Path-Based Statistical Static Timing Analysis for Large Integrated Circuits in a Weak Correlation Approximation

In recent years, there has been a lot of research into statistical static timing analysis (SSTA) to compute the critical path delay of a circuit under timing variation. In order to compute the true critical path delay, however, false paths that cannot be sensitized by any input vector must be identified first. Since SSTA is unable to capture the dynamic timing behavior of a circuit, it is completely blind to false paths, and thus it overestimates the circuit timing. In this paper, we propose a new concept of timing analysis approach called statistical dynamic timing analysis (SDTA), which is able to precisely express the statistical behavior of dynamic transitions at the output of gate into a compact form and to directly evaluate and propagate the expressions throughout the circuit, by which the false paths can be cleaned effectively. In addition, to be practical, we propose a couple of techniques that enable a fast computation of the SDTA. We tested the proposed approach on ISCAS benchmarks and carry skip adders under timing variation to show its accuracy in computing the distribution of the true critical path delay of a circuit. In summary, compared to the previous approach of false path-aware statistical timing analysis, our timing analysis technique is able to reduce the accuracy error in the mean and standard deviation of true critical path delay distribution from 9.8% to 1.9% and from 29.4% to -3.4%, respectively.

In this paper, we propose a calculation method of gate delay for SSTA (Statistical Static Timing Analysis) considering MIS (Multiple Input Switching). In SSTA, statistical maximum/minimum operation is necessary to calculate the latest/fastest arrival time of multiple input gate. Most SSTA approaches calculate the distribution in the latest/fastest arrival time under SIS (Single Input Switching assumption), resulting in ignoring the effect of MIS on the gate delay and the output transition time. MIS occurs when multiple inputs of a gate switch nearly simultaneously. Thus, ignoring MIS causes error in the statistical maximum/minimum operation in SSTA. We propose a statistical gate delay model considering MIS. We verify the proposed method by SPICE based Monte Carlo simulations. Experimental results show that the neglect of MIS effect leads to 80% error in worst case. The error of the proposed method is less than 20%.

In this paper, we explore the implementation of Monte Carlo based statistical static timing analysis (SSTA) on a Graphics Processing Unit (GPU). SSTA via Monte Carlo simulations is a computationally expensive, but important step required to achieve design timing closure. It provides an accurate estimate of delay variations and their impact on design yield. The large number of threads that can be computed in parallel on a GPU suggests a natural fit for the problem of Monte Carlo based SSTA to the GPU platform. Our implementation performs multiple delay simulations at a single gate in parallel. A parallel implementation of the Mersenne Twister pseudo-random number generator on the GPU, followed by Box-Muller transformations (also implemented on the GPU) is used for generating gate delay numbers from a normal distribution. The μ and σ of the pin-to-output delay distributions for all inputs and for every gate, are obtained using a memory lookup, which benefits from the large memory bandwidth of the GPU. Threads which execute in parallel have no data/control dependencies on each other. All threads compute identical instructions, but on different data, as required by the Single Instruction Multiple Data (SIMD) programming semantics of the GPU. Our approach is implemented on a NVIDIA GeForce GTX 8800 GPU card. Our results indicate that our approach can obtain an average speedup of about 260x as compared to a serial CPU implementation. With the recently announced quad 8800 GPU cards, we estimate that our approach would attain a speedup of over 785x. The correctness of the Monte Carlo based SSTA implemented on a GPU has been verified by comparing its results with a CPU based implementation.

Existing statistical static timing analysis (SSTA) techniques suffer from limited modeling capability by using a linear delay model with Gaussian distribution, or have scalability problems due to expensive operations involved to handle non-Gaussian variation sources or non-linear delays. To overcome these limitations, we propose a novel SSTA technique to handle both nonlinear delay dependency and non- Gaussian variation sources simultaneously. We develop efficient algorithms to perform all statistical atomic operations (such as max and add) efficiently via either closed- form formulas or one-dimensional lookup tables. The resulting timing quantity provably preserves the correlation with variation sources to the third-order. We prove that the complexity of our algorithm is linear in both variation sources and circuit sizes, hence our algorithm scales well for large designs. Compared to Monte Carlo simulation for non-Gaussian variation sources and nonlinear delay models, our approach predicts all timing characteristics of circuit delay with less than 2% error.

Statistical static timing analysis (SSTA) methods, which model process variations statistically as probability distribution function rather than deterministically, have been thoroughly performed on traditional zero clock skew circuits. In the traditional zero clock skew circuits, the synchronizing clock signal is designed to arrive in phase with respect to each register. However, designers will often schedule the clock skew to different registers in order to decrease the minimum clock period of the entire circuit. Clock skew scheduling imparts very different timing constraints that are based, in part, on the topology of the circuit. In this paper, SSTA is applied to nonzero clock skew circuits in order to determine the accuracy improvement relative to their zero skew counterparts, and also to assess how the results of skew scheduling might be impacted with more accurate statistical modeling. For 99.7% timing yield (3σ variation), SSTA is observed to improve the accuracy, and therefore increase the timing margin, of nonzero clock skew circuits by up to 2.5×, and on average by 1.3×, the amount seen by zero skew circuits.

Statistical Static Timing Analysis has received wide attention recently and emerged as a viable technique for manufacturability analysis. To be useful, however, it is important that the error introduced in SSTA be significantly smaller than the manufacturing variations being modeled. Achieving such accuracy requires careful attention to the delay models and to the algorithms applied. In this paper, we propose a new sparse-matrix based framework for accurate path-based SSTA, motivated by the observation that the number of timing paths in practice is sub-quadratic based on a study of industrial circuits and the ISCAS89 benchmarks. Our sparse-matrix based formulation has the following advantages: (a) It places no restrictions on process parameter distributions; (b) It embeds accurate polynomial-based delay model which takes into account slope propagation naturally; (c) It takes advantage of the matrix sparsity and high performance linear algebra for efficient implementation. Our experimental results are very promising.

Near-threshold computing has emerged as one of the most promising solutions for enabling highly energy efficient and high performance computation of microprocessors. This paper proposes architecture-level statistical static timing analysis (SSTA) models for the near-threshold voltage computing where the path delay distribution is approximated as a lognormal distribution. First, we prove several important theorems that help consider architectural design strategies for high performance and energy efficient near-threshold computing. After that, we show the numerical experiments with Monte Carlo simulations using a commercial 28nm process technology model and demonstrate that the properties presented in the theorems hold for the practical near-threshold logic circuits.

We present a new technique for statistical static timing analysis (SSTA) based on Markov chain Monte Carlo (MCMC), that allows fast and accurate estimation of the right-hand tail of the delay distribution. A ¿naive¿ MCMC approach is inadequate for SSTA. Several modifications and enhancements, presented in this paper, enable application of MCMC to SSTA. Moreover, such an approach overcomes inherent limitations of techniques such as importance sampling and Quasi-Monte Carlo. Our results on open source designs, with an independent delay variation model, demonstrate that our technique can obtain more than an order of magnitude improvement in computation time over simple Monte Carlo, given an estimation accuracy target at a point in the tail. Our approach works by providing a large number of samples in the region of interest. Open problems include extension of algorithm applicability to a broader class of synthesis conditions, and handling of correlated delay variations. In a broader context, this work aims to show that MCMC and associated techniques can be useful in rare event analyses related to circuits, particularly for high-dimensional problems.

This paper introduces a novel compact implicit model for a probabilistic set of waveforms (PSoW) which arise as representations for uncertain signal waveforms in Statistical Static Timing Analysis (SSTA). In traditional SSTA tools, signals are just represented as (distributions of) arrival time and slew. In our approach, to increase accuracy, PSoW's are used instead. However, to represent PSoW's explicitly, a very large amount of data is necessary, which can be problematic. To solve this problem, a compact implicit model is introduced, which can be characterized with just a handful of parameters. The results obtained show that the implicit model can generate real-life PSoW's with high accuracy.

We propose a scalable efficient parameterized block-based statistical static timing analysis (SSTA) algorithm incorporating both Gaussian and non-Gaussian parameter distributions, capturing spatial correlations using a grid-based model. As a preprocessing step, we employ an independent component analysis to transform the set of correlated non-Gaussian parameters to a basis set of parameters that are statistically independent. Given the moments of the variational parameters, we use a Pade acute-approximation-based moment-matching scheme to generate the distributions of the random variables representing the signal arrival times and preserve correlation information by propagating arrival times in a canonical form. Our SSTA procedure is able to generate the circuit delay distributions with reasonably small prediction errors. For the ISCAS89 benchmark circuits, as compared to Monte Carlo simulations, we obtain average errors of 0.99%, 2.05%, 2.33%, and 2.36%, respectively, in the mean, standard deviation, and 5% and 95% quantile points of the circuit delay. Experimental results show that our procedure can handle as many as 256 correlated non-Gaussian variables in about 5 min of runtime. For a circuit with |G| gates and a layout with g spatial correlation grids, the complexity of our approach is O(g|G|) .

Near-Threshold Computing (NTC) is shown to be a promising approach for improving the energy efficiency of VLSI circuits. Nevertheless, by reducing the supply voltage the delay impact of process variation significantly increases, leading to up to 20× performance variation compared to the nominal voltage. As a result, it is wasteful of energy and performance to deal with such variation by increasing the timing margins, which is common in nominal voltage. Therefore, considering the impact of process variation during the near-threshold circuit design phase is of decisive importance. In this paper, we propose a variation-aware synthesis flow for NTC to address this problem. The objective is to improve the performance and energy efficiency of a circuit during design time by considering statistical variation information. This is done by providing variation information to the synthesis tool, evaluating the performance of the synthesized circuit by Statistical Static Timing Analysis (SSTA), and adjusting the timing constraints accordingly in an iterative manner. Simulation results for a set of benchmark circuits show that our proposed flow reduces the variation by 86.6% and improves the performance and energy by 24.9% and 7.4%, respectively, at the expense of 4.8% area overhead.

Variability continues to pose challenges to integrated circuit design. With statistical static timing analysis and high-yield estimation methods, solutions to particular problems exist, but they do not allow a common view on performance variability including potentially correlated and non-Gaussian parameter distributions. In this paper, we present a probabilistic approach for variability modeling as an alternative: model parameters are treated as multi-dimensional random variables. Such a fully mul-tivariate statistical description formally accounts for correlations and non-Gaussian random components. Statistical characterization and model application are introduced for standard cells and gate-level digital circuits. Example analyses of circuitry in a 28 nm industrial technology illustrate the capabilities of our modeling approach.

Statistical static timing analysis (SSTA) tools have mostly addressed the process variations of devices and lumped interconnect RC effects. This paper provides an overview of interconnect process variations of capacitive coupling and its effect on crosstalk delay and noise. The correlations among parallel plate, lateral and total capacitance is shown. Correlations between resistance and capacitance are illustrated to enable development of a simple and efficient model for delay and noise analysis. Experimental results are shown to validate the assumptions on the linearity of sensitivity of delay and noise to process variations. A methodology to account for process variations in crosstalk delay and noise is proposed.

Automated cell placement is a critical problem in very large scale integration (VLSI) physical design. New analytical placement methods that simultaneously spread cells and optimize wirelength have recently received much attention from both academia and industry. A novel and simple objective function for spreading cells over the placement area is described in the patent of Naylor et al. (U.S. Pat. 6301693). When combined with a wirelength objective function, this allows efficient simultaneous cell spreading and wirelength optimization using nonlinear optimization techniques. In this paper, we implement an analytic placer (APlace) according to these ideas (which have other precedents in the open literature), and conduct in-depth analysis of characteristics and extensibility of the placer. Our contributions are as follows. 1) We extend the objective functions described in (Naylor et al., U.S. Patent 6301693) with congestion information and implement a top-down hierarchical (multilevel) placer (APlace) based on them. For IBM-ISPD04 circuits, the half-perimeter wirelength of APlace outperforms that of FastPlace, Dragon, and Capo, respectively, by 7.8%, 6.5%, and 7.0% on average. For eight IBM-PLACE v2 circuits, after the placements are detail-routed using Cadence WRoute, the average improvement in final wirelength is 12.0%, 8.1%, and 14.1% over QPlace, Dragon, and Capo, respectively. 2) We extend the placer to address mixed-size placement and achieve an average of 4% wirelength reduction on ten ISPD'02 mixed-size benchmarks compared to results of the leading-edge solver, FengShui. 3) We extend the placer to perform timing-driven placement. Compared with timing-driven industry tools, evaluated by commercial detailed routing and static timing analysis, we achieve an average of 8.4% reduction in cycle time and 7.5% reduction in wirelength for a set of six industry testcases. 4) We also extend the placer to perform input/output-core coplacement and constraint handing for mixed-signal designs. Our paper aims to, and empirically demonstrates, that the APlace framework is a general, and extensible platform for spatial embedding tasks across many aspects of system physical implementation.

To exploit the benefits of throughput-optimized processors such as GPUs, applications need to be redesigned to achieve performance and efficiency. In this work, we present techniques to speed up statistical timing analysis on throughput processors. We draw upon advancements in improving the efficiency of Monte Carlo based statistical static timing analysis (MC SSTA) using techniques to reduce the sample size or smart sampling techniques. An efficient smart sampling technique, Stratification + Hybrid Quasi Monte Carlo (SH-QMC), is implemented on a GPU based on NVIDIA CUDA architecture. We show that although this application is based on MC analysis with straightforward parallelism available, achieving performance and efficiency on the GPU requires exposing more parallelism and finding locality in computations. This is in contrast with random sampling based algorithms which are inefficient in terms of sample size but can keep resources utilized on a GPU. We show that SH-QMC implemented on a Multi GPU is twice as fast as a single STA on a CPU for benchmark circuits considered. In terms of an efficiency metric, which measures the ability to convert a reduction in sample size to a corresponding reduction in runtime w.r.t a random sampling approach, we achieve 73.9% efficiency with the proposed approaches compared to 4.3% for an implementation involving performing computations on smart samples in parallel. Another contribution of the paper is a critical graph analysis technique to improve the efficiency of Monte Carlo based SSTA, leading to 2--9X further speedup.