Embed Size (px)
Citation preview
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Sto
ch
ast
icPro
pe
rtie
sa
nd
Infe
ren
ce
for
Re
pa
ira
ble
Sy
ste
ms
Nu
ria
Torr
ad
o
De
pa
rtm
en
to
fSt
atist
ics
an
dO
pe
ratio
ns
Re
sea
rch
Un
ive
rsid
ad
Pu
blic
ad
eN
ava
rra
,Sp
ain
Jun
e4,2012
Sem
ina
ra
tTh
eB
asq
ue
Ce
nte
rfo
rA
pp
lied
Ma
the
ma
tic
s(B
CA
M),
Bilb
ao
,Sp
ain
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
1
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Ou
tlin
e
1In
tro
du
ctio
nM
otiva
tio
nR
elia
bili
tym
ea
sure
sSt
oc
ha
stic
co
un
tin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
sN
etw
ork
s
2St
oc
ha
stic
co
mp
ariso
ns
of
spa
cin
gs
ba
sed
on
ord
er
sta
tist
ics
De
finitio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
3B
aye
sia
nIn
fere
nc
eP
relim
ina
rie
sA
ne
wa
pp
roa
ch
toSR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
4C
on
clu
sio
ns
an
dc
on
trib
utio
ns
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
2
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Pre
limin
ary
De
fin
itio
ns
Re
liab
ility
isd
efin
ed
as
the
pro
ba
bili
tyth
at
asy
ste
mw
illp
erf
orm
its
in-
ten
de
dfu
nc
tio
nu
nd
er
spe
cifi
ed
de
sig
nlim
its.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
3
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Pre
limin
ary
De
fin
itio
ns
Re
liab
ility
isd
efin
ed
as
the
pro
ba
bili
tyth
at
asy
ste
mw
illp
erf
orm
its
in-
ten
de
dfu
nc
tio
nu
nd
er
spe
cifi
ed
de
sig
nlim
its.
Asy
ste
mc
an
be
de
fine
da
sa
co
llec
tio
no
ftw
oo
rm
ore
pa
rts
wh
ich
is
de
sig
ne
dto
pe
rfo
rmo
ne
or
mo
refu
nc
tio
ns.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
3
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Pre
limin
ary
De
fin
itio
ns
Re
liab
ility
isd
efin
ed
as
the
pro
ba
bili
tyth
at
asy
ste
mw
illp
erf
orm
its
in-
ten
de
dfu
nc
tio
nu
nd
er
spe
cifi
ed
de
sig
nlim
its.
Asy
ste
mc
an
be
de
fine
da
sa
co
llec
tio
no
ftw
oo
rm
ore
pa
rts
wh
ich
is
de
sig
ne
dto
pe
rfo
rmo
ne
or
mo
refu
nc
tio
ns.
Asy
ste
mis
are
pa
irab
lesy
ste
mw
he
nit
ca
nb
ere
sto
red
tofu
llysa
tisf
ac
tory
pe
rfo
rma
nc
eb
ya
ny
me
tho
d,o
the
rth
an
rep
lac
em
en
to
fth
ee
ntire
syst
em
,a
fte
rfa
ilin
gto
pe
rfo
rmo
ne
or
mo
reo
fits
fun
ctio
ns
satisf
ac
torily
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
3
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Pre
limin
ary
De
fin
itio
ns
Re
liab
ility
isd
efin
ed
as
the
pro
ba
bili
tyth
at
asy
ste
mw
illp
erf
orm
its
in-
ten
de
dfu
nc
tio
nu
nd
er
spe
cifi
ed
de
sig
nlim
its.
Asy
ste
mc
an
be
de
fine
da
sa
co
llec
tio
no
ftw
oo
rm
ore
pa
rts
wh
ich
is
de
sig
ne
dto
pe
rfo
rmo
ne
or
mo
refu
nc
tio
ns.
Asy
ste
mis
are
pa
irab
lesy
ste
mw
he
nit
ca
nb
ere
sto
red
tofu
llysa
tisf
ac
tory
pe
rfo
rma
nc
eb
ya
ny
me
tho
d,o
the
rth
an
rep
lac
em
en
to
fth
ee
ntire
syst
em
,a
fte
rfa
ilin
gto
pe
rfo
rmo
ne
or
mo
reo
fits
fun
ctio
ns
satisf
ac
torily
.
Ac
om
pu
ter
syst
em
co
nsi
sts
oftw
om
ajo
rc
om
po
ne
nts
:h
ard
wa
rea
nd
soft
-w
are
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
3
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
Tod
ay,
co
mp
ute
rsy
ste
ms
are
em
be
dd
ed
ina
irtr
affi
cc
on
tro
l,n
uc
lea
rre
ac
tors
,a
ircra
ft,
rea
l-tim
ese
nso
rn
etw
ork
s,in
du
stria
lp
roc
ess
co
ntr
ol,
au
tom
otive
me
ch
an
ica
lan
dsa
fety
co
ntr
ol,
an
dh
osp
ita
lhe
alth
ca
re,a
mo
ng
oth
ers
.
With
inth
ela
std
ec
ad
eo
fth
e20th
ce
ntu
rya
nd
the
first
few
ye
ars
of
the
21st
ce
ntu
ry,
the
de
ma
nd
for
co
mp
lex
soft
wa
resy
ste
ms
ha
sin
-c
rea
sed
,a
nd
the
refo
re,
the
relia
bili
tyo
fso
ft-
wa
resy
ste
ms
ha
sb
ec
om
ea
ma
jor
co
nc
ern
for
ou
rm
od
ern
soc
iety
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
4
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
Tod
ay,
co
mp
ute
rsy
ste
ms
are
em
be
dd
ed
ina
irtr
affi
cc
on
tro
l,n
uc
lea
rre
ac
tors
,a
ircra
ft,
rea
l-tim
ese
nso
rn
etw
ork
s,in
du
stria
lp
roc
ess
co
ntr
ol,
au
tom
otive
me
ch
an
ica
lan
dsa
fety
co
ntr
ol,
an
dh
osp
ita
lhe
alth
ca
re,a
mo
ng
oth
ers
.
With
inth
ela
std
ec
ad
eo
fth
e20th
ce
ntu
rya
nd
the
first
few
ye
ars
of
the
21st
ce
ntu
ry,
the
de
ma
nd
for
co
mp
lex
soft
wa
resy
ste
ms
ha
sin
-c
rea
sed
,a
nd
the
refo
re,
the
relia
bili
tyo
fso
ft-
wa
resy
ste
ms
ha
sb
ec
om
ea
ma
jor
co
nc
ern
for
ou
rm
od
ern
soc
iety
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
4
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
Tod
ay,
co
mp
ute
rsy
ste
ms
are
em
be
dd
ed
ina
irtr
affi
cc
on
tro
l,n
uc
lea
rre
ac
tors
,a
ircra
ft,
rea
l-tim
ese
nso
rn
etw
ork
s,in
du
stria
lp
roc
ess
co
ntr
ol,
au
tom
otive
me
ch
an
ica
lan
dsa
fety
co
ntr
ol,
an
dh
osp
ita
lhe
alth
ca
re,a
mo
ng
oth
ers
.
With
inth
ela
std
ec
ad
eo
fth
e20th
ce
ntu
rya
nd
the
first
few
ye
ars
of
the
21st
ce
ntu
ry,
the
de
ma
nd
for
co
mp
lex
soft
wa
resy
ste
ms
ha
sin
-c
rea
sed
,a
nd
the
refo
re,
the
relia
bili
tyo
fso
ft-
wa
resy
ste
ms
ha
sb
ec
om
ea
ma
jor
co
nc
ern
for
ou
rm
od
ern
soc
iety
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
4
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
Tod
ay,
co
mp
ute
rsy
ste
ms
are
em
be
dd
ed
ina
irtr
affi
cc
on
tro
l,n
uc
lea
rre
ac
tors
,a
ircra
ft,
rea
l-tim
ese
nso
rn
etw
ork
s,in
du
stria
lp
roc
ess
co
ntr
ol,
au
tom
otive
me
ch
an
ica
lan
dsa
fety
co
ntr
ol,
an
dh
osp
ita
lhe
alth
ca
re,a
mo
ng
oth
ers
.
With
inth
ela
std
ec
ad
eo
fth
e20th
ce
ntu
rya
nd
the
first
few
ye
ars
of
the
21st
ce
ntu
ry,
the
de
ma
nd
for
co
mp
lex
soft
wa
resy
ste
ms
ha
sin
-c
rea
sed
,a
nd
the
refo
re,
the
relia
bili
tyo
fso
ft-
wa
resy
ste
ms
ha
sb
ec
om
ea
ma
jor
co
nc
ern
for
ou
rm
od
ern
soc
iety
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
4
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
Tod
ay,
co
mp
ute
rsy
ste
ms
are
em
be
dd
ed
ina
irtr
affi
cc
on
tro
l,n
uc
lea
rre
ac
tors
,a
ircra
ft,
rea
l-tim
ese
nso
rn
etw
ork
s,in
du
stria
lp
roc
ess
co
ntr
ol,
au
tom
otive
me
ch
an
ica
lan
dsa
fety
co
ntr
ol,
an
dh
osp
ita
lhe
alth
ca
re,a
mo
ng
oth
ers
.
With
inth
ela
std
ec
ad
eo
fth
e20th
ce
ntu
rya
nd
the
first
few
ye
ars
of
the
21st
ce
ntu
ry,
the
de
ma
nd
for
co
mp
lex
soft
wa
resy
ste
ms
ha
sin
-c
rea
sed
,a
nd
the
refo
re,
the
relia
bili
tyo
fso
ft-
wa
resy
ste
ms
ha
sb
ec
om
ea
ma
jor
co
nc
ern
for
ou
rm
od
ern
soc
iety
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
4
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
Tod
ay,
co
mp
ute
rsy
ste
ms
are
em
be
dd
ed
ina
irtr
affi
cc
on
tro
l,n
uc
lea
rre
ac
tors
,a
ircra
ft,
rea
l-tim
ese
nso
rn
etw
ork
s,in
du
stria
lp
roc
ess
co
ntr
ol,
au
tom
otive
me
ch
an
ica
lan
dsa
fety
co
ntr
ol,
an
dh
osp
ita
lhe
alth
ca
re,a
mo
ng
oth
ers
.
With
inth
ela
std
ec
ad
eo
fth
e20th
ce
ntu
rya
nd
the
first
few
ye
ars
of
the
21st
ce
ntu
ry,
the
de
ma
nd
for
co
mp
lex
soft
wa
resy
ste
ms
ha
sin
-c
rea
sed
,a
nd
the
refo
re,
the
relia
bili
tyo
fso
ft-
wa
resy
ste
ms
ha
sb
ec
om
ea
ma
jor
co
nc
ern
for
ou
rm
od
ern
soc
iety
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
4
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
hu
ma
ne
rro
r
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
5
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
hu
ma
ne
rro
rfa
ult/b
ug
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
5
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
hu
ma
ne
rro
rfa
ult/b
ug
failu
re
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
5
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
The
ne
ed
for
relia
ble
soft
wa
re
hu
ma
ne
rro
rfa
ult/b
ug
failu
re
So
ftw
are
relia
bili
tyis
the
pro
ba
bili
tyo
ffa
ilure
-fre
eso
ftw
are
op
era
tio
nfo
ra
spe
cifi
ed
pe
rio
do
ftim
ein
asp
ec
ifie
de
nviro
nm
en
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
5
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Typ
ica
lfa
ilure
his
tory
ofa
soft
wa
rep
rog
ram
Si
isth
ei’
thso
ftw
are
failu
re.
Ti
isth
etim
eb
etw
ee
nth
ei’
thso
ftw
are
failu
rea
nd
the(i−
1)’
thso
ftw
are
failu
re.
0
t 1t 2
············
t i−
1t i
s 1s 2
s i−
2s i−
1s i
We
ass
um
eth
at
at
tim
eze
roth
ep
rog
ram
isru
no
nth
ec
om
pu
ter
an
dw
ork
ssa
tisf
ac
torily
un
tilt
ime
s 1,w
he
nth
efir
stfa
ilure
oc
cu
rs.
The
pro
gra
mm
er
the
nre
pa
irs
the
pro
gra
m,it
wo
rks
satisf
ac
torily
for
tim
et 2
,it
isre
pa
ired
ag
ain
,a
nd
soo
n.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
6
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Re
liab
ility
me
asu
res
Let
Xb
eth
elif
etim
eo
fa
syst
em
or
an
un
ita
nd
let
Fd
en
ote
the
dis
trib
utio
nfu
nc
tio
no
fX
.
Re
liab
ility
Fun
ctio
n(s
urv
iva
lfu
nc
tio
n):
the
pro
ba
bili
tya
un
itsu
rviv
es
be
yo
nd
tim
et.
F(t)=
1−
F(t)=
P(X
≥t).
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
7
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Re
liab
ility
me
asu
res
Let
Xb
eth
elif
etim
eo
fa
syst
em
or
an
un
ita
nd
let
Fd
en
ote
the
dis
trib
utio
nfu
nc
tio
no
fX
.
Re
liab
ility
Fun
ctio
n(s
urv
iva
lfu
nc
tio
n):
the
pro
ba
bili
tya
un
itsu
rviv
es
be
yo
nd
tim
et.
F(t)=
1−
F(t)=
P(X
≥t).
Ha
zard
rate
fun
ctio
n(f
ailu
rera
tefu
nc
tio
n):
rep
rese
nts
the
inst
an
tan
eo
us
pro
ba
bili
tyth
at
an
ite
mw
illfa
il,g
ive
nth
at
itsu
rviv
ed
un
tilt
ime
t.
h(t)=
lim
∆t→
0
P(t<
X≤
t+∆
t|X
>t)
∆t
.
No
teth
at
h(t)≈
P(X
≤t+
∆t|X
>t),
for
sma
ll∆
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
7
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Re
liab
ility
me
asu
res
Let
Xb
eth
elif
etim
eo
fa
syst
em
or
an
un
ita
nd
let
Fd
en
ote
the
dis
trib
utio
nfu
nc
tio
no
fX
.
Re
liab
ility
Fun
ctio
n(s
urv
iva
lfu
nc
tio
n):
the
pro
ba
bili
tya
un
itsu
rviv
es
be
yo
nd
tim
et.
F(t)=
1−
F(t)=
P(X
≥t).
Ha
zard
rate
fun
ctio
n(f
ailu
rera
tefu
nc
tio
n):
rep
rese
nts
the
inst
an
tan
eo
us
pro
ba
bili
tyth
at
an
ite
mw
illfa
il,g
ive
nth
at
itsu
rviv
ed
un
tilt
ime
t.
h(t)=
f(t)
F(t)=−
∂ ∂t
lnF(t).
Inte
gra
tin
ga
nd
exp
on
en
tia
tin
gb
oth
sid
es
of
the
pre
ce
din
gg
ive
su
sth
ee
xpo
ne
ntia
tio
nfo
rmu
lao
fre
liab
ility
F(t)=
exp
{−∫
t
0h(x)d
x
}=
e−H(t) ,
wh
ere
H(t)
isth
ec
um
ula
tive
ha
zard
rate
fun
ctio
n.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
7
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Re
liab
ility
me
asu
res
Let
Xb
eth
elif
etim
eo
fa
syst
em
or
an
un
ita
nd
let
Fd
en
ote
the
dis
trib
utio
nfu
nc
tio
no
fX
.
Re
liab
ility
Fun
ctio
n(s
urv
iva
lfu
nc
tio
n):
the
pro
ba
bili
tya
un
itsu
rviv
es
be
yo
nd
tim
et.
F(t)=
1−
F(t)=
P(X
≥t).
Ha
zard
rate
fun
ctio
n(f
ailu
rera
tefu
nc
tio
n):
rep
rese
nts
the
inst
an
tan
eo
us
pro
ba
bili
tyth
at
an
ite
mw
illfa
il,g
ive
nth
at
itsu
rviv
ed
un
tilt
ime
t.
h(t)=
f(t)
F(t)=−
∂ ∂t
lnF(t).
Tim
e
Hazardrate
Ea
rly
life
Use
fullif
eW
ea
r-o
ut
(a)
Ha
rdw
are
syst
em
Tim
e
Hazardrate
(b)
Soft
wa
resy
ste
m
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
7
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Ac
ou
ntin
gp
roc
ess
{ N(t),
t≥
0}
isa
no
nh
om
og
en
eo
us
Po
isso
np
roc
ess
,N
HP
P,
with
me
an
va
lue
fun
ctio
nΛ(t)
an
din
ten
sity
fun
ctio
nλ(t)
if
a){ N
(t),
t≥
0}
ha
sth
eM
ark
ov
pro
pe
rty,
b)
P(N
(t+
∆t)=
n+
1|N
(t)=
n)=
λ(t)∆
t+o(∆
t),
n≥
1,
c)
P(N
(t+
∆t)>
n+
1|N
(t)=
n)=
o(∆
t),
n≥
1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
8
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Ac
ou
ntin
gp
roc
ess
{ N(t),
t≥
0}
isa
no
nh
om
og
en
eo
us
Po
isso
np
roc
ess
,N
HP
P,
with
me
an
va
lue
fun
ctio
nΛ(t)
an
din
ten
sity
fun
ctio
nλ(t)
if
a){ N
(t),
t≥
0}
ha
sth
eM
ark
ov
pro
pe
rty,
b)
P(N
(t+
∆t)=
n+
1|N
(t)=
n)=
λ(t)∆
t+o(∆
t),
n≥
1,
c)
P(N
(t+
∆t)>
n+
1|N
(t)=
n)=
o(∆
t),
n≥
1.
The
no
nh
om
og
en
eo
us
Po
isso
np
roc
ess
ca
nb
eg
en
era
lize
dto
wh
atc
an
be
ca
lled
an
on
ho
mo
ge
ne
ou
sp
ure
birth
pro
ce
ss,
NH
PB
.
Ac
ou
ntin
gp
roc
ess
{ N(t),
t≥
0}
isa
no
nh
om
og
en
eo
us
pu
re,
birth
pro
ce
ssw
ith
me
an
va
lue
fun
ctio
ns
Λn(t)
an
din
ten
sity
fun
ctio
ns
λn(t)
if
a){ N
(t),
t≥
0}
ha
sth
eM
ark
ov
pro
pe
rty,
b)
P(N
(t+
∆t)=
n+
1|N
(t)=
n)=
λn(t)∆
t+o(∆
t),
n≥
1,
c)
P(N
(t+
∆t)>
n+
1|N
(t)=
n)=
o(∆
t),
n≥
1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
8
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Re
latio
ns
be
twe
en
sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
an
do
rde
red
rv
Ord
er
Sta
tist
ics
d ⊂Se
qu
en
tia
lO
rde
rSta
tist
ics
d ⊃
Re
co
rdV
alu
es
No
nh
om
og
en
eo
us
Po
isso
nP
roc
ess
lim
t→∞
Λ(t)<
∞
d ⊂N
on
ho
mo
ge
ne
ou
sP
ure
Bir
thP
roc
ess
d ⊃N
on
ho
mo
ge
ne
ou
sPo
isso
nP
roc
ess
lim
t→∞
Λ(t)=
∞
d =d =
d =Λ(t)=
θF(t)
Λi(
t)=−
lnG
i(t)
Λ(t)=−
lnF(t)
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
9
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s:O
rdin
ary
Ord
er
Sta
tist
ics
Ifth
era
nd
om
va
ria
ble
sX
1,...,X
na
rea
rra
ng
ed
ina
sce
nd
ing
ord
ero
fm
ag
nitu
de
,
the
nth
ei’
thsm
alle
sto
fX
i’s
isd
en
ote
db
yX
i:n
.Th
eo
rde
red
qu
an
titie
s
X1:n≤
X2:n≤···≤
Xn:n,
are
ca
lled
ord
ina
ryo
rde
rst
atist
ics
(OO
S),a
nd
Xi:
nis
the
i’th
ord
er
sta
tist
ic.
Sin
ce
the
tim
es
toso
ftw
are
failu
re0≡
S0≤
S1≤
···≤
Si≤
···
are
ord
ere
d,
the
yc
on
stitu
tea
na
tura
lfra
me
wo
rkfo
ra
no
rde
rst
atist
ics
typ
ea
na
lysi
s.
An
oth
er
inte
rest
ing
ran
do
mva
ria
ble
sa
re
Di:
n=
Xi:
n−
Xi−
1:n
an
dD∗ i:
n=(n
−i+
1)D
i:n,
wh
en
X0:n≡
0,c
alle
dsi
mp
lesp
ac
ing
sa
nd
no
rma
lize
dsp
ac
ing
s,re
spe
ctive
ly.
Inth
eso
ftw
are
relia
bili
tyc
on
text
the
yc
orr
esp
on
dto
tim
es
ela
pse
db
etw
ee
nsu
cc
ess
ive
soft
wa
refa
ilure
s. BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
10
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Ord
er
Sta
tist
ics
an
dk-o
ut-
of-
nsy
ste
ms
Let
X1,...,X
nb
ea
co
llec
tio
no
fra
nd
om
va
ria
ble
s.If
Xi
de
no
tes
the
life
len
gth
of
the
i’th
co
mp
on
en
t,th
en
the
life
tim
eo
fa
k-o
ut-
of-
nsy
ste
mis
usu
ally
de
scrib
ed
by
the(n
−k+
1)’
tho
rde
rst
atist
ic.
Ak-o
ut-
of-
nsy
ste
mc
on
sist
so
fn
co
mp
on
en
tso
fth
esa
me
kin
d.
The
en
tire
syst
em
isw
ork
ing
ifa
tle
ast
ko
fits
nc
om
po
ne
nts
are
op
era
tin
g.
The
tim
es
be
twe
en
failu
res
ofc
om
po
ne
nts
ina
syst
em
co
rre
spo
nd
with
the
spa
c-
ing
sa
sso
cia
ted
with
ord
er
sta
tist
ics.
The
sesy
ste
ms
ha
ve
pra
ctic
al
ap
plic
atio
ns
inva
rio
us
rea
llif
esi
tua
tio
ns
suc
ha
se
lec
tric
ale
ng
ine
erin
g,
avia
tio
nin
du
stry
,a
uto
ma
tic
pa
ym
en
tsy
ste
ms
inb
an
ks,
etc
.
(c)
2-o
ut-
of-
4sy
ste
m(d
)2
-ou
t-o
f-8
syst
em
(e)
3-o
ut-
of-
6sy
ste
m
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
11
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Se
rie
ssy
ste
ms
Asy
ste
mth
at
isfu
nc
tio
nin
gif
an
do
nly
ife
ac
hc
om
po
ne
nt
isfu
nc
tio
nin
gis
ca
lled
ase
rie
ssy
ste
ma
nd
isre
pre
sen
ted
by
an-o
ut-
of-
nsy
ste
m.
Its
life
tim
eis
de
scrib
ed
by
the
sma
llest
life
tim
e,
X1:n
.Th
esu
rviv
alf
un
ctio
no
fth
issy
ste
mis
giv
en
by
F1:n(t)=
n ∏ i=1
Fi(
t),
wh
ere
the
Xi’
sa
rea
ssu
me
dto
be
ind
ep
en
de
nt
an
dF
iis
the
surv
iva
lfu
nc
tio
no
fX
i,fo
ri=
1,...,n
.
21
···
n
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
12
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Pa
ralle
lsy
ste
ms
Asy
ste
mth
at
isfu
nc
tio
nin
gif
an
do
nly
ifa
tle
ast
on
ec
om
po
ne
nt
isfu
nc
tio
nin
gis
ca
lled
ap
ara
llels
yst
em
an
dis
rep
rese
nte
db
ya
1-o
ut-
of-
nsy
ste
m.
Its
life
tim
eis
de
scrib
ed
by
the
larg
est
life
tim
e,
Xn:n
.Th
ec
um
ula
tive
dis
trib
utio
nfu
nc
tio
n(c
df)
of
this
syst
em
isg
ive
nb
y
Fn:n(t)=
n ∏ i=1
Fi(
t),
wh
ere
the
Xi’
sa
rea
ssu
me
dto
be
ind
ep
en
de
nt
an
dF
iis
the
dis
trib
utio
nfu
nc
tio
no
fX
i,fo
ri=
1,...,n
.
1 2 . . . n
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
13
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
A2-o
ut-
of-
3sy
ste
m:
X2:3
12
13
23
X2:3
ha
sth
ep
ath
sets
:P
1={ 1,2} ,
P2={
1,3}
an
dP
3={ 2
,3} .
☞A
pa
thse
tP
of
asy
ste
mis
ase
tsu
ch
tha
tif
all
the
co
mp
on
en
tsin
Pw
ork
,th
en
the
syst
em
wo
rks.
The
surv
iva
lfu
nc
tio
no
fth
issy
ste
mis
giv
en
by
F2:3(t)=
F1(t)F
2(t)+
F1(t)F
3(t)+
F2(t)F
3(t)−
2F
1(t)F
2(t)F
3(t),
wh
ere
the
Xi’
sa
rea
ssu
me
dto
be
ind
ep
en
de
nt
an
dF
iis
the
surv
iva
lfu
nc
tio
no
fX
i,fo
ri=
1,...,n
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
14
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Co
nn
ec
tivity
pro
ble
ms
inN
etw
ork
s
Ag
rap
ho
rn
etw
ork
isa
no
rde
red
pa
irG=(V
,E)
co
mp
risi
ng
ase
tV
of
no
de
sto
ge
the
rw
ith
ase
tE
of
ed
ge
s,w
hic
ha
re2-e
lem
en
tsu
bse
tso
fV
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
15
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Co
nn
ec
tivity
pro
ble
ms
inN
etw
ork
s
Ag
rap
ho
rn
etw
ork
isa
no
rde
red
pa
irG=(V
,E)
co
mp
risi
ng
ase
tV
of
no
de
sto
ge
the
rw
ith
ase
tE
of
ed
ge
s,w
hic
ha
re2-e
lem
en
tsu
bse
tso
fV
.
Ad
ire
cte
dg
rap
his
an
ord
ere
dp
air
G=
(V,E
)c
om
prisi
ng
ase
tV
of
no
de
sto
ge
the
rw
ith
ase
tE
of
ed
ge
s,w
hic
ha
ree
lem
en
tso
fV×
V.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
15
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
Co
nn
ec
tivity
pro
ble
ms
inN
etw
ork
s
Ag
rap
ho
rn
etw
ork
isa
no
rde
red
pa
irG=(V
,E)
co
mp
risi
ng
ase
tV
of
no
de
sto
ge
the
rw
ith
ase
tE
of
ed
ge
s,w
hic
ha
re2-e
lem
en
tsu
bse
tso
fV
.
Ad
ire
cte
dg
rap
his
an
ord
ere
dp
air
G=
(V,E
)c
om
prisi
ng
ase
tV
of
no
de
sto
ge
the
rw
ith
ase
tE
of
ed
ge
s,w
hic
ha
ree
lem
en
tso
fV×
V.
Let
us
ass
um
eth
at
ina
gra
ph
(dire
cte
dg
rap
h)
the
no
de
sc
an
no
tfa
ilb
ut
the
ed
ge
sc
an
fail.
Let
X1,...,X
nb
eth
ee
dg
es
life
tim
es.
Sup
po
seth
at
we
wa
nt
tost
ud
ya
giv
en
co
nn
ec
tivity
pro
ble
m,
e.g
.,th
ec
on
ne
ctio
no
fa
llth
en
od
es.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
15
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Ne
two
rkPa
thse
ts:
k-o
ut-
of-
nsy
ste
m
?
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
16
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Ne
two
rkPa
thse
ts:
k-o
ut-
of-
nsy
ste
m
{ 1,2,3}
?
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
16
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Ne
two
rkPa
thse
ts:
k-o
ut-
of-
nsy
ste
m
{ 1,2,3}
Serie
ssy
ste
mX
1:3
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
16
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Ne
two
rkPa
thse
ts:
k-o
ut-
of-
nsy
ste
m
?
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
17
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Ne
two
rkPa
thse
ts:
k-o
ut-
of-
nsy
ste
m
{ 1}
?{ 2
}{ 3
}
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
17
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Ne
two
rkPa
thse
ts:
k-o
ut-
of-
nsy
ste
m
{ 1}
Pa
ralle
lsyst
em
{ 2}
X3:3
{ 3}
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
17
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Mo
tiva
tio
n
Re
liab
ility
me
asu
res
Sto
ch
ast
icc
ou
ntin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
s
Ne
two
rks
All
no
de
sc
on
ne
ctio
np
rob
lem
ina
ne
two
rk
Wh
at
isth
eb
est
wa
yto
co
nn
ec
tth
ree
no
de
sw
ith
thre
ee
dg
es
?
Ne
two
rkk-o
ut-
of-
nsy
ste
m
?
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
18
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ou
tlin
e
1In
tro
du
ctio
nM
otiva
tio
nR
elia
bili
tym
ea
sure
sSt
oc
ha
stic
co
un
tin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
sN
etw
ork
s
2St
oc
ha
stic
co
mp
ariso
ns
of
spa
cin
gs
ba
sed
on
ord
er
sta
tist
ics
De
finitio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
3B
aye
sia
nIn
fere
nc
eP
relim
ina
rie
sA
ne
wa
pp
roa
ch
toSR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
4C
on
clu
sio
ns
an
dc
on
trib
utio
ns
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
19
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
Xis
said
tob
esm
alle
rth
an
Yin
the
usu
alst
oc
ha
stic
ord
er,
X≤
stY
(or
F≤
stG
),if
F(t)≤
G(t),
for
all
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
Xis
said
tob
esm
alle
rth
an
Yin
the
usu
alst
oc
ha
stic
ord
er,
X≤
stY
(or
F≤
stG
),if
F(t)≤
G(t),
for
all
t.
Xis
said
tob
esm
alle
rth
an
Yin
the
ha
zard
rate
ord
er,
de
no
ted
by
X≤
hr
Y(o
rF≤
hr
G),
ifh
F(t)≥
hG(t),
for
all
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
Xis
said
tob
esm
alle
rth
an
Yin
the
usu
alst
oc
ha
stic
ord
er,
X≤
stY
(or
F≤
stG
),if
F(t)≤
G(t),
for
all
t.
Xis
said
tob
esm
alle
rth
an
Yin
the
ha
zard
rate
ord
er,
de
no
ted
by
X≤
hr
Y(o
rF≤
hr
G),
ifh
F(t)≥
hG(t),
for
all
t.
☞(X
−t|X
>t)≤
st(Y
−t|Y
>t)
,fo
ra
llt.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
Xis
said
tob
esm
alle
rth
an
Yin
the
usu
alst
oc
ha
stic
ord
er,
X≤
stY
(or
F≤
stG
),if
F(t)≤
G(t),
for
all
t.
Xis
said
tob
esm
alle
rth
an
Yin
the
ha
zard
rate
ord
er,
de
no
ted
by
X≤
hr
Y(o
rF≤
hr
G),
ifh
F(t)≥
hG(t),
for
all
t.
☞(X
−t|X
>t)≤
st(Y
−t|Y
>t)
,fo
ra
llt.
Xis
said
tob
esm
alle
rth
an
Yin
like
liho
od
ratio
ord
er,
de
no
ted
by
X≤
lrY
(or
F≤
lrG
),if
f(s)
f(t)
≥g(s)
g(t),
wh
ere
s≤
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
Xis
said
tob
esm
alle
rth
an
Yin
the
usu
alst
oc
ha
stic
ord
er,
X≤
stY
(or
F≤
stG
),if
F(t)≤
G(t),
for
all
t.
Xis
said
tob
esm
alle
rth
an
Yin
the
ha
zard
rate
ord
er,
de
no
ted
by
X≤
hr
Y(o
rF≤
hr
G),
ifh
F(t)≥
hG(t),
for
all
t.
☞(X
−t|X
>t)≤
st(Y
−t|Y
>t)
,fo
ra
llt.
Xis
said
tob
esm
alle
rth
an
Yin
like
liho
od
ratio
ord
er,
de
no
ted
by
X≤
lrY
(or
F≤
lrG
),if
f(s)
f(t)
≥g(s)
g(t),
wh
ere
s≤
t.
☞(X
|s<
X<
t)≤
st(Y
|s<
Y<
t),fo
rs<
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Un
iva
ria
teSto
ch
ast
icO
rde
rs
Sto
ch
ast
ico
rde
rsb
etw
ee
np
rob
ab
ility
dis
trib
utio
ns
isa
wid
ely
stu
die
dfie
ld.
The
rea
rese
ve
ralk
ind
so
fst
oc
ha
stic
ord
ers
tha
ta
reu
sed
toc
om
pa
red
iffe
ren
ta
spe
cts
of
pro
ba
bili
tyd
istr
ibu
tio
ns
like
loc
atio
n,va
ria
bili
ty,d
ep
en
de
nc
e,e
tc.
Xis
said
tob
esm
alle
rth
an
Yin
the
usu
alst
oc
ha
stic
ord
er,
X≤
stY
(or
F≤
stG
),if
F(t)≤
G(t),
for
all
t.
Xis
said
tob
esm
alle
rth
an
Yin
the
ha
zard
rate
ord
er,
de
no
ted
by
X≤
hr
Y(o
rF≤
hr
G),
ifh
F(t)≥
hG(t),
for
all
t.
☞(X
−t|X
>t)≤
st(Y
−t|Y
>t)
,fo
ra
llt.
Xis
said
tob
esm
alle
rth
an
Yin
like
liho
od
ratio
ord
er,
de
no
ted
by
X≤
lrY
(or
F≤
lrG
),if
f(s)
f(t)
≥g(s)
g(t),
wh
ere
s≤
t.
☞(X
|s<
X<
t)≤
st(Y
|s<
Y<
t),fo
rs<
t.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
20
X≤
lrY
⇒X≤
hr
Y⇒
X≤
stY
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Exa
mp
le
Let
X1,X
2,X
3b
ein
de
pe
nd
en
ta
nd
ide
ntic
ally
exp
on
en
tia
lra
nd
om
va
ria
ble
sw
ith
ha
zard
rate
λ,fo
ri=
1,2,3
.
X1:3→
F1:3(t)=
e−3λ
t .
X2:3→
F2:3(t)=
3e−
2λ
t−
2e−
3λ
t .
X3:3→
F3:3(t)=
1−( 1
−e−
λt)
3
.
02
46
810
12
14
0.2
0.4
0.6
0.8
1.0
i=3
i=2
i=1
☞X
1:3≤
stX
2:3≤
stX
3:3
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
21
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Sto
ch
ast
icc
om
pa
riso
ns
ine
xp
on
en
tia
lo
rde
rst
atist
ics
mo
de
ls
The
ore
m(B
ola
nd
et
al.,
1998)
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tX
ih
as
ha
-za
rdra
teλ
i,fo
ri=
1,...,n
.Th
en
,
Xi:
n≤
hr
Xi+
1:n,
for
i=
1,...,n
−1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
22
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Sto
ch
ast
icc
om
pa
riso
ns
ine
xp
on
en
tia
lo
rde
rst
atist
ics
mo
de
ls
The
ore
m(B
ola
nd
et
al.,
1998)
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tX
ih
as
ha
-za
rdra
teλ
i,fo
ri=
1,...,n
.Th
en
,
Xi:
n≤
hr
Xi+
1:n,
for
i=
1,...,n
−1.
☞M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-b
les
(EO
S).
Co
rolla
ry:
Let
S1,...,S
nb
eth
efa
ilure
tim
es
of
aEO
SSo
ftw
are
Re
liab
ility
mo
de
l.Th
en
,S
i≤
hr
Si+
1,
for
i=
1,...,n
−1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
22
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Sto
ch
ast
icc
om
pa
riso
ns
ine
xp
on
en
tia
lo
rde
rst
atist
ics
mo
de
ls
The
ore
m(B
ola
nd
et
al.,
1998)
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tX
ih
as
ha
-za
rdra
teλ
i,fo
ri=
1,...,n
.Th
en
,
Xi:
n≤
hr
Xi+
1:n,
for
i=
1,...,n
−1.
☞M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-b
les
(EO
S).
Co
rolla
ry:
Let
S1,...,S
nb
eth
efa
ilure
tim
es
of
aEO
SSo
ftw
are
Re
liab
ility
mo
de
l.Th
en
,S
i≤
hr
Si+
1,
for
i=
1,...,n
−1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
22
Mill
er
(1986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-M
ille
r(1
986)
mo
de
led
failu
retim
es
of
aso
ftw
are
pro
gra
ma
so
rde
rst
atist
ics
of
ind
ep
en
de
nt
no
nid
en
tic
ally
dis
trib
ute
d(i
nid
)e
xpo
ne
ntia
lra
nd
om
va
ria
-
Are
the
spa
cin
gs
fro
mh
ete
rog
en
eo
us
ex
po
ne
ntia
lra
nd
om
va
ria
ble
so
rde
red
ac
co
rdin
gto
the
ha
zard
rate
ord
eri
ng
?
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
No
rma
lize
dSp
ac
ing
so
fEO
Sm
od
els
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gh
a-
zard
rate
λi.
Let
D∗ i:
n=(n
−i+
1)(
Xi:
n−
Xi−
1:n)
the
i’th
no
rma
lize
dsp
ac
ing
fro
mX
i’s
with
X0:n≡
0.
The
n
a)
D∗ i:
n≤
stD∗ i+
1:n
,fo
ri=
1,...,n
−1.
Ple
dg
er
an
dP
rosc
ha
n,1971
b)
D∗ 1:n≤
lrD∗ i:
n,fo
ri=
2,...,n
.K
oc
ha
ra
nd
Ko
rwa
r,1996
c)
D∗ 1:3≤
hr
D∗ 2:3≤
hr
D∗ 3:3
.K
oc
ha
ra
nd
Ko
rwa
r,1996
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
23
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
No
rma
lize
dSp
ac
ing
so
fEO
Sm
od
els
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gh
a-
zard
rate
λi.
Let
D∗ i:
n=(n
−i+
1)(
Xi:
n−
Xi−
1:n)
the
i’th
no
rma
lize
dsp
ac
ing
fro
mX
i’s
with
X0:n≡
0.
The
n
a)
D∗ i:
n≤
stD∗ i+
1:n
,fo
ri=
1,...,n
−1.
Ple
dg
er
an
dP
rosc
ha
n,1971
b)
D∗ 1:n≤
lrD∗ i:
n,fo
ri=
2,...,n
.K
oc
ha
ra
nd
Ko
rwa
r,1996
c)
D∗ 1:3≤
hr
D∗ 2:3≤
hr
D∗ 3:3
.K
oc
ha
ra
nd
Ko
rwa
r,1996
Co
nje
ctu
reo
fK
oc
ha
ra
nd
Ko
rwa
r:Le
tX
1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xih
avin
gh
aza
rdra
teλ
i,th
en
D∗ i:
n≤
hr
D∗ i+
1:n,
for
i=
1,...,n
−1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
23
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
De
nsi
tyfu
nc
tio
no
fN
orm
aliz
ed
Sp
ac
ing
sTh
ed
istr
ibu
tio
no
fD∗ i
isa
mix
ture
of
ind
ep
en
de
nt
exp
on
en
tia
lra
nd
om
va
ria
ble
sw
ith
p.d
.f.:
f i(t)=
∑ r n
∏n k=
1λ
k
∏n k=
1
( ∑n j=
kλ(r
j))·
∑n j=
iλ(r
j)
n−
i+1
·exp
{−
t∑
n j=iλ(r
j)
n−
i+1
}.
Let
β(i)
mj=
∑n ℓ=
iλ(r
ℓ)
n−
i+1
wh
ere
mjin
dic
ate
sa
gro
up
of
ind
ice
so
fsi
zen−
i+1.
The
n,
the
de
nsi
tyfu
nc
tio
nc
an
be
writt
en
as
f i(t)=
Mi
∑ j=1
∆( β
(i)
mj,n)
β(i)
mj
e−tβ
(i)
mj,
wh
ere
Mi=
(n
n−
i+1
)a
nd
∆(β
(i)
mj,n)
=∑
r i−
1,m
j
∏
k∈
Hm
j
λk
i−1
∏ ℓ=1
i−1
∑ u=ℓ
r(u)∈
Hm
j
λ(r(u))+(n
−i+
1)β
(i)
mj
−1
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
24
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
Ob
serv
ing
the
de
nsi
tyfu
nc
tio
no
fD∗ i:
n,n
ote
tha
tD∗ i:
n≤
hr
D∗ i+
1:n
ifa
nd
on
lyif
hi(
t)=
Mi
∑ j=1
∆( β
(i)
mj,n)
β(i)
mj
e−tβ
(i)
mj
Mi
∑ j=1
∆(β
(i)
mj,n)
e−tβ
(i)
mj
≥
Mi+
1
∑ j=1
∆( β
(i+
1)
mj
,n)
β(i+
1)
mj
e−tβ
(i+
1)
mj
Mi+
1
∑ j=1
∆(β
(i+
1)
mj
,n)
e−t β
(i+
1)
mj
=h
i+1(t),
wh
ich
ca
nb
ere
writt
en
as
Mi+
1
∑ j=1
Mi
∑ k=
1
∆(β
(i)
mk,n)∆
(β(i+
1)
mj
,n)
e−t( β
(i)
mk+
β(i+
1)
mj
)( β
(i)
mk−
β(i+
1)
mj
)≥
0,
wh
ere
Mi=
(n
n−
i+1
) .
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
25
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
Ob
serv
ing
the
de
nsi
tyfu
nc
tio
no
fD∗ i:
n,n
ote
tha
tD∗ i:
n≤
hr
D∗ i+
1:n
ifa
nd
on
lyif
hi(
t)=
Mi
∑ j=1
∆( β
(i)
mj,n)
β(i)
mj
e−tβ
(i)
mj
Mi
∑ j=1
∆(β
(i)
mj,n)
e−tβ
(i)
mj
≥
Mi+
1
∑ j=1
∆( β
(i+
1)
mj
,n)
β(i+
1)
mj
e−tβ
(i+
1)
mj
Mi+
1
∑ j=1
∆(β
(i+
1)
mj
,n)
e−t β
(i+
1)
mj
=h
i+1(t),
wh
ich
ca
nb
ere
writt
en
as
Mi+
1
∑ j=1
Mi
∑ k=
1
∆( β
(i)
mk,n)∆
(β(i+
1)
mj
,n)
e−t( β
(i)
mk+
β(i+
1)
mj
)( β
(i)
mk−
β(i+
1)
mj
)≥
0,
wh
ere
Mi=
(n
n−
i+1
) .
The
ine
qu
alit
yh
old
sif
itis
po
sitive
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
25
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
D∗ i:
n≤
hr
D∗ i+
1:n⇔
Mi+
1
∑ j=1
Mi
∑ k=
1
∆(β
(i)
mk,n)∆
(β(i+
1)
mj
,n)
e−t( β
(i)
mk+
β(i+
1)
mj
)( β
(i)
mk−
β(i+
1)
mj
)≥
0.
β(i+
1)
mj
β(i)
mk
β(i)
mk
β(i)
mk−
β(i+
1)
mj
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
26
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
D∗ i:
n≤
hr
D∗ i+
1:n⇔
Mi+
1
∑ j=1
Mi
∑ k=
1
∆(β
(i)
mk,n)∆
(β(i+
1)
mj
,n)
e−t( β
(i)
mk+
β(i+
1)
mj
)( β
(i)
mk−
β(i+
1)
mj
)≥
0.
β(i+
1)
mj
β(i)
mk
β(i)
mk
β(i)
mk−
β(i+
1)
mj
The
ore
m
Let
β(i)
mk=
∑n ℓ=
iλ(r
ℓ)
n−
i+1
,th
en
Mi
∑ k=
1
Mi+
1
∑ j=1
( β(i)
mk−
β(i+
1)
mj
)=
0.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
26
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
D∗ i:
n≤
hr
D∗ i+
1:n⇔
Mi+
1
∑ j=1
Mi
∑ k=
1
∆(β
(i)
mk,n)∆
(β(i+
1)
mj
,n)
e−t( β
(i)
mk+
β(i+
1)
mj
)( β
(i)
mk−
β(i+
1)
mj
)≥
0.
β(i+
1)
mj
β(i)
mk
β(i)
mk
β(i)
mk−
β(i+
1)
mj
The
ore
m
Let
β(i)
mk=
∑n ℓ=
iλ(r
ℓ)
n−
i+1
,th
en
Mi
∑ k=
1
Mi+
1
∑ j=1
( β(i)
mk−
β(i+
1)
mj
)=
0.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
26
Ch
eb
ysh
ev
’ssu
min
eq
ua
lity
:
nn ∑ i=
1
aib
i≥
(n ∑ i=
1
ai)(
n ∑ i=1
bi)
.
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
The
ore
m
Let
X1,...,X
4b
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xih
avin
gsu
rvi-
va
lfu
nc
tio
nF
i(t)=
exp(−
λit),
t≥
0,fo
ri=
1,...,4
.Th
en
D∗ 1:4≤
hr
D∗ 2:4≤
hr
D∗ 3:4≤
hr
D∗ 4:4.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
27
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofN
orm
aliz
ed
Sp
ac
ing
s
The
ore
m
Let
X1,...,X
4b
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xih
avin
gsu
rvi-
va
lfu
nc
tio
nF
i(t)=
exp(−
λit),
t≥
0,fo
ri=
1,...,4
.Th
en
D∗ 1:4≤
hr
D∗ 2:4≤
hr
D∗ 3:4≤
hr
D∗ 4:4.
The
ore
m
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gh
a-
zard
rate
λi,
the
n
D∗ 2:n≤
hr
D∗ 3:n,
for
all
n.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
27
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofSim
ple
Sp
ac
ing
s
We
turn
toc
on
sid
er
the
sim
ple
spa
cin
gs
of
the
ord
er
sta
tist
ics
wh
ere
no
w,
β(i)
mj=
n ∑ ℓ=iλ(r
ℓ).
On
ese
es
tha
tth
ep
.d.f.o
fD
i:n
for
1≤
i≤
nis
f i(t)
=∑ r n
∏n k=
1λ
k
∏n k=
1( ∑
n ℓ=k
λ(r
ℓ))
(n ∑ ℓ=
iλ(r
ℓ)
)e−
t ∑n ℓ=
iλ(rℓ)
=M
i
∑ j=1
∆( β
(i)
mj,n)
β(i)
mj
e−t β
(i)
mj.
No
teth
at
Di:
n≤
hr
Di+
1:n
ifa
nd
on
lyif
Mi+
1
∑ j=1
Mi
∑ k=
1
∆( β
(i)
mk,n)∆
(β(i+
1)
mj
,n)
e−t( β
(i)
mk+
β(i+
1)
mj
)( β
(i)
mk−
β(i+
1)
mj
)≥
0.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
28
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofSim
ple
Sp
ac
ing
s
The
ore
mLe
tX
1,...,X
4b
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gsu
rviv
alf
un
ctio
nF
i(t)=
exp(−
λit),
t≥
0,fo
ri=
1,...,4
.Th
en
D1:4≤
hr
D2:4≤
hr
D3:4≤
hr
D4:4.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
29
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
Ha
zard
rate
ord
eri
ng
ofSim
ple
Sp
ac
ing
s
The
ore
mLe
tX
1,...,X
4b
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gsu
rviv
alf
un
ctio
nF
i(t)=
exp(−
λit),
t≥
0,fo
ri=
1,...,4
.Th
en
D1:4≤
hr
D2:4≤
hr
D3:4≤
hr
D4:4.
The
ore
mLe
tX
1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gh
aza
rdra
teλ
i,th
en D
2:n≤
hr
D3:n,
for
all
n.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
29
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
exp
on
en
tia
lIID
ca
se
An
atu
ral
qu
est
ion
isto
exa
min
ew
he
the
ra
syst
em
isb
ett
er
tha
no
the
ro
ne
inso
me
sto
ch
ast
icse
nse
.
Let
X1,X
2,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
g
ha
zard
rate
λi
i=
1,...,n
.Le
tY
1,Y
2,...,Y
nb
ea
ran
do
msa
mp
leo
fsi
zen
fro
ma
n
exp
on
en
tia
ldis
trib
utio
nw
ith
co
mm
on
ha
zard
rate
λ.
The
n
a)
C∗ i:
n≤
stD∗ i:
n,
Ple
dg
er
an
dP
rosc
ha
n,1971
b)
C∗ i:
n≤
lrD∗ i:
n,
Ko
ch
ar
an
dK
ow
ar,
1996
for
i=
1,...,n
,w
he
reC∗ i:
n=
(n−
i+1)(
Yi:
n−
Yi−
1:n)
an
dD∗ i:
n=
(n−
i+1)(
Xi:
n−
Xi−
1:n)
are
the
i’th
no
rma
lize
dsp
ac
ing
fro
mY
i’s
an
dX
i’s,
resp
ec
tive
ly,
with
Y0:n≡
0a
nd
X0:n≡
0.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
30
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
exp
on
en
tia
lIID
ca
se
The
ore
m(K
oc
ha
ra
nd
Xu
,2011)
Let
X1,X
2,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gh
aza
rdra
teλ
ii=
1,...,n
.Le
tY
1,Y
2,...,Y
nb
ea
ran
do
msa
mp
leo
fsi
zen
fro
ma
ne
xpo
ne
ntia
ldis
trib
utio
nw
ith
co
mm
on
ha
zard
rate
λ.
The
n,fo
ri≥
2,
Ci:
n≤
lrD
i:n⇔
(n−
i+1) λ
≥
∑ j∈r n
pj
(n ∑ j=
iλ(r
j)
)2
∑ j∈r n
pj
(n ∑ j=
iλ(r
j)
),
for
i=
1,...,n
,w
he
re
pj=
n ∏ k=
1
λk
n ∏ k=
1
(n ∑ j=
k
λ(r
j)
).
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
31
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
no
nIID
ca
se
Let
X1,X
2,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
sw
ith
Xi
ha
vin
gh
aza
rdra
teλ
ii=
1,...,n
an
dY
1,...,Y
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
-ria
ble
ssu
ch
tha
tY
ih
as
ha
zard
rate
θifo
ri=
1,...,n
.In
ge
ne
ral,
a)
ifθθ θ≤
mλλ λ
the
nC∗ i:
n�
stD∗ i:
n,
Ple
dg
er
an
dP
rosc
ha
n,1971
b)
ifθθ θ≤
mλλ λ
the
nC∗ 2:n≤
stD∗ 2:n
,K
oc
ha
ra
nd
Ko
wa
r,1996
c)
ifθθ θ≤
mλλ λ
the
nC∗ 2:n�
hr
D∗ 2:n
,a
lth
ou
gh
for
n=
2,
C∗ 2:2≤
hr
D∗ 2:2
,
for
i=
1,...,n
.
Let{x
(1),
x (2),...,
x (n)}
de
no
teth
ein
cre
asi
ng
arr
an
ge
me
nt
of
the
co
mp
on
en
tso
fth
eve
cto
rx=
(x1,x
2,...,x
n).
The
ve
cto
rx
issa
idto
be
ma
jorize
db
yth
eve
cto
ry,
de
no
ted
by
x≤
my,if
j ∑ i=1
x (i)≥
j ∑ i=1
y (i),
for
j=
1,...,n
−1
an
dn ∑ i=
1
x (i)=
n ∑ i=1
y (i).
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
32
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
no
nIID
ca
se
The
ore
m
Let
X1,...,X
na
nd
Y1,...,Y
nb
etw
ose
qu
en
ce
so
fin
de
pe
nd
en
tb
ut
no
tn
ec
ess
arily
ide
ntic
ally
dis
trib
ute
dra
nd
om
va
ria
ble
s.Th
en
,
Ci:
n≤
lrD
i:n⇔
C∗ i:
n≤
lrD∗ i:
n,
for
i=
1,...,n
.
Let
us
de
fine
α(i)
min=
min
1≤
mj≤
Mi
α(i)
mj,
wh
ere
α(i)
mj=
n ∑ ℓ=iθ
r ℓ.
No
teth
at
α(i)
min=
n−
i+1
∑ j=1
θ(j),
wh
ere{ θ
(1),...,
θ(n)}
de
no
teth
ein
cre
asi
ng
arr
an
ge
me
nt
of
θi,
for
i=
1,...,n
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
33
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
exp
on
en
tia
ln
on
IID
ca
se
The
ore
m
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tX
ih
as
ha
-za
rdra
teλ
ifo
ri=
1,...,n
,a
nd
Y1,...,Y
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tY
ih
as
ha
zard
rate
θifo
ri=
1,...,n
.If
α(i)
min
n−
i+1≥
λ,
the
nC
i:n≤
lrD
i:n,
for
i=
1,...,n
,w
he
reD
i:n
an
dC
i:n
are
the
i’th
sim
ple
spa
cin
gfr
om
Xi’
sa
nd
Yi’
s,re
spe
ctive
ly.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
34
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
exp
on
en
tia
ln
on
IID
ca
se
Pro
po
sitio
n
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tX
ih
as
ha
-za
rdra
teλ
ifo
ri=
1,...,n
;Y
1,...,Y
nb
ea
ran
do
msa
mp
leo
fsi
zen
fro
ma
ne
x-p
on
en
tia
ldis
trib
utio
nw
ith
co
mm
on
ha
zard
rate
λ(n),
an
dZ
1,...,Z
nb
ea
ran
do
msa
mp
leo
fsi
zen
fro
ma
ne
xpo
ne
ntia
ld
istr
ibu
tio
nw
ith
co
mm
on
ha
zard
rate
λ(1).
The
nC
i:n≤
lrD
i:n≤
lrH
i:n,
for
i=
1,...,n
wh
ere
Ci:
n,
Di:
n,
Hi:
nd
en
ote
the
i’th
sim
ple
spa
cin
gs
of
Yi’
s,X
i’s
an
dZ
i’s,
resp
ec
tive
ly.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
35
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
De
fin
itio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
The
two
sam
ple
pro
ble
m:
the
exp
on
en
tia
ln
on
IID
ca
se
Pro
po
sitio
n
Let
X1,...,X
nb
ein
de
pe
nd
en
te
xpo
ne
ntia
lra
nd
om
va
ria
ble
ssu
ch
tha
tX
ih
as
ha
-za
rdra
teλ
ifo
ri=
1,...,n
;Y
1,...,Y
nb
ea
ran
do
msa
mp
leo
fsi
zen
fro
ma
ne
x-p
on
en
tia
ldis
trib
utio
nw
ith
co
mm
on
ha
zard
rate
λ(n),
an
dZ
1,...,Z
nb
ea
ran
do
msa
mp
leo
fsi
zen
fro
ma
ne
xpo
ne
ntia
ld
istr
ibu
tio
nw
ith
co
mm
on
ha
zard
rate
λ(1).
The
nC
i:n≤
lrD
i:n≤
lrH
i:n,
for
i=
1,...,n
wh
ere
Ci:
n,
Di:
n,
Hi:
nd
en
ote
the
i’th
sim
ple
spa
cin
gs
of
Yi’
s,X
i’s
an
dZ
i’s,
resp
ec
tive
ly.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.2
0.4
0.6
0.8
1.0
HΛ1,Λ2,Λ3L=H0.9,1,4L
HΛ1,Λ2,Λ3L=H1.967,1.967,1.967L
HΛ1,Λ2,Λ3L=H0.9,0.9,0.9L
HΛ1,Λ2,Λ3L=H4,4,4L
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
35
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ou
tlin
e
1In
tro
du
ctio
nM
otiva
tio
nR
elia
bili
tym
ea
sure
sSt
oc
ha
stic
co
un
tin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
sN
etw
ork
s
2St
oc
ha
stic
co
mp
ariso
ns
of
spa
cin
gs
ba
sed
on
ord
er
sta
tist
ics
De
finitio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
3B
aye
sia
nIn
fere
nc
eP
relim
ina
rie
sA
ne
wa
pp
roa
ch
toSR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
4C
on
clu
sio
ns
an
dc
on
trib
utio
ns
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
36
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
So
ftw
are
me
tric
sin
form
atio
n
Aso
ftw
are
me
tric
isa
me
asu
reo
fso
me
pro
pe
rty
of
ap
iec
eo
fso
ftw
are
or
its
spe
cifi
ca
tio
ns.
Me
tric
sc
an
be
use
dto
me
asu
reso
ftw
are
pro
du
ctivity
an
dq
ua
-lit
y.
Co
mm
on
soft
wa
rem
etr
ics:
Nu
mb
er
of
Lin
es
of
Co
de
,LO
C.
Nu
mb
er
of
No
n-C
om
me
nt
Lin
es
of
Co
de
,N
CLO
C.
Nu
mb
er
of
Co
mm
en
tLi
ne
so
fC
od
e,C
LOC
.
LOC
=N
CLO
C+
CLO
C
So
ftw
are
scie
nc
em
etr
ics,
de
ve
lop
ed
by
Ha
lste
ad
(1977).
The
ya
rese
nsi
tive
top
rog
ram
size
bu
tn
ot
top
rog
ram
co
ntr
olfl
ow
.
Cyc
lom
atic
nu
mb
er,
de
ve
lop
ed
by
Mc
Ca
be
(1976).
This
me
tric
me
asu
res
som
ea
spe
cts
of
co
ntr
olflo
wc
om
ple
xity
an
dit
isn
ot
ne
ce
ssa
rily
rela
ted
top
rog
ram
size
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
37
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
So
ftw
are
relia
bili
tym
od
els
inth
elit
era
ture
usi
ng
soft
wa
rem
etr
ics
Mo
sto
fth
ese
mo
de
lsc
on
sid
era
Typ
eII
soft
wa
rere
liab
ility
mo
de
lwh
ere
the
nu
m-
be
ro
ffa
ilure
s,N
i=
ni,
de
tec
ted
by
tim
et i
ism
od
ele
db
ya
NH
PP,
for
i=
1,...,M
.
Ro
drig
ue
z-B
ern
ala
nd
Wip
er
(2001)
sho
wh
ow
toc
om
bin
eso
ftw
are
me
tric
sd
ata
with
inte
rfa
ilure
tim
ed
ata
toim
pro
ve
the
pre
dic
tio
ns
of
fau
ltn
um
be
rsa
nd
relia
bili
tyo
fa
pro
gra
mu
sin
gB
aye
sia
na
pp
roa
ch
.
Ra
ye
ta
l.(2
006)
ass
um
ea
na
pp
roa
ch
ba
sed
on
est
ima
tin
gth
en
um
be
ro
ffa
ilure
sb
ya
reg
ress
ion
typ
em
od
el.
Rin
saka
et
al.
(2006)
co
nsi
de
ra
pro
po
rtio
na
lh
aza
rds
typ
ea
pp
roa
ch
with
diff
ere
nt
inte
nsi
tyfu
nc
tio
ns
of
the
NH
PP.
Wip
ere
ta
l.(2
011)
de
ve
lop
an
ap
pro
ac
hto
bo
thTy
pe
Ian
dTy
pe
IIso
ftw
are
relia
bili
tym
od
els
ba
sed
on
ne
ura
lne
two
rks.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
38
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ga
uss
ian
pro
ce
sse
s
AG
au
ssia
np
roc
ess
(GP
)is
ac
olle
ctio
no
fra
nd
om
va
ria
ble
s,a
ny
finite
nu
mb
er
of
wh
ich
ha
ve
Ga
uss
ian
dis
trib
utio
ns.
AG
au
ssia
nd
istr
ibu
tio
nis
fully
spe
cifi
ed
by
am
ea
nve
cto
r,µ
,a
nd
co
va
ria
nc
em
atr
ixΣ
f=(f
1,...,f
n)T
∼N( µ
,Σ).
AG
au
ssia
np
roc
ess
isc
om
ple
tely
spe
cifi
ed
by
am
ea
nfu
nc
tio
n,
m(x),
an
dc
o-
va
ria
nc
efu
nc
tio
nC(f(x),
f(x′ )),
f(x)∼
GP( m
(x),
C(f(x),
f(x′ ))),
wh
ere
m(x)
=E[f(x)],
C(f(x),
f(x′ ))
=E[ (
f(x)−
m(x))( f(
x′ )−
m(x
′ ))].
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
39
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ga
uss
ian
pro
ce
ssre
gre
ssio
nm
od
els
Giv
en
ase
tD
={(
xi,
y i)
:i=
1,...,n}
of
no
bse
rva
tio
ns,
wh
ere
xi=
(xi1,...,x
ik)T
de
-n
ote
sa
nin
pu
tve
cto
r(c
ova
ria
tes)
of
dim
en
sio
nk
an
dy=
(y1,...,y
n)T
de
no
tes
asc
ala
ro
utp
ut
or
targ
et
( de
pe
nd
en
tva
ria
ble
),th
ere
gre
ssio
nm
od
eli
sd
efin
ed
as
y i=
f(x
i)+
ε i,
wh
ere
ε i∼
N(0,σ
2)
isa
ne
rro
rte
rm.
The
prio
rd
istr
ibu
tio
n,
p(f|θ
θ θ),
on
f=(f
1,...,f
n)T
isa
GP,
wh
ere
f i=
f(x
i),a
nd
we
will
write
the
GP
as
f| θ
θ θ∼GP(m
,C(θθ θ
)),
wh
ere
the
me
an
ve
cto
rm
will
be
an
n-e
lem
en
tc
olu
mn
ve
cto
r,a
nd
the
co
va
-ria
nc
em
atr
ixC(θθ θ
)w
illb
ea
nn×
nm
atr
ixw
ith
ele
me
nts
as
follo
ws
C(f(x),
f(x′ )| θ
θ θ)=
η2
exp
{−
1 2
k ∑ j=1
ρ−
2j
( x j−
x′ j) 2}.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
40
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Pro
po
sed
mo
de
ls
Typ
eIm
od
el
Ti|λ
i∼
E(λ
i),
lnλ
i|f
i=
f(x
i)+
ε i,
f|θ
θ θ∼
GP(0,C
(θθ θ)),
ε i| σ
2∼
N(0, σ
2),
σ2,η
2,ρ
2 j∼
IG(0.0
01,0.0
01).
Typ
eII
mo
de
l
Ni|λ
i∼
P(L
iλi),
lnλ
i|f
i=
f(x
i)+
ε i,
f|θ
θ θ∼
GP(0,C
(θθ θ)),
ε i| σ
2∼
N(0, σ
2),
σ2,η
2,ρ
2 j∼
IG(0.0
01,0.0
01).
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
41
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Pro
po
sed
mo
de
ls
Typ
eIm
od
el
Ti|λ
i∼
E(λ
i),
lnλ
i|f
i=
f(x
i)+
ε i,
f|θ
θ θ∼
GP(0,C
(θθ θ)),
ε i| σ
2∼
N(0, σ
2),
σ2,η
2,ρ
2 j∼
IG(0.0
01,0.0
01).
Typ
eII
mo
de
l
Ni|λ
i∼
P(L
iλi),
lnλ
i|f
i=
f(x
i)+
ε i,
f|θ
θ θ∼
GP(0,C
(θθ θ)),
ε i| σ
2∼
N(0, σ
2),
σ2,η
2,ρ
2 j∼
IG(0.0
01,0.0
01).
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
41
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Pro
po
sed
mo
de
ls
Typ
eIm
od
el
Ti|λ
i∼
E(λ
i),
lnλ
i|f
i=
f(x
i)+
ε i,
f|θ
θ θ∼
GP(0,C
(θθ θ)),
ε i| σ
2∼
N(0, σ
2),
σ2,η
2,ρ
2 j∼
IG(0.0
01,0.0
01).
Typ
eII
mo
de
l
Ni|λ
i∼
P(L
iλi),
lnλ
i|f
i=
f(x
i)+
ε i,
f|θ
θ θ∼
GP(0,C
(θθ θ)),
ε i| σ
2∼
N(0, σ
2),
σ2,η
2,ρ
2 j∼
IG(0.0
01,0.0
01).
No
teth
at
by
de
co
mp
osi
tio
n,
the
join
tp
ost
erio
rfo
rλλ λ
,f
an
dσ
2g
ive
nth
eo
b-
serv
ed
da
ta,D
,is p( λλ λ
,f,σ
2|D
)=
p(λλ λ
|f,σ
2,D
)·p(f|σ
2,D
)·p(σ
2|D
),
wh
ere
p( λλ λ
|f,σ
2,D
)∝
p(D
| λλ λ)·p(λλ λ
|f,σ
2).
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
41
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Infe
ren
ce
pro
ce
du
refo
rth
eTy
pe
IIm
od
el
The
like
liho
od
fun
ctio
n:
p(D
|λλ λ)=
M ∏ i=1
λn
ii ni!
e−λ
i.
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
fσ
2:
p( σ
2| λ
λ λ,f,D
)∝IG
( α+
M 2,
β+
1 2(l
nλλ λ−
f)T(l
nλλ λ−
f)) .
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
ff:
p(f| λ
λ λ,σ
2,D
)∝GP
( σ−
2A−
1ln
λλ λ,A
−1) ,
wh
ere
A=
σ−
2I+
C(θθ θ
)−1.
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
fλλ λ
giv
en
the
da
ta,
an
dth
ep
ara
me
-te
rsσ
2a
nd
f:
p( λλ λ
|f,σ
2,D
)∝
1( 2
πσ
2) M
/2
exp
( −λλ λ
T1−
1
2σ
2(l
nλλ λ−
f)T(l
nλλ λ−
f)
)(
M ∏ i=1
λn
ii ni!
),
wh
ere
1d
en
ote
sa
ve
cto
rw
ith
all
en
trie
so
ne
,th
at
is,
1=(1,...,1)T
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
42
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Infe
ren
ce
pro
ce
du
refo
rth
eTy
pe
IIm
od
el
The
like
liho
od
fun
ctio
n:
p(D
|λλ λ)=
M ∏ i=1
λn
ii ni!
e−λ
i.
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
fσ
2:
p(σ
2|λ
λ λ,f,D
)∝IG
( α+
M 2,
β+
1 2(l
nλλ λ−
f)T(l
nλλ λ−
f)) .
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
ff:
p(f|λ
λ λ,σ
2,D
)∝GP
( σ−
2A−
1ln
λλ λ,A
−1) ,
wh
ere
A=
σ−
2I+
C(θθ θ
)−1.
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
fλλ λ
giv
en
the
da
ta,
an
dth
ep
ara
me
-te
rsσ
2a
nd
f:
p( λλ λ
|f,σ
2,D
)∝
1( 2
πσ
2) M
/2
exp
( −λλ λ
T1−
1
2σ
2(l
nλλ λ−
f)T(l
nλλ λ−
f)
)(
M ∏ i=1
λn
ii ni!
),
wh
ere
1d
en
ote
sa
ve
cto
rw
ith
all
en
trie
so
ne
,th
at
is,
1=(1,...,1)T
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
42
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Infe
ren
ce
pro
ce
du
refo
rth
eTy
pe
IIm
od
el
The
like
liho
od
fun
ctio
n:
p(D
|λλ λ)=
M ∏ i=1
λn
ii ni!
e−λ
i.
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
fσ
2:
p( σ
2| λ
λ λ,f,D
)∝IG
( α+
M 2,
β+
1 2(l
nλλ λ−
f)T(l
nλλ λ−
f)) .
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
ff:
p(f| λ
λ λ,σ
2,D
)∝GP
( σ−
2A−
1ln
λλ λ,A
−1) ,
wh
ere
A=
σ−
2I+
C(θθ θ
)−1.
The
co
nd
itio
na
lpo
ste
rio
rd
istr
ibu
tio
no
fλλ λ
giv
en
the
da
ta,
an
dth
ep
ara
me
-te
rsσ
2a
nd
f:
p(λλ λ
|f,σ
2,D
)∝
1( 2
πσ
2) M
/2
exp
( −λλ λ
T1−
1
2σ
2(l
nλλ λ−
f)T(l
nλλ λ−
f)
)(
M ∏ i=1
λn
ii ni!
),
wh
ere
1d
en
ote
sa
ve
cto
rw
ith
all
en
trie
so
ne
,th
at
is,
1=(1,...,1)T
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
42
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Mo
de
lse
lec
tio
na
nd
pre
dic
tive
ca
pa
city
Mo
de
lse
lec
tio
n:
We
use
ava
ria
nt
of
the
de
via
nc
ein
form
atio
nc
rite
rio
n(D
IC)
de
fine
da
sfo
llow
s
DIC
3=−
4E[l
np(n
|θθ θ)|n
,M]+
2ln
p(n
|n,M
),
wh
ere
p(n
|n,M
)=
M ∏ i=1
p(n
i|n
,M),
an
d
p(n
i|n
,M)=
1 J
J ∑ j=1
p(n
i|n
, λi,
j,M)=
1 J
J ∑ j=1
λn
ii,
je−
λi,
j
ni!
.
Pre
dic
tive
ca
pa
city:
We
use
the
pre
dic
tio
nsq
ua
ree
rro
r(P
SE)
wh
ich
isd
e-
fine
da
s
PSE=
1
M−
r
M ∑i=
r+1
( ni−E[N
i|n
1,...,n
i−1])
2
,
wh
ere
the
ob
serv
ed
da
taa
reth
en
um
be
rso
fso
ftw
are
failu
res
N1=
n1,...,N
M=
nM
,fo
ra
tra
inin
gsa
mp
leo
fsi
zer.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
43
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
DS1
ha
ve
be
en
take
nfr
om
Rin
saka
et
al.
(2006).
Itc
on
tain
s54
failu
rec
ou
nts
ob
serv
ed
du
rin
g17
we
eks.
The
rea
reth
ree
soft
wa
rem
etr
ics:
Exe
cu
tio
ntim
e(C
PU
hr)
,Fa
ilure
ide
ntific
a-
tio
nw
ork
(pe
rso
nh
r)a
nd
Co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n(C
PU
hr)
.
Va
rio
us
mo
de
lsb
ase
do
nb
oth
line
ar
reg
ress
ion
sa
nd
on
the
use
of
Ga
uss
ian
pro
ce
sse
s,w
ith
diff
ere
nt
soft
wa
rem
etr
ics
as
inp
uts
,w
ere
co
nsi
de
red
for
DS1
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
44
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
DS1
ha
ve
be
en
take
nfr
om
Rin
saka
et
al.
(2006).
Itc
on
tain
s54
failu
rec
ou
nts
ob
serv
ed
du
rin
g17
we
eks.
The
rea
reth
ree
soft
wa
rem
etr
ics:
Exe
cu
tio
ntim
e(C
PU
hr)
,Fa
ilure
ide
ntific
a-
tio
nw
ork
(pe
rso
nh
r)a
nd
Co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n(C
PU
hr)
.
Va
rio
us
mo
de
lsb
ase
do
nb
oth
line
ar
reg
ress
ion
sa
nd
on
the
use
of
Ga
uss
ian
pro
ce
sse
s,w
ith
diff
ere
nt
soft
wa
rem
etr
ics
as
inp
uts
,w
ere
co
nsi
de
red
for
DS1
.
Tab
le:
DIC
3c
rite
rio
nfo
rd
iffe
ren
tty
pe
IIm
od
els
for
DS1
Mo
de
lD
IC3
Mo
de
lD
IC3
β0+
βC
x C88.9
387
GP
(C)
59.5
539
β0+
βE
x E86.5
568
GP
(E)
59.3
764
βF
x F67.9
521
GP
(F)
58.8
286
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
44
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
DS1
ha
ve
be
en
take
nfr
om
Rin
saka
et
al.
(2006).
Itc
on
tain
s54
failu
rec
ou
nts
ob
serv
ed
du
rin
g17
we
eks.
The
rea
reth
ree
soft
wa
rem
etr
ics:
Exe
cu
tio
ntim
e(C
PU
hr)
,Fa
ilure
ide
ntific
a-
tio
nw
ork
(pe
rso
nh
r)a
nd
Co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n(C
PU
hr)
.
Va
rio
us
mo
de
lsb
ase
do
nb
oth
line
ar
reg
ress
ion
sa
nd
on
the
use
of
Ga
uss
ian
pro
ce
sse
s,w
ith
diff
ere
nt
soft
wa
rem
etr
ics
as
inp
uts
,w
ere
co
nsi
de
red
for
DS1
.
Tab
le:
DIC
3c
rite
rio
nfo
rd
iffe
ren
tty
pe
IIm
od
els
for
DS1
Mo
de
lD
IC3
Mo
de
lD
IC3
β0+
βC
x C88.9
387
GP
(C)
59.5
539
β0+
βE
x E86.5
568
GP
(E)
59.3
764
βF
x F67.9
521
GP
(F)
58.8
286
β0+
βE
x E+
βC
x C88.6
426
GP
(EC
)58.9
651
βE
x E+
βF
x F69.2
479
GP
(EF)
58.7
668
βF
x F+
βC
x C68.5
449
GP
(FC
)59.0
388
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
44
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
DS1
ha
ve
be
en
take
nfr
om
Rin
saka
et
al.
(2006).
Itc
on
tain
s54
failu
rec
ou
nts
ob
serv
ed
du
rin
g17
we
eks.
The
rea
reth
ree
soft
wa
rem
etr
ics:
Exe
cu
tio
ntim
e(C
PU
hr)
,Fa
ilure
ide
ntific
a-
tio
nw
ork
(pe
rso
nh
r)a
nd
Co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n(C
PU
hr)
.
Va
rio
us
mo
de
lsb
ase
do
nb
oth
line
ar
reg
ress
ion
sa
nd
on
the
use
of
Ga
uss
ian
pro
ce
sse
s,w
ith
diff
ere
nt
soft
wa
rem
etr
ics
as
inp
uts
,w
ere
co
nsi
de
red
for
DS1
.
Tab
le:
DIC
3c
rite
rio
nfo
rd
iffe
ren
tty
pe
IIm
od
els
for
DS1
Mo
de
lD
IC3
Mo
de
lD
IC3
β0+
βC
x C88.9
387
GP
(C)
59.5
539
β0+
βE
x E86.5
568
GP
(E)
59.3
764
βF
x F67.9
521
GP
(F)
58.8
286
β0+
βE
x E+
βC
x C88.6
426
GP
(EC
)58.9
651
βE
x E+
βF
x F69.2
479
GP
(EF)
58.7
668
βF
x F+
βC
x C68.5
449
GP
(FC
)59.0
388
βE
x E+
βF
x F+
βC
x C70.1
835
GP
(EFC
)58.5
152
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
44
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
02
46
81
01
21
41
61
802468
10
12
14
16
18
we
ek
number of failures
ob
se
rve
d d
ata
estim
ate
d m
ea
n
Fig
ure
:Est
ima
ted
me
an
nu
mb
er
of
failu
res
an
d95%
inte
rva
lsfo
rD
S1g
ive
nth
eG
P(E
FC)
mo
de
l.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
45
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
We
pe
rfo
rmo
ne
ste
pp
red
ictio
n.
We
sha
llc
on
sid
er
tra
inin
gse
tsc
on
sist
ing
of
(ap
pro
xim
ate
ly)
the
first
50%
an
d75%
an
d90%
of
the
sam
ple
.
The
pro
po
rtio
na
lin
ten
sity
PIM
(·,·
)m
od
elp
rop
ose
db
yR
insa
ka
et
al.
(2006),
wh
ere
the
first
co
mp
on
en
tre
pre
sen
tsth
em
etr
ics
use
d(E
=e
xec
utio
ntim
e,
C=
failu
reid
en
tific
atio
nw
ork
an
dF=
co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n)
an
dth
ese
co
nd
isa
dis
trib
utio
nfu
nc
tio
n,
suc
ha
sW
=W
eib
ull
dis
trib
utio
n,
G=
Ga
mm
ad
istr
ibu
tio
na
ndE=
exp
on
en
tia
ldis
trib
utio
n.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
46
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
We
pe
rfo
rmo
ne
ste
pp
red
ictio
n.
We
sha
llc
on
sid
er
tra
inin
gse
tsc
on
sist
ing
of
(ap
pro
xim
ate
ly)
the
first
50%
an
d75%
an
d90%
of
the
sam
ple
.
The
pro
po
rtio
na
lin
ten
sity
PIM
(·,·
)m
od
elp
rop
ose
db
yR
insa
ka
et
al.
(2006),
wh
ere
the
first
co
mp
on
en
tre
pre
sen
tsth
em
etr
ics
use
d(E
=e
xec
utio
ntim
e,
C=
failu
reid
en
tific
atio
nw
ork
an
dF=
co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n)
an
dth
ese
co
nd
isa
dis
trib
utio
nfu
nc
tio
n,
suc
ha
sW
=W
eib
ull
dis
trib
utio
n,
G=
Ga
mm
ad
istr
ibu
tio
na
ndE=
exp
on
en
tia
ldis
trib
utio
n.
Tab
le:
Pre
dic
tio
nsq
ua
red
err
ors
mu
ltip
lied
by
100
for
DS1
Mo
de
l50%
75%
90%
PIM
(F,G
)143
1.0
2.0
PIM
(F,W
)7
2.0
4.0
GP
(F)
4.7
32.5
31.6
0
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
46
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
We
pe
rfo
rmo
ne
ste
pp
red
ictio
n.
We
sha
llc
on
sid
er
tra
inin
gse
tsc
on
sist
ing
of
(ap
pro
xim
ate
ly)
the
first
50%
an
d75%
an
d90%
of
the
sam
ple
.
The
pro
po
rtio
na
lin
ten
sity
PIM
(·,·
)m
od
elp
rop
ose
db
yR
insa
ka
et
al.
(2006),
wh
ere
the
first
co
mp
on
en
tre
pre
sen
tsth
em
etr
ics
use
d(E
=e
xec
utio
ntim
e,
C=
failu
reid
en
tific
atio
nw
ork
an
dF=
co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n)
an
dth
ese
co
nd
isa
dis
trib
utio
nfu
nc
tio
n,
suc
ha
sW
=W
eib
ull
dis
trib
utio
n,
G=
Ga
mm
ad
istr
ibu
tio
na
ndE=
exp
on
en
tia
ldis
trib
utio
n.
Tab
le:
Pre
dic
tio
nsq
ua
red
err
ors
mu
ltip
lied
by
100
for
DS1
Mo
de
l50%
75%
90%
PIM
(F,G
)143
1.0
2.0
PIM
(F,W
)7
2.0
4.0
GP
(F)
4.7
32.5
31.6
0P
IM(E
F,W
)11.0
2.0
3.0
GP
(EF)
8.8
91.6
3.6
5
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
46
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
We
pe
rfo
rmo
ne
ste
pp
red
ictio
n.
We
sha
llc
on
sid
er
tra
inin
gse
tsc
on
sist
ing
of
(ap
pro
xim
ate
ly)
the
first
50%
an
d75%
an
d90%
of
the
sam
ple
.
The
pro
po
rtio
na
lin
ten
sity
PIM
(·,·
)m
od
elp
rop
ose
db
yR
insa
ka
et
al.
(2006),
wh
ere
the
first
co
mp
on
en
tre
pre
sen
tsth
em
etr
ics
use
d(E
=e
xec
utio
ntim
e,
C=
failu
reid
en
tific
atio
nw
ork
an
dF=
co
mp
ute
rtim
e-f
ailu
reid
en
tific
atio
n)
an
dth
ese
co
nd
isa
dis
trib
utio
nfu
nc
tio
n,
suc
ha
sW
=W
eib
ull
dis
trib
utio
n,
G=
Ga
mm
ad
istr
ibu
tio
na
ndE=
exp
on
en
tia
ldis
trib
utio
n.
Tab
le:
Pre
dic
tio
nsq
ua
red
err
ors
mu
ltip
lied
by
100
for
DS1
Mo
de
l50%
75%
90%
PIM
(F,G
)143
1.0
2.0
PIM
(F,W
)7
2.0
4.0
GP
(F)
4.7
32.5
31.6
0P
IM(E
F,W
)11.0
2.0
3.0
GP
(EF)
8.8
91.6
3.6
5P
IM(E
FC,W
)N
C2.0
1.0
GP
(EFC
)12.5
15.1
60.2
5
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
46
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Pre
limin
ari
es
An
ew
ap
pro
ac
hto
SR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
Ap
plic
atio
ns
tore
ald
ata
sets
:D
S1
02
46
810
12
14
16
18
0
10
20
30
40
50
60
week
cumulative number of failures
observ
ed d
ata
unobserv
ed d
ata
GP
(F)
GP
(EF
)
GP
(EF
C)
Fig
ure
:P
red
icte
dn
um
be
ro
ffa
ilure
sfr
om
the
75%
ob
serv
atio
np
oin
tfo
rD
S1.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
47
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Ou
tlin
e
1In
tro
du
ctio
nM
otiva
tio
nR
elia
bili
tym
ea
sure
sSt
oc
ha
stic
co
un
tin
gp
roc
ess
es
Mo
de
lso
fo
rde
red
ran
do
mva
ria
ble
sN
etw
ork
s
2St
oc
ha
stic
co
mp
ariso
ns
of
spa
cin
gs
ba
sed
on
ord
er
sta
tist
ics
De
finitio
ns
The
on
esa
mp
lep
rob
lem
The
two
sam
ple
pro
ble
m
3B
aye
sia
nIn
fere
nc
eP
relim
ina
rie
sA
ne
wa
pp
roa
ch
toSR
Ms
usi
ng
co
va
ria
tes
Ap
plic
atio
ns
tore
ald
ata
sets
4C
on
clu
sio
ns
an
dc
on
trib
utio
ns
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
48
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Co
nc
lusi
on
sa
nd
Co
ntr
ibu
tio
ns
➨In
the
on
esa
mp
lep
rob
lem
,w
eh
ave
sho
wn
tha
tth
ec
on
jec
ture
of
Ko
ch
ar
an
dK
ow
ar
(1996)
istr
ue
for
n=
4a
nd
we
ha
ve
est
ab
lish
ed
ha
zard
rate
or-
de
rin
gb
etw
ee
nth
ese
co
nd
an
dth
irdn
orm
aliz
ed
spa
cin
gs.
We
als
oh
ave
ob
tain
ed
the
sere
sults
for
sim
ple
spa
cin
gs.
➨In
the
two
sam
ple
pro
ble
m,
we
ha
ve
de
rive
dth
ec
on
ditio
ns
un
de
rw
hic
hth
esp
ac
ing
s(b
oth
,si
mp
lea
nd
no
rma
lize
d)
are
ord
ere
da
cc
ord
ing
toth
elik
elih
oo
dra
tio
ord
er.
We
ha
ve
illu
stra
ted
the
sere
sults
with
an
ap
plic
atio
nto
mu
ltip
le-o
utlie
rm
od
els
.
➨W
eh
ave
de
fine
da
ne
wm
od
elb
ase
do
nG
au
ssia
np
roc
ess
es
wh
ich
isu
sefu
lto
pre
dic
tso
ftw
are
failu
reu
sin
gin
form
atio
nfr
om
soft
wa
rem
etr
ics.
➨Th
em
od
els
we
rec
on
stru
cte
du
nd
er
Ba
ye
sia
nfr
am
ew
ork
an
dth
ep
ost
erio
rin
fere
nc
ew
as
pe
rfo
rme
du
sin
gM
ark
ov
Ch
ain
Mo
nte
Ca
rlo
me
tho
ds.
The
co
nd
itio
na
lap
pro
xim
atio
nw
as
imp
lem
en
ted
for
Ma
tla
bw
hic
hp
rovid
es
an
effi
cie
nt
use
rin
terf
ac
ea
nd
aw
ide
va
rie
tyo
fre
ad
ym
ad
eto
olb
oxe
s.
➨So
me
rea
lda
tac
ase
stu
die
sh
as
be
en
pre
sen
ted
toill
ust
rate
the
me
tho
do
-lo
gy
de
ve
lop
ed
.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
49
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
Re
fere
nc
es
1S.
C.
Ko
ch
ar,
R.
Ko
rwa
r,Sto
ch
ast
ico
rde
rsfo
rsp
ac
ing
so
fh
ete
rog
en
eo
us
ex-
po
ne
ntia
lra
nd
om
va
ria
ble
s,J.
MU
LTIV
AR
IATE
AN
AL.
57(1
996)6
9–83.
2C
.E.
Ra
smu
sse
na
nd
C.K
.I.
Will
iam
s,G
au
ssia
nP
roc
ess
es
for
Ma
ch
ine
Lea
rn-
ing
,Th
eM
ITP
ress
,2006.
3M
.Sh
ake
d,J.
G.Sh
an
thik
um
ar,
Sto
ch
ast
icO
rde
rs,Sp
rin
ge
r,N
ew
Yo
rk,2007.
4N
.D.
Sin
gp
urw
alla
an
dS.
Wils
on
,Sta
tist
ica
lM
eth
od
sin
So
ftw
are
Re
liab
ility
,Sp
rin
ge
rV
erla
g,N
ew
Yo
rk,1999.
5N
.To
rra
do
,R
.E.Li
lloa
nd
M.P
.W
ipe
r,O
nth
ec
on
jec
ture
ofK
oc
ha
ra
nd
Ko
wa
r,J.
MU
LTIV
AR
IATE
AN
AL.
101(2
010)1
274–1283.
6N
.To
rra
do
,R
.E.
Lillo
an
dM
.P.
Wip
er,
Se
qu
en
tia
lo
rde
rst
atist
ics:
ag
ein
ga
nd
sto
ch
ast
ico
rde
rin
gs,
ME
THO
DO
LO
GY
AN
DC
OM
PU
TIN
GIN
AP
PLIE
DP
RO
BA
BIL
ITY.
Toa
pp
ea
r.
7N
.To
rra
do
an
dJ.
J.P.
Ve
erm
an
,A
sym
pto
tic
relia
bili
tyth
eo
ryo
fk-o
ut-
ofn
sys-
tem
s,J.
STA
TIS
TIC
AL
PLA
NN
ING
AN
DIN
FE
RE
NC
E.
142(2
012)2
646–2655.
8N
.To
rra
do
an
dR
.E.
Lillo
,Li
ke
liho
od
ord
er
of
spa
cin
gs
fro
mtw
oh
ete
rog
e-
ne
ou
ssa
mp
les,
J.
MU
LTIV
AR
IATE
AN
AL.
Un
de
rse
co
nd
revie
w.
9N
.To
rra
do
,R
.E.
Lillo
an
dM
.P.
Wip
er,
So
ftw
are
relia
bili
tym
od
elin
gw
ith
soft
-w
are
me
tric
sd
ata
via
Ga
uss
ian
pro
ce
sse
s,IE
EE
T RA
NS.
SO
FTW
AR
EE
NG
INE
ER-
ING
.U
nd
er
sec
on
dre
vie
w.
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
50
Intr
od
uc
tio
n
Sto
ch
ast
icc
om
pa
riso
ns
ofsp
ac
ing
sb
ase
do
no
rde
rst
atist
ics
Ba
ye
sia
nIn
fere
nc
e
Co
nc
lusi
on
sa
nd
co
ntr
ibu
tio
ns
THA
NK
YO
UFO
RY
OU
RA
TTE
NTI
ON
BC
AM
,Ju
ne
2012
STO
CH
AS
TIC
PR
OP
ER
TIE
SA
ND
INFE
RE
NC
EFO
RR
EPA
IRA
BLE
SY
STE
MS
51
