Variability-Driven Formulation for Simultaneous Gate Sizing and Post-Silicon Tunability Allocation

Abstract — Process variations cause design performance to become unpredictable in deep sub-micron technologies. Several statistical techniques (timing analysis, gate-sizing, buffer insertion) have been proposed to counter these variations during the optimization phase of the design flow to get better timing yields. Another interesting approach to improve timing yield is post-silicon tunable (PST) clock-tree. However, gate sizing and PST clock tree management have not been integrated together into a single framework for better optimization. In this work, we propose such an integrated framework that performs simultaneous statistical gatesizing in presence of PST clock-tree buffers for minimizing binningyield loss (BYL) and tunability costs by determining the ranges of tuning to be provided at each buffer. To our best knowledge, there is no work in the current literature that tries to look at such an integrated variability-driven framework. The simultaneous gatesizing and PST buffer range determination problem is proved to be a convex stochastic programming formulation under longest path delay constraints and hence solved optimally. We further extend the formulation into a heuristic to additionally consider shortest path delay constraints. We make experimental comparisons using nominal gate sizing followed by PST buffer management using [13] as a base-case. We take the solution obtained from this approach and perform 1) Sensitivity-based statistical gate-sizing while retaining the PST clock tree 2) Simultaneous gate sizing and PST buffer range determination as proposed in this work. On an average, the BYL obtained from our approach is 98 % lower than the base-case ([13]) and 95 % lower than the sensitivity-based algorithm. On an average the base-case approach ([13] gave 22% timing yield loss (YL), the sensitivity approach gave 19 % YL, where as our proposed algorithm gave only 3 % YL. The total PST tuning buffer range that is allocated through the proposed algorithm is comparable to that obtained from [13]. The proposed algorithm had a 2.2x runtime speedup compared to the sensitivity-based algorithm. I