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We consider a network operation under which multiple simultaneously activated routes are employed to distribute multimedia traffic between identified source-destination pairs. We aim to jointly determine the path selection and rate allocation, per each flow, across multi-path routes, to ensure the utility obtained by each flow is as high as feasibly possible while striving to raise the utility of all flows in a max-min fair manner. In doing so, we incorporate the communications capacity constraints that are imposed by the underlying hybrid of directional and/or multiple-access wireline and wireless communications media employed across the network system. We develop and present in this paper an optimal algorithm for solving such complex problem that involves multiple sequential convex/linear programs. As illustrative examples, we demonstrate the effectiveness of our solution in comparing its performance with that obtained under the use of a traffic regulation scheme that is not overlaid with a traffic management mechanism that serves to set-aside resources for the support of flows.