Efficient SRAM Failure Rate Prediction via Gibbs Sampling

Abstract

Abstract-Statistical analysis of SRAM has emerged as a challenging issue because the failure rate of SRAM cells is extremely small. In this paper, we develop an efficient importance sampling algorithm to capture the rare failure event of SRAM cells. In particular, we adapt the Gibbs sampling technique from the statistics community to find the optimal probability distribution for importance sampling with a low computational cost (i.e., a small number of transistor-level simulations). The proposed Gibbs sampling method applies an integrated optimization engine to adaptively explore the failure region in a Cartesian or spherical coordinate system by sampling a sequence of 1-D probability distributions. Several implementation issues such as 1-D random sampling and starting point selection are carefully studied to make the Gibbs sampling method efficient and accurate for SRAM failure rate prediction. Our experimental results of a 90 nm SRAM cell demonstrate that the proposed Gibbs sampling method achieves 1.4-4.9× runtime speedup over other state-ofthe-art techniques when a high prediction accuracy is required (e.g., the relative error defined by the 99% confidence interval reaches 5%). In addition, we further demonstrate an important example for which the proposed Gibbs sampling algorithm accurately estimates the correct failure probability, while the traditional techniques fail to work.

Fast Statistical Analysis of Rare Circuit Failure Events via Scaled-Sigma Sampling for High-Dimensional Variation Space

“…To reduce the computational cost, several statistical algorithms based on importance sampling have been proposed [4], [7], [10], [12], [14]. The key idea is to sample a distorted PDF g(x), instead of the original PDF f(x), so that most random samples fall into the failure region Ω.…”

“…Our objective is to study the relation between the scaled failure rate P g in (14) and the original failure rate P f in (4). Towards this goal, we partition the M-dimensional variation space into a large number of identical hyper-rectangles with the same volume and the scaled failure rate P g in (14) can be approximated as…”

“…Towards this goal, we partition the M-dimensional variation space into a large number of identical hyper-rectangles with the same volume and the scaled failure rate P g in (14) can be approximated as…”

“…To address this issue, a large number of statistical algorithms and methodologies have been developed in the literature [4]- [14]. Most of these traditional methods focus on failure rate estimation for SRAM bit-cells that consist of few (e.g., 6~10) transistors.…”

Adaptive Learning Based Importance Sampling for Analog Circuit DPPM Estimation

“…Unfortunately, importance sampling requires knowledge of the importance regions in the process parameter space. In [5], Gibbs sampling is used to find important regions for SRAM circuits. Finding such importance regions is not trivial for analog circuits with multiple specifications and complex relations between process and specification parameters.…”