Optimal flow and capacity allocation in multiple joint quickest paths of directed networks

Shared vertices or edges in joint paths bring difficulties to flow routing and scheduling with delay requirements in networks with consideration of both edge lengths and capacities since flow along the different paths will encounter each other in capacitated edges with time dislocation. For an amount of flow, the quickest path problem (QPP) presents a good link for path lengths and capacities with the transmission time of the flow. Extended from the QPP, we propose an edge-path form traffic model for an amount of flow through multiple joint paths with different lengths in one-source one-sink directed capacitated networks. Then, an optimization model for minimum transmission time within feasible traffic constrained by edge capacities is constructed. We then derived the vertex-edge form of the optimization model from the edge-path form. The proposed optimization models in both forms are proved to be linear fractional programming problems, which can be solved in polynomial time. A routing algorithm based on the solution of the vertex-edge form optimization is developed combined with the DFS-based route-searching method. The proposed model and algorithm could be applicable in the real-time operation and management of practical network systems.