This paper addresses two classes of reliability analysis models: a network flow model and a project scheduling model. In each model, the arcs randomly and independently take on two possible states ― an «operating» state and a «failed» state ― corresponding to two different capacity/task-time values, and the network is required to maintain a specified «threshold» max-flow/project-completion-time value. In general, this problem is NP-hard. We address the special case in which the difference between the lower and higher arc lengths is constant for every arc in the network. For these special cases, we show that if the underlying system is 1 -critical, i.e., minimally able to withstand a single component failure, then the probability that the system can maintain the required threshold for the flow and planar project scheduling model is computable in polynomial time. Both solutions are obtained by reducing the problems to the problem of determining the probability that the failed arcs in a directed acyclic graph lie on a single path or, equivalently, that the set of failed elements in a given partial order comprises a chain in that order. We also show how the basic approach can be used to generate bounds for systems that are «almost critical»
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