Synthesizing a Representative Critical Path for Post-Silicon Delay Prediction

Qunzeng Liu and Sachin S. SapatnekarDepartment of Electrical and Computer Engineering

University of Minnesota

Variations in Digital Circuits• Variations in nanoscale

technologies

– Process: across-die/within-die

– Environment: T, Vdd

• These lead to circuit performance deviations

• Timing PDF

• Power PDF

2

Poly

DiffusionS. Tyagi

Normalized LeakageNor

mal

ized

Fr

eque

ncy

1 2 3 4 50.9

1.0

1.1

1.2

1.3

1.4

30%

5X

Inte

l

Design Phase: Pre-Silicon

3

Pre-Silicon Optimization

Deterministic/StatisticalTiming/Power Analysis

…

SynthesisGate Sizing

…

Statistical Timing Analysis result

Statistical Power Analysis result

Pre-Silicon Analysis

Statistical Static Timing Analysis (SSTA)

• Spatial correlation

• Canonical form

4

( )I0ppa ,~ NRd cT

cc ++= μ

1 2

3

Design Phase: Post-Silicon

5

Post-Silicon Optimization

• Delay Analysis,

• Design-Silicon Correlation,

…

• Adaptive Body Bias(ABB),

• Adaptive Voltage Scaling (AVS),

…

Post-Silicon Analysis

[amd.com]

Adaptive Body Bias (ABB)

6

[Tschanz et al., JSSC02]

Limitations of Critical Path Replica (CPR)

7

•Representative Critical Path (RCP)•Always predicts the worst case delay

• Only the nominal critical path is replicated

• Numerous near-critical paths in modern VLSI circuits

• Nominal critical path not necessarily critical in the manufactured die (process variations)

Outline

8

Problem Formulation

9

Circuit with Gaussian process parameter variations

Original Circuit

RCP

Build the RCP to reveal most information about the original circuit.

Representative Critical Path (RCP) should be related to the original circuit

cd

pd

Mathematical Formulation

10

From SSTA (Pre-Silicon)

From measurement data(Post-Silicon)

Our goal:

ρ MinimumMaximum

Conditional PDF

Full Correlated Case

11

Fully Correlated

kba

ba

ba

n

n ==== Λ2

2

1

1( )ppcc dkd μμ −=−

Outline

12

Representative Critical Path (RCP) Synthesis (Method I)

13

Maximum improvement

Maximum improvement

Comments on Method I

• Advantage– Guaranteed to do no worse than CPR– Exact solution when there is clearly one dominating path

• Drawback– Flexibility of the solution is limited

• Runtime: O(Ks)– Saved by only updating the SSTA results of stages

adjacent to the one sized up– K: number of iterations– s:number of stages

14

RCP Generation (Method II)

15

1ρ 0ρ2ρ1−sρsρMax

Comments on Method II

• Advantage– More flexibility, not tied to a specific path

• Drawbacks– No exact solution when there is only one dominating path– Not guaranteed to be always better than CPR

• Runtime: O(kcs)– k: number of starting locations– c: number of choices for each iteration– s: maximum number of stages

16

Outline

17

Comparison Metric

18

• Average error, maximum error w.s.t. Monte-Carlo analysis

Guard band

• Guard band

Experimental Results (Method I)

19

Scatter Plots (Method I)

20

Experimental Results (Method II)

21

Number of Paths vs. Delay

22

0 500 10000

20

40

60

80

100

delay (ps)

num

ber o

f pat

hs

# Paths vs. delay for delay optimized s13207

0 200 400 600 8000

5

10

15

20

25

30

delay (ps)

num

ber o

f pat

hs

# Paths vs. Delay for delay optimized s9234

Conclusion and Future Work

• Two novel methods for synthesizing a representative critical path under process variations are presented

• Average prediction error: below 2.8%

• To ensure 99% of the predictions pessimistic, requiring guard band 30% smaller than CPR

• Future work– Test on real silicon

23

Thank you!

24

Extra Slides

25

Scatter Plots (Method II)

26

250 300 350 400 450 500250

300

350

400

450

500

true delay (ps)

pred

icte

d de

lay

(ps)

s35932 by Critical Path Replica

250 300 350 400 450 500250

300

350

400

450

500

true delay (ps)

pred

icte

d de

lay

(ps)

s35932 by Method II

An Example RCP (Method II)

0 5900

590

x direction

y di

rect

ion

RCP for s38417

27

ρ vs. iteration number (Method II)

28

0 10 20 30 40 500.7

0.75

0.8

0.85

0.9

0.95

1

Iteration

Cor

rela

tion

coef

ficie

ntCorrelation coefficient trend for s38417

Equations for Editing

29

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡2

2

,~p

T

Tc

p

c

p

c Ndd

σσ

μμ

baba

cT

cc Rd ++= paμ22

cRT

c σσ += aa pT

pp Rd ++= pbμ

22pR

Tp σσ += bb cd prp dd = ( ) ( )2,~| σμNddd prpc =

( )pprp

T

c d μσ

μμ −+=ba

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

pc

T

c σσσσ ba122 p

Tpp Rd ++= pb3μ

21

21

1

pRc

T

σσρ

+=

bbba

Parameter Variations

30

Use the Grid-Based Correlation Model ([Chang, ICCAD03])

Parameter Variations

σ