Reliability Functions

4

FIGURE 3.4 A four-propeller system.

The appropriate block diagram for the system is shown in Figure 3.4. The structure function is the product of the structure functions of the two parallel subsystems:

φ(x) = [1 − (1 − x1)(1− x2)][1− (1 − x3)(1− x4)]  To avoid studying structure functions that are unreasonable, a subset of all possible systems of n components, namely coherent systems, has been defined. A system is coher-ent if φ(x) is nondecreasing in x [e.g., φ(x1, . . ., xi−1, 0, xi+1, . . ., xn)≤ φ(x1, . . ., xi−1, 1, xi+1, . . ., xn) for all i] and there are no irrelevant components. The condition that φ(x) be nondecreasing inx implies that the system will not degrade if a component upgrades. A component is irrelevant if its state has no impact on the structure function. Many theorems related to coherent systems have been proven; one of the more useful is that redundancy at the component level is more effective than redundancy at the system level. This is an important consideration in reliability design, where a reliability engineer decides the com-ponents to choose (reliability allocation) at appropriate positions in the system (reliability optimization).

3.2.2 Reliability Functions

The discussion of structure functions so far has been completely deterministic in nature. We now introduce probability into the mix by first defining reliability. The paragraphs following the definition expand on the italicized words in the definition.

DEFINITION 3.3 The reliability of an item is the probability that it will adequately perform its specified purpose for a specified period of time under specified environmental conditions.

The definition implies that the object of interest is an item. The definition of the item depends on the purpose of the study. In some situations, we will consider an item to be an interacting arrangement of components; in other situations, the component level of detail in the model is not of interest.

Reliability is defined as a probability. Thus, the axioms of probability apply to reliabil-ity calculations. In particular, this means that all reliabilities must be between 0 and 1 inclusive, and that the results derived from the probability axioms must hold. For exam-ple, if two independent components have 1000-hour reliabilities of p1 and p2, and system failure occurs when either component fails (i.e., a two-component series system), then the 1000-hour system reliability isp1p2.

Adequate performance for an item must be stated unambiguously. A standard is often used to determine what is considered adequate performance. A mechanical part may require tolerances that delineate adequate performance from inadequate performance. The perfor-mance of an item is related to the mathematical model used to represent the condition of the item. The simplest model for an item is a binary model, which was introduced earlier, in which the item is in either the functioning or failed state. This model is easily applied to

Reliability 3-7 a light bulb; it is more difficult to apply to items that gradually degrade over time, such as a machine tool. To apply a binary model to an item that degrades gradually, a threshold value must be determined to separate the functioning and failed states.

The definition of reliability also implies that the purpose or intended use of the item must be specified. Machine tool manufacturers, for example, often produce two grades of an item:

one for professional use and another for consumer use.

The definition of reliability also indicates that time is involved in reliability, which implies five consequences. First, the units for time need to be specified (e.g., minutes, hours, years) by the modeler to perform any analysis. Second, many lifetime models use the random variableT (rather than X, which is common in probability theory) to represent the failure time of the item. Third, time need not be taken literally. The number of miles may represent time for an automobile tire; the number of cycles may represent time for a light switch.

Fourth, a time duration associated with a reliability must be specified. The reliability of a component, for example, should not be stated as simply 0.98, as no time is specified.

It is equally ambiguous for a component to have a 1000-hour life without indicating a reliability for that time. Instead, it should be stated that the 1000-hour reliability is 0.98.

This requirement of stating a time along with a reliability applies to systems as well as components. Finally, determining what should be used to measure the lifetime of an item may not be obvious. Reliability analysts must consider whether continuous operation or on/off cycling is more effective for items such as motors or computers.

The last aspect of the definition of reliability is that environmental conditions must be specified. Conditions such as temperature, humidity, and turning speed all affect the lifetime of a machine tool. Likewise, the driving conditions for an automobile will influence its relia-bility. Included in environmental conditions is the preventive maintenance to be performed on the item.

We now return to the mathematical models for determining the reliability of a system.

Two additional assumptions need to be made for the models developed here. First, then components comprising a system must be nonrepairable. Once a component changes from the functioning to the failed state, it cannot return to the functioning state. This assumption was not necessary when structure functions were introduced, as a structure function simply maps the component states to the system state. The structure function can be applied to a system with nonrepairable or repairable components. The second assumption is that the components are independent. Thus, failure of one component does not influence the proba-bility of failure of other components. This assumption is not appropriate if the components operate in a common environment where there may be common-cause failures. Although the independence assumption makes the mathematics for modeling a system simpler, the assumption should not be automatically applied.

Previously,xi was defined to be the state of componenti. Now Xi is a random variable with the same meaning.

DEFINITION 3.4 The random variable denoting the state of componenti, denoted by X_i, is

X_i=

0 if component i has failed 1 if component i is functioning fori = 1, 2, . . ., n.

Thesen values can be written as a random system state vector X . The probability that componenti is functioning at a certain time is given by pi=P (Xi= 1), which is often called

the reliability of theith component, for i = 1, 2, . . ., n. The P function denotes probability.

Thesen values can be written as a reliability vector p = (p1, p2, . . ., p_n).

The system reliability, denoted by r, is defined by r = P [φ(X ) = 1]

wherer is a quantity that can be calculated from the vector p, so r = r(p). The function r(p) is called the reliability function. In some of the examples in this section, the components have identical reliabilities (that is,p1=p2=· · · = p_n=p), which is indicated by the notation r(p).

Several techniques are used to calculate system reliability. We will illustrate two of the simplest techniques: definition and expectation. The first technique for finding the reliabil-ity of a coherent system of n independent components is to use the definition of system reliability directly, as illustrated in the example.

Example 3.5

The system reliability of a series system ofn components is easily found using the definition ofr(p) and the independence assumption.

r(p) = P [φ(X) = 1]

=P

_n



i=1

X_i= 1

=

n i=1

P [X_i = 1]

=

n i=1

pi

The product in this formula indicates that system reliability is always less than the reliability of the least reliable component. This “chain is only as strong as its weakest link” result indicates that improving the weakest component causes the largest increase in the reliability

of a series system. 

In the special case when all components are identical, the reliability function reduces to r(p) = pⁿ, where p1=p2=· · · = p_n=p. The plot in Figure 3.5 of component reliability versus system reliability for several values ofn shows that highly reliable components are necessary to achieve reasonable system reliability, even for small values ofn.

The second technique, expectation, is based on the fact that P [φ(X ) = 1] is equal to E[φ(X )], because φ(X ) is a Bernoulli random variable. Consequently, the expected value ofφ(X ) is the system reliability r(p), as illustrated in the next example.

Example 3.6

Since the components are assumed to be independent, the system reliability for a parallel system ofn components using the expectation technique is

r(p) = E[φ(X)]

=E

 1−ⁿ

i=1

(1− X_i)

Reliability 3-9

FIGURE 3.5 Reliability of a series system ofn components.

0.0

FIGURE 3.6 Reliability of a parallel system ofn components.

= 1− E

In the special case of identical components, this expression reduces tor(p) = 1 − (1 − p)ⁿ. Figure 3.6 shows component reliability versus system reliability for a parallel system ofn identical components. The law of diminishing returns is apparent from the graph when a

fixed component reliability is considered. The marginal gain in reliability decreases

dramat-ically as more components are added to the system. 

There are two systems that appear to be similar to parallel systems on the surface, but they are not true parallel systems such as the one considered in the previous example.

The first such system is a standby system. In a standby system, not all the components function simultaneously, and components are switched to standby components upon failure.

Examples of standby systems include a spare tire for an automobile and having three power sources (utility company, backup generator, and batteries) for a hospital. In contrast, all components are functioning simultaneously in a true parallel system.

The second such system is a shared-parallel system. In a shared-parallel system, all com-ponents are online, but the component reliabilities change when one component fails. The lug nuts that attach a wheel to an automobile are an example of a five-component shared-parallel system. When one lug nut fails (i.e., loosens or falls off), the load on the remaining functioning lug nuts increases. Thus, the static reliability calculations presented in this section are not appropriate for a wheel attachment system. In contrast, the failure of a component in a true parallel system does not affect the reliabilities of any of the other components in the system.