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koutofm NetworkTable of Contents:
The koutofm network is another form of redundancy in which at least к units out of a total of m active units must work normally for the successful operation of the system/network. The block diagram of a koutofш unit system/network is shown in Figure 3.4. Each block in the diagram represents a unit. The parallel and series networks are special cases of this network for к = 1 and k = m, respectively. By using the binomial distribution, for identical and independent units, we write down the following expression for reliability of koutofm unit network/system shown in Figure 3.4: where
R_{k/m} is the koutofm network reliability. R is the unit reliability. For constant failure rate of the identical units, using Equations (3.11) and (3.31), we obtain
where A is the unit constant failure rate. Rk/m (0 is •he koutofm network reliability at time t. By substituting Equation (3.33) into Equation (3.12), we get where MTTF_{k/m} is the mean time to failure of the koutofm network/system. Example 3.8 Assume that a system has five active, independent, and identical units in parallel. For the successful operation of the system, at least four units must operate normally. Calculate the mean time to failure of the system if the unit constant failure rate is 0.0005 failures per hour. By inserting the given data values into Equation (3.34), we obtain
Thus, the mean time to failure of the system is 900 hours. Standby SystemThis is another network or configuration in which only one unit operates and m units are kept in their standby mode. The total system contains (m + 1) units, and as soon as the operating unit fails, the switching mechanism detects the failure and turns on one of the standby units. The system fails when all the standby units fail. The block diagram of a standby system with one functioning and m standby units is shown in Figure 3.5. Each block in the diagram denotes a unit. Using Figure 3.5 block diagram for independent and identical units, perfect switching mechanism and standby units, and the timedependent unit failure rate, we write the following expression for the standby system reliability [10] where R_{ss} (/) is the standby system reliability at time t. A(f) is the unit timedependent failure rate/hazard rate. FIGURE 3.5 Block diagram of a standby system containing one operating unit and m standby units. For unit’s constant failure rate (i.e., A(f) = A),Equation (3.35) yields where A is the unit constant failure rate. By inserting Equation (3.36) into Equation (3.12), we obtain
where MTTF_{SS} is the standby system mean time to failure. Example 3.9 Assume that a standby system contains two independent and identical units (i.e., one operating and other on standby). The unit constant failure rate is 0.002 failures per hour. Calculate the standby system reliability for a 100hour mission and mean time to failure, assuming that the switching mechanism is perfect and the standby unit remains as good as new in its standby mode. By inserting the stated data values into Equation (3.36), we get Similarly, by substituting the given data values into Equation (3.37), we get
Thus, the standby system reliability for a 100hour mission and mean time to failure are 0.9824 and 1,000 hours, respectively. Bridge NetworkSometimes units in systems may form a bridge network/configuration, as shown in Figure 3.6. Each block in the figure represents a unit, and all five units are labeled with numerals. For independently failing units of the bridge network shown in Figure 3.6, reliability is expressed by [11]
where R_{h}is the reliability of the bridge network. R, is the reliability of unit i, for i' = 1,2,3, 4, 5. For identical units, Equation (3.38) simplifies to where R is the unit reliability. For constant unit failure rate using Equation (3.11) and Equation (3.39), we obtain where R_{h} (/) is the reliability of the bridge network at time t. A is the unit constant failure rate. By inserting Equation (3.40) into Equation (3.12), we obtain where MTTF_{h} is the bridge network mean time to failure. Example 3.10 Assume that a system has five independent and identical units forming a bridge network/configuration and the constant failure rate of each unit is 0.0005 failures per hour. Calculate the bridge network/configuration reliability for a 250hour mission and the mean time to failure. By substituting the given data values into Equation (3.40), we obtain Similarly, by inserting the specified data value into Equation (3.41), we get Thus, the bridge network/configuration reliability and the mean time to failure are 0.9700 and 1633.33 hours, respectively. Problems
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