System reliability calculations based on incomplete information

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, S_c) when only some of the individual failure probabilities, and some of the joint or conditional failure probabilities for S₁ , S₂, and S_c are known. If x_i is the state of system S_i (“available” or “failed”), maximum entropy yields “best” estimates for the probabilities of all events of the form x₁x ₂x_c. We derive almost closed-form expressions for these probabilities under various states of knowledge. 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 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