American Journal of Oper ations Research, 2011, 1, 172-179
doi:10.4236/ajor.2011.13019 Published Online September 2011 (http://www.SciRP.org/journal/ajor)
A Coherent System Component Importance under Its
Signatures Representation
Vanderlei da Costa Bueno
Institute of Mathematics and Sta tistics, São Paulo Universi ty , São Paulo, Brazil
Received April 5, 2011; revised April 29, 2011; accepted May 20, 2011
Abstract
In this paper we discuss how to measure the component importance for a system in its signature representa-
tion. The definition is given in terms of compensator transform and it can be considered as a new formaliza-
tion of the ideas presented by Bergman [1] in the context of system signature.
Keywords: System Signature, Dynamic System Signature, Coherent Systems, Component Importance, Point
Processes Martingales
1. Introduction
The signature of a coherent system with independent and
identically distributed component lifetimes, as defined by
Samaniego [2], is a vector whose i-th coordinate is the
probability that the i-th component failure is fatal for the
system. The key feature of system signatures that makes
them broadly useful in reliability analysis is the fact that,
in the context of independent and identically distributed
(i.i.d.) absolutely continuous components lifetimes, they
are distribution free measures of system quality, depend-
ing solely on the design characteristics of the system and
independent of the behavior of the systems components.
A detailed treatment of the theory and applications of
system signatures may be found in Samaniego [2]. This
reference gives detailed justification for the i.i.d. as-
sumption used in the definition of system signatures. By
the way there are some applications in which the i.i.d.
assumption is appropriate, ranging from batteries in light-
ing , to wafers or ch ips in a d igital co mputer to th e sub sys-
tem of spark plugs in an automobile engine.
Samaniego [3], Kochar, et al. [4] and Shaked and
Suarez-Llorens [5] extended the signature concept for
components exchangeable lifetimes, an interesting and
practical situation in reliability theory.
There seems to be two mains reasons for given an im-
portance measure of systems components. Firstly, it
permits the analyst to determine which component merits
the most additional research and development to improve
overall system reliability at minimum cost or effort.
Secondly, it may suggest the most efficient way to diag-
nose system failure by generating a checklist for an op-
erator to follows.
Birnbaum [6] defined the importance of a component
in a system (essentially) as follows. Let S and T denote
the random lifetimes of the component and the system,
respectively. Then the importance of S for T at time t is




,.
B
t
ISTPTtStPTtSt

 
Note, however, that this definition is meaningful for
any two lifetimes (the coherent system framework is not
essential).
This measure depends in a given point in time and it is
not quite relevant for most design or redesign decisions.
Several time independent importance measures have
been suggested, and most of then are weighted integrals
of


,
B
t
I
ST over t (e.g. Barlow and Proschan [7], Nat-
vig [8], for a su rvey see Bolland and El-Neweihi [9]).
Bergman [1] pointed out that many importance meas-
ures in reliability theory can be obtained through the
study of the change of the system expected lifetime due
to different variations of component lifetime distribu-
tions. Assume that the components are independent, and
let Fi and Gi, denote the original and the modified distri-
bution of component i with respect to this “component
improvement” is given as
   
0
,d
B
iit
GtFtISTt


,
where


1
ii
F
tF

. For example, in the case of Natvigs
importance measure, the improvement is obtained through

V. DA C. BUENO
173

nn
a minimal repair of the component in question. The Bar-
low and Proschan reliability importance is obtained from
an infinitesimal transformation.
Concerning the signature representation of the system
survival lifetime



1,
n
ii
i
PT tPTt 

 

where

are the ordered , the
system lifetime is a function of the order statistics and
the importance of the i-th failure for the system reliabil-
ity follows standard arguments. However in field opera-
tions, the system is a function of its components and it is
natural to ask about the component reliability importance
for system reliability in its signature representation, that
is: What is the reliability importance of the component j
for the system reliability at the i-th failure? The paper
motivation is to answer this main question. Recall that,
even if the components are independent and identically
distributed, the order statistics are not.

,1
i
Ti

,1
i
Ti
To detect the effects of the independent (exchangeable)
components distribution lifetimes transformation in the
ordered statistics and in the system signature representa-
tion itself, we are going to consider compensator trans-
forms.
It is well known that there exists a bijective relation
between the space of all distributions functions and the
t-compensators space characterized by the so called
Doléans exponential equation





1
c
At
tst
F
te As


 

where

c
A
t is the continuous part of


A
t and
 
c

A
tAt At is its discrete part. We consider
dynamics signatures, as in a recent work by Bueno [10],
in a general set up, under a complete information level
where the dependence (exchangeability) can be consid-
ered. In this work, in Section 2 we summarize the results
in dynamics signature from [10]. In Subsection 3.1 of
Section 3 we develop the Barlow and Proschan impor-
tance reliability under the signature representation and in
Subsection 2 we develop the component importance
through compensator transform.
2. Dynamic System Signature
In the following we are going to use the following nota-
tion;
T is the system lifetime.
j
T
T is the lifetime of component ,1 .jj 1n

i
T is the time of the i-th order statistics, .
in

ij
is the time of the i-th order statistics caused by
component ,1 ,.jijn
F
is the distribution function of the system lifetime.
j
F
is the distribution function of the j-th component
lifetime.

ij
F
is the standardized distribution function of .

ij
T
F
is the survival function of the system lifetime.
j
F
is the survival function of the j-th component life-
time.
 
1.
ij ij
F
F



1
,i
ii
TX is a marked point process.
i
X
is the Mark corresponding to .

i
T


0
tt
 is the filtration generated by .


1
,i
ii
TX





1i
iTt
Nt


 is the counting process of .
 i
T





1ij
ij Tt
Nt


 is the counting process of .

ij
T



i
A
t is the t

-compensa tor of .


i
Nt



ij
A
t is the t

-compensa tor of



ij
Nt.
In our general setup, we consider the vector


1,, n
TT
of n component lifetimes which are finite and positive
random variables defined in a complete probability space


,,P , with


1
ij
PT T

, for all ,,ijij

in


,,,
1, ,E
,
n, the index set of components. The lifetimes
can be dependent but simultaneous failures are ruled out.
To simplify the notation, we assume that relations such
as between random variables and measur-
able sets, respectively, always hold with probability one,
which means that the term Pa.s., is suppressed.

We consider, as in Bueno [10], the system evolution
on time under a complete information level. In this fash-
ion, if the components lifetimes are absolutely continu-
ous, independent and identically distributed, the expected
dynamic system signature enjoy the special property that
they are independent of both the distribution F and the
time t. Also the dynamic system signature actualizes it-
self under the system evolution on time recovering the
original coherent system signature in the set



n
Tt

as in Samaniego [2].
We denote by
  
12 n
TT T

 the ordered lifetimes
1, as they appear in time and by

,,
n
TT

:
ij i
X
jT T
the corresponding marks. As a convention we set
 
12nn
TT

 and 12nn
X
Xe
 where
e is a fictitious mark not in E. Therefore the sequence



1
,n
nn
TX defines a marked point process.
The mathematical formulation of our observations is
given by a family of sub

-Algebras of , denoted by 


0
tt
, where



1,1 ,,1,,0
i
ti Ts Ts ,
X
jinjE st 



 



satisfies the Dellacherie conditions of right continuity
and completeness, and T is the system lifetime

174 V. DA C. BUENO
1
min max,
j
j
kiK
TT
 
i
where ,1
j
K
jk are minimal cut sets, that is, a
minimal set of components whose joint failure causes the
system fail.
Intuitively, at each time t the observer knows if the
events

have either occurred
or not and if they have, he knows exactly the value

i and the mark

 
,i
i
TtXjTt 

i


TT
X
. We assumed that 1
are totally inaccessible t-stopping time. In a practical
sense we can think of a totally inaccessible t-stopping
time as an absolutely continuous lifetime. For a mathe-
matical basis in stochastic process martingale applied to
reliability theory see the book by Aven and Jensen [11].
,,
n
TT


The simple marked point pr ocess




,
1i
i
ij TtXj
Nt



is an t-submartingale and from the Doob-Meyer de-
composition we know that there exists a unique

t

-
predictable process , called the



0
ij t
At


t

-com-
pensator of , with and such that


ij
N

At
t

ij
A

00

ijij , is an t
-martingale.



Nt


ij
A
t is ab-
solutely continuous by the totally inaccessibility of i,
. We also define the lifetime through the
process
T
1in

ij
T







,
ti
ij iji
FtPTtPTtXj

,
t

j
of the i-th failure leads by the j-th component failure.
The compensator process is expressed in terms of the
conditional probability, given the available information
and generalizes the classical notion of hazards.
As can only count on the time interval


ij
Nt
,T



1ii
T
, the corresponding compensator differential


ij
dA t

Nt
must vanish outside this interval. Let
 

1
ii
j
N t


t

be the i-th failure point process
with -compensator process





1
ii
jj
A
tA


 j
Tt



1
jij
i
t. The
-compensator of , corresponding to the
j-th components lifetime, is
t


1
j
Nt
A
tAt

.
Conveniently, we define the critical level of the com-
ponent j for the i-th failure,

ij
, as the first time from
which onwards the failure of component j lead to system
Y
failure at



,i
i
TTX j. We consider the t

-com-
pensator process




0t
At

 of the point process


1Tt Nt
 

 
Nt At


, of the system lifetime T, such that
is a zero mean t

-martingale with
 


t.NtE A




PT tE
Theorem 2.1 Under the above notation, in the set


Tt, the t

-compen sator of



1Tt Nt




, is



 

 
 1
11 1,
ii
ijij ijTtT
ij
AtAt A Y







nn
where


max ,0aa
.
Theorem 2.2 Let T be the lifetime of a coherent sys-
tem of order n, with component lifetimes 1
which are totally inaccessible 1 t-stopping
time. Then, under the above notation and at complete
information level, we have
,,
n
TT
,,
n
TT







 1
11
,1,
ii
nn it
i
tTtT
ij t
i
PT TXj
PT tPT T



 

with

1n
T

.
Remarks 2.1 1) In the case of independent and iden-
tically distributed lifetimes we have





 
 1
11.
ii i
nii
tTtT
ii
PT T
PT tPT T



 

2) Clearly, it is not seemingly true to think the general
case of dependent components in the signatures context.
However, as Shaked et al., [5], asked, it is plausible to
analyse the case of dependent and identically distributed
lifetimes (any way, its holds true for exchangeable dis-
tribution). In this case we have






 

1
11.
ii i
nt
ii
tTtT
it
ii
PT T
PT tPT T



 

Clearly, in the case of exchangeability, the expression
in 1) is holding.
Corollary 2.1 Let T be the lifetime of a coherent sys-
tem of order n, with component lifetimes 1
which are independent and identically distributed with
continuous distribution F. Then,
,,
n
TT



1
1i
n
ti
Tt
i
PT t 




where









1
1
,
ii
i
ii
PT TPT T
PT TPT T






with
 
01
0, ,
n
TT


 and
11.
n
i
i




Definition 2.1 Let T be the lifetime of a coherent sys-
tem of order n, with component lifetimes 1
which are independent and identically distributed ran-
dom variables with absolutely continuous distribution F.
,,
n
TT


Bueno [10], proves
the following results:

V. DA C. BUENO
175
Then the dynamic signature vector

is defined as

1,, n

 
if









1
1
0,
ii
ii
T
T


i
PT TPT
PT TPT


 

n and the are the order statistics of .

i
T,1
i
Ti
3. Component Importance
3.1. The Barlow and Proschan Component
Importance
Considering the classical Barlow and Proschan compo-
nent importance reliability for system reliability, as in [7],
the importance of the component i is the probability that
the system failure coincides with the failure of the i-th
component, that is




0
d
i
iii
IiPT T
PTsTsPTTsF s





s
in which case, under the iid lifetimes assumption, we
have the Barlow and Proschan structural reliability, it
should be natural to measure the reliability importance of
the i-th failure by




i
i
IiPT T

 in a similar
fashion.
However we are addressing another question. As be-
fore system realization we only know the component
lifetimes (we does not know the order statistics), the im-
portant question about the component reliability struc-
tural importance for system reliability in its signature
representation is: What is the reliability importance of
the component j for the system reliability at the i-th fail-
ure? We consider the following definition.
Definition 3.1.1 Let T be the lifetime of a coherent
system of components identically distributed and let
be the i-th failure lifetime caused by the component

ij
T
j, with -compensator process . The reli-
t




0
ij t
At

ability importance of component j to system reliability at
the i-th failures is
 

 

,
ijijij ij
IEATAY







Where

ij
is the critical level of the component j for
the i-th failure.
Y
The reliability importance of the i-th failure to system
reliability is


1
n
ij
j
I
iI


Proposition 3.1.1 Let T be the lifetime of a coherent
system of components independent and identically dis-
tributed and let

ij
be the i-th failure lifetime caused
by the component , with -compensator process
Tjt





0
ij t
At
 where


 

ln
ijij ij
At FtT 

and






1
ij ij
F
tFt , (see Arjas and Yashin [12]). Then,
the reliability importance of component j to the system
reliability at the i-th failures is
 


.
ij ij
IPTT
Proof By Fubini Theorem we can write
 

 



 


























0
00
0
0
1dln
d
1d
d
d1
d
1
1
ij
ij
ij
ij
ijijij ij
ij ij
YsT
tij
ij
YsTij
ij
ij YsT
sij
ij
ij YsTij
Y
IEATAY
EFsT
Fs
EF
Fs
Fs
EFt Fs
Fs
EF sFs
E















t











 









































 









0
0
0
d
d
d
.
ij ij
sT
ij ijij
ij ij
ij
Fs
PTTTs Fs
PT TFs
PT T







 





As in the following proposition, we note that the iden-
tically and independent conditions are not essential.
Proposition 3.1.2 Let T be the lifetime of a coherent
system of components independent and identically dis-
tributed and let

ij
be the i-th failure lifetime caused
by the component j, with
T
t

-com pensator pro cess




0
ij t
At
.
Then, the reliability importance of component j to the
system reliability at the i-th failures is
 



0
d.
ijij ijij
I
PTTTs Fs

 

Proof As the process is -predictable, we


1ij
YsT t


176 V. DA C. BUENO
have









 



 



0
0
0
1d
1d
d.
ij
ij
ij ij
YsT
ij
YsT
ij ijij
ij ijij
IE As
ENs
PYTTPTT
PTTTs Fs












 



Example 3.1.1 Let T be the system lifetime given by
of three Components independent and
identically distributed lifetimes with absolutely continu-
ous distribution function F. Follows that

123
TT TT 

.
 

1112 13
23 3 2
2122 23
0; ;
;;
YYY
YT TYTYT


Therefore
 

 








111111 11
0110 11
00
123
11
1d 1d
1.
3
sT sT
IEATAY
EAsEN
PTTPT TT


 







s
  
 



  

 




23
23
21 21
0
21
0
23 21
23123
1d
1d
1
3
TTsT
TTsT
IE As
EN
PT T TT
PT T TT T













s
  




3
22 22
0
332
22
1d
1
6
TsT
IE As
PT TTPT TT







1
Using symmetry we have
 
23 22
1.
6
II We also
have
 


111
1
3
IPTTI
1
and
 

 
22
21 2223
2111
3366
.
IPTT
III


3.2. Component Importance under
Compensator Transforms
To work in dependence conditions we consider a prob-
ability space


,,P and a family of sub

-algebras


0
tt of

as in Section 2, which is increasing, right
continuous and completed (shortly: satisfies the Del-
lacheries usual conditions).
Let


k
A
t be the t

-compensator of



1k
kTt
Nt



where k
T
is a totally inaccessible t-stopping time rep- 
resenting the lifetime of the component k. Let




0
kt
t 

be a t

-predictable non-negative process, interpreted as
a hazard transformation process for .
k
Usually T


kt

k
can be associated with an improve-
ment of . Define T
 
0
d
t
ksk
Bt As  
such that


k
Bt

. Under certain conditions, it is pos-
sible to find a new probability measure k P

such that


k
Bt is the t

-compensator of under

k
N

tk P

.
Indeed, assume that the proc ess
 

  
exp
k
Nt
kkk kk
LtTAt Bt  
is uniformly integrable. Then, it follows from well known
results on point process martingales, that


k
Lt is an
t

-martingale with expectation 1, and the desired meas-
ure k P

is given by

d.
dk
t
k
PLt
P






(See Jacod [13], Prop. 4.3 and Th 4.5; simple adaptation
can be found in Ar jas a n d N o rr os [1 4].)
At this point we can ask what are the effects of the in-
dependent (exchangeable) components distribution life-
times transformation in the dependent ordered statistics
and in the system signature representation itself. We re-
mark that, in the following results, th e proofs are heavily
based in the fact that t

-martingales summation is an
t

-martingale and the t

-compensator is unique.
Theorem 3.2.1 Let


ktN and


k
A
t be as above.
Under the compensator transform
 
0
d
t
ksk
BtAs 



and under k P

, the t

-compensator of the i-th failure,


i
A
t, is transformed to



1
n
iij
j
BtB t


where

V. DA C. BUENO
177





ij ij
BtAt jk
and

 


0
d
t
k
ik ik
Bt sAs  .
Proof We observe that the t

-c
1
.
ompensator of the i-th
failure is set as



 



1
1
1
ii
nn
iijij
TtT
jj
A
tAtA





t
Also, the component -compensator can be set in
the form: t




 



 


1
1
11
1
1
1.
ii
ii
nn
jij ij
TtT
ii
n
j
TtT
i
A
tAt A
At










t
In the case of the k

-compensator transform we
have t

 






1
1
0
d
d,
l
l
Tt
tn
kkk kk
lT
lk lk
Bt sAssAs
sA s










d
. with , where

00T


 


1
1
ll
k
lk TsT
s
s 

 
Therefore, the effect of the k

compensator trans-
form in the compensator of the ith failure is through the
i-th term of the last summation.




 









11
1
1
1d
i
ii i
n
iij
j
Tt
n
ijik ik
TtTjk T
BtBt
Bt sAs




 








Assuming that is uniformly integrable,

k
Lt

1
kk
EL T



and, under k
P

with

d
d
k
t
k
PLt
P






is the -compensator of
t






1i
Tt


i
Nt and the effect
of the k

compensator transform, in the compensator
of the i-th failure is






0
d.
t
ikik ik
Bt sAs  
Based in the Bergman [1] notion of reliability impor-
tance and in Theorem 3.1.1, we can define.
Definition 3.2.1 Let T and k be the system and the
k-th component lifetimes respectively and let k
TP

be
defined as above (assuming that is uniformly
integrable). Suppose further that k is finite and T is
integrable. Then the

k
Lt
T
k

importance of Tk for T in



i
TT

is




 



,
kkk
ik
IE ETOVTL
COVTT

,
PP
k
T
L
i 
C

 

where

ij
is the lifetime of the i-th failure caused by
the j-th component failure.
T
T

Examples 3.2.1 1) Consider the compensator change
which arises from exactly one minimal repair of compo-
nent k (Natvig [8]). Intuitively this means that when k
occurs, the system is returned to the state in which it was
immediately before k occurred. The second occur-
rence of k is considered as final, and we take it as the
improved value of k T. This improvement operation
corresponds to the compensator transformation (Norros
[15])
T
T


  
At




0
dln1.
1
tkkk
k
As As At
As



kk

Bt
In this case
1
k
kk
A
s
s
A
s

t

A





ik


exp
Nt
IC
and therefore

kk
LT
 


k
kkk kkk
TABt AT 
and, therefore,
 


,
k
iik
OVTT

 .
2) Bueno and Carmo [16] propose a parallel transform
for the case of dependent components, through compen-
sator transform as
 

22e
2e
ln


0
xp d
xp
2 ex.
tk
kk
k
kk
Bt As
At At






 


p
As
As





That is, we consider the transformations where
 

22
2e
exp ,
xp
k
k
k
A
s
s 
 



As


A

22exp
kk kk
LT T


and
 



,2OVT2exp
k
ii
IC AT


  
,1
i
kik
n
.
The case of deterministic compensators
Under the assumptions that Ti are totally ,

178 V. DA C. BUENO
inaccessible t-stopping time, the t-compensators
are continuous and from Arjas and Yashin [12] we con-
clude that




 


ik
Tln .
ik
At Ft 
ik

If

ik

A
t
T
T
is a determinist increasing function of t
(except that it is stopped when

ik occurs) we could
say that

ik is dynamically independent of everything
else in the history t. This is a generalization of the
case of independent components: if the component life-
time

ik has a continuous compensator which is de-
terministic, then the lifetimes of the others components
have no causal effect in

ik. However, other compo-
nents may well dependent casually on
 ik, so that the
compone nt s n eed not be statis t i c al l y independent.
T

TT
 Lemma 3.2.1 Assume that k is a -measur-
able density L

k
T



1
k

EL . Denote





ik ik
F
tPTt
and





.
Lk
ik ik
F
tPT
t
Suppose that


ET  and Is finite. Then

ik
T











ik

,d.
kk
i
B
ikik t
LTET
0
IE
F
tFt Tt
I



 T


Proof





















0
0
0
0
d
d
d
d.
k
ik ik
ik ik
ik ik
ik ikik
ik
ELTETTELPTsTt
PTtT t
FtPTtTtt
kk
EL
Ft
F
tP
t
TtTtPTtTtt
PTTt t

























The last integral is finite by the assumptions. The re-
sults follow by applying the above formula also for
and subtracting. 1
k
L
Examples 3.2.2 1) As in example 3.3 1) consider the
compensator change which arises from exactly one
minimal repair. Follows that, the compensator transform
of


ik
A
t is
  
ln 1
kk k
t AtAtB.
As


 



ln
ikik ik
At FtT ,
in the set



ik
Tt, we have












ln1 ln
ln1 ln
ik ikik
ik ik
Bt FtFt
FtFt


 




 


and










expln .
ikikik ikik
F
tAtFtFtF

 
 t
Therefore
 





0
ln,d .
k
B
t iikikik
I
FtFtITTt



2) In the case of Example 3.3 2) we have a parallel
transform which transform


ik
A
t to
 

ln 2exp
kk k
Bt AtAt

Now, in the set



ik
Tt, we have













lnln 2
ln
ik ikik
ikik ik
Bt FtFt
F
tFtFt


 




 


Therefore








ikikik ik
F
tFtFtFt






and
 





0
,d
k
B
t
iikik ik
.
I
FtFtITTt



4. Acknowledgements
This work was partially supported by FAPESP, Proc. No.
2010/52227-0.
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