In this paper, we formalize the problem of finding a routing path for the streaming of continuous media (e.g., video or audio files) that maximizes the probability that the streaming is successful, over a network with nonuniform edge delays and capacities, and arbitrary timedependent edge reliabilities. We call such a problem the most sustainable path (MSP) problem. We address the MSP problem in two network routing models: the wormhole and the circuit-switching routing models. We present fully-distributed polynomial-time algorithms for the streaming of constant-size data in the wormhole model, and for arbitrary-size data in the circuit-switching model. Our algorithms are simple and assume only local knowledge of the network topology at each node. The algorithms evolved from a variation of the classical Bellman-Ford shortest-path algorithm. One of the main contributions of this paper was to show how to extend the ideas in the Bellman-Ford algorithm to account for arbitrary time-dependent edge reliabilities.