SYSTEM RELIABILITY EVALUATION AND ALLOCATION
4.10 MEASURES OF COMPONENT IMPORTANCE
1+0.0045 0.8625
= 0.0052.
The one-sided lower 95% confidence bound on the system reliability is R exp(0.5ˆ ˆσ2− z1−αˆσ) = 0.8625 × exp(0.5 × 0.0052 − 1.645 ×√
0.0052)
= 0.768.
4.10 MEASURES OF COMPONENT IMPORTANCE
In the preceding sections of this chapter we described methods for estimating system reliability and confidence intervals for various system configurations. The evaluation may conclude that the reliability achieved for the current design does not meet a specified reliability target. In these situations, corrective actions must be taken to improve reliability. Such actions may include upgrading components, modifying system configuration, or both at the same time. No matter what actions are to be taken, the first step is to identify the weakest components or subsystems, which pose most potential for improvement. The identification can be done by ranking components or subsystems by their importance to system reliability, then priority should be given to the components or subsystems of high importance.
An importance measure assigns a numerical value between 0 and 1 to each component or subsystem; 1 signifies the highest level of importance, and thus the system is most susceptible to the failure of corresponding component or subsystem, whereas 0 indicates the least level of importance and the greatest robustness of the system to the failure of relevant component or subsystem.
There are numerous measures of importance. In this section we present three major measures, including Birnbaum’s measure, criticality importance, and Fussell–Vesely’s importance, which are applicable to both components and sub-systems. Other importance measures can be found in, for example, Barlow and Proschan (1974), Lambert (1975), Natvig (1979), Henley and Kumamoto (1992), Carot and Sanz (2000), and Hwang (2001). Boland and El-Neweihi (1995) sur-vey the literature concerning the topic of importance measures and make critical comparisons.
4.10.1 Birnbaum’s Measure of Importance
Birnbaum (1969) defines component importance as the probability of the com-ponent being critical to system failure, where being critical means that the component failure coincides with the system failure. Mathematically, it can be expressed as
IB(i|t) = ∂R(t)
∂Ri(t), (4.59)
where IB(i|t) is Birnbaum’s importance measure of component i at time t, R(t) the system reliability, and Ri(t) the reliability of component i. The measure of importance may change with time. As a result, the components being weakest at a time may not remain weakest at another point of time. Thus, the measure should be evaluated at the times of particular interest, such as the warranty period and design life.
As indicated in (4.59) and pointed out in Section 4.9.1, IB(i|t) is actually the measure of the sensitivity of the system reliability to the reliability of component i. A large value of IB(i|t) signifies that a small variation in component reliability will result in a large change in system reliability. Naturally, components of this type should receive and deserve more resources for improvement.
Since R(t)= 1 − F (t) and Ri(t)= 1 − Fi(t), where F (t) and Fi(t) are the probabilities of failure of the system and component i, respectively, (4.59) can be written as
IB(i|t) = ∂F (t)
∂Fi(t). (4.60)
Example 4.14 A computing system consists of four computers configured acc-ording to Figure 4.29. The times to failure of individual computers are dis-tributed exponentially with parameters λ1= 5.5 × 10−6, λ2 = 6.5 × 10−5, λ3 = 4.3× 10−5, andλ4= 7.3 × 10−6failures per hour. Calculate Birnbaum’s impor-tance measures of each computer in the system at t= 4000 and 8000 hours, respectively.
SOLUTION Let Ri(t) denote the reliability of computer i at time t, where i= 1, 2, 3, 4, and t will be omitted for notational convenience when appropri-ate. We first express the system reliability in terms of the reliabilities of individual
1 2
3 4
FIGURE 4.29 Reliability block diagram of the computing system
MEASURES OF COMPONENT IMPORTANCE 101
computers by using the decomposition method. Computer 2 is selected as the key-stone component, denoted A. Given that it never fails, the conditional probability of the system being good is
Pr(system good| A) = 1 − (1 − R1)(1− R3).
Similarly, provided that computer 2 is failed, the conditional probability that the system is functional is
Pr(system good| A) = R3R4. From (4.40), the system reliability at timet is
R(t)= [1 − (1 − R1)(1− R3)]R2+ R3R4(1− R2)= R1R2+ R2R3+ R3R4
− R1R2R3− R2R3R4.
From (4.59), Birnbaum’s importance measures for computers 1 through 4 are IB(1|t) = R2(1− R3), IB(2|t) = R1+ R3− R1R3− R3R4, IB(3|t) = R2+ R4− R1R2− R2R4, IB(4|t) = R3(1− R2).
Since the times to failure of the computers are exponential, we haveRi(t)= e−λit, i= 1, 2, 3, 4.
The reliabilities of individual computers at 4000 hours are R1(4000)= 0.9782, R2(4000)= 0.7711, R3(4000)= 0.8420, R4(4000)= 0.9712.
Then the values of the importance measures are
IB(1|4000) = 0.1218, IB(2|4000) = 0.1788, IB(3|4000) = 0.2391, IB(4|4000) = 0.1928.
According to the importance measures, the priority of the computers is, in des-cending order, computers 3, 4, 2, and 1.
Similarly, the reliabilities of individual computers at 8000 hours are R1(8000)= 0.957, R2(8000)= 0.5945,
R3(8000)= 0.7089, R4(8000)= 0.9433.
The importance measures at 8000 hours are
IB(1|8000) = 0.173, IB(2|8000) = 0.3188, IB(3|8000) = 0.4081, IB(4|8000) = 0.2975.
Thus, computers 3, 2, 4, and 1 have a descending priority.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0 1000 2000 3000 4000 5000 6000 7000 8000
3 2
computer 1 4
t (h) IB(i|t)
FIGURE 4.30 Birnbaum’s importance measures of individual computers at different times
Comparison of the priorities at 4000 and 8000 hours shows that computer 3 is most important and computer 1 is least important, at both points of time.
Computer 4 is more important than 2 at 4000 hours; however, the order is reversed at 8000 hours. The importance measures at different times (in hours) are plotted in Figure 4.30. It indicates that the short-term system reliability is more sensitive to computer 4, and computer 2 contributes more to the long-term reliability.
Therefore, the importance measures should be evaluated, and the priority be made, at the time of interest (e.g., the design life).
4.10.2 Criticality Importance
Birnbaum’s importance measure equals the probability that a component is critical to the system. In contrast, criticality importance is defined as the probability that a component is critical to the system and has failed, given that the system has failed at the same time. In other words, it is the probability that given that the system has failed, the failure is caused by the component. Mathematically, it can be expressed as
IC(i|t) = ∂R(t)
∂Ri(t) Fi(t)
F (t) = IB(i|t)Fi(t)
F (t), (4.61)
where IC(i|t) is the criticality importance and the other notation is the same as those for Birnbaum’s importance. Equation (4.61) indicates that the criticality importance is Birnbaum’s importance weighed by the component unreliability.
Thus, a less reliable component will result in a higher importance.
Example 4.15 Refer to Example 4.14. Determine the criticality importance me-asures for the individual computers at 4000 and 8000 hours.
MEASURES OF COMPONENT IMPORTANCE 103
SOLUTION By using (4.61) and the results ofIB(i|t) from Example 4.14, we obtain the criticality importance measures for the four computers as
IC(1|t) = R2(1− R3)(1− R1)
1− R , IC(2|t) = (R1+ R3− R1R3− R3R4)(1− R2)
1− R ,
IC(3|t) = (R2+ R4− R1R2− R2R4)(1− R3)
1− R , IC(4|t) = R3(1− R2)(1− R4)
1− R .
The reliability values of the four computers at 4000 and 8000 hours have been worked out in Example 4.14. The system reliabilities at the specified times are R(4000)= 0.9556 and R(8000) = 0.8582. Then the criticality importance mea-sures at 4000 hours are
IC(1|4000) = 0.0597, IC(2|4000) = 0.9225, IC(3|4000) = 0.8515, IC(4|4000) = 0.125.
Computers 2, 3, 4, and 1 have a descending priority order.
Similarly, the criticality importance measures at 8000 hours are IC(1|8000) = 0.0525, IC(2|8000) = 0.9116, IC(3|8000) = 0.8378, IC(4|8000) = 0.115.
The priority order at 8000 hours is the same as that at 4000 hours. Figure 4.31 plots the criticality importance measures at different times (in hours). It is seen that the priority order of the computers is consistent over time. In addition, computers 2 and 3 are far more important than computers 1 and 4, because they are considerably less reliable.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1000 2000 3000 4000 5000 6000 7000 8000
3 2
4 computer 1
t (h) IC(i|t)
FIGURE 4.31 Criticality importance measures of individual computers at different times
4.10.3 Fussell–Vesely’s Measure of Importance
Considering the fact that a component may contribute to system failure with-out being critical, Vesely (1970) and Fussell (1975) define the importance of a component as the probability that at least one minimal cut set containing the component is failed given that the system has failed. Mathematically, it can be expressed as
IFV(i|t) = Pr(C1+ C2+ · · · + Cni)
F (t) , (4.62)
where Cj is the event that the components in the minimal cut set containing component i are all failed; j = 1, 2, . . . , ni, and ni is the total number of the minimal cut sets containing component i; F (t) is the probability of failure of the system at time t. In (4.62), the probability, Pr(C1+ C2+ · · · + Cni), can be calculated by using the inclusion–exclusion rule expressed in (4.45). If compo-nent reliabilities are high, terms with second and higher order in (4.45) may be omitted. As a result, (4.62) can be approximated by
IFV(i|t) = 1 F (t)
ni
j=1
Pr(Cj). (4.63)
Example 4.16 Refer to Example 4.14. Calculate Fussell–Vesely’s importance measures for the individual computers at 4000 and 8000 hours.
SOLUTION The minimal cut sets of the computing system are{1, 3}, {2, 4}, and {2, 3}. Let Ai denote the failure of computer i, where i= 1, 2, 3, 4. Then we have C1= A1· A3, C2= A2· A4, and C3 = A2· A3. Since the reliabilities of individual computers are not high, (4.63) is not applicable. The importance measures are calculated from (4.62) as
IFV(1|t) = Pr(C1)
F (t) = Pr(A1· A3)
F (t) = F1F3
F , IFV(2|t) = Pr(C2+ C3)
F (t) = Pr(A2· A4)+ Pr(A2· A3)− Pr(A2· A3· A4) F (t)
= F2(F4+ F3− F3F4)
F ,
IFV(3|t) = Pr(C1+ C3)
F (t) = F3(F1+ F2− F1F2)
F ,
IFV(4|t) = Pr(C2)
F (t) = F2F4
F ,
where Fi = 1 − Ri and F = 1 − R. Substituting into the equations above the values ofRi andR at 4000 hours, which have been worked out in Examples 4.14
MEASURES OF COMPONENT IMPORTANCE 105
and 4.15, we obtain the importance measures as
IFV(1|4000) = 0.0775, IFV(2|4000) = 0.9403, IFV(3|4000) = 0.875, IFV(4|4000) = 0.1485.
Ranked by the importance measures, computers 2, 3, 4, and 1 have a descending priority order.
Similarly, the importance measures at 8000 hours are
IFV(1|8000) = 0.0884, IFV(2|8000) = 0.9475, IFV(3|8000) = 0.885, IFV(4|8000) = 0.1622.
The priority order at 8000 hours is the same as that at 4000 hours. Figure 4.32 plots the importance measures of the four individual computers at different times (in hours). It is seen that computers 2 and 3 are considerably more important than the other two at different points of time, and the relative importance order does not change with time.
Examples 4.14 through 4.16 illustrate application of the three importance mea-sures to the same problem. We have seen that the meamea-sures of criticality impor-tance and Fessell–Vesely’s imporimpor-tance yield the same priority order, which does not vary with time. The two measures are similar and should be used if we are concerned with the probability of the components being the cause of sys-tem failure. The magnitude of these measures increases with the unreliability of component (Meng, 1996), and thus a component of low reliability receives a
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1000 2000 3000 4000 5000 6000 7000 8000
3 2
4 computer 1
t (h) IFV(i|t)
FIGURE 4.32 Fessel-Vesely’s importance measures of individual computers at different times
high importance rating. The two measures are especially appropriate for systems that have a wide range of component reliabilities. In contrast, Birnbaum’s mea-sure of importance yields a different priority order in the example, which varies with time. The inconsistency at different times imposes difficulty in selecting the weakest components for improvement if more than one point of time is of interest.
In addition, the measure does not depend on the unreliability of the component in question (Meng, 2000). Unlike the other two, Birnbaum’s importance does not put more weight on less reliable components. Nevertheless, it is a valuable measure for identifying the fastest path to improving system reliability. When using this measure, keep in mind that the candidate components may not be eco-nomically or technically feasible if the component reliabilities are already high.
To maximize the benefits, it is suggested that Birnbaum’s importance be used at the same time with one of the other two. In the examples, if resources allow only two computers for improvement, concurrent use of the measures would identify computers 3 and 2 as the candidates, because both have large effects on system reliability and at the same time have a high likelihood of causing the system to fail. Although Birnbaum’s measure suggests that computer 4 is the second-most important at 4000 hours, it is not selected because the other two measures indi-cate that it is far less important than computer 2. Clearly, the resulting priority order from the three measures is computers 3, 2, 4, and 1.