Probabilistic standard cell modeling considering non-gaussian parameters and correlations

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.

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|) .

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 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.

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.

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.

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.

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.

This work is aimed at the development of a path-based approach to Statistical Static Timing Analysis. Timing Analysis is an absolutely essential step in the verification of Very Large Scale Integration (VLSI) designs. We propose a novel analytical methodology for the fast calculations of VLSI delay. The problem is stated in such a way that becomes equivalent to finding the maximum of a large set of correlated random variables (RVs). For this purpose, a corresponding extension of extreme value theory of weakly-correlated RVs is developed. Results of simulations showing a comparison of our approach with Monte Carlo simulations are presented. Possible applications, extensions of our methodology and future steps are discussed.

In this article, we present a critical path selection method that efficiently finds true (sensitizable) critical paths of a circuit in the presence of process variations. The method, which is based on the viability analysis, tries to select the least number of true critical paths that cover all of circuit critical gates. Critical gates are those that make a path critical with a probability higher than a predefined threshold value. Selecting fewer critical paths leads to less computation time for the algorithm and shorter test time of fabricated chips. For this purpose, an efficient Statistical Static Timing Analysis– (SSTA) based technique is suggested. This technique tries to find circuit-critical gates whose process parameter variations cover a major part of the process space. Improving the process space coverage using fewer paths is achieved by considering both spatial (proximity of gates) and structural (having common gates) correlations in the analysis of choosing the critical paths. In the selection process, paths with low similarities in their characteristics are preferred. In addition, only true paths whose delays affect the maximum delay of the circuit are included. The selected paths can be used in the test process of the fabricated chips to determine if the chip meets its timing requirements. Also, a modified viability analysis that incorporates statistical computations is used in the SSTA. The efficacy of the proposed method is evaluated by comparing its results for combinational and sequential ISCAS benchmarks with those obtained by exhaustive search. Results indicate although, on average, only 4.38% of all the critical paths found by the exhaustive search are selected by the proposed method, the maximum probability of criticality for the paths that are not considered in our method is, on average, less than 4%.

Leakage power dissipation and parameter variations are main topics in current research. The problem of parameter variations leads to modified timing analysis. Traditionally, this is done with corner-case simulations, which are quite conservative and pessimistic approaches. This paper proposes a new statistical static timing analysis (SSTA) to improve performance predictions. Furthermore, the developed SSTA and a dual-V/sub th//dual-T/sub ox/ CMOS (DTTCMOS) design technique are combined to reduce the total leakage within designs. Compared to traditional corner-case timing analyses the proposed approach reduces leakage by an average of 70% for raw designs and by an average of 25% in pre-optimized DTTCMOS designs.

The timing closure problem (e.g., meeting timing/clock period constraint) is one of the most important problems in the design automation. However, the rapid increase of the impact of the process variation on circuit timing makes the problem much more complicated and unpredictable to tackle in synthesis. This paper addresses a new problem of high-level synthesis (HLS) that effectively takes into account the timing variation. Conventional HLS may simply avoid the timing variation problem by considering the worst case or constant delay model of hardware resources, which are certainly not viable solutions. To overcome the impact of timing variation in HLS, this paper addresses the following two problems: 1) how can the statistical static timing analysis (SSTA) used in logic synthesis be modified and effectively applied to the delay and yield computation in HLS? and 2) how can scheduling and resource binding tasks effectively solve the HLS problem with the objective of minimizing latency under yield constraint? Through an extensive experimentation, we confirm that the proposed variation-aware HLS algorithm is able to reduce the latency by 19.7% under 90% performance yield constraint, compared with the result by conventional timing variation-unaware HLS .

Accurate Estimation of the Worst-case Delay in Statistical Static Timing Analysis

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.