RELIABILITY:CALCULATION OF EXACT SYSTEM RELIABILITY

CALCULATION OF EXACT SYSTEM RELIABILITY

A representation of the structure of a system can be expressed in terms of a network, and some of the material presented in Chap. 10 is relevant. For example, consider the system that can be represented by the network in Fig. 25.1. This system consists of five components, connected in a somewhat complex manner. According to the network diagram, the system will operate successfully if there exists a flow from A (the source) to D (the sink) through the directed graph, i.e., if components 1 and 4 operate successfully, or components 2 and 5 operate

successfully, or components 1, 3, and 5 operate successfully. In fact, each arc can be viewed as having capacity 1 or 0, depending upon whether or not the component is operating. If an arc has a 0 attached to it (the component fails), then the network would lose that arc, and the system would operate successfully if and only if there is a path from the source to the sink in the resultant network. This situation is illustrated in Fig. 25.2, where the system still op- erates if components 3 and 4 fail but becomes inoperable if components 2, 3, and 4 fail. This suggests a possible method for computing the exact system reliability. Again, denote the per- formance of the ith component by the binary random variable Xi. Then Xi takes on the value 1 with probability pi and 0 with probability (1 - pi). For each realization, X1 = x1, X2 = x2, X3 = x3, X4 = x4 and X5 = x5 (there are 25 such realizations), it is determined whether or not the system will operate, i.e., whether or not the structure function equals 1. The network con- sisting of those arcs with Xi equal to 1 contains at least one path if and only if the corre- sponding structure function equals 1. If a path is formed, the probability of obtaining this configuration is obtained. For the realization in Fig. 21.2a. a path is formed, and

P{X1 = 1, X2 = 1, X3 = 0, X4 = 0, X5 = 1} = p1p2(1 - p3)(1 - p4)p5.

Because each realization is disjoint, the system reliability is just the sum of the probabil- ities of those realizations that contain a path. Unfortunately, even for this simple system, 32 different realizations must be evaluated, and other techniques are desirable.

Another possible procedure for finding the exact reliability is to note that the relia- bility R(p1, p2, . . . , pn) can be expressed as

R(p1, p2, . . . , pn) = P{maximum flow from source to sink > 1}.

This identity allows the concept of paths and cuts presented in Chap. 10 to be used. In re- liability theory, the terminology of minimal paths and minimal cuts is introduced. A minimal path is a minimal set of components that, by functioning, ensures the success- ful operation of the system. For the example in Fig. 25.1. components 2 and 5 are a min- imal path. A minimal cut is a minimal set of components that, by failing, ensures the fail- ure of the system. In Fig. 25.1, components 1 and 2 are a minimal cut. For the system given in Fig. 25.1, the minimal paths and cuts are

If we use all the minimal cuts, there are also two ways to obtain the exact system re- liability. Because the system will fail if and only if all the components in at least one of the minimal cuts fail, the system reliability can be expressed as

Taking expectations on both sides leads to the desired expression for the reliability. Again, this method requires essentially the same amount of calculation as required for the first procedure using cuts.

Although the results presented in this section were based upon the example, an ex- tension to any system can be easily obtained. All minimal paths and/or cuts must be found and one of the four methods presented chosen.

As previously mentioned, if there are r paths and s cuts in the network, then calculating the exact reliability using paths will involve summing 2r - 1 terms, and using cuts will involve 2s - 1 terms. Hence, the method using paths should be used if and only if r < s. Generally, however, it is simpler to find minimal paths rather than minimal cuts, so that the method using paths may have to be used because finding all cuts may be computationally infeasible. It is evident that finding the exact reliability of a system is quite difficult and that bounds are desirable, provided that the calculations are substantially reduced.