Spread Sheet Studies on the Impact of Variability Chandu Visweswariah IBM Thomas J. Watson Research Center Yorktown Heights, NY http://www.research.ibm.com/people/c/chandu © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 1 of 23 Increasing and inevitable variability 249,403,263 Si atoms, 68,743 donors, 13,042 acceptors* *D. J. Frank et al, Symp. VLSI Tech., 1999 © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 2 of 23 Foregone conclusions • Variability is important • Variability is getting proportionately worse • Exhaustive corner checking is tedious and overly conservative • Statistical timing can help: – reduce pessimism – cover process space in a single timing run – provide diagnostics to target robust design • Statistical timing must take correlations into account – global/systematic – spatial – independently random • For optimization purposes, statistical timing must be incremental • Methodologies will evolve and change profoundly © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 3 of 23 This talk • Spread sheet studies – corner-based vs. statistical timing – impact of n equally critical paths (“slack wall” problem) – benefit of separating out systematic from independent variability • penalty for n equally critical paths – impact of path depth • Rethinking of conventional wisdom © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 4 of 23 3σ BEOL Check front-end corners: possible escapes 2σ 1σ -3σ -2σ -1σ 1σ 2σ 3σ FEOL -1σ 1 GHz -2σ 900 MHz 800 MHz -3σ © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 5 of 23 3σ BEOL Check all corners: no escapes, pessimistic 2σ 1σ -3σ -2σ -1σ 1σ 2σ 3σ FEOL -1σ 1 GHz -2σ 900 MHz 800 MHz -3σ © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 600 MH6zof 23 3σ BEOL Statistical timing: no escapes, less pessimism 2σ 1σ -3σ -2σ -1σ 1σ 2σ 3σ FEOL 1 GHz -2σ 900 MHz 800 MHz -3σ © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 700 MHz -1σ 600 MH7zof 23 1. Corner-based vs. statistical timing • n = # independent sources of variation (say 9) • σ = total variability in critical path delay (say 5%) • Fractional increase in frequency with a 3σ sign-off instead of 3σ√n sign-off • Assumes sources of variation are roughly equally significant © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 8 of 23 80 Corner-based vs. statistical 8 70 % frequency difference 7 9 60 10 50 11 12 40 13 30 14 15 20 16 10 17 0 18 1% 2% 3% 4% 5% 6% 7% 8% Sigma as fraction of cycle time © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 9% 10% 19 20 9 of 23 0.8 Probability 1 2. Equally critical signals (ρ=0) 0.6 1 30 2 3 0.4 0.2 0 -3.0 Delay -2.0 © Chandu Visweswariah, 2005 -1.0 0.0 1.0 2.0 Spread Sheet Studies on the Impact of Variability 3.0 10 of 23 0.8 Probability 1 Equally critical signals (ρ=0.5) 0.6 1 30 2 3 0.4 0.2 0 -3.0 Delay -2.0 © Chandu Visweswariah, 2005 -1.0 0.0 1.0 2.0 Spread Sheet Studies on the Impact of Variability 3.0 11 of 23 0.8 Probability 1 Equally critical signals (ρ=1.0) 0.6 1 30 2 3 0.4 0.2 0 -3.0 Delay -2.0 © Chandu Visweswariah, 2005 -1.0 0.0 1.0 2.0 Spread Sheet Studies on the Impact of Variability 3.0 12 of 23 0.8 Probability 1 Thirty equally critical signals 0.6 ρ=1 ρ=0.5 ρ=0 0.4 0.2 0 -3.0 Delay -2.0 © Chandu Visweswariah, 2005 -1.0 0.0 1.0 2.0 Spread Sheet Studies on the Impact of Variability 3.0 13 of 23 u d e n u t n unc awa ertai re t ntyune d tuned #paths Slack histogram +20 ps © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability slack 14 of 23 2. Separating out independent randomness 0.6 New: systematic variability N(20,2/3) 0.5 0.4 Old: N(20,1) 0.3 New: independent variability N(0,1/3) 0.2 0.1 0 17 © Chandu Visweswariah, 2005 18 19 20 21 Spread Sheet Studies on the Impact of Variability 22 15 of 23 Case study • Consider a critical path of 50 identical gates • Old: assume delay of each gate is N(20,1) ps (corner delays are 17 and 23 ps) • New: assume delay of each gate is N(20,2/3) ps + N(0,1/3) ps (same corner delays) • Old: critical path delay (3σ) = 23×50 = 1150 ps • New: critical path delay (3σ) = 22×50 + 3×1/3×√50 = 1100 + 7.1 = 1107.1 • Improvement in critical path delay = 3.7% • This case study can be generalized © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 16 of 23 Generalization of case study • As N , the benefit • As v , the benefit • As f , the benefit © Chandu Visweswariah, 2005 Rule of thumb: for 50 stages and 5% variability (σ), each percent of independent variability buys 0.1% of critical path delay improvement Spread Sheet Studies on the Impact of Variability 17 of 23 Plot of benefit for N=50 v=5% v=10% v=15% v=20% v=25% v=0% v=30% 40 35 % benefit 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 © Chandu Visweswariah, 2005 f = % independent variability Spread Sheet Studies on the Impact of Variability 18 of 23 What about shorter paths (σ=5%)? 10 9 % path delay change 8 7 6 5 4 3 2 1 0 8 10 12 14 16 18 20 22 24 26 N = # stages of logic © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 28 30 f=0% f=5% f=10% f=15% f=20% f=25% f=30% f=35% f=40% f=45% f=50% f=55% f=60% f=65% f=70% f=75% f=80% f=85% f=90% 19 of 23 M equally critical paths • Basic issue – suppose there are M equally critical paths – each of these paths has already received RMS credit, so the delay of each path consists of • a constant which is the nominal/intrinsic delay of the path plus the corner-based systematic variability • an independent variability part for which we have received an RMS credit, so there is a small probability that the delay is beyond the 3σ limit – with a large number of equally critical paths, the 3σ of the MAX delay of all M paths is not equal to the 3σ delay of each of the M paths – question: how big is this sigma shift? © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 20 of 23 A little analysis • Example: with M=50, we have to use a 4.037σ value on the random part instead of 3σ to get a “true 3σ” delay on the maximum delay of 50 paths; this diminishes RMS credit • The benefit after taking this into account is plotted in general versus M and N on the next page, assuming f=1/3, v=5% © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 21 of 23 Benefit of RMS credit + equally crit. paths N=8 3.6 N=10 N=12 N=14 N=16 3.4 Percent benefit 3.2 N=18 N=20 N=22 3 2.8 N=24 N=26 N=28 N=30 2.6 2.4 N=32 N=34 N=36 2.2 Equally critical paths © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 990 920 850 780 710 640 570 500 430 360 290 220 150 80 1.8 10 2 N=38 N=40 N=42 N=44 N=46 N=48 22 of 23 Conventional wisdom revisited • Conventional wisdom says, “Use sizing and multiple Vts to tune the circuit aggressively, creating a wall of critical paths – the new wisdom is to optimize the expected value of the critical path delay, which in turn means reducing the wall of critical paths • Conventional wisdom says, “Make pipeline stages short and crank up clock frequency” – the new wisdom is to take advantage of RMS/ effects in moderately longer pipelines © Chandu Visweswariah, 2005 Spread Sheet Studies on the Impact of Variability 23 of 23