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& Pf =
. * 0K > ? J & N Y* C , BH \ &
: + 2 = 2
+-/0 D4& +> ? J + N
NMR
.
A Systematic Approach to Sensitivity Analysis of Fault Tolerant
Systems in NMR Architecture
Kourosh Aslan Sefat(1) – Gholamreza Latif Shabgahi(2)
(1) MSc. – Departmant of Electrical Engeneering, Shahid Beheshti University, Tehran, Iran
(2) Assistant Professor - Departmant of Electrical Engeneering, Shahid Beheshti University, Tehran, Iran
A fault tree illustrates the ways through which a system fails. It states different ways in which combination of faulty components result in an undesired event in the system. Being used in phases such as designing and exploiting industrial systems, and the designers able to evaluate the dependability attributes such as reliability, MTTF and sensitivity. In addition, in the mentioned ability, the fault tree is a systematic method for finding systems bottlenecks and weakness point. In spite of its extensive use in evaluating the reliability of systems, fault tree is rarely used in calculating sensitivity. In the last decade, few researches has been conducted in this field, however these methods are not applicable to large scale systems and are not systematic. This paper provides a systematic method for evaluating system sensitivity through fault tree. Then, it introduces sensitivity of NMR architecture as one of the common structures of fault tolerance which is used for enhancing systems’ reliability, safety and availability in industry. This article presents a comprehensive and parameterized formula for NMR structure's sensitivity. The presented method can be a great help for designing and exploiting reliable systems engineers in systematic and instant calculation of sensitivity by means of fault tree.
Index Terms: Sensitivity, fault tree, redundancy, NMR architecture, fault tolerance.
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Fig. (3): An OR gate with three input [33]
B= & C ( B )
1 ( - C* & & ) P*& N >* S & ,
w & ) & G_ B
B= & [ B ( )
2 ( 0 BH \
] 5 [ .
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34 [.
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Fig. (4): An AND gate with three input [33]
w & ) & BH \ ` A & N >* S A - ( B= & [ B & K & , Error! Reference
source not found. 0 BH \
] 5 [.
(
)
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A
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A B C A
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(
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) `1 5 4 > ? J B* & * & -w<w * ~? Y , :(
Fig. (5): Fuzzy triangular membership function as a basic event in Fault Tree
) `1 6 BX0 y * ~? Y , :(
B* & * & 4 > ? J
Fig. (6): Fuzzy Trapezoidal membership function as a basic event in Fault Tree
( ) ( ) ( )
OR
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1 1 a , 1 b , 1 c
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= = =
= − − − −
∏
∏
∏
(7)AND
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= = =
=
∏ ∏ ∏
(8)(
)
(
)
(
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(
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OR
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i 1 i 1
1
P 1 a , 1 b ,...
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= = = = −
= − −
− −
∏
∏
∏
∏
) 9 (
AND
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P
,
= = = =
=
∏ ∏ ∏ ∏
(10)* X & C* & -_ N& * X 4 • & B( -* 0K & a
+
b + c d - B 4 ) T0 B0 )& E [ B -
B( V) & ,
) & N >* S
AND
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) N >* S C* & `HA OR
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12 ( * ~? Y , &
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j
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, ,
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≠ ≠ ≠
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∏
∏
∏
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i j i j i j i j
, ,
K
a , b , c , d
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,
− − − −
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∏
∏
∏
∏
) 12 (
• & P*& B( & (y B I r
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& P*& & J -
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- & A BH \ BH \ & P*& & B*& & & B )
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& B
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"
"
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"
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& * N
TOP
)
TE
& B* & * & C* B H 0 (
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1(
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5 6 A #) # ? P' 1 G
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8
(
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X
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r & J E & & B …h & P * B
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K
. 0&
[
]
T
12 21 111 112 221 222
X = P ;P ;P ;P ;P ;P (14) & Y ) & C* -E J /0 0K . & > ? J
& Y - >, & B …h & P * B r & B & P*& . )
B= & [ ) 15 ( - B 0 .
T
TE 1 2 11 22
Y
=
P ;P ;P ;P ;P
(15)
) `1 7 B0 0 :( -1 , & > ? J C* & & ]
5 [
Fig. (7): An example of static Fault Tree [5]
(8 & ` 1., & a B= & N 8) a* ,
) 16 ( &
I ) C ( B > ? J - B 0 I &
.
T
TE 1 2 11 22 TE
1 2 11 22
T
12 21 111 112 221 222 Y
11
A X
12
221 222
B
P P P P P
P 1 1 1 0 0
P 0 1 0 k 0
P 0 0 1 0 1
P 0 0 0 1 0
P 0 0 0 0 1
P P P P P P
0 0 0 0 0 0
k 0 0 0 0 0
0 1 0 0 0 0
0 0 1 1 0 0
0 0 0 0 k k
− −
−
) 16 ( a* , P &
A
+
B
& 0 x \_
X Y
[ B
B= & & N a* , =A )
13 ( B - . *K T B
k = B* & * N& B( ( ‡ 4 B<f j E
) 1 ( . 0 ) ‡ 4
Table (1): Probabilistic values of basic events E ) 1 ( & * -,r N& * X : B*
12
P =0.01 P21= 0.02 P111= 0.05
112
P = 0.06 P221= 0.04 P222 = 0.03
B= & & V & )
6 (( ) &
AND
B= & ) 4 ( ) &
OR
) & C* N& [ B
) E
2 ( - BH \
. )
) 1 (
AND i = 1
i =
P
∏
P
) 2 (
( )
OR i
i =1 = 1 - 1 - P
P
∏
E ) 2 ( ) -,r N& * X :
Table (2): Probabilistic values of intermediate events 22
P = 0.0688 P = 0.00311
1
P = 0.01297 P = 0.0013762
TE
-5
P = 1.17846×10
B B= & ) ( )
3 ( ) &
OR
(8 > ? J
* X
K
) • & [ B )
17 ( , ) 20 ( - BH \ ` A ( .
) 17 (
(
)
11 12 11
1 = P 1 - P
K = 0.22899
P
) 18 (
(
)
12 11 12
1
= P 1 - P
K = 0.768697
P
) 19 (
(
)
221 2 22 22 1
22
= P 1 - P
K = 0 .564
P
) 20 (
(
)
222 221 222
22 = P 1 - P
K = 0.419
P
* X P , & a
K
) &
OR
> ? J
B= & k = N 8) a* , +- * & )
21 ( J & &
: & & J B 0 > ?
TE 1 2 11 22 12 21 111 112 221 222
T E
1
2
11
22
A B
P P P P P P P P P P P
P 1 -1 -1 0 0 0 0 0 0 0 0
P 0 1 0 -0.299 0 0.769 0 0 0 0 0
P 0 0 1 0 -1 0 1 0 0 0 0
P 0 0 0 1 0 0 0 1 1 0 0
P 0 0 0 0 1 0 0 0 0 0.564 0.419
) 21 (
B= & & V & )
13 ( N a* , B= & [ B
) 22 ( B
- ) & , B N a* , = & , . *K & , > ? J
& * & , B K B*
- J W & = a* , P*& .
)
9
(
TOP
- B* & * & C* B H 0 a* , .
) N I = B<f P*& BH \ N & * B H 0 C*
- . & J B*
& * B B H 0 ) P*& N B( 21
+ 222 221 V
H 0 . & V Z & X & * /* B
12 21 111 112 221 222
TE
1
2
11
22
-3 -4 -5 -5 -4 -4
-2 -3
-2 -2
P P P P P P 1.372 × 10 8.923 × 10 8.173 × 10 6.811× 10 2.516 × 10 2.49 × 10 P
0.997 0 5.940 × 10 4.950 × 10 0 0 P
S = P 0 6.880 × 10 0 0 1.940 × 10 1. P
P
-2
-2
940 × 10
0 0 5.999 × 10 0.500 0 0
0 0 0 0 0.970 0.960
) 22 (
? P' 2 :
) `1 8 C* > ? J (
TMR
- d* 0 & .
P*&
M, & ‡ 4 k<= w(& N @ 0 & 0 ) +
v & d ; B 0
- ( & . 4&
) `1 8 > ? J :( TMR
Fig. (8): Fault Tree of TMR architecture
& & & X
Y • & [ B )
23 ( ) 24 ( - ` 1., . 0 )
[
]
T
a b c
X = P ;P ;P (23) T
TE 1 2 3
Y = P ;P ;P ;P (24)
* X
K
) & &
OR
) • & [ B )
25 ( , ) 27 ( BH \ (
- X B( & (y B I r . 0 ) *
K
) &
AND
C*
- B 4 ) T0 .
(
)(
)
G 1 G 2 G 3 G 1
T E
= P 1 - P 1 - P
K
P (25)
(
)(
)
G2 G1 G3
G2
TE = P 1 - P 1 - P
K
P (26)
(
)(
)
G 3 G 2 G 1 G 3
T E
= P 1 - P 1 - P
K
P (27)
& - P*& > ? J B & N 8) a* , & ,
B= & [ ) 28 ( | B*& a* , P*& . K B
B*& * X . & & & A C* 7 J ) |
) * ) P , C* =A r -V U?
) | * X & *
LG. Hw U? B*
a* , B K B N 8) a* , . 0&
A B
B= & & B 1 )
13 ( a* , N - BH \
. )
TE 1 2 3 a b c
TE
1
2
3
G1 G2 G3
B A
P P P P P P P
P 1 -K -K -K 0 0 0
P 0 1 0 0 1 1 0
P 0 0 1 0 1 0 1
P 0 0 0 1 0 1 1
) 28 (
& BH \ N & & = TMR
c B(
& * N &D TOP
& * & C* B B + & B*
B= & [ ) 29 ( B - . *K
(
)
(
)
(
)
TMR G1 G2
TMR G1 G3
TMR G2 G3
a
b
c
T
P K + K
P
P K + K
S =
P
P K + K
P
)
10
(
6 6 7 ' NMR?+' G J K
B= & ƒ^& & B dG P*& B &
N BH \ &
NMR
- . &
B= & ) 0 ‡ 4 & /* 1* B* & J N& )& )
30 ( (
& N • &
TMR + 5MR + 7MR 9MR B
& [ B - . k* O ) • ) 31 ( , ) 34 ( - B ( . *K ) 30 (
a b c 1 2 3
P = P = P , P = P = P = P ) 31 (
(
)
TMR AND AND
TMR event
a
2
2P P 1- P
= P S ) 32 (
(
)
5MR AND AND
5MR event
a
9
6P P 1- P
= P S ) 33 (
(
)
7MR AND AND
7MR event
a
34
20P P 1- P
= P S ) 34 (
(
)
9MR AND AND
9MR event
a
125
70P P 1- P
=
P
S
- 4 • & . - * & & B Y E -_ 4 & ,
N
NMR
B= & [ B )
35 ( . 0 B^& &
(
)
NMR event
NMR AND AND
a N -1 N+1 2 = N -1
P P 1- P
N -1 2 P S ) 35 (
4 B= & PNMR
B= & & ) 36 ( B< E & YE& B(
] 35 B + & K [ - . *K
(
)
N-1 2 i N-i NMR i=0 NP = P 1- P
i
∑
(36)- & B= & P*& K [ Hq& B( ( [ Hq& -S * & X & 7 & & ,
] YE 36 ) B= & . & K [ 35
P _ & & ( B_ X P*&
N - * & E
NMR
. & B^& &
1987 ] YE 37 B= & [ N - * & & & &
TMR
) B= & - \ & . ( B^& & * N 35
(
B= & K B( & -4 ( N
& 3 * X & & B ; G 0&
v - & J N& 1* ] YE & B= &
37 [
& , B( - / ) ` N + ( ; G 0& V K
) B= & - .0 B( & J 1* (8 B= & 35
(
> ? J . & B_ X P*& B^& &
TMR
)
AND
& . &
5MR
& B ) P*&
& > , , P B
NMR
)
AND
& &
N+1/2
B= & &8_ . & )
35 ( + PAND v & , B B
&
B= & ) 37 ( . & BH \ ` A
N +1 2
AND i
i=1
P =
∏
P) 37 (
- _ B B( & (y B I r +-*& X & [ Hq& C ( B & ,
4 -. ( [ Hq& & B^& &
7 6 +K S # T G
& ? [ B B_ X K • & dG P*& - *
- ) -/0 D4& &D [& o, q& . 0 ) N
Y* ,) - & J „ 0 @ 0 (
- - N (- & J -,r N&
-)R* N .
&
- B( & < A [& o, *) - & J N& [& o, B H 0 0& ,
v ( O& 4 & [& o, P*& o
N& v -,r . & B( 0K & . 0 q , -
G v - & J „ 0 &D4& +-* 0 Y* , &
I 0 0 * & - & J „ 0 &D4& H*& Y* , &
G H, & @ 0 & - & J „ 0 -,r N& @ 0 & &D4&
- B &8_ ( - ; G 0& & H*& +-* 0 Y* ,
0
N TOP NMR
v - & J „ 0 B( & & K
- B * X - 4 -,r N& Y* , @ 0 B . 0
[ B & H*& +-* 0 Y* , Y & , ' & O& < A • & ) 38 ( , ) 40 ( - . ) ) 38 (
( )
-λtR t = e
) 39 (
( )
β t-γ -ηR t = e
) 40 (
( )
( )
x n -λtx=0
λt e R t =
x!
∑
7 6 1 6 / , U C V $% 7 #? W '
N dG P*& TOP
B= & B() O& < A B H 0
H*& +-* 0 Y* , & - A + ( & & E& - & J „ 0 X - H, &
- - + (
. )
) `1 9 O& < A d*&D4& B H 0 N [& o, (
- & J „ 0 -* 0 - & J Y* , Y , - A + 0.01
& +
- .0
- . .
v O& < A Bh B(
NMR
& * N .
TOP - & J B H 0
v - ( P n .
- `1 P*& - / V) & ,
)
11
(
v O& - (
v & , d*&D4& /* [ H? B .
- v O& < A H ( 0& , P*& ( & HE N , &
n + +B *D d*&D4& B @ S ` HA P*& & & -)
1 & . & *8
) `1 9 N - \ :( NMR
v O& < A H 0 Y , x \_ )
(-* 0 Y* ,
Fig. (9): Sensitivity analysis of NMR architecture vs. modules' reliability (exponential distribution function)
BG 0 & B V) B( =0 - & J 4 &D4& I 0
- H*& -,r N& Y* , .
) `1 10 [& o, - \ (
N
NMR
+ H*& @ 0 & -,r N& Y* , B( - A +
& ) & & ,& & + V
100 & & 1.8
+
- .0 O& < A d*&D4& B H 0 & ` <\, .
- \ & ) `1
9 [ V, . & D 0 `1 P*& (
N& Y* , H*& -,r N& Y* , `1 P*& -* 0 -,r
- . & * ) LG. < A * X B( - / B(
C* D0 C* B O& d*&D4& B H 0 N 4& /0K ,
. & . O& < A
) `1 10 N - \ :( NMR
v O& < A B H 0 Y , x \_ )
(C* & ) D H*& Y* ,
Fig. (10): Sensitivity analysis of NMR architecture vs. modules' reliability (Weibull distribution function withβ >1)
) `1 11 N [& o, ( O& < A d*&D4& B H 0
„ 0 & & J Y* ,) ,r N& 4 K & J B( - & J 0.01 x & 4 - .0 & + & ( `HA `1 0 .
` <\, 4& B( [ V, P*& & D 0 `1 P*& &
& . O& < A B( - / N .
P*&
N - \
9MR
d*&D4& - \ P*& . & V
. & . `HA - \ B H 0 N 4& O& < A
.0 \ B P*& -& Y* , N B(
I& K [& o, < A B H 0 ,D , [& o, H*& Y* , ,
& O& . - N& B €* 0 C* D0 A& B P n .
- & * & C* &DE& O& < A B( - / 0&
- d ( K - & J N * d*&D4& . *
) `1 11 N - \ :( NMR
v O& < A B H 0 , x \_ )
Y
( & Y* ,
Fig. (11): Sensitivity analysis of NMR architecture vs. modules' reliability (Poisson distribution function)
7 6 2 6 / 'G H 7 #
N dG P*&
TOP
K q , d*&D4& B H 0
&DE& DE& -,r N& 4 - A +
H*& +-* 0 Y* , & &
- H, & - & .0 + (
- \ . dG P*&
) -0 0 , 200 , -0 N& ( . 0&
) `1 12 - \ (
N [& o,
NMR
v - & J B( - A + Y* , &
-* 0 „ 0 0.01 - .0 8) B H 0 & + ( H, .
- .
* - d*&D4& N 8) B(
. & B O& < A d ( K s E ,
- . P n
v & , d*&D4& B( N 4&
- ( & (8 P*& + (
& , d*&D4& B( & K ` _ B 4
v O& < A 4& „ 0 K >H B r &
- d ( 8) A& N . *
)
12
(
) `1 13 x \_ & 8) B H 0 N [& o, - \ (
0 `HA `1 [ S 4 - .
& ) D H *& B( [ V, P*&
& C* 1.8
‡ 4 - \ B( & (y B I r . & ‡ 4
K (y & ` _ P B & J -* 0 Y* , 0 C* & ` <\, . & T 4 dG P*& & - \ & B( -*
& s S , (-* 0 Y* ,) & D 0 - \ P*& &
& , K [ V, B( & ( B @ B _ &
-* 0 Y* , B H 0 [& o, P*& & & .0 B( =0 . & ( { K
) `1 12 N - \ :( NMR
\_ ) [& o, B H 0 Y* , Y , x
(-* 0
Fig. (12): Sensitivity analysis of NMR architecture vs. time (exponential distribution function)
) `1 13 N - \ :( NMR
Y* , Y , x \_ ) [& o, B H 0 (C* & ) D H*&
Fig. (13): Sensitivity analysis of NMR architecture vs. time (Weibull distribution function withβ >1)
- & J „ 0 & Y* , x \_ 0.01
x & 4 [& o, - \
[ B 8) B > 0 N ) `1
14 ( B - . *K
- .0 - \ P*& & * N d*&D4& >E 8) B(
TOP
v & , d*&D4& - d ( & 4& P*&
> . - \ P*& . & ( `HA - \ B H 0 - \ P*&
N
9MR
. & V 8) B H 0
) `1 14 N - \ :( NMR
Y* , Y , x \_ ) [& o, B H 0 ( &
Fig. (14): Sensitivity analysis of NMR architecture vs. time (Poisson distribution function)
Y* , C* & ) D H*& Y* , B W - \ & +-* 0 . N [& o, > B _ & [ | \_ -* 0 Y* , Y* , B W - \ *& I , cUJ & ,D , K - \
I& K 4 & - & J N 8) B & ,
TE
B
v ( d*&D4& - &
- ( N V) & ,
& Y* , w P * s=
N& )& . & ( [& o, ,
v - & J q [ B - & J Y* , @ 0 P 4 ) T0
- x \_
` <\, & , - & A }\ B & & B( -*
& 0 )
. 0 B^& & ~? C* - & J N& [& o, B H 0 N [& o, - \
- * /_ [ B
) `1 15 - \ P*& . & & .0 (
[ V & , _ N h & 3
+ 5 + 7 9 T0 B0 D4& v
- . . & B 4 ) +- & J N& d*&D4& B(
v - & J B H 0 N n + d*&D4&
P
v & , d*&D4& q , * N d ( B0 D4&
- - T0 B v & , d*&D4& B(
- B0 D4& & ,
+ & d ( & DE C* - & J B H 0 -<( - & J N - I , -) n +B *D d*&D4& B ( P*& .
YA& & B\_ M B_ X P*& B^& & C* & • &
- * M, B & N &D (8 . .(
0 20 40 60 80 100 120 140 160 180 200 10-250
10-200 10-150 10-100 10-50 100
Sensitivity Analysis of NMR Sys. with Exp. Distribution Func
λ = 0.01
Time
S
e
n
s
iti
v
ity
)
13
(
) `1 15 v C* - & J N& [& o, B H 0 N [& o, - \ :(
Fig. (15): Sensitivity analysis of NMR architecture vs. module's probability of failure
B &DE& - & J B H 0 - & J N [& o, -/0 /h - * /_ [ ) `1
16 \ - \ P*& . & K (
N& /0 -X4& \ N ( ?
\ ) v C* - & J
X
v & , (
NMR
- N d ( -/0 D4& d*&D4& q , - \ B H .
- * v C* - & J B H 0 - & J .
- \ BE , ` A B 10 ( E B( & P*& B^& &
v & ,
- d ( N B0 D4&
. * *
- & J „ 0 /n ` ? B( & BE , &D P*& B
- 0 C* D0 - \ P*& & B _ N YA&
d (
v - & J N& - B -/0 D4& & , d*&D4&
. *K
) `1 16 H 0 N [& o, B - \ :( v C* - & J N& [& o, B
v & ,
Fig. (16): Sensitivity analysis of NMR architecture vs. module's probability of failure and number of redundant modules (in 3
dimensions)
8 6 Y # H
=A I ? ( x \_ - * &
Pf =
& -1* & ? B & N - * & B_ X P*& . & - & -._ h 7 =A I ? BE& & & A BE ,
B + N - * & B I 0& [ X X\, - & a & - * 0 7 U . J& K BH \ & C ,
BH \
N
NMR
7 & -1* & ? B &
B= & B^& & 7 & V & . * ) B^& & -/0 D4& &
&D4& I 0 C ( B B^& & P*& N BH \ & B
MATLAB
[& o, &8) q , B _ = & B= & P*& ? €* 0
& * & v [ MG. B*
& *
TOP
. B^& &
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