Energy harvesting is a key technology to enable long-term wireless sensor network applications. In the case of multi-hop networks, each node both performs measurements to produce data to be sent to a sink, and relays data packets from other nodes. In this letter, we propose a distributed algorithm for computation of fair packet rates for multi-hop energy harvesting wireless sensor networks. The packet rate computation problem is formulated as a convex optimization problem, and using the fast alternating direction method of multipliers, the original problem is decomposed into smaller subproblems that can be solved in parallel. Simulations using real indoor light energy traces show that the algorithm computes high accuracy solutions, even with a low median number of iterations (10 or less). By setting the stop criteria parameter, a compromise can be set between the accuracy of the solution and the number of iterations required.