Toward PDN Resource Estimation: A Law of General Power Density

1. Toward PDN Resource Estimation: A Law of General Power Density Kwangok Jeong and Andrew B. Kahng (kjeong@ucsd.edu, abk@ucsd.edu) VLSI CAD Laboratory University of California San Diego

2. Outline • Power and technology trend • Early stage PDN estimation • Power law of power density • Data collection: random activity distribution • Model 1: empirical activity density model • Model 2: analytical power density model • Toward PDN resource estimation • Conclusion

3. Power and Technology Trends • Power trend and required impedance scaling • Frequency increases • Power increases • Vdd decreases • Current increases • Required impedance decreases ITRS 2010 MPU

4. Need for Early PDN Estimation and Synthesis • When we have enough information… • Placement information • Power consumption of gates • Stimuli for realistic worst operation • PDN can be designed with detailed information • At the early design stage, when we do not have enough information… • Expected (average/worst) power of a functional block • Rough estimate of area of the functional block • Still need to estimate PDN resources toplan interconnect stack, chip size, bump pitch, etc.considering routing congestion increase due to PDN

5. PDN Design at the Early Design Stage • Traditional procedure • Decide mesh pitch (and/or width)  Assign current sources  Refine mesh width • Current source distribution • Given a current consumption (Iexpected) • Evenly distributed sources • A single source at center Iexpected / N N: #nodes What is realisticcurrent distribution? Iexpected Too pessimistic! Too optimistic!

6. Trend of Power Density • Exponential dependency of power density to area • What is the general property of the activity density? 1.4e+3 1.2e+3 1.0e+3 0.8e+3 Current density (A/cm2) 0.6e+3 0.4e+3 AF=7.5% 0.2e+3 0.0e+0 0 5 10 15 20 25 30 35 40 45 50 Device level Block level Chip level unit area count (J)

7. Activity Density • Activity density = activity count within a window • Correlated with power density • What is the maximum activity density? Die (N x N) Activity density changes with respect to window size and locations Activity density changes with respect to time Sampling window (m x m) Activity at t=T Activity at t=3 Activity at t=2 Activity at t=1

8. Normalized Maximum Activity Density • Artificial activity distribution • Given NN grids and average activity factor p, • Randomly assign activity 1 to pNN grids out of NN grids • Maximum activity density for m = 1 to N c(m)  maximum #activity enclosed by mm area d(m)  c(m) / m2 for m = 1 to N dnorm(m) d(m) / d(N) • Activity density is not a constant N= 5, p=0.2 c(1) = 1  d(1) = 1.00  dnorm(1) = 5.00 c(2) = 2  d(2) = 0.50  dnorm(2) = 2.50 c(3) = 3  d(3) = 0.33  dnorm(3) = 1.32 c(4) = 4  d(4) = 0.25  dnorm(4) = 1.25 c(5) = 5  d(5) = 0.20  dnorm(5) = 1.00

9. Normalized Power Density in a Real Design • Testcase: sparc_exu_alu (in OpenSparcT1) • #instance: 3k • Core size: 130um x 130um • Power estimation flow • Synthesis: Synopsys Design Compiler • Place&route: Cadence SoC Encounter • System-level simulation: VirtuTech SIMICS • Gate-level simulation: Cadence NC Verilog • Power estimation: Synopsys PrimeTime-PX • Among 1M cycles, top-1000 high power cycles are collected • Power assignment • Divide core area into 100 x100 grids • Assign the power of each cell to the corresponding grid, based on the cell location Linear in a log-log plot  Existence of power law dnorm(m) m (a) Single cycle dnorm(m) m (b) Multiple cycles (for w-timeframes)

10. Activity Count Calculation • Massive data collection for activity count statistics using a dynamic programming approach for x = 1 to N for y = 1 to N CA[ x ][ y ][ 1 ] = CountInBox1(x,y) for m = 2 to N for x = 1 to N for y = 1 to N if ( m == even ) lb  CA[ x ][ y ][ m/2 ] lt  CA[ x ][ y+m/2 ][ m/2 ] rb  CA[ x+m/2 ][ y ][ m/2 ] rt  CA[ x+m/2 ][ y+m/2 ][ m/2 ] CA[ x ][ y ][ m ]  lb + lt + rb + rt else lb  CA[ x ][ y ][ m/2 ] lt  CA[ x ][ y+m/2 ][ m/2+1 ] rb  CA[ x+m/2 ][ y ][ m/2+1 ] rt  CA[ x+m/2+1][ y+m/2+1 ][ m/2 ] cn  CA[ x+m/2 ][ y+m/2 ][ 1 ] CA[x][y][m]  lb + lt + rb + rt - cn • Parameters to change: • Switching activity (p): 0.05, 0.10, 0.15, 0.25, 0.50, 0.90 • Trial (k): 50 • #Timeframes (w): 1, 2, 4, 8, 32 • Window size (m): 1, 2, …, 100 m = 3 (x,y) CA[x][y][3] = 1 lt rt rb lb lt rt rb lb cn

11. Activity Density from Random Distributions • Single timeframe • Multiple timeframes P = 0.05 Normalized activity density: dnorm(m) Log (dnorm(m)) Log (m) Sample window size: m P = 0.90 • Summary of characteristics • Exponential decay with m • m=1 and w = 1  dnorm(w) = 1/p • Large m  dnorm(w) = 1 • Large w  small dnorm(w) • Large p  small impact of w • Small p  large impact of w Log (dnorm(m)) Log (m)

12. Empirical Activity Density Model • Model function is determined to capture the characteristics of the observed activity density • Model coefficients are found using fitting with measured data • Summary of characteristics • Exponential decay with m • m=1 and w = 1  dnorm(w) = 1/p • Large m  dnorm(w) = 1 • Large w  small dnorm(w) • Large p  small impact of w • Small p  large impact of w For small p (~ 0), impact of w is large, andfor large p (~1), impact of w is small Exponential decay due to m Large m  dnorm(m) = 1 Slow decay due to w m = 1 and w =1  dnorm(m) = 1/p

13. Model Validation – Empirical Model dnorm(m) • Model accuracy for all data points • Average error: 6.14%, Maximum error: 40.15% dnorm(m) m m dnorm(m) dnorm(m) m dnorm(m) m dnorm(m) m dnorm(m) m m

14. Analytical Modeling from Chernoff Bound • Fact 1 (Chernoff Bound): • Let x1,…,xn be mutually independent 0/1 random variables, each equal to 1 with probability p. • If for any 0 <  < 1, then • Let sij be a 0/1 random variable denoting activity at (i,j) • We find maximum activity count in rectangles of size m  m whose top-right corner is (u,v) •  (of in Chernoff bound) = pm2 ,

15. Analytical Activity Count Model • From Chernoff bound, maximum activity count is bounded as: • For a single timeframe: • For a multiple timeframes: with probability at least 1 -  with probability at least 1 - 

16. Model Validation – Analytical Model • Model • We find model coefficient  for various  values • Validation using all data points used in empirical model construction • For all  values,  is near 1  tight estimate • When cases m < 10 are removed, avg (max) error  3.17%~ 4.88% (23.77%~41.60%)

17. Preliminary Results for PDN Design • IR-drop comparison for a fixed power mesh • 1mm2 region consuming 100mA • Mesh width = 2um, mesh pitch = 10um • PDN resource comparison for a fixed 5% voltage drop • 1mm2 region consuming 100mA • Mesh pitch: 10um Center Uniform Power law (p=0.001) I = 100mA 100mA (100x100) I = Max drop: 73.0mV Max drop: 31.7mV Max drop: 37.4mV

18. Conclusions and Ongoing Works • We have presented a general law for power density • We provide closed-form activity density models from empirical data analysis and probability theory • These can be used to improve the accuracy and efficiency of early-stage PDN resource prediction • Our ongoing work • Further simplification of models • Validation against large industry designs • Development of fast and accurate PDN design and optimization methodologies for early stages of IC design

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21. Power Estimation Flow power report (dynamic & leakage power) benchmark binary - bzip, equak, ... (SPEC-2000) benchmark analysis SimPoint power estimation PrimeTime-PX reprehensive execution phase switching activity (VCD) wire load (SPEF) system-level simulation Simics + Transplant functional simulation NC-Verilog input patterns (stimuli) gate-level netlist delay info. (SDF) RTL design OpenSPARC T1 design implementation DC, SOCE