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Network associated with the set of elements and linearly ordered subset of elements, known as paths, satisfying the switching property is an abstract network. Due to the switching property, flows crossing at intersections are diverted to the non-crossing sides. Each element of an abstract network is equipped with two types of integral capacities: one is movement capacity which transships the flow from an element to its adjacent element and another is the storage capacity which holds the flow at the element. Due to insufficient movement capacity of intermediate elements, flow out from the source may not reach at the destination. If the flow out from the source is more than the minimum cut capacity, then the problem associated with the settlement of excess flow at appropriate intermediate elements is termed as network flow with intermediate storage. In this paper, we discuss the static and dynamic flow models with intermediate storage in an abstract network using temporal repetition of flow. We solve abstractmaximum dynamic flow and contraflow problems with intermediate storage.

Contraflow is one of the best and widely accepted techniques in evacuation planning problems, where the reversal of arcs is made to increase the amount and decrease the time of ow transmission. At the time of evacuation, if the flow value leaving the source node exceeds the bottleneck capacity of the network, then the storage of excess flow at comparatively safer intermediate nodes can be a milestone to save the life of evacuees. In this paper, we introduce the maximum dynamic contraflow problem in a general network and earliest arrival contraflow problem in two-terminal series-parallel network with asymmetric transit times on anti-parallel arcs. We present polynomial time solution strategies with orientation dependent transit times by allowing the storage of flow at intermediate nodes.