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Markov MethodThis is a widely used method for performing reliabilityrelated analysis of engineering systems and is named after a Russian mathematician, Andrei A. Markov (18561922). FIGURE 4.3 A fault tree with the given and calculated fault event occurrence probability values. The method is used quite commonly for modeling repairable engineering systems with constant failure and repair rates and is subject to the following assumptions [14]:
The application of this method is demonstrated by solving the example shown below. Example 4.3 Assume that a system can be either in an operating or a failed state. The system constant failure and repair rates are A. and fi_{s}, respectively. The system state space diagram is shown in Figure 4.4. The numerals in box and diamond denote the system states. Develop equations for the system timedependent and steadystate availabilities and unavailabilities, reliability, and mean time to failure by using the Markov method. FIGURE 4.4 System state space diagram. By using the Markov method, we write down the following equations for the system states 0 and 1 shown in Figure 4.4, respectively.
where t is the time. A_{s} At is the probability of system failure in finite time interval At. H_{s}At is the probability of system repair in finite time interval At. (1А_{л}.Д?) is the probability of no failure in finite time interval At. l  [i_{s} An is the probability of no repair in finite time interval At. Pj(t) is the probability that the system is in state j at time t, for j = 0, 1. P_{0}(f + A?) is the probability of the system being in operating state 0 at time (/ +ДГ). P_{1} (? + At) is the probability of the system being in failed state 1 at time (t + At). From Equation (4.3), we get
From Equation (4.5), we write
Thus, from Equation (4.6), we obtain
Similarly, using Equation (4.4), we get
At time t = 0. P_{0} (0) = 1 and P_{{} (0) = 0. By solving Equations (4.7) and (4.8), we obtain the following equations [2]:
Thus, the system timedependent availability and unavailability, respectively, are and where A V_{s} (?) is the system timedependent availability. UAV_{S} (?) is the system timedependent unavailability. By letting time t go to infinity in Equations (4.11) and (4.12), we get [2]
and
where A V_{s} is the system steadystate availability. UAV_{S} is the system steadystate unavailability. For ii_{s} = 0, from Equation (4.9), we get
By integrating Equation (4.15) over the time interval [0,oo], we get the following equation for the system mean time to failure [2]:
where MTTF_{S} is the system mean time to failure. Thus, the system timedependent and steadystate availabilities and unavailabilities, reliability, and mean time to failure are given by Equations (4.11), (4.13), (4.12), (4.14), (4.15), and (4.16), respectively. Example 4.4 Assume that the constant failure and repair rates of a system used in industry are 0.0004 failures per hour and 0.0008 repairs per hour, respectively. Calculate the system steadystate availability and availability during a 100hour mission. By substituting the given data into Equations (4.13) and (4.11), we get and
Thus, the system steadystate availability and availability during a 100hour mission are 0.6666 and 0.9623, respectively. Network Reduction ApproachThis is probably the simplest approach for determining the reliability of systems composed of independent series and parallel systems. However, the subsystems forming bridge networks/configurations can also be handled by first using the delta star method [15]. Nonetheless, the network reduction approach sequentially reduces the series and parallel subsystems to equivalent hypothetical single units until the whole system under consideration itself becomes a single hypothetical unit. The example presented below demonstrates this approach. Example 4.5 An independent unit network representing a system is shown in Figure 4.5 (i). The reliability R_{;} of unit j for j = 1,2, 3, ..., 7 is given. Calculate the network reliability by utilizing the network reduction approach. First, we have highlighted subsystems А, В, C, and D of the network as shown in Figure 4.5 (i). The subsystems В and C have their units in series; thus, we reduce them to single hypothetical units as follows:
and
where R_{B} is the subsystem В reliability. R_{c} is the subsystem C reliability. The reduced network is shown in Figure 4.5 (ii). Now, the network is made up of two parallel subsystems A and D. Thus, we reduce both these subsystems to single hypothetical units as follows:
and
where R_{a} is the subsystem A reliability. R_{d} is the subsystem D reliability. Figure 4.5 (iii) shows the reduced network with the above calculated values. This resulting network is a twounit series system and its reliability is given by
The single hypothetical unit shown in Figure 4.5 (iv) represents the reliability of the whole network shown in Figure 4.5 (i). More clearly, the whole network is reduced to a single hypothetical unit. Thus, the whole network reliability, R_{s}, is 0.5036. 
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