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Criticality and sensitivity analysis of the components of a system
Reliability Engineering and System Safety 68 (2000) 147–152 www.elsevier.com/locate/ress
Criticality and sensitivity analysis of the components of a system V. Carot*, J. Sanz Department of Statistics and Operations Research, Politechnic University of Valencia, Camino de Versa s/n, 46022 Valencia, Spain Received 17 September 1998; accepted 25 January 2000
Abstract In the design of complex systems there is a great interest to know the relative importance of each of their elements. In this paper, we define a new method for measuring the relative importance of each element of the system. We have to specify that this paper concerns only nonrepairable systems and components. We present a way of calculating the criticality of each component for a complex system no matter what the random distribution of the life of the component is. The paper also demonstrates a simple way of calculating how the system life improves when the life of a component is improved. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Criticality; Reliability of systems; Reliability importance of components
1. Introduction The relative importance of the components of a system has been widely studied. The measures of importance are basically of two kinds: functional importance (that is, importance in relation to the reliability of the system) and structural importance. We consider only non-repairable systems and components. Barlow [1], Boland [2] and Tong [3] have studied the structural importance of the components of a system and more recently Meng [4,5] has presented new measures of that kind. Natvig [6] and Bergman [7] propose other measures of the importance of the components of a system. It is also interesting to consult Wall [8], Cheok [9] and Dutuit [10] about measures of importance and some others factors of importance. In this paper a new method is proposed for the importance of components from a functional point of view by studying how the system life improves when the mean life of a component is improved. With this knowledge one can highlight which component (or components) must be given greater attention, when all the components are independent from each other. 2. Sensitivity measure The importance of the k-component for the life of the system is calculated by studying the consequence for the mean life of the system by increasing the mean life of a component in Dmk. This calculation of 2m=2mk ; gives a * Corresponding author.
measure of the system sensitivity in relation to the kcomponent. The calculation of 2m=2mk is generally impossible to do analytically. A simulation method can be used to calculate the mean lives of the system for two values of the mean life of a k-component (for example, mk and mk ⫹ Dmk : When Dmk is small enough, we obtain the partial derivation as a forward finite-difference equation 2m Dm ⯝ 2mk Dmk
1
This means doing two runs of the simulation per component, but one run of any couple of runs may be the same for all the components. Thus it will be necessary to do nv ⫹ 1 runs (nv number of nodes), where in each run we generate, a large number of random lives of each component, for example, for each component we can generate 100,000 random lives, using a Monte Carlo method. For series and parallel systems when the variables are exponential probability distribution function (PDF) variables, this paper demonstrates that 2m p0 m0 k k mk 2mk
2
where p 0k is the probability of criticality and can be estimated as the number of times the system life matches the lifetime of the specified k-component divided by the number of simulations done in a run of the programme. m 0k is the critical mean value of the k-component and can be estimated as the sum of the lives of the k-component when it matches the system life/number of times there has been a match between a component life and a system life. mk is the
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V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
and so Nomenclature Life of the i-component Life of the system Mean value of the life of the system Mean value of the k-component life Sensitivity measure or partial variation for the k-component Dm=Dmk Approximation to the theoretical value of 2m=2mk lk Failure rate of the k-component Probability of criticality for the k-component p 0k m 0k Critical mean life value for the k-component Number of nodes of the system nv xi T m mk 2m=2mk
mean value of the k-component and it can be estimated as the sum of the lives of k-component/number of simulations done in a run of the programme. Consequently, the method proposed in this paper will only need one run of simulation to calculate the measure of the system sensitivity 2m=2mk as the calculation of the parameters pk, m 0k ; and mk requires only one run assuming that in this run, we generate for example 100,000 random lives of each component, saving much time in order to obtain the same results. For mixed networks, the same formula (2) will be used and compared to the theoretical results for a simple system since the theoretical demonstration is impracticable in more complex systems.
A series and a parallel network can be considered as extreme configurations. Thus the system life for any system of nv components is, min xi ⱕ T ⱕ max xi
3
1ⱕiⱕnv
nv X
the expressions at the left and right being the lives for series and parallel systems, respectively. The general propositions valid for both systems, series and parallel, will be applicable for any other mixed system. Consequently, a demonstration of formula (2) for a series system and a parallel system is presented.
li
5
i1
Therefore,PT follows an exponential distribution with a n li ; and the mean life of the system is, failure rate ni1 m
1 nv X
6
li
i1
Deriving the formula (6) with respect to mk, we obtain for series systems,
l2k !2 nv X li
2m 2mk
7
i1
3.2. Parallel In this case, the distribution function of the system life is, P T ⱕ y
nv Y
P Xi ⱕ y
i1
nv Y
1 ⫺ e⫺yli
8
i1
from which we obtain the mean life of the system, m
nv nv nv X X X 1 1 1 ⫺ ⫹ l l ⫹ l l ⫹ l j j ⫹ lk i1 i ij i⬍j i ijk i⬍j j⬍k i
⫺ … ⫺1nv ⫹1
3. Theoretical calculation of sensitivity measure for series and parallel systems
1ⱕiⱕnv
FT t 1 ⫺ e
⫺t
1 li ⫹ lj ⫹ …lnv
9
Deriving the formula (9) with respect to mk, we obtain for parallel systems, nv X 2m 1 1 l2k 2 ⫺ 2mk lk l ⫹ l k 2 i i1 i苷k
⫹
nv X ij i苷k j苷k
1 li ⫹ lj ⫹ lk 2
⫺ … ⫺1nv ⫹1
1 li ⫹ lj ⫹ … ⫹ lnv 2
! 10
4. Critical component: probability of criticality 3.1. Series In this case, the system reliability function is
P T ⬎ t
nv Y i1
P xi ⬎ t e
⫺t
nv X i1
li
4
For any kind of system, parallel, serial or a mixture, when the systems fail, the life of the system will be the same as that of a component of the system, identifying this component as the critical one, because it is the reason for the failure of the system. The probability that a k component is critical will be the probability of criticality of k, p 0k .
V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
149
Eq. (15) becomes
For a series network, it will be given by p 0k P Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv ! \ nv P 1 i 苷 k Xk ⱕ xi
Fk a= Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv " !# nv X 1 ⫺ exp ⫺ li a
17
i1
dxk
0
Z∞
Z∞
Z∞
xk
dx1 …
xk
dxnv f x1 ; x2 ; …; xn
11
m 0k E Xk =xk critical
hence 12
i1
For a parallel system
Z xk
Zxk
Z∞ 0
dxk
0
dx1 …
0
dxnv f x1 ; x2 ; …; xnv
13
0
Z xk
Zxk
Za dxk
0
dx1 …
1 1 ⫺ lk l ⫹ lk i i1 i苷k nv X ij i⬍j j苷k i苷k
⫹ ⫺1nv ⫹1
⫹ ⫺1
1 ⫺… li ⫹ lj ⫹ lk
1 li ⫹ lj ⫹ … ⫹ lnv
nv
14
" !# nv X lk 1 ⫺ exp ⫺a li a : nv X i1 li
P Xk ⱕ a 傽 Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv 15 p 0k
since
⫹ ⫺1 exp xk nv
P Xk ⱕ a 傽 Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv Z∞
Z∞
"
… xk
xk
p 0k 1 ⫺ exp ⫺
dxnv f x1 ; x2 ; …; xnv
nv X i1
!#
li
and the critical mean life m 0k E Xk =k critical " nv X lk 1 1 ⫺ ⫹… 0 2 2 p k lk i1 i苷k li ⫹ lk 1 l1 ⫹ l2 ⫹ … ⫹ lnv 2
21
#
6. Demonstration of the formula for obtaining the sensitivity measure
!#
li a
nv X
20
i1
⫹ ⫺1nv
dxk
19
The density function will be fxk xk =k critical " nv X lk ⫺lk xk ⫺ e⫺ lk ⫹li xk ⫹ … 0 e pk i1 i苷k
Fk a= Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv
Z∞
lk 1 ⫺ e⫺ lk ⫹li a ⫹ … lk ⫹ li
i1
!
The critical mean life is the mean life value for the kcomponent, if k is a critical component. For a series system, the conditional distribution of Xk ; Fk a= Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv will be:
0
li
dxnv f x1 ; x2 ; …; xnv
i1 i苷k
5. Critical mean life
0 nv X
1 ⫺ e⫺lk a ⫺
nv X
⫹
18
This equation shows that for a series system all the components have the same critical mean life, which is the mean life of the system. For a parallel system
hence p 0k lk
E T m
P Xk ⱖ a 傽 Xk ⱖ x1 傽 Xk ⱖ x2 … Xk ⱖ xnv
p 0k P Xk ⱖ x1 傽 Xk ⱖ x2 … 傽 Xk ⱖ xnv
1 nv X i1
lk nv X li
p 0k
and then it gives,
16
From Eqs. (7), (12) and (18), it can be obtained for a
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V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
Fig. 1. System with exponential lives for all components.
series system
10
0
7. Application of the method proposed to general systems
1
CB C B B lk C B 1 C l2k CB Clk p 0k m 0k 1 B ! C C B B nv nv 2 n X X v mk X A@ A @ li li li
2m 2mk
i1
i1
i1
22 Similarly from Eqs. (10), (14) and (21), it can be obtained for a parallel system 2m l2 m 0 k k p 0k 2mk lk
23
Therefore, for both a series system and a parallel one, as for a variable exponential PDF mk 1=lk ; we obtain the next equation 2m p0 m0 k k mk 2mk
24
is valid and it allows the calculation of 2m=2mk once the parameters p 0k ; m 0k and mk have been obtained from a single run of the simulation programme. The saving of time and cost is thus evident. On the other hand, from Eqs. (6), (12) and (18) for a series system, the following can be obtained 1 0 nv X lk C nv B X B lk 1 1 1 C C B k1 m nv C B X nv nv nv nv X X X X A k1 @ li li li li li i1
nv X
p 0k m 0k
i1
i1
I1
i1
25
k1
similarly from Eqs. (9), (14) and (21) for a parallel system, the following can be obtained m
nv X
m 0k p 0k
26
k1
Consequently, no matter whether it is a series or a parallel system, the mean life of the system can be calculated from the probability of criticality and the critical mean lives. In the summations (25) or (26), each term p 0k m 0k represents the contribution of a k-node to the mean life of the system and, taking into account Eqs. (24) and (26), 2m=2mk represents the unitary contribution of a k-node.
To establish the utility of the method proposed in this paper, it has been used for a mixed system and results have been compared to theoretical ones. The following parameters have been calculated: the probability of criticality, the critical mean life and the measure of the system sensitivity 2m=2mk for each component of the system, as well as the mean value of the component-life. A simple mixed system has been used to make the comparison intended since the theoretical calculations become impractical for more complex ones. The system studied is given in Fig. 1, where each component follows an exponential distribution with the mean lives indicated. The theoretical value of 2m=2mk for a component is calculated with the expression " 2m 1 1 2 l2 ⫹ 2 2m2 l2 ⫹ l5 l2 ⫹ l4 ⫹ l6 2 ⫺
1 1 ⫺ l2 ⫹ l3 ⫹ l5 ⫹ l6 2 l2 ⫹ l3 ⫹ l4 ⫹ l5 2
1 1 ⫺ l2 ⫹ l4 ⫹ l5 ⫹ l6 2 l2 ⫹ l3 ⫹ l4 ⫹ l6 2 # 2 ⫹ (27) l2 ⫹ l3 ⫹ l4 ⫹ l5 ⫹ l6 2 ⫺
and the value obtained for the component 2 in the system of Fig. 1 is 2m 0:33075 2m2
28
Similarly, the values calculated theoretically for the other components are 2m 0:37019; 2m3
2m 0:046; 2m4
2m 0:07458; 2m5
2m 0:08253 2m6
29
Five runs of our simulation programme have been performed and an estimation of p 0k ; m 0k and mk for each component of Fig. 1 have been obtained in each run. From these and using formula (24), we obtain the partial variation for each component (see Table 1). In the last
V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
151
Table 1 Comparison between the results from a program simulation and the theoretical values Component number
Probability of criticality (1) Critical mean value (2) Partial variation (3) Mean value of the component-life (4) Run 1
2
3
4
5
6
Run 2
Theoretical values
Run 3
Run 4
Run 5
Mean
(1) 0.2957 (2) 22.418 (3) 0.3285 (4) 20.181
0.2966 22.239 3299 20.051
0.2955 21984 3293 19.727
0.2988 22241 3347 19.862
0.3021 22251 3392 19.821
2977 22227 3321 19928
3308 20
0.3707 25.266 3731 25.101
0.3653 25.283 3704 24.932
0.3711 25.013 3742 24.807
0.3659 25.384 3693 25.146
0.3667 25.122 0.3686 24990
3679 25215 3711 24995
3702 25
0.0601 23.325 468 29.971
0.0567 23.646 449 29840
0.0607 22.361 452 30.021
593 23.197 460 29882
460 30
1224 23844 732 39878
746 40
1527 27104 828 49.963
825 50
0.0591 23.483 464 29.945
600 23170 469 29.635
0.1219 23.881 729 39910
0.1243 23.659 737 39.896
1190 23.859 715 39.718
0.1244 23.685 746 39.521
0.1222 24.138 731 40.346
0.1526 27.187 820 50.589
0.1538 26.329 820 49.382
0.1544 26940 0.0837 49.693
0.1542 27.293 835 50.375
0.1483 27770 827 49.776
column are the theoretical values according to formula (27) for each component. The values obtained for the partial variation, thanks to p^ 0 ·m^ 0 2m k k 2mk m^ k
30
is a good estimation of the theoretical ones. The theoretical calculation of p 0k and m 0k is rather more difficult, but it can be expected that the values obtained in our simulation method are a good estimate of the theoretical ones, as they give a good estimation of the theoretical partial variation.
When we consider that the components follow a distribution different from the exponential, the same relation from Eq. (30) can be extracted. For example, when we have two components in serial following a uniform distribution between 0 and 10 for the first component and between 0 and 15 for the second component, we can obtain proof that the formula (30) works in an analytical form. For this data, the results for five runs of our simulation program are shown in Table 2. When the same uniform components are arranged as parallel, we have demonstrated that the same result of Eq. (30)
Table 2 Comparison between the results from a program simulation and the theoretical values for system with uniform distributed components Component number
Probability of criticality (1) Critical mean value (2) Partial variation (3) Mean value of the component-life (4)
Theoretical values
Run 1
Run 2
Run 3
Run 4
Run 5
Mean
2
(1) 0.6703 (2) 4.150 (3) 0.55816 (4) 4.984
0.6692 4.172 0.55912 4.993
0.6704 4.198 0.56109 5.017
0.6629 4.181 0.55201 5.021
0.6575 4.158 0.54593 5.008
0.66606 4.17180 0.55526 5.00460
0.66667 4.16667 0.55556 5.00000
3
0.3297 3.342 0.14648 7.521
0.3308 3.349 0.14736 7.517
0.3296 3.338 0.14627 7.521
0.3371 3.385 0.15237 7.488
0.3425 3.303 0.15284 7.402
0.33394 3.34340 0.14906 7.48980
0.33333 3.33333 0.14815 7.50000
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V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
Table 3 Comparison between the results from a program simulation and the theoretical values for a parallel system with uniform distributed components Component number
Probability of criticality (1) Critical mean value (2) Partial variation (3) Mean value of the component-life (4)
Theoretical values
Run 1
Run 2
Run 3
Run 4
Run 5
Mean
2
0.3314 6.651 0.44114 4.997
0.3331 6.657 0.44427 4.991
0.3325 6.698 0.44682 4.985
0.3318 6.695 0.44386 5.005
0.332 6.711 0.44258 5.035
0.33216 6.68240 0.44373 5.00260
0.33333 0.66667 0.44444 5.00000
3
0.6686 9.567 0.85315 7.497
0.6669 9.551 0.85117 7.483
0.6675 9.568 0.85136 7.501
0.6682 9.586 0.85126 7.524
0.668 9.567 0.85142 7.505
0.66784 9.56780 0.85167 7.50200
0.66667 9.58333 0.85185 7.50000
applies, and the values obtained from our simulation program are shown in Table 3. For general distribution, and more complex systems the analytical demonstration of the formula (30) is very complicated, and we obtain the validation of that expression by the use of Monte Carlo methods. These results can be generalized for more complex systems in which the life for each component may follow a different distribution function and so the partial variation 2m=2mk is impossible to calculate theoretically. Thus, the use of the formula (30) deduced in this paper can be of great use in the design process of complex systems since it easily allows evaluation of how much the system life improves when the lives of its components are improved. It gives a measure of the importance of the components of a system. We consider that all the components of the system are independents. The case for a component’s dependence will be the object of future work. 8. Conclusions In this work, we have defined a new measure of the relative importance of an element of the system for its mean life, when we deal with non-repairable systems and components. The measure of the relative importance of the component is made by means that we have defined as sensitivity for a component, expressed as the partial variation of the mean life of the system in respect to the mean of the component. We have demonstrated that for serial, parallel or complex systems, with the components distributed as exponentially and independents, we obtained the partial variation of the mean life in respect to the component as a product of the probability of criticality and the critical mean life of the
component and divided this product by the mean life of that component. This form of obtaining the measure of relative importance of a component has been demonstrated analytically when components follow a uniform distribution. For more general components and forms of the graph of the system, we carried out a demonstration using the Monte Carlo method.
References [1] Barlow RE, Proschan F. Statistical theory of reliability and life testing. To begin with (Silver Spring, MD), 1981. [2] Boland PJ, EL-Neweihi E. Measures of component importance in reliability theory. Computer and Operations Research 1995;22(4):455–63. [3] Boland PJ, Proschan F, Tong YL. Optimal arrangement of components via pairwise rearrangements. Naval Research Logistics 1989;36:807–15. [4] Meng FC. Comparing criticality of nodes via minimal cut (path) sets for coherent systems. Probability in the Engineering and Informational Sciences 1994;8(1):79–87. [5] Meng FC. Comparing the importance of system components by some structural characteristics. IEEE Transactions on Reliability 1996;45(1):59–65. [6] Natvig B. New light on measures of importance of system components. Scandinavian Journal of Statistics 1985;12:43–54. [7] Bergman B. On Reliability theory and its applications. Scandinavian Journal of Statistics 1985;12:1–41. [8] Wall I, Worledge D. Some perspective on risk importance measures. Proceedings of the International Conference on Probabilistic Safety Assesment, PSA’96. [9] Cheok M, Parry G, Sherry R. Use of importance measures in risk informed regulatory applications. Reability Engineering and Systems Safety 1999;60:213–26. [10] Dutuit Y, Rauzy A. Clew Algorithms to compute importance factors Cpr, MIF, CIF, DIF, RAW and RRW. Proceedings of ESREL’99, Mu¨nich, September 1999, p. 1015–9.
Criticality and sensitivity analysis of the components of a system V. Carot*, J. Sanz Department of Statistics and Operations Research, Politechnic University of Valencia, Camino de Versa s/n, 46022 Valencia, Spain Received 17 September 1998; accepted 25 January 2000
Abstract In the design of complex systems there is a great interest to know the relative importance of each of their elements. In this paper, we define a new method for measuring the relative importance of each element of the system. We have to specify that this paper concerns only nonrepairable systems and components. We present a way of calculating the criticality of each component for a complex system no matter what the random distribution of the life of the component is. The paper also demonstrates a simple way of calculating how the system life improves when the life of a component is improved. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Criticality; Reliability of systems; Reliability importance of components
1. Introduction The relative importance of the components of a system has been widely studied. The measures of importance are basically of two kinds: functional importance (that is, importance in relation to the reliability of the system) and structural importance. We consider only non-repairable systems and components. Barlow [1], Boland [2] and Tong [3] have studied the structural importance of the components of a system and more recently Meng [4,5] has presented new measures of that kind. Natvig [6] and Bergman [7] propose other measures of the importance of the components of a system. It is also interesting to consult Wall [8], Cheok [9] and Dutuit [10] about measures of importance and some others factors of importance. In this paper a new method is proposed for the importance of components from a functional point of view by studying how the system life improves when the mean life of a component is improved. With this knowledge one can highlight which component (or components) must be given greater attention, when all the components are independent from each other. 2. Sensitivity measure The importance of the k-component for the life of the system is calculated by studying the consequence for the mean life of the system by increasing the mean life of a component in Dmk. This calculation of 2m=2mk ; gives a * Corresponding author.
measure of the system sensitivity in relation to the kcomponent. The calculation of 2m=2mk is generally impossible to do analytically. A simulation method can be used to calculate the mean lives of the system for two values of the mean life of a k-component (for example, mk and mk ⫹ Dmk : When Dmk is small enough, we obtain the partial derivation as a forward finite-difference equation 2m Dm ⯝ 2mk Dmk
1
This means doing two runs of the simulation per component, but one run of any couple of runs may be the same for all the components. Thus it will be necessary to do nv ⫹ 1 runs (nv number of nodes), where in each run we generate, a large number of random lives of each component, for example, for each component we can generate 100,000 random lives, using a Monte Carlo method. For series and parallel systems when the variables are exponential probability distribution function (PDF) variables, this paper demonstrates that 2m p0 m0 k k mk 2mk
2
where p 0k is the probability of criticality and can be estimated as the number of times the system life matches the lifetime of the specified k-component divided by the number of simulations done in a run of the programme. m 0k is the critical mean value of the k-component and can be estimated as the sum of the lives of the k-component when it matches the system life/number of times there has been a match between a component life and a system life. mk is the
0951-8320/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0951-832 0(00)00011-9
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V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
and so Nomenclature Life of the i-component Life of the system Mean value of the life of the system Mean value of the k-component life Sensitivity measure or partial variation for the k-component Dm=Dmk Approximation to the theoretical value of 2m=2mk lk Failure rate of the k-component Probability of criticality for the k-component p 0k m 0k Critical mean life value for the k-component Number of nodes of the system nv xi T m mk 2m=2mk
mean value of the k-component and it can be estimated as the sum of the lives of k-component/number of simulations done in a run of the programme. Consequently, the method proposed in this paper will only need one run of simulation to calculate the measure of the system sensitivity 2m=2mk as the calculation of the parameters pk, m 0k ; and mk requires only one run assuming that in this run, we generate for example 100,000 random lives of each component, saving much time in order to obtain the same results. For mixed networks, the same formula (2) will be used and compared to the theoretical results for a simple system since the theoretical demonstration is impracticable in more complex systems.
A series and a parallel network can be considered as extreme configurations. Thus the system life for any system of nv components is, min xi ⱕ T ⱕ max xi
3
1ⱕiⱕnv
nv X
the expressions at the left and right being the lives for series and parallel systems, respectively. The general propositions valid for both systems, series and parallel, will be applicable for any other mixed system. Consequently, a demonstration of formula (2) for a series system and a parallel system is presented.
li
5
i1
Therefore,PT follows an exponential distribution with a n li ; and the mean life of the system is, failure rate ni1 m
1 nv X
6
li
i1
Deriving the formula (6) with respect to mk, we obtain for series systems,
l2k !2 nv X li
2m 2mk
7
i1
3.2. Parallel In this case, the distribution function of the system life is, P T ⱕ y
nv Y
P Xi ⱕ y
i1
nv Y
1 ⫺ e⫺yli
8
i1
from which we obtain the mean life of the system, m
nv nv nv X X X 1 1 1 ⫺ ⫹ l l ⫹ l l ⫹ l j j ⫹ lk i1 i ij i⬍j i ijk i⬍j j⬍k i
⫺ … ⫺1nv ⫹1
3. Theoretical calculation of sensitivity measure for series and parallel systems
1ⱕiⱕnv
FT t 1 ⫺ e
⫺t
1 li ⫹ lj ⫹ …lnv
9
Deriving the formula (9) with respect to mk, we obtain for parallel systems, nv X 2m 1 1 l2k 2 ⫺ 2mk lk l ⫹ l k 2 i i1 i苷k
⫹
nv X ij i苷k j苷k
1 li ⫹ lj ⫹ lk 2
⫺ … ⫺1nv ⫹1
1 li ⫹ lj ⫹ … ⫹ lnv 2
! 10
4. Critical component: probability of criticality 3.1. Series In this case, the system reliability function is
P T ⬎ t
nv Y i1
P xi ⬎ t e
⫺t
nv X i1
li
4
For any kind of system, parallel, serial or a mixture, when the systems fail, the life of the system will be the same as that of a component of the system, identifying this component as the critical one, because it is the reason for the failure of the system. The probability that a k component is critical will be the probability of criticality of k, p 0k .
V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
149
Eq. (15) becomes
For a series network, it will be given by p 0k P Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv ! \ nv P 1 i 苷 k Xk ⱕ xi
Fk a= Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv " !# nv X 1 ⫺ exp ⫺ li a
17
i1
dxk
0
Z∞
Z∞
Z∞
xk
dx1 …
xk
dxnv f x1 ; x2 ; …; xn
11
m 0k E Xk =xk critical
hence 12
i1
For a parallel system
Z xk
Zxk
Z∞ 0
dxk
0
dx1 …
0
dxnv f x1 ; x2 ; …; xnv
13
0
Z xk
Zxk
Za dxk
0
dx1 …
1 1 ⫺ lk l ⫹ lk i i1 i苷k nv X ij i⬍j j苷k i苷k
⫹ ⫺1nv ⫹1
⫹ ⫺1
1 ⫺… li ⫹ lj ⫹ lk
1 li ⫹ lj ⫹ … ⫹ lnv
nv
14
" !# nv X lk 1 ⫺ exp ⫺a li a : nv X i1 li
P Xk ⱕ a 傽 Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv 15 p 0k
since
⫹ ⫺1 exp xk nv
P Xk ⱕ a 傽 Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv Z∞
Z∞
"
… xk
xk
p 0k 1 ⫺ exp ⫺
dxnv f x1 ; x2 ; …; xnv
nv X i1
!#
li
and the critical mean life m 0k E Xk =k critical " nv X lk 1 1 ⫺ ⫹… 0 2 2 p k lk i1 i苷k li ⫹ lk 1 l1 ⫹ l2 ⫹ … ⫹ lnv 2
21
#
6. Demonstration of the formula for obtaining the sensitivity measure
!#
li a
nv X
20
i1
⫹ ⫺1nv
dxk
19
The density function will be fxk xk =k critical " nv X lk ⫺lk xk ⫺ e⫺ lk ⫹li xk ⫹ … 0 e pk i1 i苷k
Fk a= Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv
Z∞
lk 1 ⫺ e⫺ lk ⫹li a ⫹ … lk ⫹ li
i1
!
The critical mean life is the mean life value for the kcomponent, if k is a critical component. For a series system, the conditional distribution of Xk ; Fk a= Xk ⱕ x1 傽 Xk ⱕ x2 … Xk ⱕ xnv will be:
0
li
dxnv f x1 ; x2 ; …; xnv
i1 i苷k
5. Critical mean life
0 nv X
1 ⫺ e⫺lk a ⫺
nv X
⫹
18
This equation shows that for a series system all the components have the same critical mean life, which is the mean life of the system. For a parallel system
hence p 0k lk
E T m
P Xk ⱖ a 傽 Xk ⱖ x1 傽 Xk ⱖ x2 … Xk ⱖ xnv
p 0k P Xk ⱖ x1 傽 Xk ⱖ x2 … 傽 Xk ⱖ xnv
1 nv X i1
lk nv X li
p 0k
and then it gives,
16
From Eqs. (7), (12) and (18), it can be obtained for a
150
V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
Fig. 1. System with exponential lives for all components.
series system
10
0
7. Application of the method proposed to general systems
1
CB C B B lk C B 1 C l2k CB Clk p 0k m 0k 1 B ! C C B B nv nv 2 n X X v mk X A@ A @ li li li
2m 2mk
i1
i1
i1
22 Similarly from Eqs. (10), (14) and (21), it can be obtained for a parallel system 2m l2 m 0 k k p 0k 2mk lk
23
Therefore, for both a series system and a parallel one, as for a variable exponential PDF mk 1=lk ; we obtain the next equation 2m p0 m0 k k mk 2mk
24
is valid and it allows the calculation of 2m=2mk once the parameters p 0k ; m 0k and mk have been obtained from a single run of the simulation programme. The saving of time and cost is thus evident. On the other hand, from Eqs. (6), (12) and (18) for a series system, the following can be obtained 1 0 nv X lk C nv B X B lk 1 1 1 C C B k1 m nv C B X nv nv nv nv X X X X A k1 @ li li li li li i1
nv X
p 0k m 0k
i1
i1
I1
i1
25
k1
similarly from Eqs. (9), (14) and (21) for a parallel system, the following can be obtained m
nv X
m 0k p 0k
26
k1
Consequently, no matter whether it is a series or a parallel system, the mean life of the system can be calculated from the probability of criticality and the critical mean lives. In the summations (25) or (26), each term p 0k m 0k represents the contribution of a k-node to the mean life of the system and, taking into account Eqs. (24) and (26), 2m=2mk represents the unitary contribution of a k-node.
To establish the utility of the method proposed in this paper, it has been used for a mixed system and results have been compared to theoretical ones. The following parameters have been calculated: the probability of criticality, the critical mean life and the measure of the system sensitivity 2m=2mk for each component of the system, as well as the mean value of the component-life. A simple mixed system has been used to make the comparison intended since the theoretical calculations become impractical for more complex ones. The system studied is given in Fig. 1, where each component follows an exponential distribution with the mean lives indicated. The theoretical value of 2m=2mk for a component is calculated with the expression " 2m 1 1 2 l2 ⫹ 2 2m2 l2 ⫹ l5 l2 ⫹ l4 ⫹ l6 2 ⫺
1 1 ⫺ l2 ⫹ l3 ⫹ l5 ⫹ l6 2 l2 ⫹ l3 ⫹ l4 ⫹ l5 2
1 1 ⫺ l2 ⫹ l4 ⫹ l5 ⫹ l6 2 l2 ⫹ l3 ⫹ l4 ⫹ l6 2 # 2 ⫹ (27) l2 ⫹ l3 ⫹ l4 ⫹ l5 ⫹ l6 2 ⫺
and the value obtained for the component 2 in the system of Fig. 1 is 2m 0:33075 2m2
28
Similarly, the values calculated theoretically for the other components are 2m 0:37019; 2m3
2m 0:046; 2m4
2m 0:07458; 2m5
2m 0:08253 2m6
29
Five runs of our simulation programme have been performed and an estimation of p 0k ; m 0k and mk for each component of Fig. 1 have been obtained in each run. From these and using formula (24), we obtain the partial variation for each component (see Table 1). In the last
V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
151
Table 1 Comparison between the results from a program simulation and the theoretical values Component number
Probability of criticality (1) Critical mean value (2) Partial variation (3) Mean value of the component-life (4) Run 1
2
3
4
5
6
Run 2
Theoretical values
Run 3
Run 4
Run 5
Mean
(1) 0.2957 (2) 22.418 (3) 0.3285 (4) 20.181
0.2966 22.239 3299 20.051
0.2955 21984 3293 19.727
0.2988 22241 3347 19.862
0.3021 22251 3392 19.821
2977 22227 3321 19928
3308 20
0.3707 25.266 3731 25.101
0.3653 25.283 3704 24.932
0.3711 25.013 3742 24.807
0.3659 25.384 3693 25.146
0.3667 25.122 0.3686 24990
3679 25215 3711 24995
3702 25
0.0601 23.325 468 29.971
0.0567 23.646 449 29840
0.0607 22.361 452 30.021
593 23.197 460 29882
460 30
1224 23844 732 39878
746 40
1527 27104 828 49.963
825 50
0.0591 23.483 464 29.945
600 23170 469 29.635
0.1219 23.881 729 39910
0.1243 23.659 737 39.896
1190 23.859 715 39.718
0.1244 23.685 746 39.521
0.1222 24.138 731 40.346
0.1526 27.187 820 50.589
0.1538 26.329 820 49.382
0.1544 26940 0.0837 49.693
0.1542 27.293 835 50.375
0.1483 27770 827 49.776
column are the theoretical values according to formula (27) for each component. The values obtained for the partial variation, thanks to p^ 0 ·m^ 0 2m k k 2mk m^ k
30
is a good estimation of the theoretical ones. The theoretical calculation of p 0k and m 0k is rather more difficult, but it can be expected that the values obtained in our simulation method are a good estimate of the theoretical ones, as they give a good estimation of the theoretical partial variation.
When we consider that the components follow a distribution different from the exponential, the same relation from Eq. (30) can be extracted. For example, when we have two components in serial following a uniform distribution between 0 and 10 for the first component and between 0 and 15 for the second component, we can obtain proof that the formula (30) works in an analytical form. For this data, the results for five runs of our simulation program are shown in Table 2. When the same uniform components are arranged as parallel, we have demonstrated that the same result of Eq. (30)
Table 2 Comparison between the results from a program simulation and the theoretical values for system with uniform distributed components Component number
Probability of criticality (1) Critical mean value (2) Partial variation (3) Mean value of the component-life (4)
Theoretical values
Run 1
Run 2
Run 3
Run 4
Run 5
Mean
2
(1) 0.6703 (2) 4.150 (3) 0.55816 (4) 4.984
0.6692 4.172 0.55912 4.993
0.6704 4.198 0.56109 5.017
0.6629 4.181 0.55201 5.021
0.6575 4.158 0.54593 5.008
0.66606 4.17180 0.55526 5.00460
0.66667 4.16667 0.55556 5.00000
3
0.3297 3.342 0.14648 7.521
0.3308 3.349 0.14736 7.517
0.3296 3.338 0.14627 7.521
0.3371 3.385 0.15237 7.488
0.3425 3.303 0.15284 7.402
0.33394 3.34340 0.14906 7.48980
0.33333 3.33333 0.14815 7.50000
152
V. Carot, J. Sanz / Reliability Engineering and System Safety 68 (2000) 147–152
Table 3 Comparison between the results from a program simulation and the theoretical values for a parallel system with uniform distributed components Component number
Probability of criticality (1) Critical mean value (2) Partial variation (3) Mean value of the component-life (4)
Theoretical values
Run 1
Run 2
Run 3
Run 4
Run 5
Mean
2
0.3314 6.651 0.44114 4.997
0.3331 6.657 0.44427 4.991
0.3325 6.698 0.44682 4.985
0.3318 6.695 0.44386 5.005
0.332 6.711 0.44258 5.035
0.33216 6.68240 0.44373 5.00260
0.33333 0.66667 0.44444 5.00000
3
0.6686 9.567 0.85315 7.497
0.6669 9.551 0.85117 7.483
0.6675 9.568 0.85136 7.501
0.6682 9.586 0.85126 7.524
0.668 9.567 0.85142 7.505
0.66784 9.56780 0.85167 7.50200
0.66667 9.58333 0.85185 7.50000
applies, and the values obtained from our simulation program are shown in Table 3. For general distribution, and more complex systems the analytical demonstration of the formula (30) is very complicated, and we obtain the validation of that expression by the use of Monte Carlo methods. These results can be generalized for more complex systems in which the life for each component may follow a different distribution function and so the partial variation 2m=2mk is impossible to calculate theoretically. Thus, the use of the formula (30) deduced in this paper can be of great use in the design process of complex systems since it easily allows evaluation of how much the system life improves when the lives of its components are improved. It gives a measure of the importance of the components of a system. We consider that all the components of the system are independents. The case for a component’s dependence will be the object of future work. 8. Conclusions In this work, we have defined a new measure of the relative importance of an element of the system for its mean life, when we deal with non-repairable systems and components. The measure of the relative importance of the component is made by means that we have defined as sensitivity for a component, expressed as the partial variation of the mean life of the system in respect to the mean of the component. We have demonstrated that for serial, parallel or complex systems, with the components distributed as exponentially and independents, we obtained the partial variation of the mean life in respect to the component as a product of the probability of criticality and the critical mean life of the
component and divided this product by the mean life of that component. This form of obtaining the measure of relative importance of a component has been demonstrated analytically when components follow a uniform distribution. For more general components and forms of the graph of the system, we carried out a demonstration using the Monte Carlo method.
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