A microeconomic model for simultaneous gate sizing and voltage scaling for power optimization

We investigate the problem of dynamic power optimization through gate sizing and voltage scaling under a given delay constraint. Several algorithms have been proposed in the literature to handle gate sizing and voltage scaling independently or together with the goal of satisfying certain power budget constraints without affecting the timing constraints. Decentralized algorithms have been proposed in the literature for distributing a divisible resource among the components of the system. We formulate the problems as economic models that attempt to distribute the delay among the gates of the circuit such that the dynamic power of the circuit is optimized. Since, optimizing all the gates in the circuit at the same time can be computationally intensive, the gates in a given path are handled together. The circuits are represented as economic models and mathematical formulations are developed which are further transformed as game theoretic models for which Nash equilibrium based solutions are investigated. Thus, the main contribution of this work is the application of microeconomic models and game theory for these VLSI CAD problems. Models are developed for the gate sizing, voltage scaling and simultaneous gate sizing and voltage scaling problems. The algorithms are iterative, fast, simple and can lead to rapid convergence. Competition among the gates can provide the best overall optimization. In the proposed algorithms, the gates compete against each other to optimize their power consumption and hence that of the entire circuit. The proposed solutions yield better power optimization than other methods as shown in the experimental results for MCNC '91 benchmark circuits.