MULTI VARIAUi DEPENDENT RELEVATIONS
WIlli APPLICATIONS, AND RELATED TOPICS
by
Nonnan L. Johnson
University of North Carolina at Chapel Hill
and
SanRlel Kotz
University of Maryland, College Park
Abstract
"Re1evation" is the name given by Krakowski (1973) (Rev. Pmm;aise
Automat.
Info~. Reohe~ohe
qp.,
7, Ser. V-2, 108-120) to the distribution of
failure time of a replacement from an aging stock.
The present authors
(Johnson and Kotz (1979; 1981) (IEEE Tmn8. ReUabiUty, R-28 , 292-299;
Amenoan JouzrnaZ of MathematioaZ and Management Soienaes, 1 (2))) have extended
this concept to include (i) hierarchal replacement systems and (ii) dependence
between lifetimes of original and replacement i terns.
In this paper, we present
some further developments,- including first steps towards a synthesis of (i) and
(ii) .
Hierarchal Systems.
I.
Introduction.
'Ine concept of a "relevation" was introduced hy Krakowski (1973) to
correspond to replacement from aging stocks.
Suppose that when an element with
survival distribution flIDction (SnF) S1 (t) fails, it is replaced by an element
wi th SDF S2 (t) taken from stock.
If failure occurs at time t , then the
l
conditional snp of the replacement element is
and the snp of time of failure (T) of the replacement element is
(1)
e
where f 1 (t) = -dS1 (t)/dt is the probability density flIDction (PDF)
corresponding to Sl (t) •
Sr(T) is called the reZevation of Sl (t) with S2(t).
This concept has been extended to situations wherein there is a hierarchal
system of replacement, and wherein the lifetimes of elements are not
independent (Johnson and Kotz (1979, 1981)).
In the present paper, we study
combinations of extensions of this kind, and also give examples in which
differential rates of aging in storage and in service are allowed for.
2.
Dependent Re1evations in a Hierarchal System.
Consider a set of n elements
{~l}'
with a simple replacement element t,;2.
This might occur, for example, when F;Z is called into service as a back-up
only when there is a failure among
{~1}.
This system (comprised of
{~l}
and
;2) would quite possibly be only a small part of a complete hierarchal system;
study of such relatively simple systems is an essential prerequisite to study
of more elaborate systems of which they could be a part.
We denote the Failure times of the elements of {~l} by Tll,T12,H.,Tln'
and of (.2 by 1'2'
'111e joint survival distribution (SDF) of these (n + 1)
fnilurc times is
(2)
Ct.]
=
(t n ,···, tIn)) .
lhe joint SOF of '1'1 = minCT11' H . , TIn) - - the failure
time of the first member of {~1} to fail -- is
(3)
where
l'
=
(1,1, ... ,1); and the
snp of Tl is.
(4)
lhe
snp of the replacement element is then
(5)
where f (t) = -dS l (t)/dt and S211 (T It) = P(T2 >T ITI =t) .
1
In general, explicit evaluation of (5) leads to rather complex formulas.
As a compromise between realism and sirnplicity, we will (similarly as in
Johnson and Kotz (1981)) use a generalized Par1ie-Gumbe1-Mbrgenstern
distribution (Johnson and Kotz (1975)) as a special case for detailed
examination.
We suppose
4
n
=
(SZ(t Z) .n
)=1
S(t 1).)}[1
+
a{l
n-I n
+
8
L L
j <j
I
{l -
S(tl)·)}(1 - S(t l )· I)}] .
(6)
ll1e parruneters a antI f3 have to satisfy certain conditions for the distribution
to be a proper SDF.
I (2r-n)al
These conditions can be stDnmarized as
- ~{(2r-n)2 - n}S ~ 1
(r
= 0, I, ... ,n)
,
or equivalently,
(7)
r: : 0,2, ••. , n for n even]
[r - 1,3, .•. , n for n odd
'The marginal snp of each Tlj is Set); the marginal snp of T2 is S2(t).
joint snp of Tn'.'" TIn is
5.r-1 (11)
=
n
{II S(tl·)}(l +
j=l
J
s
The
n-l n
r r {l - S(t1J·)}{l - S(t1]·,)}]
j <j ,
which is an exchangeable distribution.
,
(8)
This is likely to be a reasonable
assumption if there are no systematic differences among the conditions to which
the n elements of
{F;1}
are exposed.
The SDP of T1 (time to first failure in {F;l}) is
(9)
and the corresponding density function (PDF) is
'The joint SDP of T}
SlZ(tl,t Z)
and T
is
Z
= SZ(tZ){S(tl)}n
+
[1 + na{l - S(t 1 )}{1 - SZ(t Z)}
~n(n-1HHI - S(t1)}z] ,
(11.1)
with joint PDF
f 12 (tl't )
Z
= nfl (tl)fZ(t Z) {S(t 1)} n-1
[1 +a{n - (n+l)S(t l )H1 - ZSZ(tZ)}
+ ~(n .1)f3{l - S(t ) Hn - (n +Z)S(t )}]
1
1
(11.2)
The conditional PDP of T ' given T , is
1
Z
fZll(tzltl)
=
f 12 (tl't Z)
f (t )
1 1
- f (t )
- Z Z
z)}
1 + a{n - (n+1)S(t )} {l- 2S2 (t
+ ~(n-1)f3{l- S(t )} {n - (n+2)S(t )}
1
l
l
____;:..~----=.,
1. + ~(n-l) f3{l - S(t )} {n - (n+Z)S(t )}
l
1
------.=.~~---:-~~~_:__--,-~
(lZ.1)
and the conditional SDP is
1 + a{n - (n+l)S(t ) Hl- S2(t )} + ~(n-1)f3{l - S(t )} {n - (n+Z)S(t )}
l
1
1
Z
SZII (tzl t 1 ) = Sz (t z)
1 + ~(n-1)a{l - S(t )} {n - (n+Z)S(t )}
1
1
(lZ.Z)
6
lienee from (5) the snF of time to failure (1') of the
~Z
(replacement) element
is
+ n
J; f(t){S(t)}n-l
[1 + ]:z(n-l)f3U - S(t)}{n - (n+Z)S(t)}]
52 (-r)
x
SZ(t)
1 + o;{l - SZ(T)} {n ~ (n+1)S(t )} + ~(n-1)S{1- S(t )} {n - (n+Z)S(t )}
1
1
1
1 + o;{l - S2 (t)} {n - (n+1)S(t )} + "(n~l) a{I - S(t)} {n - (n+2)S(t)}
1
x
(13.1)
As a further specialization, we will take S2 (t) = S(t).
substitution s
= Set)
Then making the
in the integral leads to
ST(T) = {S(T)}n ['1 + ~ n(n -1) sU ~ SeT) l2]
1
n-2
+ nS(T) /.SeT) s
[1 + ~(n-l)B(l~s){n - (n+2)sl]
x 1 + o;{l- S(T)} {n - (n+1)s} + ~(n-l)a(l-s){n - (n+2)sl
1 + o;(l-s) {n - (n+1)s} + ~(n-1) 8(1-5) {n - (n+2) 5 }
ds
.
(13.2)
Note that Sr(-r) is always the same flUlction of S(T), whatever the flUlctiona1
form of the latter.
The integral in (13.2) can be evaluated in terms of elementary flUlctions
(see Appendix), though the expression is, in general, rather complicated.
the special case
0;
= 0 (so that replacement lifetime is independent of the
lifetimes of the initial elements),
For
~
7
,I
+ nS(T) JS(T)
5
11-
2
[1 + li(n-l)I3(l-s){n - (n+2)s}]ds
= SeT) [n - {SeT) }n-l]
n-l
= An
n8 .
(1.4)
+ B
Some munerica1 values of An and B are shown in Table 1.
n
TABLE 1
Values of
e
n
\t
and Bn for ST(T) ..
2
~
+ Bn l3 when ex .. O.
5
4
3
S(T~ =
An
B
n
An
B
n
A
n
Bn
A
n
B
n
0.2
0.360
0.0341
0.296
0.0410
0.266
0.0377
0.250
0.0328
0.4
0.640
0.0288
0.568
0.0475
0.524
0.0546
0.497
0.0547
0.6
0.840
0.0128
0.792
0.0269
0.757
0.0381
0.731
0.0487
0.8
0.960
0.0021
0.944
0.0054
0.930
0.0094
0.918
0.0132
(NOTE:
Possible values of 13 are limited by - 2n-1 (n -1) -1 s 13 :s 2n-1.)
If a
;t
0, the value of ST(T)
1
naSCr) ~S(.,.)
•
~ceeds
that given in, .(14) by
~[1+ ~(n-1)a(1-s){n - (n+2)s}J{s - SCr)} {n - (n+1)s}
1 + aCI-s) {n - (n+l)s} + ~(n-1) S(l-s) {n - (n+2)s
If S(l) > n/(n + 1), this will be of opposite sign to ex since
{s - S(l)}{n - (n+l)s} < 0 for SeT) <
S
< 1 .
}
ds •
8
TIlis gives some qualitative appreciation of the effect of non-zero
et.
To see how these results can be used in dealing with more complicated
systems, consider the subsystem set out schematically in Pigure 1.
The
~2
~1
PIGURE 1
e
component f,;3 is related to the two E;2 's in the same way as each E;2 is
related to the two
~1' s
belonging to it.
first failure of the two E;2 '5.
The re1evation
plays the role previously played by the
the resulting fonnula for the
E;3 comes into operation at the
snp of the
snp
snp of each E;l (S(t)). Of course, if
E;3 replacement were to be written out
explicitly, it would be very lengthy indeed.
numerical values is quite straightfoIWard.
However, the computation of
It can be carried out in stages:
first finding &rC't') and then combining it with the
~3
components.
Sr(T) given by (13.1) now
snp -- S3(t), say -- of the
(The case of independent components has been treated by
Johnson and Kotz (1979).)
3.
Differential Rates of Aging in Storage and in Service.
It is likely that an element will be under less stress while in storage
than in service.
This can be allowed for by regarding a period t in storage as
equivalent to a period get) in service (usually with get) < t).
Equation (1)
9
is then replaced by
(15)
and (2) by
(16)
One might reasonably take get)
equivalent to y years in service.
not essential.
e
= yt -- so
that 1 year in storage is
Usually one would take y < 1, but this is
Even if it is assumed that SI(t)
longer true that STC'r) depends only on S(T).
= SZ(t) = Set),
it is no
As an example, we take y < 1
and
Set)
=1
- Kt
Then from (IS),
S (T) = 1 - KT + 1.T K 1 - KiT - (l-y)t} dt
TO.
1 - Kyt
=1
- KTy -1 - (1 - KTY)y -2 10g(1 - KTY) .
(17)
]0
Some munerical values are given in Table 2.
TABLE 2
Values of ST ('() when S(1:) = 1 - K1: has specified values
(y = service years equivalent to one year in storage).
S(T)\Y
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1.0
0.661
0.812
0.918
0.980
0.639
0.804
0.915
0.979
0.611
0.794
0.913
0.979
0.562(5)
0.781
0.910
0.979
0.522
0.767
0.906
0.979
(1he last column corresponds to no difference between aging in storage and in
service.)
The effect of differential aging is not very great, especially at
the longer durations.
Acknowledgements
Norman L. Johnson's work was supported by the National Science Foundation
under Grant MCS-8021704.
Samuel Kotz' s work was supported by the U.S. Office
of Naval Research under Contract NOOOl4-81-K-0301.
References
Johnson, N.L. and Kotz, S. (1975). On some generalized Farlie-Gumbe1Morgenstern distributions. Comm. statist., 4, 415-427.
Johnson, N.L. and Kotz, S. (1979).
~ns.
Models of hierarchal replacement.
IEEE
ReZiabiZity, R-28, 292-299 •
.Johnson, N.L. and Kotz, S. (1981). Dependent re1evations: Time-tC?-failure
under dependence. AmePioan Journal of Mathematioal and Marzagement Soienoes,
1 (2).
Krakowski, M. (1973). The re1evation transfonn and a generalization of the
gamma distribution ftmcHon. Rev. Franc;aise Automat. Inform. Reohe!'ohe
Op., 7, SeT. V-2, 108-120.
JJ
~pendix
The integrand in (13.2) can be expressed in the fom
I
[a {(s-b )2 + C }]-l
d .sn+j-Z
n
n
n
j=O n,]
Using the change of variable y = s - b , the integral becomes
n
=
a-I
n
4
I
j=O
d
.
n,]
n-Z+j
I
I-b
(n-h2+j)bnn-2+j-h f,S(T)n_
h:20
bn
yh(yZ+cn)-1 dy
4
. n-2+j
n Z I d .b J
I (n-hz+j)bn-h[lh(l- bn ) - Ih(S(T) - bn )] ,
n
j=O n,J n h=O
= a-Ib -
n
where
~(-C )~(h-l) 10g(y2 +c )
n
=
[(h-3)/2] (-c )i
I
n
+
h-I-2i
i=O
(_l)~h c~(h-l) tan-ICy/ie)
n
~(-c )~Ch-l) log
[For example, IO(Y) =
(h odd)
n
y-r-cn
n
Y+ v'-cn
l/~ tan- l Y/~ if
c > 0;
n
[1/(2/-cn ))log (ey-/-c )/(y+v'-c )) if c n < 0; 11 (y) =
n
n
IZ(Y) = Y - cnIO(Y); 13 (y) = ~y
on. ]
2·
(h even,
n
C
n
> 0)
(h even, C < 0) •
n
~ log(yz+~n);
3
- cnl! (y); 14 (y) = (1/3)y
- CnIZ(Y); and so
12
The constants a , h , and care:
n
n
a
(Note that if a = 0,
c
n
=
(n+1)a +
~(n-l)(n+2)f3
n
= ~ {(Zn + 1) a
n
= [an -
h
c
n
n
Z -Z
b
n
d 0
n,
=
d
= -Z(n Z -1){l
d
n,2
d
n,
d
n,
=
(n+l)/(n+Z);
- (n+Z)-Z.)
The expressions for the coefficients d
n,1
-1
n
1/4fa + (n-l)f3} ]a
n
an~' ~(n-l)(n+2)13;
= 2{(n-l)(n+2)(3}-1
2
+ (n - 1) 13} a
;
. are:
n,]
{l + ~n(n-l)(3}[1 + naU - S('r)} + ~n(n-l)13]
+
~n(n-l)6}8
= (n -l)(n + 2) B +
- (n+l)a{l +
~(n -.1)(3n 2 + 6n + 2)[a{l
3
= -(n2-1)(n+2)[~a{l
4
= (1/4)(n -1) 2
- S(-r)} + (n-l)6]13
(n +2) Z (32 .
(3/Z)n(~-I)(3}{1
- S('r)}
- S('r)} + (n -1) 13] 13
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TEGINICAL
Multivariate Dependent Re1evations With
Applications, and Related Topics
.
-b-piRmRMiNGQ;G~flE:P'()R"l-Nu-M-BE;;--
Nonnan L. Jolmson and Samuel Kotz
NSF Grant MCS-8021704
ONR Contract N00014-81-K-0301
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DISTRIBUTION STATEMENT in/ 'hu .bal,eel "nle'ud III IUo"k 20, /I dlfl.,,,nll,,,m Nupntl)
17.
Ill. SUPPI.EMENTARY NOTES
19.
KEY WORDS (Continue 0'1 reva,se
R,d~
JI
'1f"t·to.'1.'~III)·
Itnct ic1tmrll}' ,))' block "umla''')
Re1evation; Dependence; Aging in Stock; Aging in Service; Far1ie-Gumbe1Morgenstern Distribution; Hierarchal Systems.
~-------20. ABSTRACT (Co/ltlm/C'
011
(('ve,,,,, sidl" II
.
'HH'(!~Sdl)'
-----------------_._----1
und IdentUy hy blf't'lf numhttr)
"Re1evation" is the name given by Krakowski (1973) (Re-". F'!'O.m;aise
Automat. Inform. Reaherahe Op., 7, Sere V-2, 108-120) to the distribution of
failure time of a replacement from an aging stock. The present authors
(.Tolmson and Kotz (1979; 1981) (IEEE T.ztan8. Re'LiabiUty, R-28, 292-299;
Ameriaan Journal of Mathematiaal and Management Scienae8, 1 (2))) have extended
this concept to include (i) hierarchal replacement systems and (ii) dependence
between lifetimes of original and replacement items. In this paper, we present
some further developments, including first steps towards a synthesis of (i) ~
DO FORM 1473 EDITION OF , NOV 6S IS OtlSOLETE
UNCLASSIFIED
I JAN 73
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