American Journal of Oper ations Research, 2011,1,8-15
doi: 10.4236/ajor.2011.11002 Published Online March 2011 (http://www.scirp.org/journal/ajor)
Copyright © 2011 SciRes. ..............................................................................................................................................................AJOR
Minimal Repair Redundancy for Coherent System
in its Signatures Representation
Vanderlei da Costa Bueno
Institute of Mathematics and Statistics, São Paulo University,Cx. Postal 66.281 05389-970, São Paulo, Brazil
Received February 24, 2011; revised March 20, 2011; accepted March 23, 2011
Abstract
In this paper we discuss how to maintain the signature representation of a coherent system through a minimal
repair redundancy. In a martingale framework we use compensator transforms to identify how the compo-
nents minimal repairs affect the order statistics in the signature representation.
Keywords: System signature; Dynamic system signature; Coherent systems; Point processes martingales.
1.Introduction
The signature of a coherent system, as in [1], with in-
dependent andidentically distributed component life-
times, as deifned by Samaniego, [2], is a vector whose
i-th coordinate is the probability that the I-th compo-
nent failure is fatal for the system.
The key feature of system signatures that makes
them broadly useful in reliabilityanalysis is the fact that,
in the context of independent and identically distrib-
uted (i.i.d.) absolutely continuous components lifetimes,
they are distribution free measures of system quality,
depending 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.
assumption used in the definition of system signatures.
By the way there are a host of applications in which the
i.i.d. assumption is appropriate, ranging from batteries
in lighting, to wafers or chips in a digital computer to
the subsystem of spark plugs in an automobile engine.
Formally the definition is:
LetT be the lifetime of a coherent system of order n,
with components lifetimes T1,…,Tn, which are inde-
pendent and identically distributed random variables
with absolutely continuous distributionF. Then the sig-
nature vectorαis defined as
1
(, , )
n

 (1)
whereαi= P(T= T(i)) and the

are the or-
der statistics of

n
()
, 1
i
Ti


, 1
i
Tin

 .
Clearly, under such conditions,


,1
i
TT in
is
an almost sure (P-a.s.) partition of the probability space,
with probability P, and



 


1
n
i
ii
t PT
PT 

 




ii i
PTTPTtT T
t
 
 (2)
Samaniego [3], Kochar, et al. [4] and Shaked and
Suarez-Llorens [5], extended thesignature concept to
the case where the components lifetimes T1,…,Tn, of a
system are exchangeable (i.e. the joint distribution
function, F(t1,...,tn), of (T1,…,Tn) is the same for any
permutation of (t1,...,tn), an interesting and practical
situation in reliability theory.
Concerning an improvement to system reliability, in
its signature representation, through a redundancy opera-
tion of its components, and in view of the identically
(exchangeable) distribution component lifetimes condi-
tions, to maintain a system with its structural relation



ii
PT t  PTt


, we choose to apply the
minimal repair redundancy. Intuitively the minimal re-
pair redundancy gives to component i an additional
lifetime as it had just before the failure. Clearly, in the
case of independent component lifetimes the whole
system is returned to the state it had just before the
failure.
A minimal redundancy of a lifetimeT produces the
sum T + S whereS is called spare lifetime and




PStT sPTtsTs 
(3)

V.C. BUENO 9
If the distribution function ofT is 



1
F
tF t
, the
resulting lifetime T hasthe distribution function S




1PT S tSt PT , where
  
 
0
d
ln
t
PT StPTtPT StTs Fs
Ft FtFt
 

(
4)
However in the context of system signature the ap-
proach of minimal repairs is not so clear: What are the
effects of the independent (exchangeable) component
lifetimes minimal repair in the ordered statistics and in
the signatures itself? To answer such a question we
consider dynamics signatures, as in a recent work by
Bueno [6], in a general set up, under a complete infor-
mation level, where the dependence (exchangeability)
can be considered and the redundancy operations can
be set through a compensator transform.
2. Dynamic System Signature
We consider, as in [6], the system evolution on time
under a complete information level. In this fashion, if
the components lifetimes are absolutely continuous, in-
dependent and identically distributed, the expected dy-
namic system signature enjoy the special property that
they are independent of both the distributionF and the
time t. This fact has significance beyond the mere sim-
plicity and tractability of the signature vector, reflect
only characteristics of the corresponding system design
and may be used as proxies for system designs in the
comparison of system performance. Also the dynamic
system signature actualizes itself under the system evo-
lution on time recovering the dynamical system signa-
ture in the set
 



1ii
TtT Tt

 


, as in Sama-
niego et al. [7] and the original coherent system signa-
ture in the set


n
Tt as in [2].
In our general setup, we consider the vector T1,…,Tn
of n component lifetimes which are finite and positive
random variables defined in a complete probability
space


,,P , with P(Ti≠Tj) = 1, for all in
,,ijij


1, ,En, the index set of components. The life-
times can be dependent but simultaneous failure are
ruled out.
In what follows, to simplify the notation, we assume
that relations such as between random
variables and measurable sets, respectively, always
hold with probability one, which means that the term
P-a.s., is suppressed.
,,,,
The evolution of components in time define a
marked point process given through the failure times
and the corresponding marks.
We denote by T(1)<T(2)<…< T(n) the ordered lifetimes
T1,T2,…,Tn, as theyappear in time and by



:
i ij
X
jT T the corresponding marks. As a con-
vention we set
 
12nn
TT


 and 12nn
X
Xe


where e is a fictitious mark not inE. Therefore the se-
quence



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,,0
11
i
Tss i tT
j
jnjE st
X



 

(5)
satisfies the Dellacherie ([8]),conditions of right conti-
nuity and completeness, and T is the system lifetime
1
min max
j
i
jkiK
TT
 
 (6)
where Kj, 1 ≤ j ≤ k are minimal cut sets, that is, a mi-
nimal set of components whose joint failure causes the
system fail.
Intuitively, at each time t the observer knows if the
events







,
ii
tjTt
TX
 
have either occurred
or not and if they have, he knows exactly the value
T(i)(T)and the mark Xi. We assumed that T1,…,Tn 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.
The simple marked point process 




,
,1ii
t
ij Xj
T
t
N


is an t

-submartingale and from the Doob-Meyer de-
composition we know that there exists a unique t

-
predictable process




,0 ij t
At
, called the t

-
compensator of



t

,ij
N, with 

,ij
A00 and such
that






,ij ,ijt
NA
tt

is an -martingale.



,ijt
A
is absolutely continuous by the totally inaccessibility of
Ti, 1 ≤ i ≤ n. We also define the lifetime T(i)j through its
process







,
ijiji i tt
tPt Ptj
FT TX


(7)
The compensator process is expressed in terms of
the conditional probability, given the available infor-
mation and generalize the classical notion of hazards.
Intuitively, this corresponds to producing whether the
failure is going to occur now, on the basis of all obser-
vations available up to, but not including, the present.
As



,ijt
N
 
can only count on the time interval

1, ii
TT

, th
e corresponding compensator differen

V.C. BUENO 10
tial



ij t
dA
failu
must vanish outside this interval. To count
thei-thre we let






1
iij
j
tt
NN

 with t

-
compensator process






1
iij
j
tt

.
AA
The t

-
compensator of



1jt
T
, correspond
e is

jt
Ning to t-
th components lif
hej
etim




1
.
ij
i
t
A


Follows that the -com
j
t
A
t
pensator o
is
(8)
which is an-predictable process and therefore
unique ( see Bremaud ([9])).
y,
re,Y, as the first time
fr
f



11
nn
ij
ij
Nt t
N





 
 1
11
1
ii
nn
ij TtT
ij
AtA t



t

Convenientlwe define the critical level of the
componentj for thei-th failu(i)j
om which onwards the failure of component j lead to
system failure at


,
ii
Tj
TX

. We consider the
t
-compensator process




0t of the point
cess

At

pro


1Tt
t
N
 
, of thefetimeT, such
 
tt
NA
 
 is an zero mmartingale
system li
that ean t
with
-






.Pt Et
NA
 


Bueno [6],
T tE
prove
Theorem 2.1.
s the following results:
notation, in the set Under the above


Tt, the t

-
compensator of



1Tt Nt
 
, is


nn
t


  
 
 1
11
1ii Tt T
ij
ijij ij
t
AA
AY




 


(9)
where


max, 0.a
a
Theorem 2.2.
time of a coherent system of ordern,
with component lifetimes T,…,T which are totally in-
LetT be the life
1n
accessible T1,…,Tn -stopping time. Then, under the
above notation and at complete information level, we
have




 
 1
1
11 1
,
1ii
nn t
i
i
TtT
ij t
i
PT TXj
PT T


 




(10)
with
Remarks 2.3.
i) In the case of independent and identically distributed
lifetimes we have

1n T.





 

1
1
11
1ii
ni
TtT t
ii
PT T
PTt
PT T







(11)
rro et al., [10], asked, it is plausible
to analyse the case of dependent and identically dis-
tributed lifetimes ( any way, its holds true for ex-
changeable distribution). In this case we have
ii) Clearly, it is not seemingly true to think the general
case of dependent components in the signatures context.
However, as Nava






 
 1
1
11
1ii
nit
TtT t
iit
PT T
PTt
PT T







(12)
Clearly, in the case of exchangeability, the expression
in i) is holding.
Corollary 2.4.
LetT be the lifetime of a coherent system of order n,
with component lifetimes T1,…,Tnwhich are independ-
ent and identically distributed with continuous distribu-
tionF. Then,



1i
1i
n
Tt ti
PTt 

 (13)
where












1
1
ii
i
ii
PT TPTT





(14)
PT TPTT
with T
 
01
0,, 0
ni
T



 and
Definition 2.5.
Let T be the lifetime of a coherent system
omponent lifetimes T1,…, which are indepen-
nd identically distributedom variles with
absolutely continuous distribution F. Then the dynamic
1
1.
n
i
i




of order n,
with cTn
dent ad ranab
signature vector

is defined as


,, 
1n

 (15)
where










1
1
ii
i
ii
PT TPTT
PT TPTT






and the
 i
T are
the order statistics of ,1 .
i
Tin



V.C. BUENO 11
Remarks 2.6.
We observe that


(16)
3. Minimal Repair and System Signature
It is well known that there exists a bijective relation be-
tween the space of all distributions functions and the
-compensators space characterized by the so called
Doléans exponential equation





 
  


 

1
1
11
1
i i
ij ij
n
TTXj
i
n
YTT ijij
j
n
ij ij
PT TE
E PYTT
FTFY





 










1
ij
j
1j
j
n
t







1
eπ
ct
tst
A
Ft As


(17)
where


ct
A is the continuous part of


A
t and






c
A
tAt t
A
  is its discrete part. Therefore, to
detect the effects of the independent (exchangeable)
times component minimal repairs in the ordered sta-
tistics and in the signatures βi
life
itself, we are going to
consider the minimal repair operation throug
sator transform, as in Bueno [11].
. he Firs
h compen-
3.1Tt Minimal Repair Operation
We are concerning with an improvement of the com-
ponent lifetime Ti through a transformation of thet

-
compensator process Ai(t) of the counting process


1i
Tt it
N
. The compensator process transform is
in the form
(18)
 
d,1
t
iii
tsAsin
B 


0
where 

0
iss
  is an t

-predictable procs.
Clearly, if

01s  
, the hazard process

es
i



d
ii
s
As 
is lower thanprocess

d
the hazard i
A
s and we under-
stand such improvement of the components i lifetime
as a redundancy operation.The main tool in
proach is the Girsanov Theorem which proof is in
2. in Appen
r particular cainima
r transfor
this ap-
[9](see Adix).
In ouse of ml repair (see [11]) the
compensatom is in the form
 

  

t
is
A

0
dln1
1i
ii i
i
B
tAstAt
A
As



in which case




(19)
 
1
1
ii
ii
A
t
LAT





(20)
it
N
i
AT
t

i
It is remarkable (Norros,[12]) that the continuous
components compensator processes at its final points,


,1
jj jn
AT

, are independent and identically dis-
m variables with standard exponential
distribution. This holds no matter how dependent the
actual lifetimes are and what the history, as lo
multaneous failures are ruled out. Therefore
tributed rando
ng as si


1
ii
EAT





and we have

d
iii
di Q
L
AT
P



(21)
It is well known (Arjas et al.[13]) that



ln
jj
tt
AF

where






1
jj t
ttPt
FT
F
  and the sur-
j
vival function of the component j after the cr
transform is
ompensato




 


ln 11ln
ee
jjj
ttt
jj
BAA
tt
FF

 
j
t
F
re ction.
s point we can ask how the in
ble) lifetimes component minim
. We remark that, in the following results,
the proofs are heavily based in the fact that
(2
2)
covering the expression of the first Se
At thidependent (ex-
changeaal repairs af-
fects the dependent ordered statistics and the signa-
turesβi i itself
t

-
martingales summation is a-martingale and th t
et

-
compensator is unique.
Lemma 3.1.1.
Let


k
A
t be the t

-compensator of



1k
Tt
k
Nt


where Tkis a totally inaccessible t
-stopping time rep-
resenting the lifetime of the componentk. Under the
minimal repair transform
 

 
0
d
1
t
k
As
t
Bk
k
k
A s
As

(23)
the t

-or cmpensatoof thei-th failure,



i
At
, un-
der k Q

, is transformed to
n




1
ii
j
tt
BB

j (24)
where






ij ij
tt
BA
 if and jk








,
0
d
1
tik
ik ik
i
As
k
B
tAs
A

 (25)
s

V.C. BUENO 12
Proof
We observe that the -compensator of thei-th
is set as:
t (26)
Also, the component-compensator c
form:
j
t
 failure








1
11
1ii
nn
iTtT
ij ij
jj
tAtA
A




t
an be set on the






1
1
1ii
nn
TtT
j ij
1ij
A
tAt A



 t (27)
e of a minimal repair transformation of the
component k, through its-compensator we have:

In the cas
t

 
    










1
1
1
d
1
l
l
l
l
t
tn
t
nlk
lk
llk
T
T
T
T
As As
As As
As







Therefore, the effect of the componentk minimal re-
pair compensator transform, in the compensator of the
i-th failure is through the i-th term of the last summa-
tion.
1
0
dd
11
kk
kk k
l
kk
Bt As As
As As





(28)
with

00T.















 
 
 


1
1
1
1,
d
11
i
i
ii
i
t
nk
Tt Tk
ij
jjk k
T
T
1
1,
d
1
i
ii
i
t
nik
Tij ik
jjk ik
T
T
t
B
As
1Tt
A
tAs
As










(29)
As
As
At As
As


















1
kk
EAT

, by Girsanov Theorem, under the
measure

d
d
k
kk
Q
AT
P

, B(i)(t) is the -compensator
of
t






1i
Tt
t  and the eff
i
Nect of a minimal repair
compensator transform, of the component k, in the
compensator of the i-th failure is

 






d
1
ik
ik
ik
ik
As
B
tAs
As

 (30)
In viewally (exchangeility) distribu-
tion compones conditions in the signatures
defist consider the minimal repair opera-
of the identic ab
nt lifetime
nition we mu
tions in all component lifetimes under the measure
Qδdefined by the Radon Nikodym derivative

1
d Qdπn
kk
k
P
A
T


 (31)
and we have, using Girsanov Theorem, the
resu
following
lt:
Corollary 3.1.2.
Let


j
A
t be the t

-compensator of



1j Tt jt
N


where Tj is a totally inaccessible t
-stopping time rep-
resenting the lifetime of the componentj. Under
minimal repair transform
the



 
d,1
t
j
As
BtAsj n
01
jj
j
As


 (32)

thet

-compensat



i
At
or of the i-th failure, , under
ans Qδ, isformed in
n
tr




1
ii k
k
tt
BB

 (33)
where








d
1
ik
ik i
ik
As

k
B
tA
As


otally inaccessible-stopping
time representing the lifetimes of ancomp
ent system with lifetimeT, which are abso
independent and identically distributed. Then,
under the minimal repair transformation of all compo-
nent lifetimes we have:
s
(34)
Theorem 3.1.3.
Let T1,T2,…,Tn be t t

onent coher-
lutely con-
tinuous



*
1
n
i
ii
Tt t QQ
T





 (35)
where










1 *
1
;
ii
ii
i
TT TT
QQ
TT TT QQ








(36)


 

 
 

 

 

k (37)

1
1ln
1ln
ii kikikik
k
ik ikik i
Y FY
FT FT




n
TTF Q








1
11
n
iik ik
k
tt
QTFF


 
t
(38)
and


1
ddπn
kk
k
P
Q
A
T



Proof
Firstly, we note that, underQδ, the lifetimes are in-
dependent:







 
11
11
11
1
,...,
ππ
11
ππ
1
π
kkkk
k
kk
nn
nn
kk kk
kk
n
nn
kk
kkk
k
tt
TT
kt
T
TtTt Q
EE
AT AT
tt QQ
TT
E

















k 
(39)

V.C. BUENO 13
),of
Also, as,1
j
Tjn are identically distributed, the
t
-compensatorsAj(t

1
j
Tt

are identical and we
conclude that theTi lifetimes are identically distributed
under Qδ. Therefore the signature decomposition under
Qδremains true.
Furthermore

(40)
The equivalence in the third equality is justified in
Norros [12] which defines theP a:s: inverse of
Aj(t),.As, under the hypothesis,




 









 



 
1
1
1
1,
1
1
π
π1
π1
1
i
i
i
jjij
jjij
jjij
iT
n
k
kkT
k
n
kk t
k
j
nn
kkj jt
kkj
j
jj t
j
QT
T
Tij
AAT
AAT
AAT
T
QTE
EAT
E
EAT
EE
AT AT
EAT

































1
n



11
k
T
k
n






 

1
1
π11
π1
ii
kk
T j
k
j
nn
TX
AT
EAT







1j
n








ln
jij
AtFt
where



 ijij t
tpt
FT

 we have:




 

 



 

 







1l n 1ln
ij
ij
ij ij
ijij ijij
TT
YY
FF FF
 
ln
ln
1
d
1
ij
ij
ij ij
jjT
Tx
jj T
T
F
YT Y
F
EAT
Exx
e
AT


 






(41)
Therefore

 

 







1
1ln 1ln
i
n
ij ijijijijij
j
TT Q
FYFYFTFT




 
(42)
Also, we have


(43)
And

i


















1
1
1,
1
1
1π1
π1
1
i
ii
jj ij
jj ij
nn
kk Tt
Tt k
j
nn
kkj jAT At
kkj
j
n
jj AT At
j
Q
Tt EAT QE
EATEAT
EAT









 
 
 



























ln
0
1
ed 11ln
jj
ij
ij
AT At
jj
tx
ij ij
F
EAT
x
xFt Ft







(44)
and







1
11
ln
n
ij ij
ij
itt
QTFF


 
t (45)
uccessive Minimal Repairs
More generally, we intend to define a measure
3.2. S
i
n
Q

n the which is obtained whenTi is deferred n times i
sense of minimal repair. The measures 0
i
Q

,
…,i
n
Q

1 are defined successively by 0
i
Q

P

and
i Q

,






1.. d
ii
i
nn
T
Pw QQ





w
bability measure
(46)
It can be proved that, for any n, the pro
i
n
Q

i is absolutely continuous with respect to P, with
Radon Nikodyn derivative

1
dd ! i
n
n
ii
P Q
A
T
n

 (47)
reasons follows:
pposhat we choose an w with pro dis-
We a
Sue tbability
tributionP and starting proceeding at time 0. Su
Ti occurs. In order to make a minimal repair, we have
fies
istribution among the candi-
dates satisfying these conditions. Indeed we choo
accordingto
ddenly
to change our w to another, say w
0 which is indistin-
guishable fromw strictly before the timeTi(w) and satis-
Ti(w0) >Ti(w). Moreover, w0 should be chosen ac-
cording to an appropriated d
se w0


. ,the value of the proc-
i
T
P


ess


.t P


at Ti, where




,,0
iit
TAtA t
T



 

(48)
is the history strictly before T
i Thus choosingw0 ac-
cording to


.i
T
P

 we may proceed further as if
nothing had happened.
Intuitively, ifS is an t

-stopping time, the difference
between theσ-algebras
s


and S
 is that, in
s


it is
known when but its not known what else
hap ime S. For ex
of a sys component, i
S occurs,
pen at tample, if S is the failure time
stemt is known in
s

, but at this
time, we do not known what component will fa
for some
il.
Indeed, as in Section 3.1, its holds forn = 1. Suppose
that its holds n fixed.We have to prove that:


1
1
1
1! i
Qi
SESAT
En



(49)
n
i
n



S,for any random variablei
T
-measurable.

V.C. BUENO 14







 



11 1
!
nn
ii
ii
n
TT
i
i
QQ
SESE ES
AT
EE n





 







0
1
1
!
1
1
1!
i
i
i
i
n
n
i
it
t
i
T
n
EES T
A
n












(50)
As the process
0!
1d
i
n
n
EE
SAt
t
A






0
d
1! i
EE
SAt
n




1d1
i
n
itTt
EE S
t
A







1
!i
n
it
ES
t
A
n

is -predictable,
the forth above equality is true.The sixth equality fol-
lows from Dellacheries integration formula.
t


  
11 11
1! 1!
i
nn
iiii
T
EES ES
AT AT
nn





 



 



(51)
Furthermore we have

 

  




1
0
e 1d
e
1
!!
i
i
t
nn
x
il
t
A
i n
iAT At
ii iA
t
A
x
Ex
AT
nni






 




1
11
!
!
n
i
ii
n
Tt
T
l
n
i
E QAT
n







(52)
and we conclude that the number of minimal repairs
occurring beforeTi is modeled by a doubly stochastic
Poisson process.
e realization of an equal and fi-
nite number of minimal repairs of each totally inacces-
sible stopping time representing the components
lifetes. Asthe continuous components compensator
processes at its final points,
ii
iit
Q
Tt E





Next we consider th
t
-
im

,1
ii
A
Ti
stributed random v
stribution, in the
n
, are inde-
pendt and identically diariables
withard exponential di case
en
stand of a
fixed configuration


,...,mmm, wherem is the num-
ber of minimal repairs of componenti, we can define a
product probability measure in ...

 
1
πi
m
mm
i
QQ


 (53)
where

1
dd i
m
m
i
i
P Q
A
T
m

.
Follows that, under the corrempensator
transform we can enunciate the Theorem.
Theorem 3.2.3.
Let T1,…,Tn be totally inaccessible t
-stopping time
representing the lifetimes of at c
spondin co
n componen ohere
system with lifetime T. Then, under the minimal
configuration m and under the probability measu
Where
gt
-
nt
repair
re Qm,
we have
 
1
i
i
Tt t
QQ
T


 (54)
n
mm
i










1
m
TT
Q
TT


In this paper we get resultsin a general set up in which
a coherent system can be set as a stochastic process in-
cluding the order statistics in the signature contex
fference between this results and the previous
one in the signature field is the dynamic aspect. In this
setting we can realize redundancy oprations and ask
about the reliability component importan
question is to clear out how a component operation af-
tics in the signature representation.
ansforms can help to answer such a
Applied probability. 25, 630 - 635.doi:10.2307/3213991
999). Stochastic Models in-
New York.
doi:10.1007/b97596
Life Testing: Probability models. Hold,
Inc. Silver Spring, MD.
981). Point Processes and Queues: Mar-
tingale Dynamics.Springer-Verlag, New York.
re of a coherent
ao PauloUniversity, S~ao Paulo,
*
1
ii
imm
ii
TT
QQ
TT




(55)
4. Conclusions
m
Q
t. The
main di
ce. The main
fect the order statis
The compensator tr
question. Note that, even if the components are inde-
pendent and identically distributed, the order statistics
are not. We conclude that, in the general setting, we
can develop some classical properties in reliability the-
ory.
5. Acknowledgements
This work was partially supported by FAPESP, Proc.
No. 2010/52227-0
6. References
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Reinhart and Wiston ,
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1. Stopping time
n extended and positive random variable τis an
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A-compensators


,1
iti
An. Let




0
it
t


with t

,
be n negative ont

-pre
all dr
all
dictable processes s
, an
uch that for
fo 0t
in1

,
 
0
t
ii
Bt s 


i
Asd, With








1,..., n
tt t   and


  .

t
L

Then


 
1
πe
iii
nttt
i i
i
NAB
T 



 (56)
is a noen negativ t

-l andocal martingale a non nega-
tive t

-sup
herm
d
er ma
ore,
by
rtin
if E
t
gale.
[L] =probabilFurt
define
δ 1, the
e Radon Ni
ity measur
m deriva-
A
A t

-
stopping time if, and only if,


t  
edictabl
for all
imeτis called pre if an i
0t
n
; an
creasing
t
-stopping t
sequence


0
nn

 of
ists such that
-stopping time, τn<τ, ex-
t

n


as ; an-stopping tim t
e

n
is totally inaccessible if


0P 

  for all p-
dictable t
-stopping time
re

. Fathematibasis
stochastic processes applied to reliability theory
or a mcal
of
of Avnde
Qδ
tive
h kody
ddPL
Q



is such that, underQδ,Bi(t) is
seethe booken a Jensen [14].
the
A2. Theorem A.2 (Girsanov)
Let Ti, 1 ≤ i ≤ n be totally inaccessible t

-
stoppingtimes, the point processes


unique t

-compen- i(t).
In particular, we denote
sator process of N




1, ,1,,1, ,1
ii




it
1
iT
t
N

, and dd i
iP QL



.