Title: Generalized reliability methods for systems with standbys
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Title: Generalized reliability methods for systems with standbys
Physical Description: x, 82 leaves. : illus. ; 28 cm.
Language: English
Creator: Linton, Darrell Glen, 1944-
Publication Date: 1972
Copyright Date: 1972
 Subjects
Subject: Reliability (Engineering)   ( lcsh )
Industrial and Systems Engineering thesis Ph. D
Dissertations, Academic -- Industrial and Systems Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 78-81.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
Statement of Responsibility: Darrell G. Linton.
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Bibliographic ID: UF00097626
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577169
oclc - 13923698
notis - ADA4863

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GC--ncral.ed i 1,a1 baility .Tech6ds
for C'" Le:r.s wi 'Jh S cvrndbys














By


PJtL'ULT- G. LITONC


.. DI SE L'^ .T C vD?2 :?: U TiDU OLA."TE C ;1.';C:L OF
: A F -f :
.-... -. 21f. .. DE LE . OY






,,ii..- .. i 1 ., .D L' '- "






-j -























































Copyright by
Darrell G. Linton
1972






























TO IMY FATHER,

LEO LTNTON











ACK.;01 :LEDCEME NTS


The author is particularly indebted to the members of his

supervisory committee, Dr. R. N. Braswell, Dr. J. F. Burns,

Dr. Z. R. Pop-Stojanovic and Dr. J. G. Saw, for their aid, advice

and encouragement. Special thanks go to Dr. Braswell, who served

as committee chairman, and to Dr. Pop-Stojanovic, who served as

comm-iittee co-chairman.

In addition to the members of the supervisory committee,

Drs. O. I. Elgerd, D. R. Miller, E. J. tluth, B. D. Sivazlian and

Mr. R. H. Wessels gave willingly of their time in the author's

behalf. Thanks also go to the author's friend and colleague,

Mr. Phiroze Mehta, for his excellent illustrations, and to

Mrs. Karen Walker,who had the unenviable task of typing the

dissertation--both rough and final copies.

The author would like to express his sincere gratitude to

his parents for their guidance in years past and present, and to

Miss Corinne E. Bowling for her unselfishness and constant encourage-

ment during these past few years.

This research has been partially supported by the Army Research

Office under Contract DAHC04-68-C-0002 uith the University of Florida.










TABLE OF CONTENTS




ACKN; OULEDG E; E TS ............................................

LIST OF TABLES..............................................

LIST OF FIGURES.............................................

ABSTPA CT ....................................................

CHAPTER:

1. ANl OVERVIEW. .......................................

InEroducion...................................

Previous Research Pesul s ....................

Problem Statemen .............................

Research Objeccives ..........................

Research Profile. .............................

2. THE k-OUT-OF-n SYSTEH. .............................

Incroduccion.................................

Assumptions, Definicions and Ilocation ........

The Case of = and F (*) = F(*),
m m
1 m n: n ....................................

The Case of F () Discinc 1 m: i n....
m r --

A Comparison Bec':een che Semi-Markov' Model
and the Supplemencary Variable Technique .....

3. SOME RELIABILITY CHARACTERISTICS OF THE 2-OUT-OF-n
SYSTEM AND THE TWO-UNIT STANDBY REDUNDANT SYSTEM..

Introduction............................... ...

The 2-Out-of-n System.........................

The Two-Unit Standby Redundant System.........


Page













1

1


2

6

6



















37


40

40

40

51
51












TABLE OF CONTENTS (Continued)


Page

4. APPLICATIONS ........................................ 63

Introduction................................. 63

An Example................................... 63

numerical Methods Applicable to Chapters 2
and 3.... ...................................... 6

5. CONCLUSIOtS AND AREAS FOR FUTURE RESEARCH......... 69

Conclusions.................................. 69

Areas for Future Research.................... 70

APPENrDIX ..................................................... 74

LIST OF REFERENCES.......................................... 7S

BIOGRAPHICAL SKETCH......................................... 82










LIST OF TABLES


Table Page

1 E[T2,3] for Deterministic, 2-Erlang and
Exponential Repair................................ 65

2 E[T ,] for Deterministic, 2-Erlang and
E:ponential Repair........................... ...... 65

3 The Standard Deviation of T,3 for the Four Cases
Considered in Table 1 ....... ...................... 66

E[t2,3] for the Four Cases Considered in Table 1... 67










LIST OF FIGURES


Figure Page

2.1 Sample function of the process X(t) = (i,y),
i = 0, 1, 2, y 0.................................. 12

3.1 A sample function of Z(t) in terms of the random
variables i., R., Y and T ,n....................... 2

3.2 A sample function of Z(t) in terms of the random
variables X., P. and T, when an odd number of
1 1
repairs (3) are completed.......................... 53

3.3 A sample function of Z(t) in terms of the random
variables X., P.. and T, when an even number of
1 1
repairs (2) are completed .......................... 53

5.1 A typical realization of the processes Y (t),

Y (t) and Z(t) in terms of the random variables X.

(the time to failure of unit i), P.. (the repair

time for unit i) and T............................. 72

A.1 A sample function of Z(t) in terms of the random

variables X, D and ............................. 76
.,n


viii









Abstract of Dissertation Presented to the
Graduate Council of the University C& Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy


GE:;ERALIZED RELIABILITY METHODS
FOR SYSTEMS WITH STANDBYS


By


Darrell G. Linton

June, 1972


Chairman: Dr. R. N. Eraswsell
Co-Chairman: Dr. Z. R. Pop-Scojanovic
Major Department: Industrial and Systems Engineering


For repairable systems with standbys, several candidates for

measuring reliability are (1) the time to system failure, (2) the

number of repairs completed before failure and (3) the total time

spent on repair before failure. In this research, some general

procedures are developed for studying these and other reliability

characteristics of systems with standbys. Emphasis is placed on a

class of repairable systems knon.m as k-out-of-n systems, 1 k < n,

where n units in parallel redundancy are serviced by a single

repairman and system failure occurs 'hen k units are simultaneously

inoperable for the first time.

Assuming and F. () are, respectively, the failure rate and

general repair distribution for unit i, the supplementary variable

technique is used to find the transform of the time to system failure

distribution for the k-out-of-n system. Using the principle of

regeneration, an alternative derivation is also obtained for the

2-out-of-n system.











A conditional transform approach is applied to the 2-out-of-n

system and the two-unit standby redundant system. For each system,

transforms of distributions are obtained for the time to system

failure, the idle time of the repairman and the tine spent on repair.

In the case of exponential failure and Erlangian repair capabilities,

the generating function for the distribution of the number of renewals

occurring during the life of each system is found.

Numerical results are presented for the 2-out-of-3 and

3-out-of-4 systems for several different repair disciplines. Numerical

methods for evaluating moments of distributions expressed in terms of

complex integrals are also discussed. Although specific results are

not derived, the value of stochastic integrals for investigating

attributes of redundant systems is clearly demonstrated.











CHAPTER 1

ANi OV L RV \IEl


Introduction


High speed computers, sophisticated weaponry and faster

means of coruriunication and transportation have focused attention in

the scientific community on the design and manufacture of systems

which are both proficient and reliable. Most engierL.e-rs .uld agree,

however, that unless reliability is incorporated in the design phase,

the final product will suffer regardless of correctivG maintenance

policy. ihus, it is the ability to measure reliability that concerns

the engineer and designer of today.

For repairable systems with standbys, several caiijidates for

measuring reliability are (1) the time to system failure, (2) the

nu: ber of repairs completed before failure and (3) the total time

spei-nt on repair before failure. In this dissertation, some general

procedures r 'e developed for studying these and other reliability

ri;arac:eris:ics of systems with standbys. Emphasis is placed on a

cl;a~s :f reicira-ble sssterns kno'.',n as k-out-of-n systems, 1 < k < n,

'r-. 'i its (or machines) in parallel redundancy are serviced by a

r.1 T,.irr...c. -nn- s'"stem failure cccurs when k units are simul-

:.cu pra, ;,-.:ie for the first time. The often referred to series

'ai lle rran :anents .are the special cases k = 1 and k = n,

re1- secti.' E l. Eecause 'k-aLt-of-n systems (or ccr, nations thereof)

I-l- : a i.-.rt e cLi s- of redur.dani sys:er., thi. research encompasses a

s'ucr, ja tr., aLtrihuces of sjch systems.










Previous Research Results

The body of methods and ideas used in studying characteristics

related to the life of a system is referred to as mathematical

reliability theory. Although a relatively new area of study, reliability

theory has already been applied to problems involving power systems

[1], [2], [3], [1 ], [5] and the regulation of traffic [6], [7], as

well as to multicomponent structures [8], [9], [10], [11] and [12].

Takacs [13] in 1957 and Morse [14] in 1958 discussed properties

of a system where several machines were serviced by a single repairman,

and each machine was subject to exponential failure. Limiting

distributions for the number of machines working werederived by Takacs

for general repair and by Morse for exponential repair.

One of the earliest to treat first passage times [15) in the

context of reliability theory was Gaver [16] in 1960. For a birth-

death process with constant transition rates, Gaver derived a formula

for E[T i], j i, where T. is the first passage time to go from
ij iJ
state i to state j. In 1962, Belyayev [17] found the mean time to

system failure (MTSF) for a system composed of two units in parallel

redundancy, both of which are operating initially. Belyayev assumed

that failures were exponential, repair was general, and the system

failed when both units were inoperable for the first time. In 1963,

Gaver [18] solved the same problem considered by Belyayev but used a

modified approach. Both Belyayev and Gaver used the Cox [19] method

of supplementary variables to analyze the process. Later in 1963,

McGregor [20] derived formulas for the MTSF for the more general

k-out-of-n system where both failures and repairs were exponential.











Thiruvengadam and Jaisw.al [21] in 1964 and Jaisval [22] in

1968 used discrete transforms to investigate e the k-out-of-n system with

general repair capabilities. In addition to several other system

attributes, Thiruvengadam and Jaiswal found the distribution of a busy'

period generated by i units, but they did not consider properties of

the time to system failure.

In 1966, Muth [23:, Htun [20] and Srinivasan [25] discussed a

standby redundant system composed of two units. Mutl found the IMTSF

when the repairman has the capability of completing n repairs. As

n -+ c, Muth showed that the MISF agreed with equation (20) in Gaver

[18]. Htun assumed repair and failure rates were constant but different

for each unit and derived the MISF. Srinivasan assumed general failure

and repair capabilities and derived the transform of the time to

failure distribution.

Also in 1966, Liebowitz [26] and Dow.nton [27] obt.ir.ed some new

results for parallel redundant systems. Considering the two-unic

redundant system treated also by Belyayev and Gaver, Liebowitz plotted

the ratio (called the improvement factor)


MISF
SIr S F
.-1


-]
for various repair disciplines, where is thle e:.pected time to

failure for each unit. He Loncluded thaU t.oi- difference between MTSF

with repair to that without was independent of the repair distribution

itself and depended only on the quantity E[R], where E[R] is the expected

r..-eir time for each component. Downtown used the properties of semi-











Markov processes to analyze a parallel redundant system composed of

n units. Assuming exponential failure and general repair, Downton

obtained the Laplace transform of the distribution of Tk,n, the time

to failure for a k-out-of-n system, and E[Tk,n for n k, k = 2,3,4.

The role of priorities in a slightly more complex system than

those described above was treated by Natarajan [28) in 1967. Two

paralleled radars working in conjunction with two paralleled computers

describe the system which is considered operative so long as at least

one computer and one radar are working. Natarajan derived the distri-

bution of the time to failure and the MTSF assuming failures and

repairs were exponential. Values of the MTSF were compared for the

case of no priorities and when the preemptive-resume discipline was

assumed. In his doctoral dissertation [29] completed in 1968, Natarajan

treated a large class of reliability models. Both single and multiple

repair facilities were discussed for a system composed of a basic unit

supported by n-i standbys, where the standbys may deteriorate in

storage (sometimes called warm standbys).

Also in 1968, Mine, Osaki and Asakura [30] derived the MTISF for

a 2-out-of-n system by considering first passage times. The

improvement factor was evaluated and its asymptotic behavior was

studied.

In 1969, Epstein [31] considered a two-unit parallel redundant

system where failure was exponential (but with a different failure rate

for each unit) and repair was general. Transforms of distributions

were obtained for the time to system failure, the total time spent on

repair and the free time of the repairman.










In 1970, Rao and Natarajan [32] presented a survey paper on

the reliability of systems with hot, warm and cold standbys for both

parallel redundant systems and standby redundant systems. Rao and

Natarajan remarked that the problem of finding the distribution of the

time to system failure for a k-out-of-n system has not been solved when

either the repair or the failure distribution is not exponential.

Also in 1970, Osaki, nuth and Mazumdar analyzed some two-unit systems.

Osaki [33] treated a two-unit standby redundant system and found the

HISF using a state transition diagram (or signal flow graph) approach.

Osaki [34] applied the integral equation of renewal theory to four two-

unit redundant systems (parallel redundancy, standby redundancy with

and without priorities and standby redundancy with noninstantaneous

switchover). The transform of the time to system failure w-as derived

in each case. Muth [35] took advantage of the regenerative properties

(discussed extensively by Smith [36]) of the two-unit parallel redundant

system, and derived the time to failure distribution and the NTSF in

an elegant manner. Mazumdar [37] considered a two-unit redundant

system where the operating unit and the standby may have different

failure rates. Mazumdar found the HTSF when the detection of failure

for the standby and the operating unit may not be found instantaneously.

In 1971, Gaver and Luckew [38] treated what they termed an

accumulation model, w-here a system experiences accumulating degrading,

but not fatal, failures. Joint transforms were derived relating total

time spent on repair and the number of repairs made. Later in 1971,

Branson and Shah [39] considered a two-unit parallel redundant system

where each unit had a different but constant failure rate, and repair











distributions were different but general for each unit. Using

properties of semi-Markov processes, Branson and Shah dervied the

MTSF for the two-unit system and discussed some of the difficulties

encountered when the semi-Markov model was applied to a three-unit

system under similar assumptions.


Problem Statement


Although many authors discuss properties of T, the time to

system failure, other measures of system reliability have been

generally neglected in the literature. This dissertation will exhibit

techniques for finding transforms of distributions for the time spent

on repairs during T, the time that the repairman is idle during T and

the number of breakdowns prior to system failure. In addition, assuming

random (exponential) failure, general repair capabilities and one

repairman, the supplementary variable technique of Cox [19], Belyayev

[17] and Gaver [18] will be extended to allow analysis of the k-out-of-n

system. A comparison will be made between the supplementary variable

approach used in this research and the semi-Markov model employed by

Downtown [27] and Branson and Shah [39].


Research Objectives


Specifically, the objectives of this dissertation can be

summarized as follows:

(1) For the k-out-of-n system, to find the transform of the

distribution of T assuming that .X and F (-) are, respectively, the
m m
failure rate and general repair distribution for machine m, 1 < m < n.










(2) For the n-out-of-n system with = and F (*) = F(-),
m m

1 < m < n, to find a general expression for the nITSF.

(3) For the 2-out-of-n system and the two-unit standby

redundant system, to derive transforms of distributions for the time

spent on repairs during T, the time that the repairman is idle during

T and the number of repairs completed prior to system failure. Also,

assuming exponential failure and Erlangian repair capabilities, to

analyze the distribution of I for each system and apply the results to

a variation of the accumulation model.

(4) lo provide the reader with insight into solutions of

problems which are as yet unsolved; namely, the analysis of the 2-out-

of-2 system when failures and repairs follow general distributions and

the application of stochastic integrals to redundant systems.


Research Profile


In Chapter 2, the transform of the time to failure distribution

is obtained for some k-out-of-n systems and a comparison is made

between the supplementary variable approach and the seni-Markov method.

Chapter 3 concerns the 2-out-of-n system and the two-unit

standby redundant system. A conditional transform approach is applied

and the distribution of the time to system failure for both systems

is analyzed.

Chapter 4 deals with applications of the research.

Conclusions and areas for future research are discussed

in Chapter 5.










CHAPTER 2

TFE k-OUT-OF-n SYSTDI


Introduction


In thi.; chapter, the supplementary variable technique is used

to find the transform of the time to system failure for the k-out-of-n

system. After introducing some preliminaries, the k-out-of-n systemm

is treated first, when = \ 0, F (*) = F(), 1 < m n, and second,
m m -

when \ and F (*) are distinct for each m. A comparison between the

supplementary variable method and the semi-Markov approach is also

discussed.


Assumptions, Definitions and Notation


The following will be assumed:

(1) n units (machines) are in parallel redundancy and the

time to failure of each machine is independently and exponentially

distributed. The failure rate of machine m is 0, 1 m < n.

(2) At t = 0, all units are operative.

(3) There is only one repairman (with unlimited service

ca:.:.i cities) ,;nd failed units queue up for service on a "first-come,

ljit::-servc-d" basis. Machine m is serviced according to general

re.pai dis-rihution, F (*), and repair times are independent random

varicX..s *..hiic!. are also independent of the failure times.

(.,) Repaired units are put back into operation if k-l or

iess unit., -..e in a failed state. lWhen k units are simultaneously

inoperabic f-..r the first time, the system fails.










The following notation will be used:

(a) Tk a continuous random variable which for any

k-out-of-n system represents the length of time elapsed before k

machines are simultaneously inoperable for the first time, assuming

all n machines are running at t = 0; 0 < T, < c.
k,n
(b) E continuous random variables which represent the
mj
time needed Lo repair the j-thbreakdown of machine m; 0 < RP. < ,
IJ
e=l,2,...,n; j=l,2 ....

(c) y a continuous random variable which at any moment t

represents the time already spent on the repair of the machine under-

going service at that moment; 0 < y t < .

(d) i a discrete random variable which at any moment t

represents the number of machines broken do'.n at that moment;

i=0,l,...,k n.

(e) E the event "T > t."

(f) Let F (-) denote the common distribution function for che
m
random variables R .; i.e., F (r) = P[P. r], m=l,2, ,n; j=l, ....
mj m mj -
(g) Put Pk, (t) = P[E] and hence, Pk (0) = 1 by assumption (2).
kn k ,n p

(h) Let ,n (s) denote the Laplace transform of


k (t); i.e., k,n (s) = e kn(t)dt.
n, n.k ,n
0 *
(i) Let h-n(t) and ,(s) represent the density of T,
e nt) kn %n k,n
a:nd i.s Laplace transform, respectively.

(j) Let X(t) represent the state of the k-out-of-n system it

any roor;ent t; e.g.,

(i) X(t) = (i,z), 1 < i < k < n, z ;_ 0, means that at










moment t, there are i machines awaiting repair (including the machine

presently being serviced) and an amount of time z, 0 < z < t, has

already been spent on the repair of the machine presently being serviced.

(ii) X(t) = (0,0) means that at moment t, there are no

machines broken down (i.e., no machines awaiting or being repaired) and,

consequently, no time has been spent on the repair of a machine.

(k) Put p0(t) = P[X(t) = (0,0)]; hence, p (t = 0) = 1 by

assumption (2).

(1) Let Pi(t,z) represent continuous and differentiable

functions such that pi(t,z)dz = P[X(t) = (i,y)], 1 < i < k < n,

z < y z + dz.

(m) Define


VO(s) = f eS-p0(t)dt

and
oo t

V.(s) = e ( p.(t,z)dz)dt, 1 < i < k-1
0 0

(n) By definitions (k), (1) and (g)


k-1 t
Pk,n(t) = P(t) + Z pi(t,z)dz
i=1

and hence, using (m) and (h)


k-1
W (s) = V.(s), 1 < k < n
k,n i0 1
i=0











The Case of : = and F (.) = F(*), 1 < m < n
m m


Preliminaries

From the above assumptions and definitions, transition

probabilities for X(.) during the time interval (t, t dt) depend

only on the state of the system at moment t. In particular, when


= : 0


and


F (') = F(-), 1 m < n
in


(2.1)


the transition probabilities can be described as follows:


(i) P[X(t + dt) = (l,0)IX(t) = (0,0)] = n dt + o(dt)


(ii) P[X(t + dt) = (0,0)IX(t) = (0,0)] = 1 n'dt + o(dt)


(iii) P[X(t + dt) = (j + 1, z + dt)IX(t) = (j,z)]


=(n j)Xdt{(l F(z + dt))/(l F(z))} + o(dt), 1 j -_ k-1


(iv) P[X(t + dt) = (j,z + dt)IX(t) = (j,z)]


= (1 (n-j)Xdt){(l F(z t dt))/(l F(z))} + o(dt), 1 j < k-1


(v) P[X(t + dt) = (j-l,0)IX(t) = (j,z)]


= (1 (n-j)Xdt){(F(z + dt) F(z))/(l F(z))) + o(dt), lj-k-1


The 2--Out-of-n System

Consider the 2-out-of-n system. For n=2, a typical realization

of the p:icss :Xt) in terms of the random -.ariLables y and i is

presented in Figure 2.1.








y
rl "





0 t
tl t+rI t2 t3



2

1


t1 tl+r t2 t3

Figure 2.1. Sample function of the process X(c) = (i,y),
i = 0, 1, 2, y > 0. t. is the moment at which the j-th
breakdot:n occurs and r1 is the time needed to complete the
first repair. The second repair is not completed before
the absorbing state (2,z) is reached, and thus, T2,2 = t3.

Two quantities of interest are the distribution of the random

variable T2,n and its expectation, E[T2,n]. In this regard, from

definitions (g) and (h) it is seen that


d d
h2, (t) = (1 P ()) =- P (t) (2.2)
2,n dt 2,n dt 2,n


and since P ,n(t=0) = 1, from (2.2)



h2,n(s) = 1 sW2,n(s) (2.3)


Furthermore, assuming E[T2,n] < m, from (2.3)



E[T, ] = (1 h (s))/s = W (s) (2..)
s=0 s=0

To find W2,n(s), first consider the state transition equations.

Using definitions (k) and (1), transition probabilities (i)-(v) and the










fact that (2,z), z > 0, are absorbing states, the following relation-

ships hold for n > 2:



p0(t + dc) = p0()(l n-dt) +

t
F(z + dt) F(z)
+ p (t,z)(1 (n l),dt) F(z + dt F(z) dz + o(dt)


(2.5a)
(2.5a)


pl(t, + dt,z + dt) = p (t,z)(1 (n l)',dt) 1 F( z d) + o(dt)

(2.5b)


Pl(t


+ dt,0)dt


= p (t)n-dt + o(dt)


(2.5c)


(2.5d)


PO(O) = 1


Defining gl(t,z) by the relationship


pl(t,z) =
1'


gl(t,z)e (-1 (1 F(z))


(2.6)


and using the fact that for g(') continuous and F(') absolutely

continuous


(2.7)


SF(x + h) F(x) dx = (x)dF(x)
g(x) dx = g(x)dF(x)
h0
0


equations (2.5a) (2.5d) may be rewritten as





iThis can be shown by using the properties of continuous and
absolutely continuous functions and appealing to the Lebesgue Dominated
Convergence Theorem.










t0
d-p = -n\p(t) + e -n-1)z gz (t,z)dF(z),p(0) = 1 (2.8a)
t 0 00

+ -= 0 (2.8b)


pl(C,O) = n\!P0() (2.Sc)


To obtain (2.8a), substitute (2.6) into (2.5a), use (2.7)

and take che limit as dt 0. To obtain (2.8b), first substitute (2.6)

into (2.5b). Noting that e-(n-)dt = 1 (n l)dt + o(dt) obtain

(1(t + dr, z + dt) gl(t,z) + o(dt). Adding in and subtracting out

the quantities gl(t,z + dt), gl(t + dt,z) and gl(t,z) in the preceding

ocuation, dividing by dt and taking the limit as dt 0 yields (2.8b).

Equations (2.Sa), (2.?b) and (2.8c) will nov be used to find

W, (s). Before doing so, note chat by definition (n)



W2,,)n = V (s) + Vl(s), n 2 (2.9)


Also, let the laplace-Stieltjes transform of F(z) be .4(s) = e-sdF(z)

and define 0

',. (s) = ..(s + j \,), j 1


how, .-/ (s) can be found from equations (2.8a), (2.8b) and

( c) as follows. From (2.,b)


g,(t,z) = gl(t z) (2.10)


Combi-ine (2.6) and (2.10) Wv Lh (2.8c)










(2.11)


-Sg (t)dt, take the Laplace transform of (2.8a)


0
and (2.11) and obtain respectively


1 + cn (s) G (s)
n-+ n
V (s) = 7
0 s + nA


Gl(s) = n"Vo(S)


V (s) Is n-,(l --1(s)))
V (s) = {s + n"(1 in (s))}
0n1


(2.12)


From (2.6) and (2.10)


-St
\V (-) = j e (
0


-(n-1)z t z)(1 F(z))d
e gl(t z)(l F(z))dz)dt


G (1)
s (n-) (1 ())
s + (n-1).! n-1


(2.13)


n'(l ,nn(s))
n-n-
(s + (n-1)1){s + n.(1 n- (s)))
n-1


fhus, from (2.2), (2.3), (2.4), (2.12) and (2.13) it follows

c.iat for the 2-ct-of-n system


nIn-
J, :s) is + n"(1 (s)) +n-
',n n- (s + (n-l).)1 s n.(1 I, (s)))


(2.1")

(s) = 1 sU (s)
2,n 2,n


El(t) = n."P0(t)


Defining G (s)










1 1
E[T2 J = W (0) + (2.15)
2,n!\ 2,n (n-1) n' ( ..
n-I

.'ere n- = .:((n-1).). P.elation (2.15) agrees .'ith equation (5.19a)

in D ..wntcn [27J.


The 3-Out-of-n System


For the 3-out-of-n system, W3,n(s) can be found in a similar

manner. From definition (n)


n (s) =
53n


2
v V.(s)
i=0


(2. 16)


and as before


h3,n(s) 1 sW3,n(


E[T3,n = 3,n()


Defining g l(,z) and q2(t,z) by the relationships


p (t,z) = e-(n-1).z g1(t,z)(1 F(z))


q2(C,z) = p2(c,z)/(1 F(z))


the s:aca transition equations for the 3-out-of-n system may be

r-written as


do



d- 0
d 3z


-(n-1):z ( )( ) = 1
e gl(c.z)dF(z),po0(0) = 1


(2.19a)



(2.19b)


(2.17)



(2.18)










q q> -(n--1)!
= -(n-2)qq,(t,z) + (n-l)).e g.(t,)
.t oz
t

p (t,O) = n\p0(t) + q29(t,z)dF(z)
0
p1(t,0) = 0


(2.19c)



(2.19d)


(2.19e)


Again, equation (2.7) was used in obtaining equations (2.19a) and

<'.19d). From (2.19a) (2.19e), (3) can be obtained as follows.
rom 2.13,n
From (2.19b)


gl(t,z) = gl(t z)


L bstituting (2.20) into (2.19c) and solving yields


q ) -(n-2)z g, z) -(n-)
q(t,z) e (t z) (n-l)e gl (t z)


(2.20)


(2.21)


:here g.(-) is an arbitrary function of t z. Equations (2.18) and

,2.19e) imply that q,(t,O) = 0, and thus, (2.21) becomes


q,(t,z) = (n-l)gl(t z){e- (n2)z


S-( n-l)\z
- ea


Us-in (2.17) and (2.20)


pl(t,0) = gl(t,0) = gl(t)


-Ci.i 2.2 2) and (2.23) 3 ith (2.19d-) yields

t
S+ -n-2) .z
1)= n Po(t) + (n-1) 91gl(t z)je
o


-(n-i) zl
- e (dF(z)

(2 2 )


'.(s) r-jst be four.d from equations (2.19a) -rnd (2.2-'). To this end,
ing te Lapl ransfor o ( a) nd ( otair.
after taking the Laplace transform of (2.igsa) and (2.:'"), obtain. respectively


(2.22)


(2.23)










1 + G (s) I* (s)
V (s) = 1 n
0 "s+n,

and
n;,Vo(S)
cG(s) =n (
S 1 (n-1),_, (s) + (n-1),k (s)
n- n-

.;here again


GI(s) = es 1(t)dt

and

(s) = e(S+)tdF(t) = (s + j),


Fromu (2.25) and (2.26)


(2.z5)




(>.26)


j > 1


1 (n-2)j ,(s) + (n-1), _s)
(S) = (s + n'){ (r.-l).n (s) + (n-1) (s) n'" (s)
n-- n-1 n-I


.A.so, from (2.17) and the definition of V (s)


V1(s) = J
0


t
-st -(n-1)" z

0


g1(t z)(i F(z))dz)dt


C,(s)
G C (1 C (s))
s + (n-1) -n-l


SiJ'ilarly, from (2.18), (2.22) and the definition of V,(s)


r -
: n-i) e-0
0

= "n-1)G. (s)
4.


I-

g1(t z)Ie


-(n- F())d)dt
- e ;(i F(z))dz)dt


I (s) 1 ()
n- 2n-1
s + (n-2)1 s-- (k-1)


(2.29)


'using (2 27), (2.28), (2.29) and (2.16). one finds after simplification


(2. 27)


\'V(s)


(2.28)


t(










W3,n (s) = [(s + n')(n-l)(n-2)\
3,n

{. (s + (n-l)")i' -,(s) + (s + (n-2).)';n, (s)J]/
n-, n-I


/[(s + n0){ (n-l),, -(s) + (n-l),: (s)) n'n (s)] (2.30)
n-L n-i n-I

and

h3 (s) = 1 sW, (s)
3,n .,, n


1 -q + v
1 1 'n-l n-2
F[]3 ] = i (0) = + + -i-
3,n 3,n (n-1) (n-2) nil (n-1)n 2 + (n-2)r, j
n-2 n-i

(2.31)


where q. = (0) = (j.) j 1. Equation (2.31) agrees ..'ith
J J
equation (5.19b) in Dow.nton [27].

It should now be clear, however, that although E[Tk,n] can be

found for any fixed k, a general expression for E[Tkn] cannot be

obtained by this approach; the reason of course being that the

set of differential-integral equations '..hich result from the state

transition equations is different for each value of k. A similar

difficulty encountered by Downton will be discussed belo:'.


The n-Out-of-n System


Although unsuccessful in obtaining a general expression for

E[Ik,n] it will now be shown that the problem of finding E[T n,n can

be reduced to solving a system of (n-l) x (n-l) linear equations.

First, for n > 4 (the n=2 and n=3 cases follow from equations (2.15)

and (2.31) above), the state equations are








t
p0(t + dt) = p0(O)(1 n'dt) + pl(t,z)(1 (n-l).,dt)
0
F(z + drt) F(z)
dz + o(dt) (2.32a)
1 F(z)

1 F(z + dr)
pl(t + dL, z + dt) = p (t,z)(1 (n-1),.dt) 1 F(z) + o(dt)

(2.32b)

1 F(z + dr)
pk(t + dt,z + dt) = pk(t,z)(1 (n-k).dt) 1 E(z)


1 F(z + dt)
+ p, (t.z)(n k + 1)'dt F(z ) + o(dt), 2 < k < n-i
r- F(z)
(2.32c)

1 F(z + dt)
p (t + dc.z + dt) = p (t,z) + p (t,z).'.dt 1 F(z + d + c(dc)
n n n-1 1 F(z)
(2.32d)
t
p (t + dt,0)dt = p0(t)n.dt + p2(t,z)(1 (n-2),.dt)
0
F(z + dr) F(z)
d-: + o(dt) (2.32e)
1 F(z)
t
F(z + dL) F(z)
?,(t + dtO,)dt = (tz)(1 (n-k-l)..dt) F(z dz +
j< k+l 1 F(z)
0
+ o(dt), 2 < k < n-2 (2.32f)


p.(- + d:,0)d- = o(dt), j=n-l,n (2.32g)


0(0) = 1 (2.32h)

Sn-1 b
Proceeding si-.ilarly s above, define g (t,z) and {q (t,z)} = b

the relaticrships


(t,z) = g (t,z)e-(n-l) z( F(z)) (2.33)


(2.34)


p (t,z) = qk(tz)(1 F(z)), 2 < k < n-I










Now, using again the fact that for g(') continuous and F(') absolutely


continuous


h f (,F(:x + h) F(x) d:-
lim g(x) h
h- 0
00


g(x)dF(::)


equations (2.32a) (2.32h) nay be re:riccen with the aid of (2.33)

and (2.34) as


d p
dr


= -nAp(t) +


-(n-l) .z
e


g1(c,z)dF(z),p0(0) = 1


*l 1
.C JZ


1 2q -



'" -----


(2.35b)


(n-2)\q (c,z)
2


+ (n-l).Ie (- 1 g1 (t,z)


+ (n k + l)'q (t,z)
k-l


(2.35c)


3 < k < n-i


(2.35d)


'n Pn
+ --


".q (c,z)(l F(z))
L


pl(t,0) =



S(t,0) =
l:


n' p (t) + I


q,(c,z)dF(z)


q 1 (t,z)dF(z), 2 kI < n-2
k+1


(2.35e)



(2.35f)


(2.35g)


p (:,F) = 0, j = n-l,n


Equations (2.35a) (2.257) will now be solved and the results

u'sendto find 1n(s). irst, let the solution of
nn


SqI 'qk
-- + - + cq (t,z) = f(t,z), 2 < k : n-i
"- Z .' .i


(2.35a)









be of the form


qk(t,z) = e- gk(t z) + q (t,z)


where the gk(t z) are functions whose existence and uniqueness

follow from the existence and uniqueness of the (p (t,z)}nk
k k=2
and equation (2.34), and qP(t,z) is the appropriate particular solution.

Theorem 2.1. The solution of (2.35c) and (2.35d) in terms of the
n-1
{gk(t z). is
Sk=2 s


k-1 n-k+i .
z_ (n-k+i)'z
q (tz) = (- ) g (t z) (2.36)
i=0O i

for 2 k < n-1, n > 4.

Proof. The proof consists of substituting (2.36) into (2.35c) and

(2.35d) directly. I

Now, from (2.35e) (2.35g), (2.33) and (2.34), it follows

that
t
g1(t) = pl(t,O) = nip0(t) + [ q2(t,z)dF(z) (2.37)

t
qk-l(t,) = pk-l(t,O) = qk(t,z)dF(z), 3 k n-i (2.38)
0

qnl(t,O) = Pnl(t,0) = 0 (2.39)


Substituting (2.36) into (2.37), (2.38) and (2.39), and taking

Laplace transforms of the resultant equations and (2.35a) yields a

system of n-1 equations in terms of the n-1 unknowns, V\(s), Gl(s),

C2(s),. ..,G (s) n > where










G.(s) = e g.(c)dt, 1 < j < n-2
0

Remembering that '..(s) = exp{- (s + j*)t)dF(t), the set of n-1
b
equations in n-1 unkno-wns can be written in matrix notation as


D (s) g (s) = e, (2.40)
n n.


.,here

gns) = ( s), G1s) G2s),...,Gn-_(s)


e' = (1,0,0,.. .,0)
-i


and, dropping the s's from the i;.'s, reference the matrix on page 24.

Thus, for every fixed n e4, equation (2.40) may be solved

for V (s), G1(s),. ..,G ,2(s) in terms of the .s), < j n- .
n-1
Using (2.36) and the definitions of {Vk )k=l

G (s)
1
G Is S) (1 'I (s))
1 =s + (n-1), n-l


k-1 n-k+i Gki(s)(l (s))
k- k-i n-k+i
V (s) = (-1) < k < n-2
k 7 i s + (n-k+i) -- --
i=0 inki G s)(

n-1 1 i (s) 1 '.(s)
1 11
(si) i(-1) G .(s)
n-L s + s + in



loe', by, the defi.niLion of -1 (s) and the above
n,n
n-2
.. (s) = V (s) + -V (s) + V (s) + V (s)
n,n 0 1 k2k n-I


G1 (s)(1 in (s))
= 0 ,) + n-
0' 3 + (n-1)A






24









I
















o 0
1-1
C--6




























I -
C-4
* * I I -
















C44
r-
























CI
-3




















C4







C r_
I c I
a r.












S- -








a I I -3
- '- I ."4 ---- I
SC C I I



S- c
.


v- I










I I<








c I '.


















a
Cr









n-2
+ =
k=2


k- 1 n-k+i

i=O0


Cki (s)(1 nk+i (s))
s + (n-k+i)'


+


n-1 1 ,(s) 1 is)
+ i(-i) Gn i s)
i=n-1 s + s + i,
i=2


or, after simplification

n-2
U (s) = Vo(s) + (-1)n-j G.(s) *
n,n j=
j=1


(n-j)(l + (s))
s + '.


1 (s) ]
n-j
s + (n-j) '.
Ij


(2.41)


where, for each value of n V (s, G (),...,G2(s) are computed

from (2.40). Hence, from (2.40) and (2.41) one may compute


hn, (s) = 1 sW (s)
n,n n,n


E[T n ] = (0)
n,n n,n


for the n-out-of-n system.

Taking n = 4, for example, (2.40) becomes


s + 4


"-.


-'3(s)


-( 13~ (s) )


'2 (s)


0 3(l-., (s)+ (s))


Vo()

Gl(S)



G2(s
Gs)]


(2.42)


Solving [2.42) for V (s), C1(s), G2(s) and evaluating them at s = 0

yic lds










(1 + )(1 2,1 + 22) 3;2(1 + )
V\(0) = ( + (1 + + (2.43)
0(++ 4) 1 + 32(1 21 1)) +2


1 2?, + 2,t
G (0) = ( 21 2)- 3(1 (2.44)
1 + 2h )(l +t ?.t


3(1 + ;3 (2.5
C2(O) = + 2. 3)(1 2,1 22) 3.2(1 1 + ) 2
= '. (1 3 1 22 1 '3

where :. = 0) = (i), i > 1. Evaluating ,4 (s) from (2.41)
1 1 4,4
at s = 0 and substituting (2.43) (2.45), by algebra



E[T ,] = W 4,(0)

25 321 + 1 + 63 + 19.1 3 82.: 3
-, 1 2 (>2.46)
12.11 2 2 + 2 4::3 + :233 + 34 } 6)
1 3 1'3 3 1'

Equation (2.46) agrees with equation (5.19c) in Downton [27].


The Case of ) F (*) Distinct, 1 < m -< n


Preliminaries

Then the restriction imposed by equation (2.1) is relaxed and

'5ni F (') are different for each machine, let T be the time to
L: k,n
sysLem failure. It will be shown that the problem of finding

-sT -*
E[e k,n] h (,ns)


and the MTSF, denored now by E[Tk,n], remains solvable for fixed values

of k, k n. Tne supplementary variable technique is still applicable

but a redefinition of states is necessary.










In order to avoid new, or at best cumbersome, notation, the

subscript i and the functions p (tc), p.(t,z), g.(t,z), V (s), G (s),

'.(s) and ;. will continue to be used, even though they may carry
1 1
slightly different meanings than in previous sections. The reader is

cautioned to make note of this whenever these quantities are defined

be Low.


The 2-Out-of-n System

For this more general 2-out-of-n system, let X(t) be the state

of the system at any moment t; e.g.,

(i) X(t) = (i,z), 1 i < n, z 0, means that at moment t,

machine number i is down, and an amount of time z, 0 < z < t, has

already been spent on the repair of machine i.

(ii) :(t) = (0,0) means, as before, no machines are down.

Letting


PO(t) = P[X(t) (0,0)]


p.(:,z)dz = P[:(t) (i,y)], 1 i < n, z < y c z + dz

n n
(i) = .., 1 < i < n
S 'l- J' --
j=1 j=1
j/i

the state transition equations can be written as follows


p (t + dt) = p0(t)(1 adt) +

n -t F (z -' dt) F.(z)
p+ i(t,z)(l t) z + o(dt) (2.7La)
I F (z)
i=0 i









S1 F.(z + dt)
i(t -r dt,z + dt) = pi(t,z)(i dt) -F + o(d
1 F.()
1


Pi(t + dt,O)dt = p 0(t)..dt + o(dc),


1 < i < n


PO(O) = 1


Defining g. (c,z) i= by the relationships
S i =1


Pi(i)
p.(t,z) = e


gi(r,z)(l F.(z)), 1 < i < n


equations (2.47a) (2.47d) car be rew'.ritten v'ith the aid of relation

(2.7) as


dpO
dc


- -O0(t) +


n

0


3g 5g
1 i
F--r i < u< i n


g.(t,0) = P(tO) = iP, (t))


Fcr 1 i < n, put


VO s) 1
o
0






C.(s) =

CO

' (s) = |
0


_-(i)
gi (t,z)dF i(),p (0) = I


1 < i < n


* o( )dC


-st
e ( p.(t,z)dz)dt
0


e g-t(i)(t)



e tdF.(t)
I


(2.47b)


(2.47c)


(2.47d)


(2.48)


(2.49a)




(2.49b)


(2.49c)











,.(s) = ;..(3 + )
I i


U, (s) =
,n


n
V V (s)
i =O


Equacions (2..49a) (2.49c) can be used to find W,. (s), and hence
(On) a allows. From (2.9b)
E[T, j = W, (0), as follows. From (2.49b)
2,n 2 ,n


g (t,z) gi(t z), 1 : i n


(2.50)


Combining (2.50) which (2.49c) and taking Laplace transforms yields


G.(s) = ".V (s), 1 : i 1 n
1 i -


(2.51)


Fr.--i (2.50), (2.51) and (2.49a), it follows that


'v(s) = (s + Ca -
tj


n
n -1

l 1
i=1


Using the definitions of V.(s), (2.48), (2.51) and (2.52)
1


V(s) = /
1 J
0


st pi(c,z)dz)dt
0


.(i (1 i(s))
% -1

A(1 .i(s)) Litn
(i is
(i) + C -
(: + )({s + a I .' (s)}
i=L


y (2.:22) a:nd (2.53)

.1)
j s) V (s. )
i-AU


= 4- c -


n (s) I +
i- -1
S'. .(s)) 1 +
i=-1


n .(1 .(s))
- 1 1
" (i)
i-l s + '


(2.52)


(2.53)


(2.5.4)









and hence the transform of the density of T2,, h (s), and
1,n' 2,n
E[T2,n] can be computed from (2.54) as


h ,n(s) = 1 s 2,n(s) (2.55)

and

E[Tin] = U (0)

n n ). (1 (i) .
= {a Y ).1 1 + (2.56)
i-l )ii i-1 .(i)


(i)
where q. = (. 1 ),) i 1. As a check, when = ), and F.(') = F(-),
i 1 1 1
Yi(s) = c,(s), A(i) = (n-l)), C (s) = .(s + (n-1),), a = n) and (2.54)

becomes

-1 + 1 + (n-i))]
W2 (s) = {s + n)[1 o(s + (n-1).)]} 1 + n [1 + (n-1)
2,n s + (n-l)?,


1 nl[ ..(s + (n-1):)]
s + n)41 ((s + (n-l)))] (s + (n-l)))(s + n\[l .(s+(n-l)..)]}

= W2,n(s)


as in equation (2.14) above. For n=2, equations (2.55) and (2.56) agree

with equations (8) and (9) in Osaki [34]. Using the principle of

regeneration, an alternative derivation of (2.55) is treated in the

Appendix.


The 3-Out-of-n System

For the 3-out-of-n system, finding the expectation of T3n

again necessitates an expanded state space. As before, let X(t) be

the state of the system at any moment t; e.g.,










(i) X(t) = (i.z) L < i n, z > 0, means chat at moment

t, machine number i is down and an amount of time z, 0 : z t, has

already been spent on the repair of machine i.

(ii) :(c) = (i,j;z), 1 i, j n, i / j, 0 : z < t, means

thet at moment t, machine number i and machine number j are down, and

an amount cf cime z, 0 z : c, has already been spent on the repair

of unit i.

(iii) Again, X(t) = (0,0) means that at moment t, no machines

are down.

For 1 K i, j : n, i / j, let


P (r) = P[X(t) = (0,0)]


p.( ,z)dz = P[X(c) = (i,y)], z z + dz


pij ( ,z)dz = P[X(c) = (i,j;y)], z : z + dz

n n (ij) n
j L .(ij)
= .1 "A = i ", -A A
j=i j-1 k=l
jii i j/k


With S = S(i,j) = {i,j: i,j = 1,2,...,n; i $ j1, che scace transition

equatic.is are


pt- dr) = 0(t)(l acdt) +

n .. F.(z + dc) F (z)
S I .(t.z)(I 1 dz + o(dL) (2.57a)
i1 1 F.(z)

1 F. (z t)
(L.z dt.z -d ) = p.(,z)(i id ) + C- o(dc), 1 i : n
S1 F (z)

(2.57b)









( ) 1 F (z + dt)
p. (t + dt,z + dt) = p (t,z)(1 (iJ)d) 1 +
ij ij 1 F.(z)
1


1 F.(z + dt)
+ p.(t,z). .dt +
j 1 F.(z)
1

Pi.(C + dt,0)dt = p0(t) .dt +
1 0-


p (t,z)(l kidt)
Pji


o(dt), i,j E S


F.(z + dt) F.(z)
S F.(z)
1 F.(z)


n
+
j
J 0
Jr-


+ o(dt), I < i < n


pj. ( + dt,O)dt = o(dt), i,j C S


PO(O) = 1


Defining the functions {g(t,z) i=1 and qij(t,z), i,j E 5, by the

relationships


_(i)
--I


Pi(t,z) = e


g(t,z)(l F.(z)), 1 1 i < n


uio (t,z) = q (t,.)( F e (z)), itj ( S


equations (2.57a) (2.57f) may be rewritten using (2.7) as


(2.53)



(2.59)


dr
(- -cp(t) +


n

i


e (i)
e gi(t,z)dFi(z),po(0) = 1


1i i
+ --- i < n
Cj E z.


Sj (t. ) + e


. (i)


" i(tz), i,j i- S


(2.57c)


(2.57d)


(2.57e)


(2.57f)


(2.60a)


(2.60b)


(2.60c)









. (t,O) = Pi(t,O)


ni
=.p0(t) +


qij(t,O) Pij(t,O) = 0, i,j c S


Now, for i,j L 5, put


'A,! (s L
) =


qji(t,z)dF.(z), 1 i < n


pj (t)dt

t


V.(s) = e ( p.(t,z)dz)dt
0 0


v ..(s) =
13



;.(s) = I e
0


st( pi.(t,z)dz)dt
'-.


-st
Sg.i(t)dt
1


(s) = e dF.(t)



i.1(s) = r.(s + (i))


i.(s) = .i(s + (ij)
1i] +


n
r Si
1: :s) = V (.) + 7 V (s)
i='J i,j C S


(2.60a) (2.60e) .'ill no'.' be used to find (3,n ) and
3, n


h., (a) = [c 3,n], as follc's. From (2.60b)
,,n


i(tt) = gi(t z), 1 i : n
.1 i~


(2.60d)



(2.60e)


Equat ions


(2.61)










Using (2.61) and (2.60e) the solution of (2.60c) becomes


S(ij) (i)
qij(t,z) = e 'gi(t z) e Zgi(t z), i,j E S (2.62)


Substituting (2.61) and (2.62) into (2.60d) and taking Laplace

transforms of the resultant equations, obtain

n
C (s) = \ (s) + ( ..(s) (s)) G.(s), 1 i n (2.63)
2 I 0 i 3
j=l1 -
ji

Transforming both sides of (2.60a) with the aid of (2.61) yields

n
(s + O)V (s) = 1 + 7 .(s) G.(s) (2.64)
i=l1

Equations (2.63) and (2.64) represent n+1 relationships between the

n+l functions Vo(s), Gl(s), G2(s),...,G (s).

From (2.58) and the definitions of the V.(s)
1

G.(s)
V (s) = (1 .(s)), 1 i n (2.65)
i (i)
s + \

and from (2.59), (2.62) and the definitions of the V..(s)
13

GC.(s) G (s)
( = (1 ..(s)) (1 .(s)), i'j c S
i + (ij) 1 +(i) 1
S + Cs +

(2.66)

Hence, using (2.65) and (2.66), for n 3

n
w (s) = V (s) + (s) + T V.(s)
,zn i i J i
i=l i,j CS

G (s)(1 ..(s)) n G.(s)
o(S) + (n-2)
iOs)+ (i (n-2) ( (1-..(s))
i,j S i=l s+\.
(2.67)










h (s) = i sW, (s) (2.68)
3,n ,,n
and

E[T n] = 3,n(0) (2.69)
3,n 3,n


where V (s) and {G.(s)) must be found from equations (2.63) and (2.64).

As a check on (2.67), let i = \ 3nd F.() = F(.), 1 < i < n.

Then


.(s) = .:.s), .(s) = ..(s + (n-1)k), .. (s) = .3 + (n -2))),
S1 j


= n., = (n-i)\, \ ) = (n-2)\, i,j cS


For conveniences, let


i n-(s) = ( + (n-l)))


En-2 (s) = *(s + (n-2)))


'F (s) = (s) (s)
n n- n.-1


In terms of the above notation, equations (2.63) and (2.64) can be

written in matrix notation as (equation, page 36). Using induction

and Cramer's rule on the above, for n 3

1 (n-1)y, (s)
v -- )"n (0 70)
0 (s + n,),l (n-1) n (s)) n -e 1, (S)
n n-i

nrd for 1 < i < n


,. ( \) - (2.71)
S ( s, + n).)(l (n-1), (s)) n\'i~n- s)
n n-i


Mfter substituting (2.70) and (2.71) into (2.67) and simplifying, it

.- :-en that










































-O Cr C









U, U ,





II C
C :


U, U,
C


Ul
I


.f: C<*












w, (s) = W1 (s)
,n ,n


as in equation (2.30). Thus, for the special case of i = and

F.(') = F(-), equation (2.69) agrees ..ith equation (5.19b) of Domncon
1
[27].

Again, however, since the state equations change for each

value of k, a general expression for


W k,n(s)
k,n


cannot be found with this technique. Nevertheless, for any fixed

value of k, 1 < k < n, an expression is obtainable for k, (s) and,

hence, for
-sT
E[e ,n] = 1 sWk(s)

and

E[Tk,n] k,n(O)



A Comparison Between the Semi-Markov Model and
the Supplementary Variable Technique


General


Since both Downton [27] and Branson and Shah [39] use semi-

Markov processes to model repairable systems with standbys, a comparison

is in order between the semi-Markov approach and the supplementary

variable technique used in this research.


The Approach of Branson and Shah


In the two-unit system considered by Branson and Shah, a unit











may operate on-line or off-line. The on-line unit, no matter which

unit it happens to be, always fails with rate and is repaired

according to distribution function F (*). The same is true of the

off-line unit with respect to '2 and F,2(). This is not quite the

same as the 2-out-of-2 s5steri, but it is similar.

By a judicious choice of states, Branson and Shah modeled their

two-unit system as a semi-:larkov process. Then, using a result

(Earlow aid Proschan [9], Theorem 2.5, p. 135) which does not depend

on the distribution of the time to system failure itself, the MirSF was

found, given that the process began in any of the possible up-states.


The Semi-Markcov Model af Downton


Dovnton used a semi-Markov process to model the k-out-of-n

system where .' and F(-) are, respectively, the failure rate and repair

distribution for each of the n units.

For n > k and k = 2,3, Downton derived


-sTk
E[e 1|0 units are down initially] (2.72)


but these results are special cases of equations (2.55) and (.'.63)

with = and F.(*) = F(-). In fairness to Do'.'ntn, however, although

(2.72) is often the case of interest, the semi-Mlarkov approach also

yields

-FT
E[e i units are initially dc':n, 0 i < k] (2.73)


hlicih cannot be obtained using the supplementary variable approach

above.










For each valuc of k, the approach used by Downton necessitated

a matrix in. arsion to find the transform of the appropriate first

passage time distribution. Since the macri:-: to be inverted was

different for each k, a general expression for E[Tk ] was not obtained.

In the supplementary variable approach, it was the change in the state

transition equations for each k which prevented a generalization of

F. T ,nJ


Summarizing Remarks


Because the semi-larkov process is defined on a denumerable

number of states, transforms of the distribution of T, the time to

system failure, can be found when the process is in any of the up-

sr.nLes initially as indicated in (2.73). Although this is not the case

.wi.h' the supplementary variable approach, results implied by (2.72)

can be obtained for a more general class of k-out-of-n systems than

those treated by Downton.

Since Eranson and Shah's approach to finding the MTSF is

independent of the distribution of T itself, other moments of the Lime

co system. failure are not readily available. In this sense, the

technique employed by Eranson and Shah is less general than that of

Dcv'n:on or the author.











CHAPTER 3

SOME RELIABILITY CHARACTERISTICS OF THE 2-OUT-OF-n
SYSTEM AND THE T\VO-UIIT STANDBY PEDU::DANT SYSTEM


Introduction


Reliability attributes of the 2-out-of-n and two-unit standby

redundant systems are treated in this chapter. Besides the distri-

bution of the number of repairs completed prior to system failure,

transforms of distributions are derived for the time the repairman is

idle during T (the time to system failure) and the time spent on

repair during T. Assuming random failure and k-Erlang repair

capabilities, the distribution of T for each system is analyzed and

the generating function of the distribution of the number of renewals

occurring during T is obtained.


The 2-Out-of-n System


Assumptions and Definitions


The following will be assumed:

(1) The time to failure of each unit is independently and

exponentially distributed with failure rate \ 0.

(2) At t=0, all n units are operating properly.

(3) There is only one repairman (with unlimited service

capabilities) and the repair times are independent random variables

which are also independent of the failure times.

Define:

Z(t) the number of units down at moment t (Z(t) = 0, 1 or 2).

T the time to system failure for the 2-out-of-n system. As
2 ,n










before, the density of T, and its Laplace transform will
-,n
be denoted by hr, (t) and h (s), respectively.
,n 2,n

the time elapsed between the completion of the (i-l)-st

repair and the next breakdown, i 1 (the zeroth repair is

assumed to be completed at t=0). By assumptions (1) and

(2), the {X.}, i > 1, are independent and identically

distributed (i.i.d.) as a random variable X with density

function
-\0x
f (x) = X e x > 0


where X0 = nX, n=2,3,....


R. the time required to perform the i-th repair, i > 1. By

assumption (3), the {R.}, i > 1, are i.i.d. as a random

variable with distribution function F('). Let ((s) be

the Laplace-Stieltjes transform of F('); i.e.,


s) = e-'stdF(t)
0
Y -the time to the next failure when only n-1 units are

operative. By assumptions (1) and (3), the density

function of Y is

-i y
fy(y) = Xie y 0

where 1X = (n-l)X, n=2,3,....

N -the number of repairs completed prior to system failure.

R the total time spent on completed repairs during T2,n










R tha cota. time spent on all repairs during T2,n"


I the total time that the repairman is idle during 'T2,n


A Cor.ditional Transform Approach


Before discussing the distributions of N, P. and I, it will

be instructive to consider a typical realization of the process Z(c)

(Figure 3.1).

Note that


P[a repair interval contains no failures]

I 1
= P[R < Y] = J e dF(c) = .(:1)

0
Clearly, the distribution of N is geometric and


P[N = i 1 (1 0 1)), i=0,l,..


z(t)


(3.1)


*- 1 2- : v
"1 "2


4 -- --- y


Figure 3,1. A sa'ple- function of Z(t) in terms of the
rar.dc- variables Xi, R., Y and T2
1 1 2,n


~3










Given that N = i


R R + R? + ... + P i= ,2,...
R, =i
0, i=O


where the {R.), j -
J


1, are i.i.d. as a random variable R with


-' t
'1
e 1dF(t)
dP[R < t] d
1
and

I = X + X + ... + Xi+.


Hence, from (3.1), (3.2) and (3.3)


-sR
E[e L] =
i=0


-sR
iCN = i]P[n = i]


(s +

i=1

1 ((1.)


I
1 ;(s + \ )


1 )


{.,(,) (1 0 ( 0 ))
1 1


and by (3.4) and the


E[E-sI] =
E[ -


definition of the {X.}, j 1
3


E[e-SI I = i]P[W = i]


i+1


i=0


0
S + 0
'0


i (' 1) ii 0.'1 ))
1 '1


(1 (i))
00 1
s + .C(1 (i ))


As a check, ncte that (see Figure 3.1)


T n =Y + R I
2,n c


(3.2)


(3.3)


(3.4)


(3.5)


(3.6)


(3.7)











where
1-
S'e (1 F(,))
dPY ] = 1 (. dv, v > 0
1


Now, by (3.2), (3.3) and (3.4)

-s.R -sI
E[e ce ]


-sR -sI
= E[e e 2 I
i=0


S'1( 1
S-, ( )
i=0 1


= i].P[[I = i]


i+1


.I
0
2 0


(.* (. )) (1 -


o(1 .(. ))
0 1
s 4- (1 ,s+ F))


Putting s s.s = s in (3.8)


-s(R +) (1 ( ))
0 + ( -(s +


By definition of dP[Y < t] above


*
-sY
E[e =


t ( 1 .(5 + ))
t 1 '
dP[Y t] =
(s + "1)(1 .(> )
1.+


Cotbining (2.9) and (3.10) with (3.7)

-sT -s(R +1)
Els 'n] = E[e s ] E[e c


1 i[I ';(s + 1 )]
0 A 1 1
(3 + )(s + [I s + "1 )])


and equation (3.11) agrees with (2.14).


Clearl-, the iimoments


.! (3 ))
1



(3.8)


(3.9)


(3.10)


(3. 11)











E[{R J }J1k], j,k=0,l,2,...,


are available by differentiating (3.3). Also, since R, the total time

spent on all repairs before failure, satisfies


R = R +Y
c


by (3.5) and (3.10)

-sR 13. 12
Ele ] = +(3.12)
s + '



From (3.12) it is seen that R is distributed ex::ponentially with mean

i/' regardless of the repair distribution. For n=2, relations (3.6)

and (3.12) agree with equations (17) and (21) in Epstein [31].


An Analysis of the Distribution of T,
__ ____,n

In this section, the density of T h, (t), 'ill be
",n 2,n

invesLigated -.:hen repair times follow the k'-Erlang distribution with

parameter Iu 0; i.e., when
k
s) = e dFt) = k 1 (3.13)
) e-dFt) s +
0

First, applying the shifting property of Laplace transforms to equation

(i.11) an recalling that )' = (n-i)", i-0,1, obtain


h2,n (t) = exp{-(n l) t)g(t) (3.14)

-S.
"n C -st
g (s) = e g(r)dt
0
= h2 (s (n ).\)
z,n










n(n 1) 2[1 .(s)]
s(s + ; n,.S(s))

or, using (3.13)

g (s) = C(s)/D(s)

where

C(s) = n(n 1)\(s + ) k"- k

k .k
D(s) = (s + )(5 + u) n


and k 1, n .- 2; ',u > 0.

Note, however, that for 0 <- u < and k > l,.n 2



E[repair time] = k/u > ((n 1):)- = E[Y]


(3.15)


(3.16)


In words, the expected repair time is greater than the expected time

to the next failure, given a repair has just begun. Hence, for

S< ', the 2-out-of-n system will, "a priori" be unreliable and the

interesting case is when j \, > 0. Under this latter condition, the

following theorem will be proven:


Theorem 3.1. Let p,

and n 2. If u '

-n'p i are distinct.


Proof. Assume s = r

tL'e-l'


S> 0 and let k and n be integers with k 1

> 0, then the k+1 zeros of D(s) = (s + ,)(s + p)'




is a root of multiplicity mM 2 of D(s) = 0;


0j-1
ds-i D(s)
ds- S= C


S0, j=1,2,...,m; m > 2


Taking j=2,


D'(s)
s=r


= (r + ,)k(r + u)k-1 + (r + = 0










or

(r + u)k-(k(r + 1) + r + u) = 0


But r i -u since D(-u) # 0, and hence


k(r + A,) + r + u = 0


k) + u
S k+ 1


However,



D(s)

k' + u
s = k+


k+l k k, k+1 k+l
= -A) k+k k + n'u i(k + 1) k /(k+ + 1)k

< 0, when u : '. :; 0


Therefore, D(s) cannot have any root of multiplicity m :- 2 when

u 0. 11
k+l
Consequently, if {r.) j. are the k+1 zeros of D(s) by using

(3.14), (3.15), Heaviside's expansion and Theorem 3.1


h (t) = exp{-(n-l) t)
2,n


k+l
S e:.p(r.t)C(r.)/D'(r.)
j=1 j J J


(3.17)


where C(s) and D(s) are defined by (3.16).


The Number of Renewals During T,n
2,*n


Suppose that at time t=0, a renewal process begins

which generates secondary failures. These secondary events are not











considered serious enough to necessitate immediate repair (Gaver and

Luckew [38]) and it is assumed that total failure will not be caused

by them alone (e.g., an oil leak or a fault', valve). However, when

failure of the 2-out-of-n system does occur, all secondary failures

are also repaired before the system is restarted.

Let the independent times between secondary failures have the

distribution function B('). Then, conditional upon T2,n it follows

from renewal theory that the distribution of the number of secondary

failures during T say MI, is
2,n'


F[ = iT. = t] = B.(t) B (t), i > 0 (3.18)
,n i i+
where t

EPi (t) = B (t :.)dB(:.), i 0 (3.19)

0
and E (t) is the unit step function at the origin.

Now, if
-s0

B (s) = e dB(t)

0
and if it can be shown that


Re(r.) < (n -- 1)., 1 < j k+i


it follows chat
T*

Sexp{-[(n )) r.]t)3.(t)dt
J 1i
0
[B ((n I)\ r.)]
--- (3.20)


for 1 I j < k-:-!, i 0. Using ('.17), (3.18) and (3.20), one may

calculate











G (z) =
i=O


= !
i=0

k+1

j=l


where 0 < z : 1.


i
z P[. = i]


cX,

Z i (B.(t) B.i (t))h, (t)dt
t=O
C(r.)[1-B ((n-l)'. r.)]

D'(r.)((n-1) ,-r .) [-zB ((n-l) .-r.)]
J J J


Thus, in order for (3.21) to hold, there remains to show

}k+l
Theorem 3.2. If u 0, k : 1, n 2 and if {r.j jl are the k+1
J j=1
k k
zeros of D(s) = (s + I)(s + ) nu then


(3.22)


Re(r.) < (n 1)*, j=1,2,...,k+l
J


Proof. By Descartes'sLaw of Signs, D(s) has precisely one positi'.'e

(real) zero, say r. Note that D(0) 0 and D((n 1)) )- 0 and

hence, it must be that

0 < r < (n 1)'


and all real zeros of D(s)

Now, with i = -1,~ -

such that Re(z) = t '. (n -


satisfy (3.22).

let z = r + 6i be any comple:-: zero of D(s)

1) and let


Q = (z + ) (z -+ j)k


Since z is a zero of D(s), it follows that


k 1' k
I(z + )(z + j ) = |nu = n



kQ
Q = (o + + i)(a + (c + 4i) I = n'u


(3.21)










But since I l z = j. I |z2 and z k1| = I|z k for any two complex

numbers zi, z2, it follows chat


Q = | + + kil i + u + ki k

[( + + )+ t2 1?. [( + ") + k2 (3.23)


However, for .: > (n 1)I, ', L > 0, k > 1, n -. 2 and for all real

values of 6,


[( + 2+ 2E-] [(n)-) + -] 1 n\ > 0 (3.24)

and
S2 k/2 ? k/2
[(a + )- )+ '] k/] [((n 1)\ + u)~ + /] 2

)l, k
S((n 1)\ + )" v > 0 (3.25)


Ca~mbning (3.24) and (3.25) wich (3.23)


2 i /2 J / k/2 k
Q = [(a + \] + [(a + t)- + -] n.,


whenever a i (n 1)'.. Hence, when a = Re(z) : (n 1)', the complex

nuriber z = ui + Bi cannot be a zero of D(s) and the Theorem is proven. I

From (3.21), all moments of ', the number of secondary

failures during T1 are available by differentiating G (z). In
,n '
par cic'llar

k 1 C(r.)E ((n 1)', r.)
J[l 7--- --- ( --r---I- B-i
j 1
3 3 J











The Two-Unit Standby Redundant Systemr


Assumptions and Definitions


The following will be assumed:

(1) The time to failure of unit 1 is a random variable which

is independent of the time to failure of unit 2, and switchover time

is instantaneous.

(2) At t=O, unit 1 is operating and unit 2 is a "cold" back-up.

(3) There is only one repairman with unlimited service

capabilities. The repair time for unit 1 is a random variable which

is independent of the repair time for unit 2, and che repair times and

failure times are independent of each other.

Comment: Once again, to avoid cumbersome or new notation,

the random variables X., Ri, R.i, P., I, and !1 .ill continue to be

used, even though their meaning may be different than in previous

sections. The reader is cautioned to make note of this whenever these

quantities are defined below.

The following notation will be used:


{'} the Laplace transform operator; i.e., Jff(t) = e f(t)dt.


Z(t) che number of units down at moment t (Z(t) = 0, 1 or 2).


T the time to system failure for the two-unit standby

redundant system. The density of I and its Laplace

transform will be denoted by h ,(:) and h (s), respectively.


1- subscript denoting unit i, j=1,2.


X. ime to failure of unit i.
L











g.(') probability density function of X..


G.() distribution function of X..
1 1

G() = G(.
1 1

R. time needed to repair unit i.
1

f.() probability density function of R..
1 1

F. () distribution function of R..
i 1

F.() = 1 F (*)
1 1

N the number of repairs completed prior to system failure.


R the total time spent on repair during T.


R the total time spent repairing unit i during 1.


I the total time that the repairman is idle during T.


A Conditional Transform Approach


Before deriving the distribution of N, consider some typical

realizations of the process Zft) (Figures 3.2 and 3.3).

Now, let
oc, cc,

p = P[R X = j g2(t)Fl(t)dt = f(t)G,(t)dt

0 0


q = P[R < X1] = f g1(t)F2(t)dt = f2(t)Gft)dt

0 0
and note that (see Figures 3.2 and 3.3)


(n+1),/ (n-1)/2
P[LI = n, 1N odd] = p q( /2 (l-q), n=1,3,5,... (3.26)


P[N = n, N even] = (pq) ( p), n=0,2,4,... (3.27)













Z(t)



--"------- "2 ------ --1 --v..2----
2!


1 2 1 "2 1


1

SR- PR2 2 I- R2-







Figure 3..2. A sample function of ?(c) in terms of the random
v'riables X., R. and T, when an odd number of repairs (3)
1 1
are completed.







Z(t)



<-- ---------------- T ------ ---------- --
-T



2

"'- 2 "' 1 -2


1 --- ---- -----
R 2 R1






Figure 3.3. A sample function of Z(t) in terms of the random
.acriable v., R. and T, :..hen an even number of repairs (2)

are completed.












1n] (-_ l) + p(n+l)/2 (n-l)/2
?[ = n] + 1p q (1 q) +


+ {(-1) + 1)(pq)n/21 p),
+ )(q I )


n=0,1,2 ....


As a check, using (3.28)


GN(z) =
n 0


n P[N = n]


S1 p + p(l q)z
2pq
1- z pq


0 < z < I


and from (3.29)

GN(1) = 1


To find the transform of the distribution of T, a conditional

transform approach is used:


-sT
E[e j =


L
n odd


+
n even


E[e sT N = n, N odd] FP[N =



e-sT n, N een P
e IdN n, N even] P[N


n, N odd] +



= n, N even]


(3.330)


For N = n ard N odd


(r.-l)/2
I = X +
1 j=
j=1


xj
1 ,j


where, the {X1 .j j 1, are i.i.d.
l*J '


(n+l)/2
L "2,j
j=i


as a random variable N


gl(t)F (t)
dP[X < L d
1 d


(3.28)


(3.29)


+ X


wi th










and { ), j 1, are i.i.d. as a random variable X, with
2 ,j


JF[t< t t] =



dP[:' t] =
1


g2(t'FlF1(t)
gl(t)F (t)



1 q


Also, for I; = n and I; even


n/2
r = v +
1j-1


"1,j


n,2
+ j
j=1


+
"',j


where
g,(t)F1 (t)
dP[rx t] = dt
S1 p

Defining
g1(s) = Z{g (t)}


and combining (3.26) and (3.27) with (3.30) and the above discussion,

it follows that


h (s) = E[e s'


= g1(s)


t () (n+l)/2

n odd P


1(n-lY2
{gl(t)F2(t))
q


{gl(t)F2(t)}
S_- q


+ g (s)
n even


S{g2 (t)F (t)}
21-


I (n+l)/2 q(n-l)/2 (


Z{g2(t)Fl(t)}
p


In/2


- q)


(t)F,(t) n/2
t q


(pq)n ( p)


and









g,(t)F (c)} + {g (t)F,.(t)} Z {g,(t)F (t)}
= g (s) -----
1 1 ig2(t)F (t)} Qi{g (tt)F (t)


(3.31)

Equation (3.31) agrees with equation (90) in Srinivasan [25] and

equation (14) in Osaki [34].

Using a similar approach, transforms of the distributions of

R, R1i R, and I can be found. In particular, for N = n and N odd

(n+l)/2 (n-l)/2
R = R + 7 R2 + X1
j=l 1 j=l 2'j 1
and for I = n and N even

n/2 n/2
R = R' + P R + "X
j=1 j=

where, the {(R1}, j 1, are i.i.d. as a random variable R1 with


fl(t)G,(t)
dP[R :J = dt
1 p

and R, .1, j 1, are i.i.d. as a random variable R2 with


f (t)G1(t)
dP[R, t] = dt
q

and dP[X<. t], j = 1,2, are as defined previously. Hence, from

(3.26), (3.27) and the above discussion


-sR -sR
Ee-R ] = [e I N n, N cdd]rP[: = n, N odd] +
n odd


+ E[e I; = n, N e-.en]P[N n, N even]
11 2ven










C.{gl(t)F,(t)) i{ (()(t)G (t)} + ::g (t)Fl(t)

1 .,.{fi(t)G (t) f.'f2(t -- l(t)}


Similarly,


-R
E[e ] =


n odd


(n+l)/2 (n-1)/2
p q (1 q) +


f (t) (t) (n+l)
P


( n/2 -
(pq) (1 p)


-sP.R
El[e ']


n even


:{f, (t)G1(t)
q


C.f~(t)G (t)}
+ q
n odd


(pq)n 2(1 p) +


*(n- )/2 {g (t) (t)

1-q


S(n+l)/2 (n-l)/2
p q (1 q)


p fig(t(t)(t)} + (1 p)

1 p t. f2(t)G (t))
&. 1


A check reveals that


-sP
= E[e ]


-sR,
= E[e s]
s=0


-sP. =
E[e- ] i
s=0o


U'ing the above approach, it is easy to calculate


(3.32)


+
n eve
n even


Sfl(t)G,(t)}

p


n/
n/2 : {g2(t) 1(t)}
i p


and


(1 q) f (t)G,(t)) + ;.{g2(t)Fl(t))

1 q;.{fl(t)G,(t)}









-sRI -s2R2
E[e e ] =



+ y
n even


L
n odd


-SR1 -s2R2
E[e e IN = n, II odd]P[l; = n, :; odd] +


-sR -s2R2
Ele e IN = n, N even]P[I] = n, N even]


and show that


-s R1 s22
E[e e ]

1 2~


-s(RI+R2 -R
= E[e ] = E[e- ]


as expected, since R = R1 + R2'

The same type of conditional transform approach is used to

find the transform of the distribution of I, the idle time of the

repairman during T.

Referring to Figures 3.2 and 3.3, it is clear that some

conditional probability density functions (CPDF's) must be calculated:

namely, the CPDF of X2 RI given that X2 > RI, say g ..2RI )
2 2 1 2 -R X2 -R1

and the CPDF of X R2 given that X1 > R2, say g. -R >R2(.).
1 2 1 2' .1 1-R I 2

Using the transformation of variables technique and the definitions

of p and q, it can be shown that


sI' R (v) = F

"2- 1 "2 R1
x=y




v -iR2I- (Y2 ) =2
x=v


fl(x y)g2(x)dx/p, 0 < y <




f2(x y)gl(x)dx/q, 0 < y c


g21(s) = e-sy
'=0 x=v


f (x y)g2(.:)dx)dy


Letting











g12(s) =
v=O
and remembering thac


e-Sy
e (j


f2(x V)g ('.:)dx)dy


g (s) = {gl(y))


it follo,.'s that



E[ 1 = g (s)


(n+)/2 (n-l)/2
21(S) g12( )
odd P q ~


(n+l)/2 (n-l)/2
p q (1 q) +


+
n ev en


g (s)
p


n/2


q( )
q


I
*n/-


(pq)/2(1 p)


Gl(s)[(l q)g21(s) + 1 p]

1 gl (s)g, (s)
la 1


Since g1(0) = I, g21(0) = p and g12(0) = q, from (3.33)


-SI
E[e- ]
s=0


= 1


as ic should be.

thoughh E[e ](R+ is not readily available, using (3.31),

(3.32) a.nd (3.33), by algebra


d -sr
E[T] = e [e- ]
s=0

- {E[e-S ] + -.[e-S ]}

s=0

E[R] + E[I]


?s expected, since T = R + I.


(3.33)










-sR
-sT -sR -SR
Although inverse transforms of E[e- S, E[e ], E[e ],

-2 -sI
E[e ] and E[e ] are difficult to calculate for particular failure

and repair distributions, moments of T, R, RI, R and I are available

by differentiation and possible' with the aid of numerical techniques

(reference Chapter 4).


An Analysis of the Distribution of T


In this section, the density of T, hT(t), will be investigated

w, hen

g (t) = 'e A -. 0, t 0

and
k-l
f.(t) = p.( t)k le:.p(- t)/(k-l)!, 0, k 1, t 0


fo. i = 1,2. Defining



..(s) = e f.(t)dt, i=l,2

0
iL follows that

Sk
.(s) = k -_ 1 (3.34)
s +


Fr2m (3.31) and the above


*, [1 ,'As + a)]
h (s) = (3.35)
(S + .(s + ),1(S + )


A- before, by the shifting property of Laplace transforms and equation

(3. 35)

h () T e "tk() (3.36)


where












k (;) = e k(t)dt
0


hT( .) =
s s,:S (s)


or, using (3.3.'1)

k (s) = c(s)/d(s) (3.37)

where
1 k k
c(s) = {(s + uj U

,k k
d(s) = s(s + ) "- (3.3S)


The following theorem will be useful:


Theorem 3.3. If t 0C, the zeros of

k k
d(s) = s(s + u) ,


are distinct for k=l,2,....

Proof. The proof of Theorem 3.3 follo..'s the same line of reasoning

as Theorem 3.1.11
k+l
Thus, if {s.}j=1 are the k+l zeros of d(s), then by (3.36),
j j=l
(3.37), H3aviide's expansion and Theorem 3.3

Sk+l s .t
h (t) = e a c(s.)/d'(s.) (3.39)
j-= 1

where _(s) .:nd ds) are defined b:, (3.38).


The :uiiner of Peie'w l s During T


A.:in, properties of rene'..al theor.' may-', be usad to find the












generating function of the distribution of M, the number of renewals

(or secondary failures) during T.

As before, let the independent times between renewals have the

distribution function B(-). Then

I
P[M = i IT = t] = B.(t) B i (t) i 0 (3.40)


:,'here the B.(-) are defined as in equation (3.19) above.
1

It is asserted that:

k+l 1
Theorem 3.4. If 0, \ > 0, k 1 and if {s} j1are the k+1 zeros of
_- j j=l
k k
d(s) = s(s + u) .i then


Re(s.) < j, j=1,2,... ,k+l
J

?roof. The proof is almost identical to that of Theorem 3.2 and will

not be repeatedly!

Hence, with


B (s) = e dB(t)

0
and using (3.39), (3.40) and Theorem 3.4, one may calculate
cc
7 1 i
GC (z) = i z P[M = i] = 3 2 (Bi(c) Bi+ (t))lh (c)dt
i=0 i=0
t=O
k+l c(s.) [l B (; s.)]
I -I
j=l d'(s )(;' s [l zB (, s.)]
J J J




i.
\.-'iere 0 < z I.



See Sivazlian [40] for an analysis of a special case of the
polyncrmial d(s) which arises in inventory theory.










CHAPTER 4

APPLICATIONS


Introduction


In this chapter, applications of the theory in Chapters 2 and

3 are treated. In particular, numerical results are obtained for the

2-out-of-3 and 3-out-of-4 systems with respect to an airport limousine

problem and some numerical techniques applicable to Chapters 2 and 3

are mentioned.


An Example


A city po''er plant system or an airport limousine (also

referred to as car) service are two practical situations which can be

modeled as a k-out-of-n system. For instance, if a city has say three

power plants of varying sizes, it is likely that if any two of them

are in a failed state, the third power station will become overloaded

and the entire system will fail; i.e., a 2-out-of-3 system, assuming

one repair crew. A more realistic and intuitive example is that

of an airport limousine service and, hence, discussions will be

confined to the latter.

Consider the manager of an airport limousine service composed

of three limousines. Assume that so long as at least two of the

limousines are operative, airport customers will be inclined to use

the limousine service. However, if at any time only one limousine is



According to Professor O. I. Elgerd, Department of Electrical
Engineering, University of Florida, the k-out-of-n system is too
simplified a model for power plant systems in general; however, the
above discussion does provide the reader with a feeling for what
k-out-of-n systems are.










operative, customers will lose patience and choose other means of

transportation (e.g., busses or taxis). Thus, the manager would be

interested in knowing the mean time to failure (one limousine working)

for his 2-out-of-3 system; i.e., E[T2,3] in the notation of Chapter 2.

r. addition, the manager mieht also be interested in how much an

extra limousine would be worth to him. In other words, how does

E[Ti ] compare with E[T. ,].

Assuming limousine i has an expected time to failure of

I/., weeks (I week = 7 days), and the mean time to repair limousine i

is weeks, 1 i < 4, calculations of



S,(0) = E[T ] and ,(0) = E[T, ]
-,-, -3 ,4


were made using equation (2.56) (for I.', (0)) and equations (2.63),

( '.4), (2.67) and (2.69) (for U 3,4(0)). The results for several sets

of values of i'..) and ti.) are shown in Tables 1 and 2 for three
1 1
situations, when all repair times are (1) deterministic, (2) 2-Erlang

and (3) exponential.

As indicated in Table 1, for the 2-out-of-3 system exponential

repairr appears to be better than deterministic repair capabilities

when the mean repair times are the sa.me. This peculiarity of the 2-out-

of-, system was also mentioned by Downcon [27]. Ho'.ever, the standard

de'vi'ion of T. is slightly larger for exponencial repair than it

is for either determir.istic or 2-Erlang repair (see Table 3).

A comparison of Tables 1 and 2 w.:oLld aid the limousine manaEer

:r. deciding the value of an extra limousine. Comparing cases 1 and 5

for example, the manager presently (Table 1) has three :ars with mean








TABLE 1


LE. ., ] FOR DEIERMINISTIC, 2-EPL%':G .'dD EXPO:E::,TIAL rEP'AI?

Failure RaLes and ------
Case ;leai 1 pair rimes Deterministic 2-Erlarg E:,pon.:ntial
,No. oor Uni i, i=1,2,3 I Peppir __Pe.ir Repair
1 = = l/" I .. 19.80 20.19
1. .. JN .





S= I, 1/7, L u = 2/7
1 2

S- = = 1/ 80.00 .0. 67 C.1.33
1 = = Vu 1/7
_1 _

. = 1/4 12.01 12.3" 12.66

= = u. = 2/7


U L E 2
2-EF:.LA;;G -;D E'PO:E':T'[ITAL RE'A:1F


E[(T FOR DETRPII.ISTIC,


---- ----i----- --1- -- - -


Failure :ates and
,lean Repair Times
: U: Unit i. i=1,2.3,4

* = 1/3, '= -" ,=1/"

, = 1/7, -~=u ,=2/7


S1 = 1/8, 23=X 4=1/4

1l = 1/7, 2=13=2/7,


4 = 1


DetermIinistic
Repair


-, -U,
-t 118..~


32.88


EiT. ,J
1t2- I
2-Erl in
P epai


87.7 7


28.41


E:.-p on n t i a.
F P.e air

72.61


26.32


-7 4' = ==1/8, = 1/2 49.53 40.18 35.68
1 '' = 1/3=l



8 = = = = 1/4 62.33 47.58 40.33
1 2 3 4
u1 -= u0 = v. = u, = o2/7
L i. > a4


Cas.2
No.

5




6


87.77i


I I


--











TABLE 3

THE STANDARD DEVIATION OF 7, FOR THE FOUR CASES
-,3
CONSIDERED IN TABLE 1


Standard Deviation of T2,3
Deterministic 2-Erlang Exponential
Case No. Repair Repair Repair

1 19.28 19.64 19.98
2 36.35 36.81 37.27

3 79.93 80.57 81.20
4 11.89 12.17 12.45







times to failure of S weeks, 4 weeks and 4 weeks and mean repair times

are 1 day, 2 days and 2 days, respectively. For discussion purposes,

a ".ew car" has 8 weeks and 1 day as its mean failure and repair time,

and an "old car" has 4 weeks and 2 days as its mean failure and repair

time. Thus, if the manager adds an old car to his original fleet of

one new and tvo old cars, he gains with respect to the MTSF almost 100

weeks under deterministic repair, about 67 weeks under 2-Erlang repair

and 50 weeks under exponential repair. If, however, after purchasing

an extra old car, it suddenly starts taking a full week (on the average)

to repair it (e.g., parts must be ordered), from Table 2, case 6, the

MTSF shows a marked decrease, but is still better than the MTSF for

his original fleet.

Even if the manager originally has a fleet of three old cars

(Table 1, i:ase 4), the purchase cf an extra old car (Table 2, case 8)











adds '0 weeks under deterministic repair, 35 weeks under 2-Erlang

repair and 2.S weeks under exponential repair, to his initial MTSF.

Although a more complete study would be necessary to draw

legitimate conclusions, it is clear that the addition of one extra

limousine of relatively good quality substantially increases the

difference between E[T, ,] and E[T, ]j.

Using (2.56), the benefit gained by having a repairman can

also be seen. In particular, if t, is the time to system failure
_,n

for the 2-out-of-n system without repair, as 4. 0 it follows from
1

(2.56)

n n
E[t] = Xi 1 + x n 2
2,n I (i)
i= i=l


hnen n=3, values of E[t2,3] for the four cases in Table 1 are shown

in Table 4. A comparison of E[T2,3] from Table 1 and E[t, ] from

Table 4 illustrates the benefit gained by having a repairman.



TABLE 4

E[t, 3] FOR TIE FOUR CASES CONSIDERED lii TABLE 1


Case No. E[t, ]
,3

1 4.13
2 5.83
3 6.66
4 3.33











::urerical Methods Applicable to Chapters 2 and 3


Except for particular cases (e.g., when all units fail

exponentially and repair time distributions are of the Erlang or

gamma type), the inverse transforms of many of the results in Chapters

2 and 3 cannot be found by elementary methods. The reader is referred

to Linton [41] for a numerical approach which .an be used to plot the

density of a random variable which is characterized by an irrational

Laplace transform.

However, as long as the failure and repair distributions are

expressable in closed form, moments of the random variables treated

above are available. For example, the integrals necessary to find

E[T] from (3.31),

cc

g2(t)F (t)dt and gl(t)F2(t)dt
0 0

can in some cases be computed directly using Gauss-Laguerre quadrature

formulas. In any event, one may also use a transformation like
-t
S= e to transform (0, -) to a finite interval, and then apply

Simpscn's rule or the Trapezoidal rule.

iie.-ce, although moments of the random variables discussed in

Chapters 2 and 3 are obtainable, finding the distributicn of a random

;variable (cr a plot of its density) from an irrational Laplace

-rr.sforn will depend on the structure of the case in question.










CHAPTER 5

CONCLUSIONS A:D AREAS FOR FUTURE RESEARCH


Conclusions


The research reported in this dissertation has broadened both

The class of systems and system characteristics which can be treated

by mathematical reliability theory. Although theoretical aspects were

em-Chasized, applications \were also considered.

'.n investigation of methods for analyzing systems with standbys

resulted in generalized theory for finding the transform of the

distribution of T, the time to system failure for the k-out-of-n
k, n'

system. The main contribution of the research was the computation of

the transform of the distribution of Tk, via the supplementary variable
,n

technique, under a raore general set of assumptions than has been

cI_;.-1*iJrc to date. Using the principle of regeneration, an alternative

di:-ri,.'ion of the transform of the distribution of Tn was also
2,n

obtained.

To investigate syTstem characteristics which have becr generally

neglected in the literature, a conditional transform approach c-.s applied

co :tie :'.:c-;iitr standby redundant system and the 2-out-of-n a:','LeT.

T.:ai:sfori-s of distribution were derived for the time spent on repair,

rin :Lee time of the repairman and the number of repairs completed. For

'Thu raise of e:eponential failure and Erlargian repair capabilities, the

ri'e to systcT. failure of each system \vas analyzed end the generating

function tor thi- number of renewals occurring during :he lifr. of each

systeL wa.a found.

in the conre:-:t of an airport limousine problem, numerical











results were calculated for the 2-out-of-3 and 3-out-of-4 systems. The

value of an extra limousine as well as the benefit gained by having a

repairman were discussed. Numerical methods applicable to finding

moments of random variables characterized by complex integrals were

also mreanioned.


Areas for Future Research

The k-out-of-n System


In Chapters 2 and 3, whenever a k-out-of-n system was treated,

it was always assumed that failures were random and repair was general,

but onl one repairman was permitted. Hence, two obvious extensions

are tc allow both failures and repairs to follow general distributions

and to allow several repairmen.

For the case of several repairmen and general repair distri-

bu ions, the ability to analyze the system will depend, as always, on

hvu the state are defined. If the supplementary variable approach

;.ire used, two supplementary variables would most likely be required.

When general failure and repair times are assumed, the problem

of finding the distribution of T (or its Laplace transform) for

eve. tl.he 2-our-of-2 system has not been solved. One approach in

pr:rticular is wor:h mentioning because of the insight it provides. For

T-: 2-our-of-2 system with one repairman, let Z(t) be the number of

units do'-..n (0, 1 or 2) at moment t. For reasons which will become

c-lear ii a momtrnt, assume thai each unit is repaired by a different

repairman, and for i = 1,2, define





71






( 0, if unit i is working at moment t
Y.g (t =
1 t 1, if unit i is under repair at moment t



Now, if T is the time to system failure, then


T = min{t;Y'1(t) = Y?(t) = 1) (5.1)


A typical realization of the processes Yl(t), Y'(t) and Z(t) (see

Figure 5.1) helps validate the above relationship.

Although meaningful results were not obtained in this investiga-

tion, equation (5.1), or some equivalent form, could perhaps be used

Lt find the transform of the distribution of 1 for this general

2-out-of-2 system.


The 2-ouc-of-n System and the Two-Unit Standby Redundant System

In Chapter 3, the transform of the total time spent on repair

(R) .'as calculated for both the 2-out-of-n system and the two-unit

stnndby' redundant system using a conditional transform approach. From

Figures 3.1 and 3.2, however, it is clear that computation of the

,?ppropria:e stochastic integrals (Parzen [42]) would also yield the

distribution of R for each system. In particular, letting

22,n
d I(t,y) = d P[ | Z(u)du Il, = t]
v y j ,n
u=0

T

dI (t,y) = d P[ Z(u)du u yJT = t]
u=0

it fcllo.s that fcr the 2-out-of-n system






72



11(t)






1 -----




















R- ?2 -R 2
t



., (C )










T




1----- -^^- -- -- -
? 1 2
t
























failure of unit i), Ri (the repair time for unit i) and T.












e d It,v)h, (t)dt)dv = E[e

y =0 t=O

and for the two-unit standby redundant system



J Se ( d i(t,y)h (t)dt)dy = E[e ]
v=0 t=0

Hence, even though d I(t,y) and di(t,:,y) are not easily obtained,

properties of the random variables



Z(u)du and Z(u)du
0 0
-sR
are available by means of ELe ].

Although a powerful tool, the application of stochastic

integrals to reliability models has not as yet been researched.


Priority Models

A third major research area concerns the use of priorities.

In this dissertation, it was always assumed that units were repaired

in the order in which they failed (i.e., "first-come, first-served").

By clasnifyinn each unit with a priority index, the preemptive-resume

or !-h, a-of-lhe-line disciplines could be imposed and new results might

be obtained (see Jaiswal [22] anJ Natarajan [28], [29]).



































AP'PENDIX











A. APPLICATION OF THE PRINCIPLE OF RECENEIRATION


In the spirit of Muth [35], the principle of regeneration

will be used to find the transform of the distribution of the time

to system failure for the 2-out-of-n system.

Define


T The time to system failure for the 2-out-of-n system.
2,n
~*
As before, let h2,n(t) and h2,n(s) denote the density of

T2,n and its Laplace transform, respectively.
2,n

U. -The time to failure of unit i, 1 < i < n. The density
-X.u.
I 1
function of U. is assumed to be A.e u. > 0, 1 < i n.


Y. The time to the next failure when unit i is under repair.
(1 (i)

The density function of Y. is e y 0,

1 < i < n, where

n
)(i) -
j-l J
j i


X The time elapsed until the first failure occurs. The

-ctx
density function of X is ae x > 0, where


n
a = X X.
j=l

D = T2,n -X


R.. The time needed to perform the j-th repair of unit i,

1 < i < n, j=l,2,.... The {R..}, j > 1, are assumed to be

i.i.d. as a random variable with distribution function

F.(*), 1 < i < n.











Z(t) The number of units down at moment t (Z(t) = 0, 1 or 2).


Assume that the failure times are independent random variables

which are alsc independent of the repair times, and all units are

operative at t=O.

Unless otherwise stated, density functions will be denoted

by a lower case f and Laplace transforms of densities by f each with

an appropriate subscript for the random variable. The convolution

operator will be denoted by an asterisk (*).
-*
Before deriving h2,n(s), consider a typical sample function

of Z(t) (Figure A.1). Since T = X + D,
2 ,n


h2,n(t) = f (t)*fD(t) (A.1)


and note that for 1 < i < n


2[U. < U ,...,U. < U U < U ... U. < U ] (A.2)
i '1'" -1 i i+l'' i n CL





2,n


2 I. D
--- X -----






0
t


Figure A.I. A sample function of Z(t) in terms of the
rar.om variables X:, D and T2,n"











No', from the principle of regeneration, (A.2) and the laws of

probability,


n
fD(t) =
i=l


(, e- (i)


t

+ =
x=0


(i)
-" 'v i
SdF. (:) -


h (t :)
,n


From (A.1) and the definition of X


h (s) = --- f (s)
2,n s + o D


(s) = e st dF(t), (.(s) = ii(s +
0


(i) i
X ), I < i l n


and transforming both sides of (A.3), it follows that


1 n x (1 i (s))
fD(s) = +
i= + +


n
~* 1 n (
+ h2n(s) x- .(s)
i=1


From (A.4) and (A.5)


~2,
2,n(S)


(i)
n X. i.(s))
1 1
i=1 s + 1 i
n
s + Z X (1 i(s))
i=l


and equation (A.6) agrees with equation (2.55).


(A.3)


Letting


(A. )


(A.5)


(A.6)


i
Ir












LIST OF REFERENCES


[1] Billinton, P.., "Composite System Reliability Evaluation,"
IEEE Transactions on Power Apparatus and Systems, Vol. PAS-S8,
No. 4, April, 1969, pp. 276-281.

[2] Billinton, R., and Pollinger, K., "Transmission System
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[3] De Sieno, C. F., and Stine, L. L., "A Probability Method for
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[4] Gaver, D. P., Moncmeat, F. E., and Patton, A. D., "Po,.er System
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[5] Stanton, K. :N., "Reliability Analysis for Power System Applications,"
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['] Sagi, C. S., and Campbell, L. R., "Vehicle Delay at Signalized
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[7] ;aver, D. P., "A Probability Problem Arising in Reliability ard
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[i] Arro.;, K. J., Karlin, S. and Scarf, H., ed., Scudies in Apolied
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SLa-fcrd, California, 1962.

[1 .rIc-:. R. E. and Proschan, F., Halthematical Theory of
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F0] f;azo-.'svy, I., Reliability Theory and Practice, Prenticc-Hall,
Inc., Englewood Cliffs, N.J., 1961.

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[12' Lloy. D K., and Lipow,., M. Reliability.': Manaemernt. Methods,
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[14] Morse, P. M., Queues, Inventories and Maintenance, John Wiley
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[15] Paley, R. E. A., and Wiener, N., "Fourier Transforms in the
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[16] Gaver, D. P., "Fluctuations Described by Birth and Death
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[17] Belyayev, Yu. K., "Line-Markov Processes and Their Application
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[13] Caver, D. P., "Time to Failure and Availability of Paralleled
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[19] Cox, D. R., "The Analysis of Non-Markovian Processes by the
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[20] McGregor, M. A., "Approximation Formulas for Reliability with
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[21] Thiruvengadam, K. and Jaiswal, N. K., "Application of Discrete
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[22] Jaiswal, N. K., Priority Queues, Academic Press, New York, 1968.

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[24] Htun, L. T., "Reliability Prediction Techniques for Complex
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[25] Srinivasan, V. S., "The Effect of Standby Redundancy in Systems
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[26] Liebowitz, B. H., "Reliability Considerations for a T',o
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[27] Downton, F., "Reliability of Hultiple:- Systems with Repair,"
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[2S] Natarajan, R., "Assignment of Priority in Improving System
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[29] Natarajan, R., "Some Stochastic Models in Reliabilitv," Ph.D.
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[34] Osaki, S., "Renewal Theoretic Aspects of Two-Unit Redundant
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BIOGRAPHICAL SKETCH


Darrell Glen Linton was born December 29, 1944, in Norfolk,

Virginia. His parents moved to Baltimore, Maryland in 1945 and in

June, 1962, he received his high school diploma from the Park School

in Brooklandville, Maryland. From 1962 through 1966, he attended

Western Maryland College at Westminster, Maryland where in June, 1966,

he received the degree Bachelor of Arts with honors in mathematics.

In 1966, the author's family moved to Florida, and in September, 1966,

he enrolled in the Graduate School of the University of Florida at

Gainesville, Florida. After a brief stay in the Department of

Mathematics, he transferred to the Department of Industrial and Systems

Engineering and in June, 1971, he received the degree Master of

Engineering in operations research. From June, 1971, until the present

he has worked as a Graduate Pesearch Assistant and a Graduate Teaching

Assistant in the Department of Industrial and Systems Engineering.

Among other honors bestowed upon the author are memberships

in Kappa Mu Epsilon and Alpha Pi Mu honor societies.











I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doccor of Philosophy.




p-. 1Bras'l Chairman
Professor of Industrial and System
Engineering

I certify that I have read this study and that in my opinion
it conforims to acceptable standards of scholarly presentation and is
full',' adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.



C$\Cj- ^--c
Z. R. Pop-Stojanovilc, Co Chairman
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.




J7 F. Eurns
Associate Professor of Industrial
and Systems Engineering

I certify that I have ro-d this study' and that in my opinion
i: conforms r.o acceptable standards of scholarly presentation and is
fully 2.-quace, in scope and quality, as a dissertation for the
degree ol Doctor of Philosophy.


Sa,.:
ssor of Statistics












This dissertation vas submitted to the Dean of the College of Engineering
and to the Graduate Council, and uas accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.

June, 1972


Dean, Graduate School




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