GCncral.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. PopStojanovic 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. PopStojanovic, who served as
commiittee cochairman.
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
dissertationboth 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 DAHC0468C0002 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 kOUTOFn 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 SemiMarkov' Model
and the Supplemencary Variable Technique .....
3. SOME RELIABILITY CHARACTERISTICS OF THE 2OUTOFn
SYSTEM AND THE TWOUNIT STANDBY REDUNDANT SYSTEM..
Introduction............................... ...
The 2Outofn System.........................
The TwoUnit 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, 2Erlang and
Exponential Repair................................ 65
2 E[T ,] for Deterministic, 2Erlang 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
CoChairman: Dr. Z. R. PopScojanovic
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 koutofn 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 koutofn system. Using the principle of
regeneration, an alternative derivation is also obtained for the
2outofn system.
A conditional transform approach is applied to the 2outofn
system and the twounit 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 2outof3 and
3outof4 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.ers .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
speint 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 reicirable sssterns kno'.',n as koutofn 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 'kaLtofn systems (or ccr, nations thereof)
Il : 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
koutofn 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 koutofn 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 twounic
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 koutofn 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 preemptiveresume 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 ni 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 2outofn system by considering first passage times. The
improvement factor was evaluated and its asymptotic behavior was
studied.
In 1969, Epstein [31] considered a twounit 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 koutofn 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 twounit systems.
Osaki [33] treated a twounit 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 was derived
in each case. Muth [35] took advantage of the regenerative properties
(discussed extensively by Smith [36]) of the twounit parallel redundant
system, and derived the time to failure distribution and the NTSF in
an elegant manner. Mazumdar [37] considered a twounit 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, where 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 twounit 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 semiMarkov processes, Branson and Shah dervied the
MTSF for the twounit system and discussed some of the difficulties
encountered when the semiMarkov model was applied to a threeunit
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 koutofn
system. A comparison will be made between the supplementary variable
approach used in this research and the semiMarkov 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 koutofn 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 noutofn system with = and F (*) = F(),
m m
1 < m < n, to find a general expression for the nITSF.
(3) For the 2outofn system and the twounit 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 2out
of2 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 koutofn systems and a comparison is made
between the supplementary variable approach and the seniMarkov method.
Chapter 3 concerns the 2outofn system and the twounit
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 kOUTOFn SYSTDI
Introduction
In thi.; chapter, the supplementary variable technique is used
to find the transform of the time to system failure for the koutofn
system. After introducing some preliminaries, the koutofn 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 semiMarkov 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 "firstcome,
ljit::servcd" basis. Machine m is serviced according to general
re.pai disrihution, 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 kl 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
koutofn 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 jthbreakdown 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 hn(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 koutofn 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 eSp0(t)dt
and
oo t
V.(s) = e ( p.(t,z)dz)dt, 1 < i < k1
0 0
(n) By definitions (k), (1) and (g)
k1 t
Pk,n(t) = P(t) + Z pi(t,z)dz
i=1
and hence, using (m) and (h)
k1
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 _ k1
(iv) P[X(t + dt) = (j,z + dt)IX(t) = (j,z)]
= (1 (nj)Xdt){(l F(z t dt))/(l F(z))} + o(dt), 1 j < k1
(v) P[X(t + dt) = (jl,0)IX(t) = (j,z)]
= (1 (nj)Xdt){(F(z + dt) F(z))/(l F(z))) + o(dt), ljk1
The 2Outofn System
Consider the 2outofn 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 jth
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 ndt) +
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)ndt + 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
dp = n\p(t) + e n1)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 laplaceStieltjes transform of F(z) be .4(s) = esdF(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)
Combiine (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
(n1)z t z)(1 F(z))d
e gl(t z)(l F(z))dz)dt
G (1)
s (n) (1 ())
s + (n1).! n1
(2.13)
n'(l ,nn(s))
nn
(s + (n1)1){s + n.(1 n (s)))
n1
fhus, from (2.2), (2.3), (2.4), (2.12) and (2.13) it follows
c.iat for the 2ctofn system
nIn
J, :s) is + n"(1 (s)) +n
',n n (s + (nl).)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 (n1) n' ( ..
nI
.'ere n = .:((n1).). P.elation (2.15) agrees .'ith equation (5.19a)
in D ..wntcn [27J.
The 3Outofn System
For the 3outofn 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(n1).z g1(t,z)(1 F(z))
q2(C,z) = p2(c,z)/(1 F(z))
the s:aca transition equations for the 3outofn system may be
rwritten as
do
d 0
d 3z
(n1):z ( )( ) = 1
e gl(c.z)dF(z),po0(0) = 1
(2.19a)
(2.19b)
(2.17)
(2.18)
q q> (n1)!
= (n2)qq,(t,z) + (nl)).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 ) (n2)z g, z) (n)
q(t,z) e (t z) (nl)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) = (nl)gl(t z){e (n2)z
S( nl)\z
 ea
Usin (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+ n2) .z
1)= n Po(t) + (n1) 91gl(t z)je
o
(ni) zl
 e (dF(z)
(2 2 )
'.(s) rjst 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 (n1),_, (s) + (n1),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 (n2)j ,(s) + (n1), _s)
(S) = (s + n'){ (r.l).n (s) + (n1) (s) n'" (s)
n n1 nI
.A.so, from (2.17) and the definition of V (s)
V1(s) = J
0
t
st (n1)" z
0
g1(t z)(i F(z))dz)dt
C,(s)
G C (1 C (s))
s + (n1) nl
SiJ'ilarly, from (2.18), (2.22) and the definition of V,(s)
r 
: ni) e0
0
= "n1)G. (s)
4.
I
g1(t z)Ie
(n F())d)dt
 e ;(i F(z))dz)dt
I (s) 1 ()
n 2n1
s + (n2)1 s (k1)
(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')(nl)(n2)\
3,n
{. (s + (nl)")i' ,(s) + (s + (n2).)';n, (s)J]/
n, nI
/[(s + n0){ (nl),, (s) + (nl),: (s)) n'n (s)] (2.30)
nL ni nI
and
h3 (s) = 1 sW, (s)
3,n .,, n
1 q + v
1 1 'nl n2
F[]3 ] = i (0) = + + i
3,n 3,n (n1) (n2) nil (n1)n 2 + (n2)r, j
n2 ni
(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 differentialintegral 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 nOutofn 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 (nl) x (nl) 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 (nl).,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 (n1),.dt) 1 F(z) + o(dt)
(2.32b)
1 F(z + dr)
pk(t + dt,z + dt) = pk(t,z)(1 (nk).dt) 1 E(z)
1 F(z + dt)
+ p, (t.z)(n k + 1)'dt F(z ) + o(dt), 2 < k < ni
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 n1 1 F(z)
(2.32d)
t
p (t + dt,0)dt = p0(t)n.dt + p2(t,z)(1 (n2),.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 (nkl)..dt) F(z dz +
j< k+l 1 F(z)
0
+ o(dt), 2 < k < n2 (2.32f)
p.( + d:,0)d = o(dt), j=nl,n (2.32g)
0(0) = 1 (2.32h)
Sn1 b
Proceeding si.ilarly s above, define g (t,z) and {q (t,z)} = b
the relaticrships
(t,z) = g (t,z)e(nl) z( F(z)) (2.33)
(2.34)
p (t,z) = qk(tz)(1 F(z)), 2 < k < nI
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) +
(nl) .z
e
g1(c,z)dF(z),p0(0) = 1
*l 1
.C JZ
1 2q 
'" 
(2.35b)
(n2)\q (c,z)
2
+ (nl).Ie ( 1 g1 (t,z)
+ (n k + l)'q (t,z)
kl
(2.35c)
3 < k < ni
(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 < n2
k+1
(2.35e)
(2.35f)
(2.35g)
p (:,F) = 0, j = nl,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 : ni
" 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
n1
{gk(t z). is
Sk=2 s
k1 nk+i .
z_ (nk+i)'z
q (tz) = ( ) g (t z) (2.36)
i=0O i
for 2 k < n1, 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
qkl(t,) = pkl(t,O) = qk(t,z)dF(z), 3 k ni (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 n1 equations in terms of the n1 unknowns, V\(s), Gl(s),
C2(s),. ..,G (s) n > where
G.(s) = e g.(c)dt, 1 < j < n2
0
Remembering that '..(s) = exp{ (s + j*)t)dF(t), the set of n1
b
equations in n1 unknowns 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 .
n1
Using (2.36) and the definitions of {Vk )k=l
G (s)
1
G Is S) (1 'I (s))
1 =s + (n1), nl
k1 nk+i Gki(s)(l (s))
k ki nk+i
V (s) = (1) < k < n2
k 7 i s + (nk+i)  
i=0 inki G s)(
n1 1 i (s) 1 '.(s)
1 11
(si) i(1) G .(s)
nL s + s + in
loe', by, the defi.niLion of 1 (s) and the above
n,n
n2
.. (s) = V (s) + V (s) + V (s) + V (s)
n,n 0 1 k2k nI
G1 (s)(1 in (s))
= 0 ,) + n
0' 3 + (n1)A
24
I
o 0
11
C6
I 
C4
* * 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
n2
+ =
k=2
k 1 nk+i
i=O0
Cki (s)(1 nk+i (s))
s + (nk+i)'
+
n1 1 ,(s) 1 is)
+ i(i) Gn i s)
i=n1 s + s + i,
i=2
or, after simplification
n2
U (s) = Vo(s) + (1)nj G.(s) *
n,n j=
j=1
(nj)(l + (s))
s + '.
1 (s) ]
nj
s + (nj) '.
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 noutofn 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 2Outofn System
For this more general 2outofn 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
Fr 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 gt(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. )
iAU
= 4 c 
n (s) I +
i 1
S'. .(s)) 1 +
i=1
n .(1 .(s))
 1 1
" (i)
il 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)
il )ii i1 .(i)
(i)
where q. = (. 1 ),) i 1. As a check, when = ), and F.(') = F(),
i 1 1 1
Yi(s) = c,(s), A(i) = (nl)), C (s) = .(s + (n1),), a = n) and (2.54)
becomes
1 + 1 + (ni))]
W2 (s) = {s + n)[1 o(s + (n1).)]} 1 + n [1 + (n1)
2,n s + (nl)?,
1 nl[ ..(s + (n1):)]
s + n)41 ((s + (nl)))] (s + (nl)))(s + n\[l .(s+(nl)..)]}
= 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 3Outofn System
For the 3outofn 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 j1 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) + (n2)
iOs)+ (i (n2) ( (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 + (n1)k), .. (s) = .3 + (n 2))),
S1 j
= n., = (ni)\, \ ) = (n2)\, i,j cS
For conveniences, let
i n(s) = ( + (nl)))
En2 (s) = *(s + (n2)))
'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 (n1)y, (s)
v  )"n (0 70)
0 (s + n,),l (n1) n (s)) n e 1, (S)
n ni
nrd for 1 < i < n
,. ( \)  (2.71)
S ( s, + n).)(l (n1), (s)) n\'i~n s)
n ni
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 SemiMarkov 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 semiMarkov approach and the supplementary
variable technique used in this research.
The Approach of Branson and Shah
In the twounit system considered by Branson and Shah, a unit
may operate online or offline. The online 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
offline unit with respect to '2 and F,2(). This is not quite the
same as the 2outof2 s5steri, but it is similar.
By a judicious choice of states, Branson and Shah modeled their
twounit 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 upstates.
The SemiMarkcov Model af Downton
Dovnton used a semiMarkov process to model the koutofn
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 10 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 semiMlarkov 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 semilarkov 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 koutofn 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 2OUTOFn
SYSTEM AND THE T\VOUIIT STANDBY PEDU::DANT SYSTEM
Introduction
Reliability attributes of the 2outofn and twounit 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 kErlang 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 2Outofn 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 2outofn 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 (il)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 ith 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 LaplaceStieltjes transform of F('); i.e.,
s) = e'stdF(t)
0
Y the time to the next failure when only n1 units are
operative. By assumptions (1) and (3), the density
function of Y is
i y
fy(y) = Xie y 0
where 1X = (nl)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[EsI] =
E[ 
definition of the {X.}, j 1
3
E[eSI 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)
) edFt) s +
0
First, applying the shifting property of Laplace transforms to equation
(i.11) an recalling that )' = (ni)", i0,1, 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 2outofn 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'el'
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;
0j1
dsi D(s)
ds S= C
S0, j=1,2,...,m; m > 2
Taking j=2,
D'(s)
s=r
= (r + ,)k(r + u)k1 + (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{(nl) 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 2outofn 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.)[1B ((nl)'. r.)]
D'(r.)((n1) ,r .) [zB ((nl) .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 = Iz 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  ( rI Bi
j 1
3 3 J
The TwoUnit 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" backup.
(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 twounit 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),/ (n1)/2
P[LI = n, 1N odd] = p q( /2 (lq), 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 (nl)/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 =
esT 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 +
1j1
"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(nlY2
{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(nl)/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 (nl)/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
EeR ] = [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 (n1)/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)
1q
S(n+l)/2 (nl)/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 1R 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) = esy
'=0 x=v
f (x y)g2(.:)dx)dy
Letting
g12(s) =
v=O
and remembering thac
eSy
e (j
f2(x V)g ('.:)dx)dy
g (s) = {gl(y))
it follo,.'s that
E[ 1 = g (s)
(n+)/2 (nl)/2
21(S) g12( )
odd P q ~
(n+l)/2 (nl)/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[eS ] + .[eS ]}
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
kl
f.(t) = p.( t)k le:.p( t)/(kl)!, 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
2outof3 and 3outof4 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 koutofn 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 2outof3 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 koutofn 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
koutofn 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 2outof3 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) 2Erlang
and (3) exponential.
As indicated in Table 1, for the 2outof3 system exponential
repairr appears to be better than deterministic repair capabilities
when the mean repair times are the sa.me. This peculiarity of the 2out
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 2Erlang 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, 2EPL%':G .'dD EXPO:E::,TIAL rEP'AI?
Failure RaLes and 
Case ;leai 1 pair rimes Deterministic 2Erlarg 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
2EF:.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
2Erl 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 2Erlang 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 2Erlang 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 2Erlang
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 2outofn 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 GaussLaguerre 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
emChasized, 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 koutofn
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 2outofn a:','LeT.
T.:ai:sforis 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 2outof3 and 3outof4 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 koutofn System
In Chapters 2 and 3, whenever a koutofn 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 2ourof2 system has not been solved. One approach in
pr:rticular is wor:h mentioning because of the insight it provides. For
T: 2ourof2 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
clear 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
2outof2 system.
The 2oucofn System and the TwoUnit Standby Redundant System
In Chapter 3, the transform of the total time spent on repair
(R) .'as calculated for both the 2outofn system and the twounit
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 2outofn 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 twounit 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., "firstcome, firstserved").
By clasnifyinn each unit with a priority index, the preemptiveresume
or !h, aoflheline 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 2outofn system.
Define
T The time to system failure for the 2outofn 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) 
jl 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 jth 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
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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. PopStojanovilc, 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 rod 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
