2. On the Need for Statistical Timing Analysis Farid N. Najm University of Toronto f.najm@utoronto.ca

3. 3 Introduction n Increased process variability leads to chip timing variability and lower timing yield n Traditionally, corner-analysis (worst-case files) has been used to manage timing variability l Corner analysis has some disadvantages n Statistical Static Timing Analysis (SSTA) has been proposed as an alternative approach l SSTA has its own disadvantages n Perhaps an alternative “best of both worlds” approach is required: l Use statistical analysis to better choose the corners or the margins to be used in a traditional STA approach

4. 4 Generation of Corner-Case Files n Principal Components Analysis de-correlates SPICE parameters and captures bulk of variations n Process parameter corners chosen to maximize MOSFET performance yield

5. 5 Disadvantages of Corner Analysis n Some disadvantages: l Corners should maximize circuit yield, not device yield u Goal is to bracket most (say, 99.73%) of what? 1. individual process variable space? 2. typical transistor strength? 3. typical gate/cell delay? 4. overall circuit performance? è Ideally, one would like #4, but traditionally go with #2 l There are too many corners l Cannot take care of within-die variations l Corner analysis is overkill u One is capturing much more yield (performance spread) than one really needs to l Cannot determine how robust the design is

6. 6 Too Many Corners n With more process parameters, the number of process corners increases exponentially n However, there have been recent proposals to reduce the number of corners to be considered l Corner clustering (Sengupta et al., ISQED-04) l This method also allows one to choose corners so as to bracket circuit performance, instead of device performance  Quadratic circuit response, RSM: g(X) = a + bX + X T BX  Solution: X vector that minimizes and maximizes g(X) u Cluster corners that are close in the parameter space

7. 7 Case Files & Intra-Die Variations n Traditional corner analysis cannot take care of within- die variations l Heuristic techniques are used within some traditional STA tools to approximately take care of within-die effects n The crux of the problem lies in the systematic within-die variations l Random within-die variations “cancel out” on a path u They don’t exactly cancel out, but their net result is reduced n The overall impact of within-die variations on circuit delay arguably remains small compared to die-to-die variations (S. Samaan, ICCAD-04)

8. 8 Too Much Guardbanding n Corner analysis becomes “overkill” when the implicit yield target becomes too large l Not always the case in corner analysis! Assume that a nominal value of yield is what covers the ± 3  of a standard normal distribution: Y 0 = 99.73% n Whether corner analysis is overkill or not depends on the performance metric If g(X) =  Xi (i = 1,…, n), then Y   (3n 1/2 ) -  (-3n 1/2 ) > Y 0  Setting Xi at ± 3  is overkill If g(X) = max (Xi) then Y   n (3) -  n (-3) < Y 0  Setting Xi at ± 3  is NOT overkill n It also depends on the shape of the acceptability region

9. 9 Assessment n The straightforward nature of corner-case analysis has made it the method of choice in industry n It has some limitations: l Need to determine corners based on circuit performance l Location of corners depends on acceptability region l Need to reduce the number of corners to be covered l Cannot determine how robust the design is n Nevertheless, criticisms do not dismiss this approach altogether

10. 10 Statistical Timing Analysis n Recently, “Statistical Static Timing Analysis” (SSTA) has been proposed l Deal with circuit timing uncertainty l An alternative to corner analysis n Basic Idea: l Propagate delay distributions, instead of deterministic delays, in the timing graph u Compute node and path delay distributions l Estimate the distribution of circuit delay as the joint distribution of path delays l Find the chip timing yield from circuit delay distribution

11. 11 Statistical Timing Analysis n How to handle different types of delay correlations ? l Within-die systematic correlation l Path sharing (reconvergent fanout) l Dependence on global sources of variations n How to propagate distributions in the timing graph ? l The statistical MAX function l Statistical SUM function n What types of distributions to use ? l Gaussian, or arbitrary distributions ? n Distinct trends: l Block-based statistical timing l Path-based statistical timing

12. 12 Block-Based SSTA n Propagate distributions of arrival times in the timing graph of the block to get circuit delay distribution n Path distributions are available only indirectly

13. 13 The MAX Operation n Arrival times are “MAX-ed” at the nodes of the graph l Circuit delay distribution is obtained on the primary outputs n The various methods differ in: l How the MAX operation is performed l Assumptions on the nature of the distributions (Gaussian/not) l Whether and how correlation is taken care of

14. 14 Overview: Block-Based Methods n A key difference among block-based methods lies in whether delays are assumed Gaussian or arbitrary n Two Gaussian approaches both use decomposition, but differ in what underlying variables are used l Visweswariah et al. derive correlations from global sources of variation l Sapatnekar et al. perform PCA on the spatial correlations n Two non-Gaussian approaches differ in the propagation algorithm of arrival times l Blaauw et al. use conservative bounds on delay distributions l Devgan et al. use piece-wise linear approximations

15. 15 Path-Based Methods n Path delay distributions are expressed as functions of the underlying sources of variation l Gate delay distributions are added to get path delay distribution l Literature by: Nassif, Jess, Orshansky, Bowman n Circuit delay distribution is obtained from the joint probability of path delays l Circuit delay = MAX(all path delays) n Flow: l Enumerate all critical paths l Estimate path delay distributions l Use multi-dimensional integration to combine all paths l Estimate the timing yield

16. 16 Assessment n The problem of propagating delay distributions along paths or through blocks is now “solved” n Yet, this does not mean that SSTA is now “solved”! n Key problems in the proposed methods of SSTA l What does one do with all these distributions?! l Unless if the full chip is “timed” flat, require change in methodology: cannot “time” a path or block in isolation l Correlation handling requires layout information, hence cannot be used pre-placement during circuit design/optimization l Not clear how to get correlation statistics from the process; a disconnect between process and EDA

17. 17 Practical SSTA n Desirable features of a practical SSTA approach: l Must require minimal statistical process data l Must account for correlated and uncorrelated variations l Must be usable pre-placement to enable design optimization l Must be applicable to “early design” with uncertain circuitry, in order to allow one to time a path/block in isolation n One can envision three types of SSTA: l Process-specific, not design-specific, during early design l Design-specific, not placement-specific, during circuit design l Placement-specific, during physical design n A mix of the three types of SSTA would constitute a practical framework for managing timing variability

18. 18 An Early Design Approach n A recent approach (Najm and Menezes, DAC-04) is applicable during early design l Employ notion of generic paths to develop an approach which is process-specific, not design-specific n The ability to handle early (uncertain) design is key to being able to time a path/block in isolation! n Shift focus from the specific design to a design type l What are a typical transistor/gate in this technology? l What is a typical path length in this class of design? l Assume the circuit or block consists of a large number of such “generic paths”

19. 19 Overview

20. 20 Conclusions n Process variability is a key factor of timing yield loss and deterioration of circuit performance n Traditional corner analysis has some limitations, but they are not insurmountable n Statistical timing analysis is being proposed as an alternative, but it has its own limitations n Perhaps one can have it both ways l Combine features of statistical analysis and corner-case files l Derive virtual corners and timing margins for a yield-aware timing verification n This continues to be an active research topic