TELKOM
NIKA
, Vol.12, No
.2, June 20
14
, pp. 437~4
4
6
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v12i2.1950
437
Re
cei
v
ed Ma
rch 7, 2
014;
Re
vised Ap
ril
22, 2014; Accepte
d
May 8
,
2014
Availability Analysis of Predictive Hybrid M-Out-of-N
Systems
Abba
s Kari
mi
1
Department o
f
Computer En
gin
eeri
ng, F
a
cult
y
of E
ngin
e
e
r
ing, Arak Bran
ch, Islamic Aza
d
Univ
ersit
y
,
Arak, Iran
2
Department o
f
Computer an
d Commu
nicati
on S
y
stems En
gin
eeri
ng, F
a
cult
y
of Engin
e
e
r
ing, Un
iversiti
Putra Mala
ys
ia
, Mala
y
s
ia
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: Akarimi@i
au-
arak.ac.ir
A
b
st
r
a
ct
In m
-
out-of-n system
, if m
-
out-of-n m
o
dules
agr
ee, system
c
a
n report c
ons
ensus; otherwise,
the
system
fails. O
n
the
other
ha
nd, in pr
edictiv
e hy
brid syst
em
if there is
no agreement,
a hist
ory rec
o
rd of
previ
ous s
u
cce
ssful resu
lt(s) i
s
use
d
to
pred
i
c
t the o
u
t
put. In or
der to
an
al
y
z
e
the
ava
ila
b
ility of
pre
d
ictiv
e
hybri
d
redu
nd
a
n
cy system, Markov mod
e
li
n
g
is utili
z
e
d. B
y
using Marko
v
mod
e
l of the
system in steady
state, the av
ailab
ility
is deri
ved
and compared w
i
th
m-out
-of-n system
. The results of si
m
u
lati
on
de
mo
nstrated t
hat the
avai
la
b
ility of pr
edictiv
e hybr
id syste
m
is
hi
gher
th
an m-o
u
t-of-n system espec
i
a
lly
for large
m
.
Key w
o
rds
:
Fault-tole
rant
System
, Red
unda
ncy, m
-
out-of-n
syste
m
, voting.
1. Introduc
tion
Red
und
ancy
is a well-kno
w
n techniq
u
e
to enhan
ce f
ault tolera
nce
of highly reli
able an
d
highly availa
ble co
ntrol
systems. Red
unda
ncy of hard
w
a
r
e m
odule
s
is p
e
r
hap
s the m
o
st
appli
c
able f
o
rm of redu
nd
ancy in
co
ntrol syste
m
s
a
nd is
appli
e
d
in thre
e forms of p
a
ssive
(stati
c), a
c
tive (dyn
amic)
and hyb
r
id.
Voter is the
main el
emen
t in pa
ssive
redu
nda
ncy t
hat
mas
ks t
he e
ff
ect of fault fr
om the outpu
t of
the syst
e
m
. Active redunda
ncy doe
s not try to hide
failure
but
det
ects fault(s) a
nd lo
cate
the
faulty
elemen
ts. In hyb
r
id
a
ppro
a
ch, the
system
ma
sks
faults
while
th
e me
ch
anism
s fo
r fault
det
ection, fa
ult l
o
catio
n
a
nd f
ault recovery
are
pe
rform
e
d to
rec
o
n
fi
gu
re t
he sy
stem in
ca
se
of fault occu
rre
nc
e [
5
]. The m
-
out
-of-n
sy
stem, as
bee
n wi
d
e
ly
applie
d in
en
ginee
ring
sy
stems, utili
ze
s static
redu
n
dan
cy on
n
parall
e
l o
r
se
ries redu
nda
nt
module
s
a
nd
function
s wh
en at lea
s
t m module
s
am
ong n mo
dule
s
in the
syste
m
wo
rk p
r
op
e
r
ly
[3][4] .
Predi
ctive hybrid
red
und
a
n
cy [6] is
hybrid
red
und
a
n
cy a
r
chitect
u
re. Thi
s
architecture
has only di
scussed
on t
r
ipl
e
mod
u
lar re
dund
an
cy (T
MR) where th
ree
pa
rallel m
odule
s
a
r
e
u
s
ed.
In fact, predi
ctive hybrid
redun
dan
cy a
s
in [6]
is the
hard
w
a
r
e im
plementatio
n
of hybrid vot
e
rs
inco
rpo
r
ating
smoothi
ng an
d predi
ction. It has been
o
r
iginally pre
s
e
n
ted for X-by
-Wi
re sy
stem
s;
however, the
numbe
r of se
nso
r
s
and
a
c
t
uators in re
al
X-by-Wi
re sy
stem
s is no
rmally more th
an
three. P
r
edi
ct
ive Hyb
r
id
m
-
out-of-n
sy
stem (P
Hmn
)
[
22] was a
ppli
ed o
n
n
redu
ndant
mod
u
l
e
s
Nom
e
ncla
tur
e
n
Number of modules
m
Number of agreed modules
i
Number of failed components in
the s
y
stem,
i
=
0
,
1,…,
n-m
+1.
()
P
t
i
First
deriv
a
tion
of
()
t
i
;
0
iF
.
i
F
a
ilure
rat
e
of
th
e s
y
s
t
em
when
t
h
ere
are
i
fai
l
ed
components;
0
in
m
or
in
m
.
()
Ls
i
Laplace tr
ansform of
()
t
i
;
0
iF
.
i
Repair
rat
e
of
th
e s
y
s
t
em
when
t
h
ere
are
i
fai
l
ed
components;
11
in
m
.
1
()
L
s
i
An Inverse
Laplace tr
ansform of
()
Ls
i
;
0
iF
()
P
t
i
The probab
ility
that ther
e
are i failed
components
in
th
e s
y
s
t
em
a
t
tim
e
t;
0
iF
.
s
A
S
t
ead
y S
t
ate
Av
ail
a
bili
t
y
of
s
y
s
t
em
.
*, Faraneh Zarafshan
, S.A.R Al-Ha
d
ad
Department o
f
Computer En
gin
eeri
ng, F
a
cult
y
of E
ngin
e
e
r
ing, Hamedan
Bran
ch, Islamic Aza
d
Univ
ersit
y
,
Hamedan
, Iran
3
1 2 3
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 2, June 20
14: 437 – 44
6
438
rathe
r
than th
ree mo
dule
s
as in [6]. For real
-tim
e appl
ication
s
in
clu
d
ing X-by-Wi
r
e system
s (e.
g
.,
Brake-by
-Wi
re, Steering
b
y
-Wire),
sen
s
ors an
d a
c
tu
ators mu
st b
e
availabl
e al
l the time. T
hey
function
imm
ediately on
ce up
on a
sensory d
a
ta
is avail
able
;
otherwise t
he result of
this
unavailability is cata
strophi
c.
Variou
s app
roache
s have been intr
odu
ced in literature to com
pute
the availability of m-
out-of-n syst
ems by
con
s
iderin
g
di
ff
erent
techniqu
es and di
ff
e
r
ent sce
nari
o
s of failures [
12]–
[17],[21]. The novelty of thi
s
work is to use
Ma
rkov
process for modeling
the
availability of P
H
m
n
system an
d calcul
ating rel
a
ted equatio
n
s
by appl
ying
the steady state conditio
n
. Moreover, the
e
ff
ect
s
of fail
ure
rate
an
d
rep
a
ir rate
on the
availa
bility of the
system have
been
taken i
n
to
accou
n
t. To the best of o
u
r kn
owl
edg
e, avail
ability analysi
s
of PHmn sy
ste
m
has not b
een
introdu
ce
d a
nd analy
z
ed
in literature. In this
study, steady state
availability
and MTT
R
a
r
e
utilized for e
s
timation of MTTF and MT
BF. While no
close
d
-fo
r
m solutio
n
wa
s repo
rted in th
e
literature for estimation of
MTTF and M
T
BF, when a
predi
ction p
h
a
se i
s
co
nsi
d
ered.
The re
st of this pa
pe
r is
orga
nized a
s
fo
llows: in S
e
ction 2, System descri
p
tion and
assumptions
are described and avail
a
bility of PH
mn systems is obtained
by m
a
thematical
and
probabili
stic methods.
The availability of PHmn
syst
ems a
nd m
-
out-of-n
systems are
compared
in se
ction 4. Finally, the concl
u
si
on an
d future wo
rks are di
scu
s
sed.
2. Predictiv
e
H
y
brid
m
-
out-o
f-
n sy
stem
(PHm
n)
A control
system is assu
med with n redun
dant
mo
dule
s
whi
c
h
work in pa
ral
l
el. This
architectu
re i
s
well-kno
w
n
as N-Mo
dul
ar R
edu
nda
n
c
y (NMR). Every module
gene
rate
s ou
tput
indep
ende
ntly and the
out
put of a m
o
d
u
le do
es
not i
n
fl
u
e
n
c
e o
n
o
ne an
othe
r. T
he mo
dule
s
a
r
e
repai
ra
ble at
any time; however, o
n
ly one mo
dule i
s
eligibl
e
to fail or repai
r in a time unit. A
deci
s
io
n ma
ki
ng mo
dule
known a
s
Vot
e
r p
e
rfo
r
ms
d
e
ci
sion
ma
kin
g
or voting o
n
the o
u
tputs of
redu
nda
nt m
odule
s
. In P
H
mn
system,
a two
-
ph
ased voter i
s
utilized. In th
e
fi
r
s
t ph
as
e, a
majority voting [11] is appli
ed on t
he voter’s in
puts. V
o
ter gen
erate
s
a
fi
rst ph
ase deci
s
io
n i
ff
at
least a m
a
jori
ty of inputs, i.e., m =
n + 1
=
2
, agree o
r
almost a
g
ree
(co
n
si
de
ring
a thre
shol
d).
A
se
con
d
pha
se inco
rp
orati
ng Predi
ction
[7]–[9] or Smoothing [1]
is used for
p
o
ssible d
e
ci
si
on
ma
k
i
ng
, wh
en
th
e
fi
rst pha
se d
o
e
s
not
make
con
s
en
su
s. The
acti
vity of
fi
nding
app
rop
r
iate v
o
ter
output in se
cond ph
ase is
based on
so
me cal
c
ulatio
ns on vote
r’s
history record
. Control
syst
em
fails inevitabl
y when the se
con
d
pha
se does n
o
t make a resul
t. The struct
ure of the PHmn
system i
s
deli
neated in Fig
u
re 1.
Figure 1. Structure of
the P
H
mn System
[22]
3. Av
ai
labilit
y
Modeling
The availabili
ty is de
fi
n
ed
as the p
r
ob
a
b
ility of a syst
em to funct
i
on co
rrectly
and be
available at t
he in
stant of
time [5]. Si
milar di
scu
s
sions fo
r relia
bility analysis as in [2
2] a
r
e
pre
s
ente
d
for the availability analysis. Combinat
o
r
ial
modelin
g and
the Markov modelin
g are
two
known techni
ques to model the av
ailability of the system; however
, Markov modeling i
s
used in
this study for
some
rea
s
o
n
s
. Markov mo
dels a
r
e
very
robu
st [23]; many system
s ca
nnot si
m
p
ly
be modeled
by combi
natorial methods
because t
hey
concentrate
on probabilist
i
c technique for
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Availabilit
y Analysi
s
of Predictive Hybr
i
d
M-Out-of-N
System
(Abbas Karim
i
)
439
cal
c
ulatin
g a
v
ailability; and modeli
ng t
he re
pair
pr
o
c
e
ss i
s
not e
a
sy by combi
natorial
mod
e
ling
[5]. For Availability analysi
s
of PHmn sy
stem,
a Markov model as i
n
Figure 2 is
pre
s
ente
d
.
Figure 2. Markov availabilit
y model for PHmn sy
stem
Based
on th
e system
de
scriptio
n in Se
ction
2,
the PHmn syste
m
wo
rks in o
ne of the
three m
ode
s:
operating, p
r
edi
ction a
n
d
failed.
Whe
n
the syste
m
wo
rks correctly an
d voter
make
s d
e
ci
si
on in t
he
fi
rst pha
se of exe
c
ution, the
system is
in
op
erating m
ode
[19]. Recall t
hat
there m
a
y be
some
faulty module
s
in th
e syste
m
; ho
wever, the
nu
mbers of the
m
are l
e
ss th
an
the majo
rity. In the oth
e
r
word
s, at the l
east
(n
- m
+
1) non
-faulty m
odule fu
nctio
n
corre
c
tly. The
system
states in operatin
g mode a
r
e
lab
e
led from 0 t
o
(n-m
) a
s
is
i
n
Fig.2. The system
swit
ches
from an op
erating state i to next operati
ng stat
e (i
+1
) with an expo
nential failure
rate of
λ
i
.
Possi
ble t
r
an
sa
ction
s
fo
r
state i a
r
e i
→
i +
1;(i
=
0, 1,
2, · · ·, n
–
m), with
the
d
epartu
re
rate of
λ
i
= (n -
i)
λ
an
d i
→
i - 1;(i
= 1,
2, ·
·
·, n
- m
+
1),
with a
de
p
a
rture
rate
of
µ
i
=iµ .
S
y
st
em
swit
che
s
f
r
om
an
ope
ratin
g
state
(n
–
m) to p
r
edi
ction
state
(pr)
with
dep
artu
re
rat
e
λ
Pr
an
d fro
m
predi
ction
sta
t
e to previous operat
in
g sta
t
e with depa
rture rate µ
Pr
.
A failed mo
d
u
le may b
e
repai
re
d with
an exp
onen
tial rep
a
ir
rat
e
. For
simpli
city, all
values
of
a
nd
in
ope
rat
i
ng
states
are con
s
ide
r
ed
as th
e
same
and t
w
o t
r
an
sa
ction
s
a
r
e
not
allowed
simu
ltaneou
sly.
Whe
n
(n-m
+1) m
odule
s
are fail
ed
or are
in th
e repair qu
eue,
the
system mi
gra
t
es to Pre
d
ict
i
on mod
e
. Th
is mod
e
ha
s
one
state whi
c
h i
s
lab
e
led
by Pr. P
i
(t) a
nd
P
Pr
(t) denote
the prob
abilit
y that the sy
stem be in stat
e i and in stat
e Pr at time t, resp
ectively.
Based u
pon t
he relatio
n
s o
f
state i with its neig
hbo
rs,
00
1
()
()
()
pt
npt
p
t
,
(1)
11
()
[
(
(
)
)
()
(
1
)
(
)
(
1
)
()
]
1
ii
i
i
pt
n
i
i
p
t
n
i
p
t
i
p
t
f
o
r
i
n
m
,
(2)
And,
(
)
()
()
ff
r
pt
n
p
t
n
p
t
(3)
The initial co
ndition
s are P
0
(0)=1 an
d P
i
(
0
)=
0 for i
≠
0 .
By taking Laplace tran
sform of
equatio
ns (1, 2, 3), a matrix
form as [c]P(s)=P(0
)
[22].
The sy
stem is unavail
able
in state F, an
d t
he unavail
ability of the system is sho
w
n by P
f
(t), and the
r
ef
ore, sy
stem a
v
ailability is given by:
()
1
(
)
f
A
tP
t
(4)
And
1
()
£
(
)
ff
Pt
P
s
Solving the
equatio
ns fo
r P
i
(t) by taki
ng Lapl
ace i
n
verse tran
sf
orm of P
i
(s
) is
too
compli
cate
d.
Therefore
th
e ste
ady
stat
e conditio
n
i
s
define
d
fo
r
availability [2
4]. Steady st
ate
availability is denote
d
by A
s
in Equation 5.
A
s
=
lim P(s
y
s
t
em is
work
ing at time t)
(5)
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 2, June 20
14: 437 – 44
6
440
0
li
m
(
)
1
li
m
(
)
1
l
i
m
[
(
)
]
sf
f
tt
s
A
At
P
t
s
P
s
(6)
By defining
Γ
Pi ,
(i=
1
,…,n
−
m) as
()
i
Pi
n
i
(7)
And
Γ
Pr
as
Pr
()
(1
)
(
)
1
(1
)
nm
n
nm
nm
nm
nn
m
(8)
P
i
and P
Pr
are simpli
fi
ed to
0
*
;
[
1
,
...,
]
iP
i
PP
i
n
m
(9)
Pr
Pr
0
*.
PP
(10)
Sinc
e
0
1
nm
iP
r
f
i
pp
p
, we have:
0
Pr
0
1
(1
)
nm
Pi
i
P
(11)
Based o
n
Eq
uation (6
), the steady stat
e ava
ilability of PHmn syst
em is obtain
e
d
as
n-
m
Pi
Pr
i=
0
n-
m
Pi
Pr
i=
0
s
A
=
i
[
1
,..n
-
m
]
Γ
+
Γ
;
λ
Γ
+(
1+
)
Γ
μ
(12)
Mean Tim
e
T
o
Failu
re
(M
TTF) i
s
si
mpl
y
de
fi
n
ed a
s
the avera
ge
uptime of the
system
[18]. In this study, Steady State Availability and
M
e
a
n
T
i
m
e
t
o
R
e
p
a
i
r
(MTT
R) are
utili
zed
f
o
r
estimation of MTTF. The a
v
erage d
u
rati
on of time
th
at the system
spend
s
for repairi
ng a faulty
module i
s
kn
own a
s
MT
T
R
. Equation
(13), sho
w
s t
he rel
a
tion b
e
twee
n MTT
F
and MTT
R
and
the steady st
ate avail
ability (Equation 1
2
).
s
A=
M
TTF
M
TTF
+
M
TTR
(13)
As only one repair i
s
allo
wed at any time, t, MTTR equals to
and
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TELKOM
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Availabilit
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s
of Predictive Hybr
i
d
M-Out-of-N
System
(Abbas Karim
i
)
441
n-
m
Pi
Pr
i=
0
n-
m
Pi
Pr
i=
0
n-
m
Pi
Pr
i=
0
n-
m
Pi
P
r
i=
0
s
s
A
=
A
Γ
+
Γ
λ
Γ
+(
1+
)
Γ
M
TTR
μ
M
TTF
=
1-
Γ
+
Γ
μ
(1
-
)
λ
Γ
+(
1
+
)
Γ
μ
(14)
Mean
Tim
e
Between
Fail
ure
s
(MTBF
)
is
d
e
fi
ned
a
s
the ave
r
a
g
e
ope
rating
ti
me of th
e
system bet
ween co
nsecutiv
e failures (excludi
ng the
time duratio
n of a system in failed state)
[21]. MTBF i
s
estim
a
ted
according
to St
eady St
ate A
v
ailability and
MTBR,
as
seen i
n
Eq
uati
on
(15
)
.
s
s
A
=
A
MT
B
R
MT
B
F
1-
(15)
E
a
ch
mod
u
le
ha
s M
T
B
R
=
1
and
MTBF
=
1
. Therefore,
the steady
stat
e availability for
each m
odule
is
.
Whe
n
the system
fails, (n - m
+
2
)
mo
dule
s
a
r
e
waiting
f
o
r
repai
r.The
r
ef
ore, MTBR of
the system is
1
.
(2
)
MT
BR
nm
4. Experimental Re
sults
In this section, the availab
ility of PHmn system i
s
compar
ed to m-out-of-n sy
stem as in
[20]. For this purpose, Matlab simul
a
tor i
s
utilized. Simulations are iterated for di
ff
ere
n
t values of
n, m,
λ
an
d µ.
Once
the system switches
from
stat
e i to
state j
by failure
rat
e
λ
, it maybe be
repai
re
d an
d
returned to
th
e previou
s
worki
ng
state
b
y
repai
r
rate
µ; except fo
r
the fail state.
It
can
be obvio
usly cl
aimed
that the larg
e
r
the rate
of failure, the
sy
stem is
more
su
sceptible t
o
fail. Becau
s
e
the pro
babili
ty of failure is mo
re th
a
n
the proba
bility of repai
r. It is al
so exp
e
ct
ed
that a state with larger rate
of repair is li
kely re
p
a
ire
d
rathe
r
than g
o
ing to the ne
xt state which
is
perceptibly cl
ose
r
to the fail state.
The
re
sults o
f
simulatio
n
are
discu
s
se
d in t
w
o
sub
s
e
c
tion
s: ba
sed o
n
vari
ation of m,
λ
and µ, and
based on vari
ation of
λ
and µ.
4.1 The effec
t
of m,
λ
and
µ.
Thre
e scena
rios
fo
r repai
r rate
a
nd
fail
u
r
e rate are
co
nsid
ere
d
:
λ
< µ, i.e. failure rate i
s
smalle
r
than repair rate;
λ
=µ, i.e. failure
rate is eq
ual
to repai
r rate;
and
λ
>
µ
, i.e. failure rate is
large
r
than re
pair rate. The
result
s of simulation
s are
also analy
z
e
d
base
d
on the hardne
ss (n/
4
< m
≤
n/2
+
1
)
and
softness (m
≤
n/4
)
of an ag
ree
m
ent [22]. F
o
r in
stan
ce, i
n
a 12
8 mo
d
u
lar
system, if th
e
agreem
ent a
c
hieve
s
i
n
th
e
fi
rst
pha
se
of voting, m i
s
a
value
bet
wee
n
1
< m
≤
65.
This di
stan
ce
is divided int
o
soft agreem
ent (1 < m
≤
33) an
d ha
rd
agre
e
me
nt if
(33 < m
≤
65).
• Experiment 1.
λ
<µ
The results
of simulation shows that
the
availabilit
y for both m
-
out-of-n and PHmn
system
s a
r
e
1, i.e., they ar
e hig
h
ly avail
able
whe
n
λ
<µ. The 100% availability
is due to t
he l
o
w
failure rate (consequ
ently small p
r
ob
abi
lity of
failure) and high re
pair rate (co
n
se
que
ntly high
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TELKOM
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Vol. 12, No. 2, June 20
14: 437 – 44
6
442
probability of
repair and system
restoration),
whi
c
h yield to lo
ng-term
operati
on and
a highly
available syst
em.
• Experiment 2.
λ
=µ
In this scen
ario the val
u
e
s
of
λ
and
µ
a
r
e
co
nsid
ered
as the
same,
i.e. 0.5. F
o
r
λ
=µ=
0.5, the avail
ability of PHmn
improved 0.64% in ov
erall, an
d 1.29% for hard
agreements
as
sho
w
n in Fig.
3. As seen in
Fig.3 the improvem
ent
for soft agre
em
ents is ne
glig
ible and is cl
ose
to 1 for
small
m. Becau
s
e
module
s
e
a
s
ily agree
wh
en m p
o
sse
s
a sm
all valu
e in comp
ari
s
on
with n.
Ho
we
ver, the avail
ability of PHmn is
hig
her than m
-
o
u
t-of-n
system
whe
n
m te
nd
s to
highe
r value
s
.
Figure 3. Availability of PHmn and m-ou
t-of-n
sy
stem
s for n = 1
28,
1 < m < 65, =0.5, and
λ
=0.5,
µ=0.5
• Experiment 3.
Λ
>µ
The results o
f
availability in PHmn
and
m-out
-of-n
are displayed i
n
Figu
re
s 4-6
whe
r
e
λ
is
re
spe
c
tively 0.9, 0.7, a
nd 0.6
an
d
µ
is resp
e
c
tively 0.1, 0.3
and
0.4. Wh
en failu
re
rat
e
is
highe
r tha
n
repair rate, th
e availa
bility is exp
e
cte
d
t
o
de
gra
d
e
g
e
nerally. Thi
s
phe
nome
n
a
is
see
n
in Figu
re 4-6.
The availa
bili
ty improvem
ent for soft agree
ment
s is negligibl
e
a
s
seen in
Fi
gure
5-6;
however, PHmn improved
the general
availability to 5.43% (F
i
gure 5), 3.03% (Figure 6) and for
h
a
r
d ag
r
e
e
m
e
n
t
s
to 13
.8
2% (
F
igu
r
e 5) a
n
d
6
.
49
%
(
F
ig
ur
e 6
)
.
W
h
en
th
e fa
ilu
r
e r
a
te
is
mu
c
h
highe
r than the rep
a
ir rate, i.e.,
λ
=0.9 and µ=0.1, the availabilit
y of both system
s deg
r
a
d
e
s
signi
fi
cantly
compared to F
i
gure 5-6.
However, t
he
av
ailability of P
H
mn i
m
proves 35.6% for hard
agre
e
me
nt.
Figure 4. Availability of PHmn and m-ou
t-of
-n sy
stem
s for n = 1
28,
1 < m < 65,
λ
=0.9, and
µ =0.1
Figure 5. Availability of PHmn and m-ou
t-
of-n sy
stem
s for n = 1
28,
1 < m < 65,
λ
=0.7 an
d µ=0
.
3
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TELKOM
NIKA
ISSN:
1693-6
9
30
Availabilit
y Analysi
s
of Pr
edictive Hybri
d
M-Out-of-N
System
(Abbas Karim
i
)
443
Table 1. Mean availability of
PHmn vs. m-out-of-n
system
Parameters
0<
λ
<1
µ
=
0.
1
0<
λ
<1
µ
=
0.
5
0<
λ
<1
µ
=
0.
9
0<
µ
<
1
λ
=0.1
0<
µ
<
1
λ
=0.5
0<
µ
<
1
λ
=0.9
PHmn
0.5105
0.9192
0.9893
0.9706
0.8342
0.7006
m-o
u
t
-
of
-n
0.3182
0.8148
0.9685
0.9499
0.7301
0.5196
Figure 6. Availability of PHmn and m-ou
t-
of-n sy
stem
s for n = 1
28,
1 < m < 65,
λ
=0.6 an
d µ=0
.
4
4.2 The e
ff
ec
t of
λ
and µ.
As m
=
65 i
s
t
he bo
und
ary
of soft an
d h
a
rd
agreem
e
n
t, the e
ff
ec
t o
f
λ
on the
availability
of a 65-o
u
t-of-128
syste
m
is discu
ssed in this
section. Th
e result
s are prese
n
ted for t
w
o
scena
rio
s
:
fi
x
ed rep
a
ir a
n
d
v
a
ried failur
e
rate (0
<
λ
≤
1) as in Fi
g. 9-11; and
fi
xed failure an
d
varied repai
r rate (0
< µ
≤
1) as in Fig.
12-14. Theoretically,
the probability of
failure increases
and the avail
ability of the system
s de
cre
a
ses fo
r highe
r
failure
rates (Figu
r
e 9-11
) an
d the
higher the
repair
rate, the availability increases for
a
fi
xed fail
ure
rate. Th
e result of sim
u
lati
on
con
fi
rmed th
e theoretical
expectatio
n
s
as seen i
n
Fi
g. 9-14. Me
a
n
availability of PHmn sy
stem
vs. m-out
-of-n system
ba
sed on
di
ff
ere
n
t values
of
λ
and µ ha
s b
e
en dem
on
stra
ted in Tabl
e 1
.
It
sho
w
s 2.17%
, 14.26% and 34.84% im
provement in the averag
e
availability of PHmn when
λ
is
respe
c
tively 0.1, 0.5, and 0
.
9. Other valu
es of
λ
have
been
also investigate
d
in
whi
c
h in
crea
sing
in the availabi
lity of PHmn is obtain
ed.
As the other
conclusion from
Table
1
and Fi
gure 9-14, the
av
ailability of both
sy
stem
s i
s
clo
s
ed to
1 for lo
w failu
re
and hi
gh re
pair
rate. Althoug
h the
small µ
and la
rge
λ
yield to
the
worst availability of
the system, large
λ
has more n
egative e
ff
ect in comparison with sm
all µ.
The avail
abili
ty has
rea
c
h
ed to m
a
ximum 1
wh
en
µ=0.1
and
0
<
λ
< 1, wh
erea
s it
ha
s
not
achi
eved 1 f
o
r di
ff
e
r
e
n
t value
s
of µ. The average a
v
ailability in Figure 9 is
0
.
9893 for P
H
mn
system while
it is 0.9685 for m-
out
-of-n system sy
stems. The
s
e scena
rio
s
are highlighte
d
in
Table 1. G
e
n
e
rally, in a large scale
high
ly avail
able a
pplication, the most imp
o
rtant assumpti
on
is utilizing the highly available modu
les. Because
as t
hey fail, thei
r repai
r and restor
ation to a
full
operational m
anne
r doe
s n
o
t likely occur.
Figure 9. Availability of PHmn and m-ou
t-
of-n sy
stem
s for n = 1
28,
m = 65, 0 <
λ
< 1, µ=0.1
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Vol. 12, No. 2, June 20
14: 437 – 44
6
444
Figure 10. Availability of PHmn and m-ou
t-of-n
systems for n = 128, m = 65, 0 <
λ
< 1, µ=0.5
Figure 11. Availability of PHmn and m-out-
of-n
systems for n = 128, m = 65, 0 <
λ
< 1, µ=0.
9
Figure 12. Availability of PHmn and m-out-of-n
systems for n = 128, m = 65, 0 < µ < 1,
λ
=0.
1
Figure 13. Availability of PHmn and m-out-of
-n
systems for n = 128, m = 65, 0 < µ < 1,
λ
=0.5
Figure 14. Availability of PHmn and m-out-of
-n
systems for n = 128, m = 65, 0 < µ < 1,
λ
=0.9
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Availabilit
y Analysi
s
of Predictive Hybr
i
d
M-Out-of-N
System
(Abbas Karim
i
)
445
5. Conclusio
n
A PHmn syst
em is an
extensi
on of trip
le pr
e
d
ictive
redu
nda
ncy t
o
larg
e scale
control
system
an
d compri
se
s n re
dund
ant
h
a
rd
ware
m
odul
e
s
. If m-out-of-n mo
dule
s
are in
ag
ree
m
e
n
t,
the syste
m
m
a
ke
s a
n
outp
u
t; otherwise, a hi
story re
cord of
previous su
cc
es
sf
ul
re
sult
s i
s
us
e
d
to predi
ct the result of current
cycle. In
order to investigate t
he av
ailability of PHmn
system,
a
Markov avail
ability model has b
een p
r
e
s
ente
d
. Then
the availabili
ty of PHmn system ha
s be
en
derived
in
st
eady-state
co
ndition
and
si
mulated i
n
di
ff
e
r
en
t sc
en
ar
io
s o
f
r
e
pa
ir rate, µ
, failure
rate,
λ
, an
d
m (qu
o
rum of
con
s
e
n
sus)
as the
e
ff
e
c
ti
ve parameters on th
e
syst
em’s
availabil
i
ty
according to
the comp
uted
availability equation.
The
result sho
w
e
d
that the PHmn system h
a
s
totally more a
v
ailability than m-out
-of-n
system. In
all
ca
se
s, the use of PH
mn sy
stem is the
b
e
st
choi
ce
espe
ci
ally whe
n
the
large
scal
e control
system
s a
r
e d
ealt wi
th. The exce
ption is fo
r th
e
situation
s
wh
ere the u
s
e
of traditional
system
i
s
favored
due t
o
the co
st preferen
ce
s if the
numbe
r of m
is
very small.In
futu
re
wo
rks, th
e
other pa
ram
e
ters in
fl
ue
nc
in
g th
e s
y
ste
m
depe
ndability
will be invest
igated.
Ackn
o
w
l
e
dg
ements
We
woul
d like to app
re
cia
t
e all the revi
ewe
r
s.
T
h
is rese
arch wa
s partially
supp
orted
by
I
s
lamic A
z
a
d
Univ
er
sit
y
,
A
r
ak B
r
an
ch
re
sea
r
c
h
Grant. The aut
hors de
cla
r
e
that there i
s
n
o
con
fl
i
c
t of interest
s re
gardin
g
the publi
c
at
ion of this arti
cle.
Referen
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03; 2
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ault-T
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