x2), they, the repairman is working on component x: and the

next transition must be to state number two (x,, x2 ). If the

system is in state number four (x 2 s x 1 ), then the repairman

is working on- component x 2 and the next transition will be

to state one (R2l, x 2 ). Ordering of the component failures

is not necessary iui the reliability transitlon diagram since

once the system enters a state with more than one railie, no

repair is attempted.

The differential equations for the state probabi±ities

are given for availability analysis by

54

[P(2) + 1 0 0 P0 (t)

(t2 + ) '0p (t)p/(t) - A2 0 (A: + u 2 ) I 0 p2 (t)

p3(t)i 0 A. 0 -'u 0 P 3 (t't

. " ( t4( : W 0 - X u tj L "- 2 P4(t)

(3. la)

where

A(t) p 0 (t) + p (t) + P2 (t) (3.1b)

and for the reliability analysis by

FPOt F 1 + 2) 2 J0P"(t) -(A2 + U) 0 0 0 1 (t)

A. t 0 1+(t) + P 2 ) (3 .2b)P 3(t)020p ~ ( ) 0A 2 0 0 ) P 3 ( t )P'4 tj 0 0 01 p 4 tJ

(3.2a)

e ee

R(t) =p(t) +0 p(t) + p2 (t) (3.2b)

Ordering would not have been necessary if there weretwo repairmen. The availability transition diagram is givenIn Figure !1. The reliability transition diagram remainsunchanged.

Tf the components are identical (x =2 X2 = ) and2= U), then states 1. and 2 can be combined and states

3 and 4 can be combined resulting in a greatly simplified

55!

vQ~ 0iE-

0 -

- E- 0

00

56 O

transition diagram as in Figure 15.

I - X'A• 1 -I x + 'A tI

C 2

(a) Availability

I - Vxt1- t +Xt0 XA t x At

0 1 2failes ure failures

(b) Reliability

Fig. 15

Trar,,itioi' Diagram3 for General System of Two identical Components

It should be noted that for the case of two repairmen,

when one component is failed one has the option of assigning

only one of the lepairmen or assigning both repairmen permit-

ting joint effort. When a Joint effort is permitted, the

repair rate •i'= av. Sandler (Ref 73) uses 3 = 1.5.

The differential equatlons for system availability

is, therefore, given b!

5 7

0[PO

p•( 0 M -- " P(t _

where

A(t) p 0 (t) + p 1 (t) (3-3b)

and the differential equations for a system reliability is

given by

"P0' 0 P0O)

Plt 1• - • + L,/) 0 1P t) (3-4a)

#2tJ0 P2(t)

where

R(t) = po(t) + pl(t) (3-4b)

Equation (3-3) is readily solved for the system availability

yielding 2_=r ]A(t) = 1 - X - XX1 rt('

e 2 •_

r1r2 r1 r r r

where r1 and r 2 are the roots of the equation

r + + + X' + /'+ ,)r + (xx' I Vpi"+ W'•/) =0

where

r- + r 2 -(X + V+ •'+ u")

r = r X + xV'U+ (3-6)

58

-A -i

The steady state availability is given by

A = rn A(t) I iss t r, r r 2

Xu"+ (3-7)=xx' + x'W"+ •'•"'

Equation (3--4) is readily solved for system reLiability

ytelding

R(t) - r 3 r 3e -r ite3 (3-8)

where r3 and r4 are the 6olutions of

r 2 + (X + X'+ W r + 0 (3-9)

The roots r, and r4 are clearly negative real since the dis-

criminant of e-uation (3-9) satisfies

b2 - 4ac = + X 2 4 x

X (x - I) 2 + 2'( X + x') + •,2 > 0

Shoorman (Ref 73) gave the solution for the two identi-

cal paralle elements ad K reoairmen. He derived the fol-

lowIng equations with respect to Figure 15. The differential

equations associated with Figure 16 are

PsD" (t) + "- 5s (t) -

"P t) + OL + x)psI(t) =,si - - 10 •

r(0) C (3)3

5A*

1- A t 1t (x + A~t

XAt XAt

s o sI -2 + Xl12 1S2 = 2 12

,' 2X for an ordinary system

X for a standby system

iiu for one repairman

w'= Kv for more than one repairman (K > 1)

Fig. 16

Markovian Reliability Model for Two Identical

Parallel Elements and K Repairmen

Taking the laplace transform of this set of eqlations yields

(s + X')p sO(s) - *sp(s) = 1

-'psO(s) + (s + .,'+ X)PsI(s) = 0(3-11)

-XP ,(s) + sP 2 (s) = 0

Solving the set of equations of (3-11) using Cramer's Rule

yields

(s + X + L')P so(s) s 2 + (X + )/+ uOs + xx'

Ps(s)= 2 + (\ ' (3-12)

S e SLS 2 + ( + X, + 'jOs + XX,]

Soi,'ution for the roots of the denominator quadratic yields

60

r (+ + + + 4

Expanding equation (3-12) in partial fractions

P(s) = (s - rl)(s r

12-

i.e.,

(A + •'+ rI)/(r 1 - r 2 ) (I + u + r 2 )/(r2 - r 2 02

Pso (s - r 1 ) + (s - r')

(3-14)

__ _ __ _ _(rlr2) rX'_(r2 r_,P(s) = (s - r,)(s r2) s 1 + s - r2

ps (S) = ~ - A (3-15

= X /r r 2 XX\/rl(rl - r 2 ) XX/r 2 (r 2 - r1 )1 + 1- 1 + 2 .2s s - r 1 s - r 2

(3-16) "°,

As a check one can sum equations (3-14) to (3-36) noting that

r XX and + ÷ r -(x + X'+ j'). The result is

PsO(s) + Psl(s) + Ps 2 (s) -- (3-17)

Taking the inverse transforms, one obtains

Ps 0 (t) + psi(t) + Ps 2 t = s2

The inverse transforms of equations (3-14) to (3-!6) yield

- + v + r1 rt + -/+ r2 r2t-

P2 r1 -2 - -1 '

61

r rlt X r 2t

PSI~ = r1 r e - r - r e (3-19) I

ps 2 (t) = 1 + r 2 et rl 2 e 2 (3-20)

From equation (3-13) r and r are always negative real num- "kbets; therefore, the time functions are decaying exponentials.

For a system composed of two series elements, the system re--2ýt ". . • . - o '

liability is unaffected by repair, and R(t) = , whichcan be obtained by setting u"= 0 in the expression for ps 0 (t).

For a parallel system (standby or ordinary), the reliability

is given by p s(t) + p l(t). By appropriately choosing coef-

ficlents X/, P, and u' (as given in Table 3-2), a large number

of systems can be mcdeled.

Table 3.2 Coefficients of X u (Ref 59)

Repair/Non-repairable Type Repair Crew X • _

non-repairable parallel 0 2X 0 0

non-repairable standby 0 % 0 0

repairable parallel 1 2X P

repairable parallel 2, no 2X P 2PJoint effort

repairable parallel 2, 2X Bi 2vJoint effort

repairable standby 1 % V

repairable standby 2, no X p 2)Joint effort

repairable standby 2, X Bu 2vjoint effort t

62

Ae

Systems with more than two components are treated also by

Messenger (Ref 58).

p..4i

41

Sit-IL

,La.

el

S5

4,,

So63

41 ....P) ___ __ __ __ ___- -_____ __ __ ___ Z.

ilk

"IV. Confidence Limits for Availabilities

of Maintained Systems

Exact Analytical Methods

40.

An estimate of system availabilty calculated from

time-to-failure and time-to-repair test data will be subject

to some degree of uncertainty due to the uncertainty associ-

ated with the sample estimates of MTTF and MTTR. This chap- .

ter presents techniques for determining a lower confidence -

limit on system availability when time-to-failure and time-

to-repair are independent, exponentially distributed vari-

ables. Also, the case of log normally distributed repair

times will be included.

The Case of Both ExponentialDistributions for the Time-to-Failure and Time-to-Repair

Mary Thompson (Ref 83) presents the techniques for de-

termining a lower confidence limit by making use of the one-

to-one correspondence between availability and the ratio of

mean-time-to-reoair and mean-time-to-failure in order to de-

fine F-distributed variables upon which the confidence limits

are based.

The availability, as defined before, is usually defined

as the probability that the system is operating satisfactor-

Ily at any point in time. This probability can be expressed -7

mathematically es

+ +

~k. e+ ~ 1 (~/8

- -- =--= - -- -~, ~ - -

where

e = system mean-time-to-failure

4 = system mean-time-to-repair

The one-to-one correspondence between availability €/8 is

obvious. The usual sample estimate of availability is

eA (4-2)0+¢

where 0, the sample estimate of 6, is calculated from

n,e = E t 1 /n1 (4-3)i= 1I

where

t = time between the (i - l)th and the i-th failures

= number of failures

and 4, the sample estimate of €, is calculated fromn 2

Z lt 2 j /n2 (4-4)

where

t = time-to-repair associated with the J-th failure

2J

n2 = number of repair actions initiated

It is assumed that t1 (time-to-failure) and t 2 (time-to-

repair) are stocastically independent random variables with

probability density functions

fl(tl) = - e (4-5)

65

and

f 2 (t 2 ) = e 2 (4-6)

If we consider a random sample of n1 times-to-failure

and n 2 times-to-repair drawn from the above populations with

random sample means 8 and 0 calculated from equations (4-3)

and (4-4>. It is well known that 2n08/6 and 2n 2 •/4 are chi-

square distributed variables with 2nI and 2n 2 degrees of

freedom, respectively. Since they are independent due to

the independence of the variables t, and t 2 , it is possible

to define two new variables:

2n 1 /0 2n2/1= n ( - (4-7)

2n2

which is F-distributed with 2nl, 2n 2 degrees of freedom, and

z (4-8)2 =z

which is F-distributed with 2n 2, 2n1 degrees of freedom. The

variable z can be used to obtain a lower confidence limit

for availability A as follows (Ref 83)

pr F 1 _ 2nI 2n ) }i.e.,

S* ~~(2n 2n) -apr { --- F1 - a2 n -

66

pr 1+. 1+4F (2n,, 2n 2 ) = 1 - a

"~I + /+ F -a (2nl, 2n 2 ) +

pr 11A+e + 0 F 1 (2n., 2n 2 ) (4-9)

Most practical cases n 1 n 2 n and equation (4-9) becomes

•r @< A =1 - (4-10){ + € F 1 -(2n, 2n)

The (I - a) lower confidence limits is found from

LCL 6 (4-11)

+ 0 F1 _ (2n, 2n-

A two-sided (1 - a) confidence interval, derived in a similar

manner, is given by

LCL A ^ (4-12)

a + * F1 - aI2(2n, 2n)

UCL = F j.2(2n, 2n) (4-13)

SF! a12( 2 n, 2n) +

Confidence intervals calculated from equations (4-11), (4-12),

a: ! (4-13) cover the true value of availability 100 (1 - a)

percent of the time. The curves of the 0.9 and 0.95 lower

confidence limits on availability calculated from equation

'4-1!) for values (€/6) ranging from 0 to 0.3 and for

6 7

I ~ -=~~ ~ - - - -=~~ ----~-~----~-= -~

for selected values of n between 2 and 50 are given in (Ref

83,.

Suppose for example that during the field testing of

a com~munication system, five T'ailures were experienced. Thie

average time-to-failure e was 125 hours. The average time-

to repair € was 3 hours. Previous experience with similar

systems indicates that time-to-failure and time-to-repair are

exponentially distributed. Independence of the two -,ariables

assumed (in practical terminology, this means that the time

required to fil a failure does not depend on how long the

equipment operated prior to the failure) a point estimate of

the system availability.

0 125 -+125

To find the 90 percent lower confidence limit, compute

-4-= 3 - 0.024e 125

From the curves given (Ref 83), we can read directly the LCL

on availability with ;/^ = 0.024 and n = 5; the 90 percerc

LCL is 0.95.

Lower Confidýence Limits Assuming Lognormally

Distributed Repair Times

Gray and Schucany (Ref 36) introduced the lower confl-

dence limits in the case of lognormally distributed repair

times using the tables of H. L. Gray and T. 0. Lewis (Ref 35).

-thc r ,ndom variables x amA y denote the reoair times and

68

times to failure, respectively, since the availability is

defined by 7

A -I--14-

WY WV

where ux is the mean-time-to-repair (MTTR) and ýy is the mean-X y

time-between-failures CMTBF). 1` is difficult to work ana-..

lytically with the assumption of lognormal x and exponential

j in trying to establish confidence limits on the ratio of

the means. Gray and Lewis (Ref 35) tabulate to some extent

the distribution of the ratio of independent lognormal and

chi-square quantities for known variance of the lognormal

distribution. These tables enable us to establish a lower

confidence limit for availability based on the assumption of

lognormal repairs.

Cray and Schucany (Ref 36) derived the following LCL

for the availability assuming lognormally distributed repair

times. If x and y have lognormal and exponential distribu-

tion, the respective probability density functions are given

by

h(x , L 2) 1 exp nx 1 , x > 0x2 s ]B I (4-19)

0 elsewhere

f(y; Uy Yexp[ , Y 0Y yy (4-16)

0 eisewhere

69

for the lognormal, distribution, the parameter a and 82 are

the mean and variance of £n(x), respectively; that is,

E[inx] - a

var~tnx] -

since

lx= Etx] = exp [a + ½B21

It can be seen that for a random sample of size n from

h(x; a, 6%, the quantity defined by

Q = x/e

where

[n 1 1/nx R(x1-/7)

is the sample geometric mean, is distributed lognormally with

parameters 0 and 6 2 /n. Also for a random sample size m from

f(y), it is known that

n• = 2my/ly

where

• m i(Y)

is distributed as chi-square with 2m degrees of freedom. The

quantities Q1 and 2 are clearly independent. Consequently,

if we let

70

w Q 2Qi 2ecmyuX2 y

then it is known (Ref 35) that w has E. probability density

function g given by

/ exp 2 n 28z

r(r) 2m(2 it)½ 0 2a2g(w)

-- ½w x~wz)r - w~z <w

elsewhere

(4-20)

The variable w can now be used to obtain a lower confidence

limit for the availability:

1

1 + exp[a + 8 12]I

as follows. For a given n/B2 and m, we may obtain a number

b such that

p~w < b C p[2emy < b C

b exp[a 2 /2] -

exp[a + 82/2] < _ _ _ _

by algebraic manipulation

1 + b exp'12/2]R/2m 3Therefore, the 100C percent lower con-fidence limit (LCL) is

given by

LCL2m (4-?2)2,n + b0 exp[82/'2]•

where oc Is obtained from the tables of Cray and Lewis (Ref

35).

Gray and Schucany (Ref 36) gave some figures which show

the lower confidence limits versus the statistic / for

several values of m and n for the single value 82 1.0.

Also, they gave some figures for different 82, but these fig-

ures are insensitive of the LCL to the number of observed

failures m and repairs n that comprise the statistic. This

insensitivity makes the figures difficult to read, since for

any giver problem, it is unlikely that m., n, and 82 would

match those figures.

7L

V. Mont Carlo Comparisons

Confidence Limits for

Availability and Reliability

Summary

Reliability and maintainability engineering are recent

and related engineering disciplines that make extensive use

of mathematical techniques. In analyzing the dynamics of

physical systems, certain probabilistic concepts have been

developed in order to account for and explain random obser-

vations (i.e., failures or repairs).. One of these concepts

embodies the theory of Markov processes, the idea that the

past has no effect on the future except through the present.

The applicability of Markov process theory and techniques to

the study of reliability and.maintainability engineering has

been shown in earlier studies.

A system is designed to achieve a given performance

and its quality is the degree to which it meets this perfor-

mance specification. Performance is normally specified in

terms of the acceptable limits of such parameters .as the

maximum permissible noise or the minimum acceptable output

power of a speech transmission channel, the maximum number

of lost calls or the maximum switching time in a telephone

exchange, the stability of a pow&• supply or the frequency

limits of an oscillator, and so on. It follows that system

failure is defined as a departure from these specified limits.

73

Failure may be defined at many levels whereas only a system

failure gives rise to complete loss of system use. A unit

or sub-system failure may or may not give rise to a system

failure depending upon the presence, or otherwise, of redun-

dancy. Redundancy is a design configuration, as in the dupli-

cation or triplication of units within a control system,

whereby the failure of some part of the system-does not re-

sult in a system failure. Reliability is frequently enhanced

by the use of redundancy, but the total number of unit fail-

ures requiring repair, hence the amount of maintenance, is

usually increased due to the additional equipment. Since the

availability is a measure of the ratio of the operating time

of the system to the operating time plus the downtime, thus

it includes both reliabillty and maintainability.

This thesis concentrates mainly on the confidence lim-

its for the asymptotic'availability of maintained systems.

Confidence limits for the availability A(t) and reliability

R(t) of maintained systems may be obtained in exactly the

same method as applied to the two cases of steady state studies

in this thesis as long as equations for the case have been

derived (see Chapter III, equations 3-5, 3-7, 3-8). Using

Table 3.2, a large number of systems can be modeled.

74

Results

Case 1. Exponential.failure time

Exponential repair time

Number of repetitions = 500

Mean-time between.failure = 100 hours

Mean time to repair = 2 hours= MTBF

Exact availability MTBF + MTTR

1= 00/100 + 2 = .98

Table 5.1 Results of the Double Mont Carlo Technique

Exponential Failure and Repair Times

Sample No. of Trials, Actual Percentaie CoverageSize :per Repetition 95%- 90% 85% 80%

10 100 100 91.16 85.8 80.2

20 100 100 89.-6 85 79.2

30 100 100 89.8 85 80.2

10 200 95 89.2 84 . 2 79.4

20 200 95.6 92.4 87.8 82.2

30 200 95.6 90.8 87 82

10 500 94.4 90.4 85 80.4

20 500 94.8 90.6 84.8 78

30 500 95.4 90.8 86 82.2

75

Case 2. Exponential failure time

Lognormally repair time

Number of repetitions = 500

MTBF = 100 hours

MTTR = 2 hours

Exact availability = .98

Table 5.2 Results of the Double Mont Carlo Technique

Exponential Failure Time and Lognormally Repair Time

Sample No. of Trials Actual Percentage of CoverageSize per Repetition 95% 90% 85% .80%

10 100 100 92.1 91.3 87.4

20 100 100 88.9 82.3 81

30 100 100 91.4 86.2 80.5

10 200 96 92.3 88.5 82.3

20 200 99.3 92.2 86.1 82.8

30 200 99.3 93.6 88.1 81.4

10 500 99.3 93.8 87.4 83.6

20 500 99.3 93.6 86.7 82.8 '

30 500 -9.3 92.8 88.1 81.5

76

We get accurate results in the case of exponential failure M)

time and exponential repair time. A set of optimistic re-

suits was obtained in the cabe of the lognormal repair time

W.ith exact availability .98; that is, when repair times were

correctly assumed to be lognormally distributed, the coverage

was greater than in the situation of exponential repair time. A:4-

The same conclusion has been obtained by Gray and Schycany

(r.ef 3 etcer results car. be obtained wiuh smaller exact

availabil"ty.

€1

*Ark

-t,

ii

B=•,,±O.•A0dH

J l;

S:It

ii@

-:E . .I• .

1. Amstadter, Bertram L. Reliability Mathematics Fundamen-tals; Practices; Procedures. New York: McGraw-Hill, Inc.

2. Anderson, D. E. "Reliability of a Special Class of Redun-dant Systems," IEEE Transactions on Rel!abillty, R-18:21-28 (February-T9 9).

3. ARINC Research Corporation. Reliability Engineering,edited by William. H. Von Alven. Englewood Cliffs NJ:Prentice-Hall, Inc., 1964.

J. Arndt, K. and P. Franken. "A Random Point ProcessesAppl ied to A-alailt Anal- I.dtoAa~lab-liy Analysis of Redundant Systems

with .Repair," IEEE Transacti ons on Reliability, R-26:26-269 (197,.

•. Barber, C. 7. "Expanding Nantainability Concepto andTechniques, " 15,Th Transactions on Reliability, 0-16:5-9(Ma •:'a"967).

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7 Barrett E. H., R. N. Miller and M. J. Smith. "Comput-r z ed 'areov System Effectlveness Mcdels," Proceedings

of the 169 ".nnual SSmposium on Reliabilitvy 535-L2January, 1969. 'IEEE Cataog No. 9-8-R)

8. Bazovsky, i. Reliability Theory and Practice. EnglewoodC s : Prentice-Hall, Inc., 1

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ETransactions on Reliability, R-2P : 199-- (uust

1m1 1r -am, J. E . and A. H. Moore. "Estimation of :MlissionneliaDi!-Ltv from ultpe . ndeendent roued Cens-red,

Tvzi., " .týELcns on Re a1j 1,i-t,- 7z- r -7 -6

1. "Estimation of R'eliability fr - ul.tie indeuen-ltroued Censored Zamo_,es W.zh n.iurc r.;K-,,TOE ... C..... .... . !'" -•= . , • .. " --

I-rhe •.c ,

_.,." erich "Applications of

13 Blakney, K. Z. and F. L. Dietr ch. nMarkev Processes to Reliability and MaintainabilityEngineering." Unpublished master's thesis. GRE/MATH/66-3114. AFIT/EN, Wright-Patterson AFB OH, December 1969.

i•. Bovaird, R. L, "Characteristics of Octimal MaintenancePolicies," Mana__ement Science, 7: 238-253 (April 1961)

15. Brender, D. M. and M. Tainter. "A Markovian Model forPredicting the Reliability of an Electronic Circuitfrom Data on Component Drift and Failure," 1961 IREInternational Convention Record, Part 6. 230-24I7March, 19.

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1• . Chowd, D. K. "Availability of Some Reoairabie ComnuterSystems," IEEE Transactions on Reliability, R-24: 64-66(April 1975-

19. Clarke, A. Bruce and Raalph L. Disney. Probability andRandom Processes for Engineers and' Scienists. NewYork: John Wiley and Sons, inc., 1970.

20. Constantinides, A. "An Oneratlcns Research Aonrcach toSystem Effectiveness," Proceedings of the 1969 AnnualS, osium on Reliability. 250-255. January 1Q99.kIEEE Catalocr No. -FQ.c . and .. -ler. The Thecry of Stochastic

Processes. New York: John Wiley and Sons, Inc., ,9b5.

2. ...•�-illon, E. S. and "har-on -'nir niering RelIability,New Technicues ana A..]ications. " York: Jo*n 1le:and Sons, Inc., 8i

23. Toan' L. R. "Solving for the .. TT of Comrriex Systems,"-rcc'eeoinzs of' the :c~9 Annual S'oosium • on'5 -, 1. January I E- -- ' a taf3a~o •o ""-"

and .... Sersenbru;:e S"Scvl.,o Comn-ýex 7e-1ia-b il!1t,' Models by Coriiutler," ' oc-se*ThflA.r~nual• Ev','rnoSlun on celi~h!!itv . / uR-1 4'3 li anur / '

,.'• =,c'c'__,.~ ~~ ,a•, _ nl u• aU t;rý-

23. Earrnest, C. A. "Estimating Reliability After CorrectiveAction: A -ayesian Viewpoint." Master's thesis.Monterey CA: U. S. Naval Postgraduate School, Mlay 1969.

26. EckIes, J. E. "Optimum Maintenance with IncompleteInformation," Ooerations Research, 16: 1058-1067(Sertember-October 1968).

27. Efron, B. "Eootstrap Method: Another Look at the Jack-knife," Annals of Statistics, 7: 1-2. (January 1979).

25. Epstein, M. and G. Weinstook. "Reliability with Allow-able Downtime," 1969 Annals of Assurance Sciences,E"ghth Reliability and MaIntainab-lity Conference.36-39. July 1969.

29. "Evaluatink h= KTI M-ont Carlo Method for System Relia-bi!!t; Calculations," IEEE Transactions on Reliability;,R-23: 363-372 (December-- 979).

3C. 7--Ferton, R. A. "Further Acplications of Markov Processes!o Reliability and Maintainability Enr•.ýeeri• n ."..ncu.-

!isned mast.er's thesis. GRE/MATH/69-2 AFT / NJ Wri Cght-

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32. Oatliffe, T. R. Accuracy; Analysis for a Lower ConfidenceLimit Procedure for Systemi ?-. ati 1-. .YMonterey CA:]iaval Postgraduate School, I " D. AO3il7. Aval nble

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3 Gra-y, H. L. and T. 0. Lewis. "A Confidence Interval fcrthe Availability," Technomet r', 9: 465- 4 1 (August

- --- ald 1La, m S cn-ca n cy. "Lower CO '0 d"'ece LimitsPor A••a'a-!t issumln; LC'gnor,,m-all' [ ' d+''it'.e ... .Reoair

..... _, ran-L act' ipPs o-eliabIt y , Co[

.Io~-..

Qi!

37. Grippo, G. "Evaluation of ReliakAlity and AvailabilityMeasures for Large Complex Systemns," Proceedings of the1969 Sprin Seminar on Reliabiltty, Maintainability -Parts, Microelectronics, Syster.s. 33-49. April 1968.

38. Hevesh, A. H. and D. J. Harrahy. "Effects of Failureon Phased-Array Radar Syste'-_," 7EE Transactions onFeliab-tlbt, R-15: 22-31 (May 17M,.

39. 'Hillier, F. S. and G. J. Liberman. Introduction toOnerations Research, Third Edition. San Francisco:Holden-Day, Inc., 1980.

43. Jeiinek, F. Probabilistic Information Theory. New York:

McGraw-Hill Book Co., Inc., 1,66

41. Kamat, S. J. and V. W. Riley. "Determination of Relia-biit' Using. Event-based Mont Carlo Simulation, IE-Transactions on Reiibitv, R-24: 73-75 (April 1 )

2. Kamien, . I. and . L. "Maintenanceand Saleage for a Machine Subject to Failure," Manage-ment Science, 17: 495-504 (April 1971).

"3. Kapur, K. C. and L. R. Lamberson. Reliability in BEngb-neering Design. New York: John Wiley and Sons, inc.,

1 7.

" "4 Kaufmann,,, A., D. Grouchko and R. Cruon. Ma•hemati*alMoidels for the Study: of the Reliability of Systems.New York: Academic Press, Inc., 1977.

.•5. Kinderman, A. J. and J. G. Ramage. "Comcuter Generatýlcnof Normal Random Variables," Jourral of the AmericanS tatirtcal Association, 71: .

ance Protlems. Technical •nozrt no. 2". New York:Statist cal E ne e r in zrc-, Clbi" a . e rsit-December A96L A 6D 6 7•7;.

Tanon, • . " "AMon-. 'arlo Technique iXr A' r aiS......e~ ±n , lence Limiti s ,

istribu~tU- " Unublished master•s t-ei 7-

A.../ET , \ro- h• t -tte-son AFB OH, 19-_7. 7'

"Levey, L• .T d, IA- ooreW "A 'Mo't a] -1or ObSyain em Reliability Con. -e

C16:onen- Tes 7ata E Transac -s cn e3 7I a.-1i1n- i : 6•-_72 ( 6 )

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50 Tocks, M. 0. Mont Carlo Bayesian System 'Reliability-and MT-BP-Confidence Assessment. Tecn~c-al report,777L-uT-75-144, Air Force Flight Dynamics Laboratory,Wright-Patterson AFB CH. AD-AQ157068.

51.---------ont Carlo Bayesian System Reliability - and611TBF-Confizdence Assessment 11, Vol. 1: Theory, Vol. I:I:SPAR'S - 2 U'sers Manual. Tech:71cal report, AFFDL-TR-75-18, Air Force Flight Dynamic Laboratory, Wright-Patterson AFB OH. ADI-A025820.

--2 --- Reliablit-;', ',aintaIn~abilit,,,, and Arail ab~li tyAssessment. HoreePakY: iayden Book Co., inc.,

1973.

T3 notor, S. N. oA no Carl o T-chnixoe o'r Acporxlmat~np-S*2-z.tem- Reliabllt,: Conf--'aence Limits from Comoonens

Fali 're est- Da-!a. Unpublished nazter's thess.G/M4A ' -9. 1 AT / E, 1 -Wright-Patterson AFS OH.

Ma. i~ntainability Specilficatlcn of ?earTime Reouire-merits," IEIEE Transactions on Reliability, R-29: 13-16(April 1-980).

~.Mann, NN. R. and F. E. Qx-rubbs. "Approximately OptimumConfidence Bounds for System Reliabil-ity Based on Cn-,cnent Test Data," Technoinetrics, 16: 335-3i47 'Augu.st 1974).

,Ray E. Schafer and N'ozer D. Singpu-rw-a-la. Netho d sfor~ ~ a, s ca!Avsls of Reliability and, Lifeý rD'a ta .

Ne;.' "' o r'. ,Th-, Wiley and Sons, Tnc. , 197,11

57. "IAý. "Anoroximation Formula t'or Fýýe'iaib;ltT .WT Tansactions on. Reliabilit T.1

6!4-97 Deceriuter ?T

_____ t_____ -ailabil-t.. Research' Recort- TL~j> ok Poly'technic lnstitut~e

cOC j--n S' ''q.L

on! Carlo -ai r ; z!ncom Var hies,"1 3E 7Tr n s-

.)U -or-e '-"Rxe~~ru or*t "arlo Techr'ique GU;;ir;

6 E cI C DaL c, crced L~ ialna I e on ean.

61,-- H. Leon Harter and Robert C. Snead. "Comparison

of Mont Carlo Techniques for Obtaining System Rella-bility Confidence Limits," .EEE Transactions on Rella-bilit, -29: 327-331 (October 1960).

Munroe, A. J. "System Effectivenesz of Military SystemsMathematics and History," presented at George WashingtonUniversity, Washington DC. Program of Continuing Educa-tion - System Effectiveness Course. June 1968.

63. Nakagawa, T. and Yasui, K. "Approximate Calculation ofSystem Availability," IEEE Transactions on Reliability,R-26: 133-134 (June 1977.

64. Natarajan, R. "Assignment of Priority in improving Sys--tem. Reliability," EET ransactIons on Reliabilit:i, R-16:m34-110 (December TE77.

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F n . " iJat A'-.I na ysIs T sin F o wgraphs."

Unpublished master's thesis. GRE/1EiE/63-3. AFIT/EN,Wright-Patterson AFB CH, August .. 63.

67. "Ootima_ Periodic Maintenance Pc! icy for Machines Sub =e-to D,ýteroraticn and Random Breakdow_'n," !EEE 'ra.nsacti.nson leliability'1 R-28: •28-30 (O.tober ....

.?apzo-lcu, 1. A. and E. P. Gy-ftopoulos. ":,':rkov Processor .Reliability Analysis of Large Systems," ____ <r ns-

actions on Reliability, -26: 232-237 (Augut •97--,

69 .Put: :e. B. "A Univarla-e Mont Oario 7Technique tocrx amt•ate RPel1abiIity Confide-ce _ -.- c Sosteins

with "or'onents Characrerized bJ tI; I,,'i., -,tzcn. Unpublished master's t GC:/A,',9D-,. t.n T.

't,*r`l_,g-h-atterson aF'3 OH.

70. T- . "Reliab"ity Systoo in 'ai- Envlror men>"

7. "Reliabi•_ly and M•ntalna-tit .1 "1 1o," ... FE .actons cn Reliability Q:3 2 60 oEb_ e r -f .

I a7 '..._.2 8 26- r:- , .... ; 71i---7? -Ot b r Qe2 I"-.liaiit Evaluat ion. 12o r Iths. Comet o. "s;.

... i. s_ of CoT-ci et= e [ .. 3 k , zi-; ? • n 9c - • •• •

aýý :i R 2ý: 2

73. Sandler, 3. H. System System R.eliabillty "Engý_neering.Englewood. Cliffs NJ: Prentice--Hall, Inc., 19-53.

7~4. Sasaki, M1. "1Slide Methods for Redundant Mission~ Avail-abillty,," -rocceedini-s of th~e 1969 Annual S~ympo-Siim onR1iat.lity. _ý7,-07- January '96. (IEEE Catalog Nc.

75.---- and T. Yana'. "A~vailabilities for Fixed PeriodIcSy'stem (ýwIth Su~pp'ement)," IEEE Transactions on Relia-bility, R-26: 300-302 (October 1977).

76. SGik . J. "markov ?:oessand Some Ae-osoaceApplications." Paper presented at the 48th annual meet-Ing of Specific Division of American Asszociation for the

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acgatlon. of Ur.niertafnties in System AvaIlatilItv ari Re-ibIIty Computat' -ons," N-aval Research Ic-7ist"c~uatory, 3: 673-6E(1T)

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7?. mth avsJ. and A.lex 11. Babb. Ma-*nrainabilit

7.n:-neei~ . Ye JoDk: ýhn Wiley and Sons, Inc. ,'Q73.

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A, S cQo

0cs - ri an 'U~" ~ En ~ ~c a -c -s c lI 1 t v _ 1 i-

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APPE-NDICES

APPENDIX A

MAJOR REFERENCE SOURCES

88

Guides tc Books

:. Card Catalog

Guides to Periodicals, Journals, Reports, etc.

I. Applied Science and Technology Index

2. Defense Documentation Center Bibliographic Search Facil-

ities

3. The Engineering Index (Engineering Index, Inc.)

4. Guide to library Research Resources (Air Force Instituteof Technology Library, August 1969)

5. international Abstracts in Operations Research (OperationsResearch Society of America)

6. International Aerospace Abstracts (American Institute of

Aeronautics and Astronautics, Inc.)

7. Mathematical Reviews (The American Mathematical Society)

8. National Aeronautics and Space Administration LiteratureSearch Facilities

9. PANDEX Current Index to Scientific and Technical Litera-ture (CCM Information Sciences, inc.)

10. Quality Control and Applied Statistics (Executive SciencesInstitute, Inc.)

11. Reliability Abstracts and Technical Reviews (NationalAeronautics and Space Administration)

12. Scientific and Techrnioal Aerospace Reports (National Aero-nautics and Space Administration)

13. Selected RAND Abstracts (The RAND Corporation)

!14. Statistical Theory and Method Abstracts (The InternationalStatistical Institute)

15. Tecnnical Abstract Bulletin (n-fense Documentation Center)

1. "•. S. uovernmernt Research and Development Renorts (Clear-ing House for Federal Scientific and Technl•cal Information)

89{

I'I

APENDIX B

GENERAL REFERENCE BIBLIOGRAPHY

90

i. Aggarwal, K. K., K. B. Misra and J. S. 3upta. "A FastAlgorithm for Reliability Evaluation," IEEe Transactionson Reliability, R-24: 83-85 (April 197--7.

2. Barlow, R. E. and F. Proschan. Mathematical Theory ofReliability. New York: John Wiley and Sons, Inc., 1965.

3. Breiman, L. Probability and Stochastic Processes. Boston:Hought on-Mifflfn, 1969.

4. Chambers, J. M., C. L. Mallows and B. . Stuck. "A NewMethod for Simulating Stable Random Variables," Journalof the American Statistical Association, 71: 340-344Apri l76.ATE)i_1 1-97, .

5. Chung, K. K. Markov Chains with Stationarv TransitionProbabilities, Second Edition. New York: Springer-Verlage, 1967.

Cramer, H. "Model Building with the Aid of StochasticProcesses," Techncmetrics. 6: 138-159 (May 196ý).

7. Endrenti, J. "Algorithms for Solving Reliability Modelsof System Exposed to a 2-State Environment," IEEE Trans-actions on Reliability, R-24: 281-285 (October 1 T

8. Feller, W. An Introduction to Probatility Theory and ItsApplications, Vol. I.. Third Edition. New York: JohnWiley and Sons, Inc., 1968.

9. Fussell, J. B. "How to Hand Calculate System Reliabilityand Safety Characteristics," IEEE Transactions on Relia-bility, R-24: 169-174 (August 1975T.

I). Karter, H. L. and A. H. Moo:e. "Asymptotic Variancesand Covariances of M... "u..um-LIkelihocd Estimates fromCensored Samples, of the Parameters of Weibull and GammaPopulations," Annals of Mathematical Statistics, 38:557-570 (April179--6•--

11. -------- "Local-Maximum-Likelihood Estimation of the Para-meters of Normal Populations from Singly and Doubly Cen-sored Samples," Biometrika, 53: 205-213 (August l966).

12. "Maximum-Likelihood Estimation from CensoredS.n.e.. .f thtI:• of a LoDLztic Di-tributilon,Journal of the American Statistical Association, 62:575777 T~ue. T--967).

13.----- "Maxmum-Likelihooa Estimation of the Parametersof Gamma and Welbull Populations from Complete anraCensored Samples," 7echnometrics, 7:639-643 (November1965).

91

14. Jayachandran, T. and L. R. Moore. "A Comparison ofReliability Growth Models," IEEE Transactions on Relia-Llity, R-25: 4 49-51 (April 1797)--

1=. Kemeny, J. G. and J L Snell. Finite narkov Cha•s.

Princeton NJ: D. Van Nostrand Co., inc., 19-66.

16. Miller, I. and J. E. Freund. Probability and Statisticsfor Fineers. IEnglewood Cliffs NJ: Prentice-Fall,Inc ., 1365.

17. Neuman, C. P. and N. M. Bonhomme. "Evaluating of Main-tenance Policies Using Markov Chains and Fault TreeAna>ysis," TEE Transactions on Reliability, R-24:37-45 (April 197-.

18. Olsen, D. E. "Estimating Reliability Growth," IEEETransactions on Re'Liabilty, R-26: 50-53 (Auril 1977).

19. O'Neil, T. S. "System Reliability Assessment from ItsCAomponerts, Apled Statistician, 21: 297-320 (172.

?0. Singpurwaila, N. D. "Estimating Reliability Growth (orDeterioration Using Tme Series Analysis)," Naval Research

-~stIcs Quarterly, 25: 1-14 (Aoril 1978).

21. Szringer, M. D. and W. E. Thompson. "Bayesian Confi-dence Limits for the Product of N Binomial Parameters,"Biometrika, 53: 611-613 (December 1966).

2.-- -"Bayesian Confidence nimits for the ReliabIlityof Cascade Exponential Systems," IEEE Transactions onReliability, R1-16: 86-89 (Septemb-er1967).

23.---- "Bayesian Confidence .imits -for the ReliabilityRedundant Syste:ns W-1hen Tests Terminated at First Fail-ure," Technometrics, 10: 29-36 (February 1968).

24. Thompson, W. E. and P. A. Palicio. "Bayesian ConfidenceL_-nits for the Availability of System," IEEE Transactionson Reliability, R-24: 118-120 (June 197-7-.

25. Vit, P. "On the Ecuivalence of Certain Markov Chains,"Journal of Apclied Probability, 13: 357-360 (April 1976).

92

APPENDIX C

SUPPLEMENTAL BIBLIOGRAPHY

93

1. Althaus, E. J. and H. D. Voegthen. "A Practical Relia-bility and Maintainability Model and its Applications,"Proceedings of the Eleventh National Symposium on Relia-bilitý, and Quality Control. 41-4b. January 195.

2. ARINC Research Corporation. Maintainability Prediction.Publication No. 267-02-6-42C. Washington DC: Aeronau-tical Radio, inc., 31 December 1963. AD 428817

3. Barlow, R. E. and E. M. Scheuer. "Reliability GrowthDuring a Testing Program," Technometrics, 8: 53-60(February 1966).

-. and R. Proschan. Statistical ThorZ of Reliabilityand Life Testing. New York: Wolt, Rinehart and Winston,

5. Barzilovich, E. Y. and V. F. Voskoboev. "Markovian Prob-lems in the Maintenance of Aging Systems (A Survey),"Automation and Control, 28: 1890-1904 (December 1967).

6. 5iondi4, E., et al. "Analysis of Discrete Markov SystemsUsing Stochastic Graphs," Automation and Control, 28:

7. Borgerson, B. R. and R. F. Freitas. "A Reliability Modelfor Gracefully Degrading and Standby Sparing System,"IEFE Transactions on Computers, EC-24: 517-525 (May 1975).

8. Bosinoff, i. and S. A. Greenberg. "System EffectivenessAnalysis: A Case Study," Proceedings of the 1967 AnnualSymposium on Reliability. 40-53. January 167.-(IEEECatalog No. 7-'50).

9. Butler, D. A. "The Use of Decomposition in the OptimalDesign of Reliable Systems," 0Oerations Research, 25:4509-468 (May-June 1977).

10. Chou, T. C. K. and J. A. Abraham. "Performance/Avail-ability Model of Shared Resource Multiprocessors," IEEETransactions on Reliability, 5-29: 70-74 (April 198t-.

11. Coleman, J. j. Evaluation of Automatic Checkout forSystem Availability, Reia-b-i-ity an-d M-intanab•!TFyConference Technical Papers. 474-482. May 1953.

12. Dal Cin, Mario. "Availability Analysis of a Fault-Toierant Computer System," IEEE Transactions on Reliabil-

_6 -2.. .. .. . I ug 'A_ ... •

9)4

3.Dermar., C. "On Optimal Replacement Rules When Changesof State Are Markovian," in Mathematical Optimiiationlechniues, pp. 201-210. Edited by Richard Bellman.Los Ang~eles CA1: universit.y of California Press, 1-963.

P4I. Eliot, C. C. and W. H. Sellers. "Autocmated MissionAnalysis by Markov Chain Techniques," IE7EE Transactionson Aerospace and Electronic Systems, S'u-PFeme-nt,- AES-2:27-209 Ziuly-7-67 T

15. Emoto, Sherrie E. "A Problem in Equipmnent, Mainternan:!e,"Manageement Science, 8' 266-2717 (April 1962).

`6.---------and Ray Z. Schafer. "Orn the Specification ofRepair Timne Requirement~s," IEEE Transactions on Relia-bifllt, R-.29,: 13-16 (April l9700- ___

1-7. Evans, I. G. and A. H. M. Nigrn. "Bayesian 1-Sample Pre-.diction for the 2-?arameter Weibull Distribution," IEEE

`rnatoni or. Reliabilit3,, R-29: 4'0-ý`4 (December

18.l F'inkelstein, Jack M."Starting and LimitIng Values forR-eliabllity Growth," IEEE Transactions on Reiiati:,R-28: 111-114 (June T9_79). ~lti

19. Plehinger, E. J. "A General %lodel for the ReliabilityAnalysis of System Under Various Preventive M-aintenancePolicies," Annals of Mathematical Statistics, 33: 137-156' (March 1962).

20. Friedman, H. D. "A Universal Stochastic Model for Cer-tain Reliability Queuing and Cont~rol Systems," Sulletinof' the Operations Research Society of America, 77 1.3--

21.Gayr, . F, Jr "'2Ime to Failure and Availability ofParalleled Systems with Repair," IEEE Transactions onReli-ability, R-12: 30-3z8 (J3une 19737.

22. Ge-ha~rt, L1. IS. and W. Wolman. "A Probabilistic Modelfor RelIabil~Ity Estimation f'or Space System Analysis,"Blulliet-n de l'Institute int~ernatioa eSaýt~e33e session (1961).

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98

VITA

,ýohamed Shib. Biltagy Hasaba'la was born on 5 December

194ý in Cairo, Egypt. He graduated from Ain Shams University,

Faculty of Engineering, Electrical Department iln 1968. He

received a Diploma in Computer Science from Cairo University

in 1976. He also received a Diploma in Operations Research

from the same university in 1978. He served as the Commander

of Electronic Warfare Battalion which has been chosen by the

Commander of the Army as the best electronic warfare unit in

-he A.rm. He served as the Commander of Electronic Warfare

Laboratory before he had come to the United States 'fn June

!,0 to obtain the Masters degree in Operations Research.

Permanent address: 52, Hader El Toni St.Nasr City, Cairo, Egypt

99

h.e

UNCLASSIFIEDSECURITY CLASSIFICATION Of TMIS PAGE (Whe~n Dalsa S.ln).

PAGE READ INSTRUCTIONSREPORT DOCUMENTATION PAEBEFORE COMIPLETING FORM'

1. P.(PORT NUMBER 17. GOVT ACCESSION NO. 3. RECIPIENTV5 CATALOG NUMBER

GOR/MA/81D-7 I f) -I// 2r4. TITLE (and Subtitle) S. TYPE OF REPORT A 0ERIO0 COVEREG

CCNNFIDENCE TNýTERVALS FOR S YSTEM?.EIIAEL-T71 AND AVAILABILITY OF MAIN,1TAINE MSThsiSYSTEMS USING MONT CAR~LO TECHNIQUEES 6. PERFORMING ORO. REPORT NUMBER

7. AU fNOA(s) 6. CONTRACT OR GRANT mumBIR(.)

Mohamned S. B. HasaballaLt . %Col. Egyptian Army

3. P11FFORMING 0R0AP4IZATION NAME ANO AOOinESS WC. PROGRAM ELEMENT. PROJECT. T ASiCAREA 6 WORK UNIT NUMBERS

Air Force lnstitute of Technology (AFIT)Wriht-?ater'son AFB OH i45L433

11. COPTROLLING, QVTICE N4 &ME ANO AOGRESS 12, REPORT GATR

December 1981I). NUMOER OF PAGES

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Unclassified15A. Otc~ ASSIFICATiON, OOWNGPAOIING

Anproved for publ-Ac release; distribution unlimited

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Approved -pbi release; 1AW AFR 1 01- 117

19. KEY WOROS (Continue on ravets, oid* It n@40984ey and ldontitf, by block number)

Confiden~ce intervals Mont Carlo Techniques

MaintainabilityAvailability

20. AST(ýACT (Continue on I.tovrs side if necmaceire and jIdentify by, block rnustbt)

,,~This thesis presents the results of an extensive literatu~resppch for finding the confidence intervals for system rel2.a-bility and availability of mnainvtainedc systemms uzing Monti Carlo

tecni~.es. -he charact-eristics of s.ystem reliability and,-ajntai'na'b'-'"-t- analys-s a-e dis-cussed. The basil Markollianfallure anrd repatr models are developed. A su-m.ary of the

oitesti1mates of avallability, reliability of maintained

IJAN71 1473 90ITION OF 1 NOVA IS 13OGSOLETZ fjNCLASSIF! E-DSECURITY CLASISIFICATION OF THIS PAGC 'WI.n D.le garieod)

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systems, and the exact analytical methods are presented. Finally,the B00-strap(Double Mont Carlo technique) is used to obtain theconfidence limits for the availability of the maintained systems.

UNCLASTC S I-:JStCjPIT " CLASSIFICATIO94 0 .', PA, ;C : at Es e,,d)