Basic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability ConfigurationsBasic Reliability Configurations•Assumptions on the reliability model:–Each component has two possible states: operational or failed.–The failure of each component is an independent event.Basic Reliability Configurations•Assumptions on the reliability model:–Component i is functioning (operational) with probability pi and is inoperational (failed) with probability 1 – pi : (These probabilities are usually known.)–The reliability R of the system is some fuction of the component reliabilities:R = f(p1, p2,…, pn )where N is the number of components.Basic Reliability Configurations•The function f(…) above depends on the configuration, which defines when the system is considered operational, given the states of the components.•Basic examples are shown in the configurations discussed below.Basic Reliability Configurations•Series Configuration:–In the series configuration the system is operational if and only if all components are functioning. This can be schematically represented by the figure below, in which the system is considered operational if there is an operationalvpath between the two endpoints, that is, all components are functioning:Basic Reliability Configurations•Series Configuration:•The reliability of the series configuration is computed simply as the product of the component reliabilities:–Rseries = p1 p2 … pNP1…P2P3PnBasic Reliability Configurations•Note: If many components are connected in series, then the reliability may be much lower than the individual reliabilities.– For example, if p = 0.98 and N = 10, then Rseries = (0.98)10 = 0.82; –significantly lower than the individual reliabilities.Basic Reliability Configurations•Parallel Configuration–The parallel configuration is defined operational if at least one of the components are functioning. This is schematically represented in the figure below:Basic Reliability Configurations•Parallel configuration:P1……PNBasic Reliability Configurations•The reliability can be computed as follows. –The probability that component i 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, that is, at least one of them works:•Basic Reliability Configurations•k out of N Configuration:–In this configuration the system is considered functional if at least k components out of the total of N are functioning.–The probability that a given set of k components are functioning ispk(1 - p)N-kBasic Reliability Configurations•k out of N Configuration:–The probability that some set of k components are functioning iswhere represents the number of ways one can choose a k-element set out of N.•Basic Reliability Configurations•k out of N Configuration:–Finally, since we need at least k operational componenets, we have to sum up the above for all possible acceptable values of k. This gives the reliability of the k out of N
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