Download Johnson, Norman L. and Kotz, SamuelSome Replacement - Times Distributions in Two-Component Systems."

SOME REPLACEMENT - TIMES DISTRIBUTIONS
IN TWO-COMPONENT SYSTEMS
by
Norman L. .Johnson
University of North Carolina
Chapel Hill, NC 27514
Key words and phrases:
Samuel Kotz
University of Maryland
College Park, MD 20742
Replacement times, relevations, revealed and unrevealed faults, Farlie-Gumbel-Morgenstern distribution
AMS 1982 subject classification 62N and 62P
ABSTRACT
We consider a system consisting of two modules, one of which cannot be
easily inspected, whde the other is monitored continuously.
A fault devel-
oping in the first modcl is called "unrevealed" (U) while the one in the
second module (which is assumed to be detected immediate ly) is called "revealed" (R).
is observed,
It is supposed that repairs are initiated as soon as an R
but not otherwise.
Phillips (Microelectron. Reliab.
1§, 495-503; Reliab. Engineer. (1981),
are always replaced.
~,
(1979)
221-231) assumed that both modules
In this paper we allow for replacement of the first
module only if a U fault is found on special inspection.
Expressions are
deri ved for the j oint distribution of time to first repair and time between
first and second repairs.
Special cases are considered in which fairly
simple results are obtaincd hy gencral arguments without recourse to analysis.
Octai led formulas are developed for a particular parametric model.
Relationships with relevation theory are indicated.
1.
Introduction
Phillips [1,2] has considered a system consisting of two modules, one
of which cannot be easi Iy inspected, whi Ie the other is monitored continuous Iy. A fault developing in the first module is called "unreveal ved (U) that is, unti 1 a special inspection is carried out - while a fault in the
second module, which it is assumed will be detected immediately, is called
"revealed" (R).
It is supposed that repairs are initiated as soon as an
R is observed but not otherwise.
As in [2] we will further assume that re-
pairs are effected instantaneously and, if carried out, result in the repaired module being "as good as new".
We will be especially concerned with the distribution of the time at
which a second (or later) repair is needed, on similar assumptions to those
of Phillips, except that we will allow for the possibility that repairs are
made only to modules l"ith faults.
This would mean that if an R occurs, the
first module is specially inspected, and replaced only if it has a U fault.
Phillips' model is based on three random variables:
X: time from repair
of R to next occurrence of R, assuming no U present.
Y: time from repair
of U to next occurrence of U, assuming no R present.
z:
to an R fault.
X and Yare assumed independent.
time from a U fault
Z is independent of X in
both [1] and [2J, but may depend on Y in [1], though not in [2].
We will
retain this possib Ie dependence.
Using the notation fW(w) to denote the density ftffiction of a random
variable W, and
00
to denote its survival function, the density function of the first replace-
page 2
mcnt timc, T
I'
IS
tllJ, cq lint ion ( i))
(1)
(We use subscripts to denote the order of the replacement period.)
If "repair" consists of complete replacement, as in [1] and [2], then
the time T
2
from first to second replacement has the same distribution as
T , and T and 1 are mutually independent.
l
I
2
Hence the time of second re-
placemcnt ('1'1+1'2) is distributed simply as the convolution of two Tl' s. Extension to later replacement times is straightforward.
The situation is different, however, if the first module is replaced
only if a U fault is found to be present on inspection after an R fault
occurs.
When such a fault is not present we start the second replacement
period with a first module already aged T , rather than a new one.
l
We note
in passing that the same situation arises when replacement of both modules
is automatic, when an R fault occurs, if the first module is replaced from
aging stock, with aging occurring at the same rate in storage as in service.
I{clt'V;ltion thcory, \vhidl
\VC
h:1VC discussed in I:;] ill1d elsewhere, is ;Ippli-
c;lhlc to this sitlliltion.
I f a LJ fault
1:8
present, then X ' Y2 and Z2 will have the same joint
2
distribution as Xl' "I ilnd =1; and the sets of variables (X 1 ,Y l ,ZI)
(X ,Y ,Z2) \,ill be Illutually independent.
2 2
given T
1
t
l
,
and
If a U fault is not present, then
wc \vi11 have
Y2 distributed ilS (Yi-ttl, nmditional on YI>t l : - density function
f (v +t ) IS (t ) and sllrvi val functi on S (Y2+ t l)
y J2
1
Y 1
Y
IS y(t.J );
page 3
Z2' given Y2' distributed as Zl given
fun~tion
Y~Y2'
so that the joint density
of Y and Z2 is
2
=
f Y,Z(Y2,z2)
f Y(Y2)
(2)
Also we have
The
~onditional
density function of T , given Tl=t , is therefore
2
l
t
f y (t +t -z)
I-PU(t )
l
2
l 2
+ Sy(t ) {fx(t2)Sy(tl+t2)+fo Sx(t 2-z)fY,z(t 2-z,z)
f (t -z) dz}
y 2
l
e
(O<t 2)
(4)
The joint densi ty
fun~tion
of T1 and T2 is
f'1'2 lT l (tzlt l ) f (t l )
Tl
.
= {f
t 2
X(t 2 )Sy(t 2 )+f o Sx(t 2-z)fY,z(t 2-z,z)dz}
t
f o1Sx(tl-z)fY,z(tl-z,z)dz
f (t +t -z)
Y 1 2
f (t _zr1z}~(~
Y
2
(5)
(Sa)
where
II(a,l»
f~;
\ (a- z) f y, Z(a- z, z) {fY(a +b- z) Ify (a- z) }d z
The overall density of T is
2
.e
(6)
page 4
00.
00
t2
f y (t l +t 2 -z)
+ fX(t2)·fofX(tl)Sy(tl+t2)dtl+!OfX(tl)fo SX(t 2-z)fY,Z(t 2-Z,z) f (t -z) dz
y 2
(7)
00
We note that
t
fo fa l Sx(tl-z)fY,z(tl-z,z)dzdt l
is simply the overall proba-
bility that a U fault is present at the first replacement time, that is,
3.
Some special cases
Before proceeding to study cases ill which the joint distribution of
x, Y and Z is completely specified, we first take note of a few results of
broader character which can be derived by general reasoning, without recourse
to analysis.
Intuitively one might feel that I - PU(t ) = Sy(t ), since the probal
l
bility of no U fault occurring up to time t, is Sy(t ), but this does not
l
take into account the differential effect of the times of occurrence of R
faults according as they are, or are not
preceded
by U faults.
If the
time to an R fault (TI=X if XsY; = Y+Z if Y<X) were independent of Y, then
f
T
I
(t ) = fX(t ) and so, we wouZd have I - PU(t l ) =
l
l
clear that this is not usually the case;
Sy(tl)~
It is, however,
indeed independence of Z and Y
does not imply independence of TI and Y; in fact it usually implies dependence.
If replacements of first modules are from aging stocks, with aging being
the same whether in storage or in service, ("relevated" using the terminology
of [37]), then it makes no difference (so far as distributions of T.'s are
J
concerned) whether or not a first module is replaced when there is no U fault.
.e
This is because after the repair we have, in either case, a first module of
age t ; and this result is valid whether replacement of second modules is
l
from aging stocks or not.
page 5
If the lifetime distribution of the first module (that is, of Y)
is exponential, then (XI,YI,Zl) and (X ,Y ,Z2) and so T and T2 will be
l
2 2
mutually independent, whether replacement of the first module is from aging
stocks or not (or is, indeed, performed or not when no
U
is present). This
follows from the lack of memory (or old-as-good-as-new) property
(see
If replacement of the second
e.g. [4]) of the exponential distribution.
module is from aging stock, this may no longer be the case, though it is
so if the distributions of X and Yare both exponential, whatever the dis-
tribution of Z, provided this depends only on time (Y) since replacement,
and not on actual age.
If the joint distribution of Y and Z is of the Farlie-Gumbel-Morgenstern
form, with
(see, e.g., [5]), then in (Sa)
a
H(a,b)
J
o
Sx(a-z)fy(a+b-z)fZ(z) [1+a{2S y (a-z)-1}{2S (z)-1}]dz
Z
(9)
When we come to consider explicit forms for the distribution functions,
numerical evaluation of the distributions of T
I
and T
2
is straightforward,
but for elegant analytical results there are technical difficulties, centering mainly on the form of Sy(y) and in
particular
fy(tl+t2-z)/fy(t2-z), and also on the form of SZ(z).
of
the
ratio
Assuming exponen-
tial forms docs, as we have seen, give simple results, but these are perhaps too simple to make good examples.
•
We will take, as an illustrative
special case, X, Y and Z to have density functions
e.-2
t
1
-1
exp (- t e.)
1
(0< 81
. , t)
·
page ()
i'-'1,2,3 respectively, with the joint dcnsity function of Y :1T10 Z of thc
Farlic-Gumbel-Morgenstern form (8).
We have
-1
fy(tl+tZ-z)/fy(tz-z)
-1
Sy(y)
=
(1+y8 z )e
=
-y8 z
(tl+tZ-z) (tz-z)
=
(1+z8
-1
3
e
-t/8 Z
-z/8
) e
;SX(x)
=
-1
(1+x8 l )e
-x8 l
3
Now from (5)
Although (11) is complicated, it involves only integrals
a
Jo
a
z exp(Bz)dz
=
of
form
-a-1 aS a -w
B OJw e dw and so can be expressed in terms of
elementary functions.
Similar analyses can be performed for other choices of distribution
for X, Y and Z (e.g. Weibu1l) but they lead to even more complicated express ions than (11).
We hope to provide some detailed analyses of such
cases, in later work, and also to introduce cost functions, on the basis
of which replacement strategies can be compared.
.e
4.
Acknowledgements
N. L. Johnson's work was supported by the National Science Foundation
under Grant MCS-8-Zl704; S. Kotz's work was supported by the US Office of
Naval Research under Contract N00014-81-K-0301.
REFERENCES
.e
[1]
Phillips, M. J. (1979) The reliability of a system subject to
revealed and unrevealed faults, Microelectron. Reliab. 1],
495-503.
[2]
Phillips, M. J. (1981) A preventive maintenance plan for a system
subject to revealed and unrevealed faults, Reliab. Eng., £,
221-231.
[3]
Kotz, S. and Johnson, N. L. (1981) Dependent relevations: Timeto-failure under dependence, Amer. J. Math. and Mgmt. Sci., 1,
155-165.
[4]
Galambos, J. and Kotz, S. (1978) Characterizations of Probability
Distributions, Lecture Notes on Mathematics, ~, SpringerVerlag: Berlin, Heidelberg and New York.
[5]
Johnson, N. L. and Kotz, S. (1972) Distributions in Statistics:
Multivariate Continuous Distributions, Wiley: New York .
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one of which cannot be easily
inspected, while the other is monitored continuously. A fault developing in the
first model is called "unrevealed" (U) while the one in the second module (which
is assumed to be detected immediately) is called revealed (R). It is supposed
that repairs are initiated as soon as an R is observed, but not otherwise.
Phillips (Microelectron. Reliab. (1979) l§, 495-503; Reliab. Engineer. (1981),£,
221-231) assumed that both modules are always replaced. In this paper we allow
for. replacement of the first module only if a U fault is found on special
I
DO I ~~~~3 1473
EDITION Of I"'liV ,,~ IS
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20.
inspection. Expressions are derived for the joint distribution of time
to first repair and time between first and second repairs. Special case.
llre considered in which fairly simple results arc obtained by qeneral
arguments without recourse to analysis. Detailed formulas are developed
for a particular parametric model. Relationships with relevation theory
are indicated.
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