A New Statistical Max Operation for Propagating Skewness in Statistical Timing Analysis

Abstract

Statistical static timing analysis (SSTA) is emerging as a solution for predicting the timing characteristics of digital circuits under process variability. For computing the statistical max of two arrival time probability distributions, existing analytical SSTA approaches use the results given by Clark (1961). These analytical results are exact when the two operand arrival time distributions have jointly Gaussian distributions. Due to the nonlinear max operation, arrival time distributions are typically skewed. Furthermore, nonlinear dependence of gate delays and non-Gaussian process parameters also make the arrival time distributions asymmetric. Therefore, for computing the max accurately, a new approach is required that accounts for the inherent skewness in arrival time distributions. In this work, we present analytical solution for computing the statistical max operation. First, the skewness in arrival time distribution is modeled by matching its first three moments to a so-called skewed normal distribution. Then by extending Clark's work to handle skewed normal distributions we derive analytical expressions for computing the moments of the max. We then show using initial simulations results that using a skewness based max operation has a significant potential to improve the accuracy of the statistical max operation in SSTA while retaining its computational efficiency