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System Reliability Calculations Based on Incomplete Information
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| IEEE Transactions on Systems, Man, and Cybernetics |
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In large computer or communication networks, there are sometimes components that do not fail independently to each other, such that the dependencies among them are only partially known. To address the problem of estimating the reliability of such groups of components, we show how the maximum entropy principle can be used to calculate the probability of failure of a 3-component system (S1,S2,Sc) when only some of the individual failure probabilities, and some of the joint or conditional failure probabilities for S1, S2, and Sc are known. If xi is the state of system Si ("available" or "failed"), maximum entropy yields "best" estimates for the probabilities of all events of the form x1x2xc. We derive almost closed-form expressions for these probabilities under various states of knowledge, that is, forms depending on the numerical solution of a single nonlinear equation, from which any probability about (S1,S2,Sc) can be calculated. We establish analytically that the maximum entropy distributions have many properties that agree with intuition, and show how the distributions illustrate Shore and Johnson's axioms for the the maximum entropy principle. We obtain closed form expressions in almost all cases for the conditional and unconditional reliabilities of series and parallel connections of S1 and S2. The results are illustrated graphically, and shown to be compatible with intuition. We conclude by discussing some questions that may be raised regarding our use of maximum entropy, namely system models with independent components, lower bounds on reliability, and whether maximum entropy estimates are conservative in any sense.
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