Rectilinear Steiner arborescences under linear delay constraints play an important role for buffering. We present exact algorithms for either minimizing the total length subject to delay constraints, or minimizing the total length plus the (weighted) absolute total negative slack.Our main theoretical contribution is the first minimum cost flow formulation for embedding Steiner arborescences at minimum length subject to delay constraints, resulting in the first strongly polynomial time algorithm for this subproblem.We use the minimum cost flow formulation to quickly compute lower bounds in a branch-&-bound algorithm for optimum Steiner arborescences. We demonstrate the benefit of our new algorithm experimentally.