Digital controllers found in many industrial real-time systems consist of a number of interacting periodic tasks. To sustain the required control quality, these tasks possess the maximum activation periods as performance constraints. An essential step in developing a real-time system is thus to assign each of these tasks a constant period such that the maximum activation requirements are met while the system utilization is minimized [1]. Given a task graph design allowing producer/consumer relationships among tasks [2], resource demands of tasks, and range constraints on periods, the period assignment problem falls into a class of nonlinear optimization problems. This paper proposes a ploynomial time approximation algorithm which produces a solution whose utilization does not exceed twice the optimal utilization. Our experimental analysis shows that the proposed algorithm finds solutions which are very close to the optimal ones in most cases of practical interest.