 # Decomposition Approach

This approach is used for determining reliability of complex systems, which it decomposes into simpler subsystems by using the conditional probability theory. Subsequently, the system reliability is determined by combining reliability measures of the subsystems.

The basis for this approach is the selection of the key unit used for decomposing a given network. The approach’s efficiency depends on the selection of this key unit. The past experience normally plays a pivotal role in its selection.

The approach/method starts w'ith the assumption that the key unit, say m, is replaced by another unit that never fails (i.e., 100% reliable) and then it assumes FIGURE 4.5 Diagrammatic steps of the network reduction approach: (i) original network; (ii) reduced network; (iii) reduced network; (iv) single hypothetical unit.

that the key unit is completely removed from the system/network. Thus, the overall system/network reliability is expressed by where

Rs is the overall system/network reliability.

P(m) is the probability of success or reliability of the key unit in.

P(in) is the probability of failure or unreliability of the key unit m.

P(.) is the probability.

The application of this approach/method is demonstrated by solving the following example.

Example 4.6

Assume that five independent and identical units form a bridge network/system as shown in Figure 4.6. The capital letter R in the figure denotes unit reliability. Obtain an expression for reliability of the bridge network/system by using the decomposition approach/method.

In this case with the aid of past experience, we choose the unit falling between nodes X and Y, shown in Figure 4.6, as our key unit, say m.

Next, we replace this key unit m with a unit that is 100% reliable (i.e., never fails). Consequently, the network shown in Figure 4.6 becomes a series-parallel network/system whose reliability is given by where

R is the series-parallel network reliability.

Similarly, we totally remove the key unit m from Figure 4.6 and the resulting network becomes a parallel-series network. This parallel-series network reliability is expressed by  where

R is the parallel-series network reliability.

The reliability and unreliability of the key unit m, respectively, are expressed by and Rewriting Equation (4.17) in terms of this example (i.e., Example 4.6), we obtain By inserting Equations (4.18) and (4.19) into Equation (4.22), we get Equation (4.23) is for the reliability of the bridge network/system shown in Figure 4.6.

# Delta-Star Method

This is the simplest and a practical method for evaluating reliability of independent units bridge networks. The method transforms a bridge network to its equivalent parallel and series form. However, it is to be noted that the transformation process introduces a minor error in the end result, but for practical purposes it should be neglected .

Once a bridge network is transformed to its equivalent series and parallel form, the network reduction approach can be used for obtaining network reliability. Nonetheless, the delta-star method can quite easily handle networks containing more than one bridge configurations. Furthermore, it can also be applied to bridge configurations composed of devices having two mutually exclusive failure modes [5,15]. FIGURE 4.7 Delta to star equivalent reliability diagram.

Delta to star equivalent reliability diagram is shown in Figure 4.7. In this diagram, the numbers 1, 2, and 3 denote nodes, the blocks the units, and R(.) the respective unit reliability.

The delta-star equivalent legs are shown in Figure 4.8.

Reliabilities of Figure 4.8 delta to star equivalent diagrams (i), (ii), and (iii), respectively, are as follows: By solving Equations (4.24)-(4.26), we obtain where  FIGURE 4.8 Delta to star equivalent diagrams (i), (ii), (iii).

Example 4.7

A five independent unit bridge network with given unit reliability Rj, for j = a, b, c, d, and e is shown in Figure 4.9. Calculate the network reliability by using the delta-star method and also use the stated data values in Equation (4.23) for obtaining the bridge network reliability. Compare both results.

In Figure 4.9, nodes 1, 2, and 3 denote delta configuration. By using Equations (4.27)-(4.32) and the stated data values, we obtain the following star equivalent units’ reliabilities:  FIGURE 4.9 A five unit bridge network with given unit reliabilities. FIGURE 4.10 Equivalent network to bridge network shown in Figure 4.9. where and Using the above results, the equivalent network to Figure 4.9 bridge network is shown in Figure 4.10.

The reliability of network in Figure 4.10 is, Rb„, By inserting the specified data values into Equation (4.23), we obtain It is to be noted that both the above reliability results are basically same, i.e.,

0.9787 and 0.9784. It basically means that for practical purposes the delta-star method is quite effective.