A general probabilistic framework for worst case timing analysis

Abstract

The traditional approach to worst-case static-timing analysis is becoming unacceptably conservative due to an ever-increasing number of circuit and process effects. We propose a fundamentally different framework that aims to significantly improve the accuracy of timing predictions through fully probabilistic analysis of gate and path delays. We describe a bottom-up approach for the construction of joint probability density function of path delays, and present novel analytical and algorithmic methods for finding the full distribution of the maximum of a random path delay space with arbitrary path correlations. General Terms: Algorithms INTRODUCTIONOver the years it has been widely acknowledged that the uncertainty about the true design and manufacturing conditions is a major cause of unnecessary over-design and resulting underperformance of circuits [1] [2]. The sources of this uncertainty are manifold, and are due to the limitations of the actual design practices, uncertainty about the environmental design characteristics (cross-talk noise, temperature and supply voltage variation), and the inherent variation of the underlying process parameters. With the advance of deep sub-micron technologies, process variability and, in particular, intra-chip variation, has been increasing. This is due to various processing and device physics factors such as random dopant placement in the channel, spatially correlated and proximity-caused Lgate variation, and interconnect metal thickness variation [2].The emergence of intra-chip parameter variability as a dominant source of uncertainty and circuit degradation requires a new set of approaches to circuit timing analysis, whose role is to guarantee that the predicted maximum clock speed is as close as possible to the actual (silicon) timing behavior. Industrial experience shows that the gap between the worst-case timing constraints predicted by the tools, and the final silicon performance is routinely greater than what can be tolerated and is sometimes as high as 30% [3].What is wrong with the existing tools and approaches? Circuitdependent parametric yield loss is predicted to become a key issue in nanometer silicon technologies [4]. The fundamental problem is that the standard timing techniques are incapable of accurately predicting parametric yield of a circuit due to their nonprobabilistic formulation. One particular result of this failing is the well-known conservatism of traditional worst-case modeling techniques. We may distinguish at least two levels of conservatism. The first is the practice of defining the worst-case timing behavior of a cell by performing circuit analysis in SPICE that simultaneously sets all the device model parameters to their worst-case values. Several approaches to reduce this type of conservatism have been proposed [5]. When we move to the level of cell-based static timing analysis, an additional level of conservatism is created by the non-probabilistic delay computation of the traditional analysis. This conservatism is relatively new but ...

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