Parameterized Timing Analysis with General Delay Models and Arbitrary Variation Sources

1. Parameterized Timing Analysis with General Delay Models and Arbitrary Variation Sources Khaled R. Heloue and Farid N. Najm University of Toronto {khaled, najm}@eecg.utoronto.ca

2. Problem • Timing verification is a crucial step • More pronounced in current technologies • Types of variations • Process variations are random statistical variations • Environmental variations are uncertain variations that are non-statistical • … cause circuit delay variations! • Parameterized Timing Analysis (PTA) • Delay is “parameterized” as a function of variations • Propagated in the timing graph to determine arrival times • Circuit delay becomes parameterized • Useful information: sensitivities, margins, distributions, yield

3. Previous Work • Statistical Static Timing Analysis (SSTA) • One type of PTA • Parameters are random variables withknown distributions • Gaussian?? • Different delay models • Linear, quadratic… • Different correlation models • Grid/Quad-tree, Principal Component Analysis (PCA) • Limitations: • Can not handle uncertain variables, i.e. nonstatistical variables • Some have difficulty in handling the Maxoperation efficiently • In nonlinear/non-Gaussian case

4. Previous Work • Multi-Corner Static Timing Analysis (MCSTA) • Is another type of PTA • Get a conservative bound on maximum(worst case) corner delay • Delay is parameterized using affine (linear) functions • Hyperplanes • Parameters can be random variables and/oruncertain variables • Limitations • Linear delay models • Does not follow well the spread of the circuit delay • Accuracy guaranteed only at the maximum corner delay • Sensitivities are not captured well

5. Our Approach • Propose a Parameterized Timing Analysis technique • Random parameters with arbitrary distributions • Uncertain non-statistical parameters • General class of delay models • Linear in circuit size (for linear and quadratic models) • Propose two methods to resolve the MAX operation • Using guaranteed upper/lower bounds • Using an approximation that minimizes the square of the error • Both methods preserve the nonlinearities of the delay model • Propose two applications: • MCSTA with linear/nonlinear models • SSTA with nonlinear models, random & uncertain variables

6. General Delay Models • To represent timing quantities, we will use a general class of delay models F • This class of nonlinear functions F has the following three properties: • F is closed under linear (and/or affine) operations • All functions in F are bounded • All functions in F can be maximized andminimized efficiently

7. General Delay Models – Cont’d • Property 1 • Property 2 • Property 3 • Guarantees overall efficiency of approach

8. Propagation • To propagate arrival times in the timing graph • SUM operation • MAX operation • SUM can be performed • By Property 1 of F • MAX is nonlinear • Bound the MAX using functions in F • Approximate the MAX using functions in F

9. MAX Operation • Let C = max(A,B) be the maximum of A and B and assume that A, B belong to F • C does not necessarily belong to F • We want to find

10. MAX Linear or Nonlinear?? • The nonlinearity of the MAX depends on the difference D, between A and B • Note that and that • MAX is linear when • Dmin ≥ 0 that is A dominates B  C = A • Dmax ≤ 0 that is B dominates A  C = B • MAX is nonlinear when Dmax ≥ 0 and Dmin ≤ 0

11. Bounding the MAX • C = B + max(D,0) and Dmax ≥ 0, Dmin ≤ 0 • Let Y = max(D,0) • Y does not belong to F since MAX is nonlinear

12. MAX Upper Bound • Yu is the best ceiling on Y and is exact at the extremes • Since Yu is a linear function of D, then

13. MAX Upper Bound – Cont’d • Since C = B + Y, then • Where

14. MAX Lower Bound

15. MAX Lower Bound – Cont’d • Lower bound on Y • Lower bound on C

16. MAX Approximation • Y = max(D,0) • Minimize:

17. MAX Approximation – Cont’d • Take the partial derivatives with respect to and • Set them to zero and solve for the variables • Simple expressions in Dmax and Dmin

18. Summary • Given a general class of nonlinear functions F with certain properties • If timing quantities • Then propagation (SUM & MAX) can be done while maintaining the same delay model • Bounds • LS Approximation • The MAX is “linearized” • Coefficients are simple functions of Dmin and Dmax • Independent of whether variables are random or uncertain • Distribution independent

19. Application 1 • Traditional STA • Need to check circuit timing at all process corners • Exponential number of runs • Multi-corner STA • Parameterize delay as a function of process/environmental parameters • Propagate once to get the maximum delay(also parameterized) • Determine the maximum/minimum cornerdelays efficiently • Apply our framework to MCSTA with linear/quadratic models

20. Linear/quadratic models • Timing quantities are expressed as follows: • Show that all properties hold • Linear/quadratic models survive linear(affine) operations • Bounded since -1 ≤ Xi ≤ 1 • Maximized efficiently (show in paper how this is done)

21. Results • 90nm library and following process parameters: • Vtn, Vtp, Ln, Lp • Characterized library to get delay sensitivities • Used ISCAS’85 circuits • Maximum delay at the maximum/minimum corners are computed using exhaustive STA • Maximum/minimum corner delays are determined using our approach (Bounds and LS-approximation) • Average errors:

22. Application 2 • SSTA with quadratic delay models • random parameters with arbitrary distributions (Gaussian, uniform, triangular, etc…) • uncertain non-random parameters varying inspecified ranges • Delay model:

23. The Three properties… • Surviving addition: • Bounded & can be maximized and minimized • The maximum and minimum of a quadratic function depends on whether the vertex is within the range or not (explained in the paper)

24. Results • In addition to Xr we use four global variables Xi • Truncated Gaussian, Uniform, and Triangular • 10%-20% deviation innominal delay • Compared our LS approach to Monte Carlo • Metrics: 95%-tile, 99%-tile, σ/μ • Avg error very small < 1%

25. CDF Comparison

26. Conclusion • Proposed the first Parameterized Timing Analysis technique • Random parameters with arbitrary distributions • Gaussian, uniform, triangular, etc… • Uncertain non-statistical parameters • Variables in ranges • General delay models (some restrictions) • Linear, quadratic, other… • Simple and accurate technique • Applied our framework to • Multi-corner STA with linear and quadratic models • Nonlinear (quadratic) SSTA with arbitrary distributions