Algorithms for minimising the lost flow due to failed components in repairable flow networks with complex topology

A number of fundamental theorems related to non‐reconfigurable repairable flow networks have been stated and proved. For a specified source‐to‐sink path, the difference between the sum of the unavailabilities of its forward edges and the sum of the unavailabilities of its backward edges is the path resistance. In a repairable flow network, the absence of augmentable cyclic paths with negative resistance is a necessary and sufficient condition for a minimum lost flow due to edge failures. For a specified source‐to‐sink path, the difference between the sum of the hazard rates of its forward empty edges and the sum of the hazard rates its backward empty edges is the flow disruption number of the path. The absence of augmentable cyclic paths with a negative flow disruption number is a necessary and sufficient condition for a minimum probability of undisturbed throughput flow, by edge failures.