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### A Importances

Importances indicate the influence of each basic event on a system parameter. There is a whole range of importances in the literature, which are often defined differently, and almost always without mentioning the system parameter for which they were defined. Usually it is the unreliability Fsys(Tmission).

Importance for the mean system failure rate hsys is practically not mentioned in the literature. This is understandable because importances are almost always defined in connection with fault trees, and the calculation of the system failure rate with fault trees is also rarely dealt with in literature. Some importances can be transferred directly to the failure rate, some analogous, and some importances cannot be meaningfully defined for the failure rate.

Although importances were mostly defined for use with fault trees, some can also be applied to other models, such as Markov models.

#### A.1 Partial Derivative (PD) and Birnbaum-Importance (BI)

The partial derivative is an obvious measure of the importance of individual base events of the system value Fsys(T), Qsys or hsys. The partial derivatives of the system unreliability Fsys to the unreliability of each basic event Fx are also called Birnbaum-Importances.

##### A.1.1 Partial derivative of the system unreliability

For fault trees, the derivative of the system unreliability Fsys to the unreliability of each basic event Fx is given by:

| (75)

Here, F denotes the vector of the unreliabilities of the basic events – either as time variant functions or at the end of system lifetime. If the fault tree contains conditions, i. e. basic events described by their unavailability Q instead of F, the derivative of the system unreliability Fsys to the unreliability of these basic events is not defined. Instead, the derivative of the system unreliability to the condition’s unavailability Qx may be defined:

| (76)

In case of Markov models, where the basic events are described by their failure rates h instead of F, the partial derivatives can be defined by:

| (77)

If the Markov model contains conditions, the derivative to their unavailability can be defined in the same way as for fault trees, see formula (76).

##### A.1.2 Partial derivative of the system unavailability

For fault trees, the derivative of the system unavailability Qsys to the unavailability of each basic event Qx is given by:

| (78)

Here, Q denotes the vector of the unavailabilities of the basic events – either as functions of time or as mean values.

In case of Markov models, where the basic events are described by their failure rates h instead of Q, the partial derivative can be defined by:

| (79)

If the Markov model contains conditions, the partial derivative can be defined by formula (78).

##### A.1.3 Partial derivative for the system failure rate

For fault trees, the system failure rate hsys is a function of both the failure rate hx and the unavailability Qx of each basic event, in general. Therefore, a partial derivative to the failure rate hx only () doesn’t make much sense. You could of course define two derivatives Ihh,xPD = and IhQ,xPD = , but Qx depends on the failure hx for most basic event models:

| (80)

Thus, it makes more sense to define Ih,xPD as derivative to the (mean) failure rate of the basic event λi:

| (81)

If you calculate the occurrence rate hMCS,i of each minimal cut-set MCS by

| (82)

you’ll get

| (83)

If basic event x is not included in MCSi the derivative is zero. If it is included, the summand with j = x is equal to ∏ k=1,k≠jmQ k (where all unavailabilities of this product are independent of basic event x), and all summands with j≠x are equal to hj ∏ k=1,k≠j,k≠xmQ k.

Thus we get

| (84)

For Markov models, the derivative of the system failure rate to the event’s failure rate is just given by

| (85)

In Functional Safety Suite the derivatives are calculated numerically. The principle described by formula (81) is implemented in a simple way: In order to calculate Ih,xPD, all basic event values are altered in parallel, i. e. each basic event value given to the model for calculating the system value will be varied, let it be hx, Qx or Fx.

#### A.2 Criticality Importance (CRI) and statistical confidence

The criticality importance is defined as the ratio of the relative change in system quantity Ψ for the relative change of basic event quantity chi:

| (86)

The criticality importance is the most interesting importance at all, because it gives a direct answer to the question of how much a (relative) uncertainty in the statistical value of a base event affects the overall result: A CRI of e. g. 0.1 means, that the system quantity Ψ will increase by 10% if the basic event’s failure rate is in fact twice as high as assumed (+100%).

Or in other words: The greater the criticality importance, the greater the impact that a relative improvement of the component has. It is therefore sometimes called Upgrading Importance.

In addition, the criticality importance is equal to the probability that component x is in failure, if the system has failed. Hence it gives a hint where to look for the failure first, if the system has failed.

If the system unreliability is given by Fsys(T) = fct(F) (e. g. by a fault tree without conditions), the criticality importance can be calculated by:

| (87)

#### A.3 Risk Reduction (RR)

The risk reduction is the difference of the system value Q, F(T) or h, given component x would never fail. For a fault tree, the risk reduction can be calculated by

| (88)

where Fsys denotes the vector of the component unreliabilities, in which the unreliability Fx of component x is set to zero.

If the fault tree contains conditions, and component x describes such a condition, the formula can be replace by

| (89)

where Fsys denotes the vector of the component unavailabilities (in general time dependent unavailability functions, in fact), in which the unavailability Qx of component x is set to zero.

Equivalent formula can be used for unavailabilities:

| (90)

The risk reduction can directly be applied to the system failure rate, but in case of fault trees, both hx and Qx must be set to zero:

| (91)

#### A.4 Risk Reduction Worth (RRW)

The Risk Reduction Worth states the relative reduction of the system value F(T), Q or h if component x wouldn’t fail:

| (92)

| (93)

| (94)

Obviously the Risk-Reduction-Worth can take any value from 0 to infinity. The higher the RRW, the higher the effect of an enhancement of component x. A value close to zero means that component x has no significant effect.

Note: In some other definitions, the summand -1 is omitted.

#### A.5 Fussel-Vesely-Importance (FV)

Even though the Fussel-Vesely importance has been defined based on minimal cut-sets of fault trees originally, it can be defined in a general manner as the quotient of the risk reduction (RR) and the the original system value:

| (95)

| (96)

| (97)

#### A.6 Risk Achievement (RA)

The Risk Achievement value is defined as the difference of the system value with an extremely bad component, i. e. Qx := 1 (component never available) or Fx := 1 (component for sure fails until end of mission/end of system lifetime), and the estimated system value:

| (98)

or

| (99)

The component failure rate is not limited to a certain maximum value, and thus also the system failure rate is not limited (see formulas for fault trees in section 7). Hence, no Risk-Achievement can be defined for the system failure rate.

#### A.7 Risk Achievement Worth (RAW)

If the Risk Achievement is divided by the original system value, you get the factor, by which the system value would increase if the component will fail for sure:

| (100)

or is always unavailable:

| (101)

Note: In some definitions, the summand -1 is omitted.

As for the Risk Achievement, the RAW is not defined for failure rates.

#### A.8 Importancies for generic basic events

The above mentioned definitions are related to single basic events. If a system includes several components of the same type, a modification of the component (or an uncertainty of the statistic values) will influence several basic events simultaneously, but not only one. Therefore, Functional Safety Suite offers the calculation of the above mentioned importancies also for generic basic events: All basic events referring to the same generic basic event will be modified simultaneously.

The CRI related to generic basic events can be greater than 1.0, if the generic basic event is used in multiple conjuncted paths. For example for the simple fault tree shown in figure 82, the CRI for the generic basic event A IF,ACRI is 2.0.

Figure 82: Importance of generic basic events

If the failure rate of A increases by 1%, the system failure rate hsys will increase by 2%. But be careful: Whenever you’ve got redundant structures, the values of thegeneric basic events are non-linear in the system values. The CRI IF,ACRI is valid only for small changes, therefore. If you double the failure rate of generic basic event A, the system failure rate hsys will be four times as high.

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