Basic Reliability ConfigurationsAssumptions on the reliability model:• Each component has two possible states: operational or failed.• The failure of each component is an independent event.• Component i is functioning (operational) with probability piand isinoperational (failed) with probability 1 − pi. (These probabilities areusually known.)• The reliability R of the system is some fuction of the component relia-bilities:R = f(p1, p2, . . . , pN)where N is the number of components.The function f (...) above dep ends on the configuration, which defines whenthe system is considered operational, given the states of the components.Basic examples are shown in the configurations discussed below.Series ConfigurationIn the series configuration the system is operational if and only if all com-ponents are functioning. This can b e schematically represented by the figurebelow, in which the system is considered operational if there is an operationalpath between the two endpoints, that is, all components are functioning:o— p1—— p2—— p3——.........—— pN—oThe reliability of the series configuration is computed simply as the productof the component reliabilities:Rseries= p1p2. . . pNNote: If many components are connected in series, then the reliability maybe much lower than the individual reliabilities. For example, if p = 0.98and N = 10, then Rseries= (0.98)10= 0.82, significantly lower than theindividual reliabilities.Parallel ConfigurationThe parallel configuration is defined operational if at least one of the compo-nents are functioning. This is schematically represented in the figure below:— p1—......o—... —o......— pN—The reliability can be computed as follows. The probability that componenti fails is 1 − pi. The probability that all components fail is (1 − p1)(1 −p2) . . . (1 − pN). The complement of this is that not all component fails, thatis, at least one of them works:Rparallel= 1 − (1 − p1)(1 − p2) . . . (1 − pN) = 1 −NYi=1(1 − pi)k out of N ConfigurationIn this configuration the system is considered functional if at least k compo-nents out of the total of N are functioning.The probability that a given set of k components are functioning ispk(1 − p)N−k.The probability that some set of k components are functioning isÃNk!pk(1 − p)N−kwhereÃNk!=N!k!(N − k)!represents the number of ways one can choose a k-element set out of N.Finally, since we need at least k operational componenets, we have to sumup the above for all possible acceptable values of k. This gives the reliabilityof the k out of N configuration:Rk/N=NXi=kÃNi!pi(1 −
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