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Probability Tree AnalysisTable of Contents:
This method can be used for performing reliabilityrelated task analysis by diagram matically representing human actions and other associated events in question. In this case, diagrammatic task analysis is represented by the branches of the probability tree. More specifically, the tree’s branching limbs represent each event’s outcome (i.e., success or failure) and each branch is assigned probability of occurrence [17]. Some of the advantages of this method are flexibility for incorporating (i.e., with some modifications) factors such as interaction effects, emotional stress, and interaction stress; simplified mathematical computations; and a visibility tool. It is to be noted that the method can also be used for evaluating reliability of networks such as series, parallel, and seriesparallel. The method’s application to such configurations is demonstrated in Dhillon [18]. Nonetheless, the following example demonstrates the application of this method: Example 4.8 Assume that a person has to perform two independent and distinct tasks m and n to operate an engineering system. Task in is performed before task n. Furthermore, each of these two tasks can be conducted either correctly or incorrectly. Develop a probability tree and obtain an equation for probability of not successfully accomplishing the overall mission (i.e., not operating the engineering system correctly) by the person. In this example, the person first performs task m correctly or incorrectly and then proceeds to perform task n. This task can also be performed either correctly or incorrectly. Figure 4.11 depicts a probability tree for the entire scenario. The symbols used in the figure are defined below. m denotes the event that task m is performed correctly. m denotes the event that task m is performed incorrectly. FIGURE 4.11 A probability tree for performing tasks in and n. n denotes the event that task n is performed correctly. n denotes the event that task n is performed incorrectly. In Figure 4.11, the term nm denotes operating the engineering system successfully (i.e., overall mission success). Thus, the probability of occurrence of events inn is given by [ 18]
where P_{m} is the probability of performing task m correctly. P„ is the probability of performing task n correctly. Similarly, in Figure 4.11, the terms mn, inn, and inn denote three distinct possibilities of not operating the engineering system correctly or successfully. Thus, the probability of not successfully accomplishing the overall mission (i.e., not operating the engineering system correctly) by the person is
where p_ is the probability of performing task n incorrectly. p_ is the probability of performing task m incorrectly. ^{1} m Pf is the probability of not successfully accomplishing the overall mission (i.e., mission failure). Example 4.9 Assume that in Example 4.8, the probabilities of the person not performing tasks m and n correctly are 0.3 and 0.25, respectively. Calculate the probability of correctly operating the engineering system by the person. Thus, we have P = 0.3 and P = 0.25. m n Since P +P_{m} = 1 and P +P_{n} = 1, we have m n
By substituting Equations (4.35)(4.36) and the specified data values into Equation (4.33), we obtain
Thus, probability of correctly operating the engineering system (i.e., the probability of occurrence of events mri) by the person is 0.525. Binomial MethodThis method is used for evaluating the reliability of relatively simple systems, such as series and parallel systems/networks. For such systems’/networks’ reliability evaluation, this is one of the simplest methods. However, in the case of complex systems/ networks the method becomes a trying task. The method can be applied to systems/ networks with independent identical or nonidentical units. The following formula is the basis for the method [16]:
where n is the number of nonidentical units/components. R, is the /th unit reliability. Fj is the /th unit failure probability. Example 4.10 Using Equation (4.37) develop reliability expressions for parallel and series networks having two nonidentical and independent units each. In this case, since n = 2 from Equation (4.37) we get
where R_{x} is the reliability of unit l. R, is the reliability of unit 2. F_{1} is the failure probability of unit 1. F, is the failure probability of unit 2. Thus, using Equation (4.38), we write the following reliability expression for the parallel network having two nonidentical units:
where Rp2^{ls} thetwo nonidentical units parallel network reliability. Since (+ Fj) = 1 and (R_{2} + F_{2}) = 1, Equation (4.39) becomes
By rearranging Equation (4.40) we obtain
Finally, the two nonidentical units series network reliability from Equation (4.38) is
where R_{sl} is the two nonidentical units series network reliability. Thus, reliability expressions for parallel and series networks having two nonidentical and independent units each are given by Equations (4.41) and (4.42), respectively. Problems
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