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

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

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

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

6. 6 General Delay Models n To represent timing quantities, we will use a general class of delay models F n This class of nonlinear functions F has the following three properties: 1. F is closed under linear (and/or affine) operations 2. All functions in F are bounded 3. All functions in F can be maximized and minimized efficiently

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

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

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

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

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

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

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

14. 14 MAX Lower Bound

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

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

17. 17 MAX Approximation – Cont’d n Take the partial derivatives with respect to and l Set them to zero and solve for the variables l Simple expressions in D max and D min

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

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

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

21. 21 Results n 90nm library and following process parameters: l Vtn, Vtp, Ln, Lp n Characterized library to get delay sensitivities n Used ISCAS’85 circuits 1. Maximum delay at the maximum/minimum corners are computed using exhaustive STA 2. Maximum/minimum corner delays are determined using our approach (Bounds and LS-approximation) n Average errors:

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

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

24. 24 Results n In addition to X r we use four global variables X i l Truncated Gaussian, Uniform, and Triangular l 10%-20% deviation in nominal delay n Compared our LS approach to Monte Carlo l Metrics: 95%-tile, 99%-tile, σ/μ n Avg error very small < 1%

25. 25 CDF Comparison

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