Stochastic Analysis of Bubble Razor Guowei Zhang Peter A. Beerel - - PowerPoint PPT Presentation

Stochastic Analysis of Bubble Razor

Guowei Zhang

Department of Microelectronics and Nanoelectronics Tsinghua University, China

Peter A. Beerel

Ming Hsieh Electrical Engineering Department

• Univ. of Southern California, USA

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Notable delay variations in IC

– Process, temperature, and voltage variations – Aging effects

• Traditional synchronous design

– Too much timing margin – Performance and energy loss

• Existing resilient designs – detect errors & change freq. or voltage

– Canary circuits [Nakai 2005, Hirair 2012]

• Delay chains mimic critical path

– Razor I & II & Lite [Ernst 2003, Park 2013, Kim 2013, Tokunaga 2009]

• In situ error detection and correction
• Recover via architectural replay

Why Bubble Razor?

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Problem: Adoption challenge

– Detection: Necessary to analyze inserted error signals – Correction: Necessary to implement replay

• Bubble Razor [Fojtik et al., ISSCC 2012] [Fojtik et al.,JSSC 2013]

– Architecture independent – Latch-based structure

• Latches don’t change together
• No architectural replay requirement

– Local correction mechanism

• Single stall recovery
• Suitable for large circuits (no global control requirement)

Why Bubble Razor?

26-Mar-14 3

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Factors effecting performance unclear

– # of stages in pipeline – Delay variance – Probability of a timing error

• Thus, effectiveness and sensitivities not quantified
• Our proposed solution

– Analyze performance using Markov Chains

• Focus on an N-stage pipeline ring
• Consider both normal and log-normal delays

– Propose simplified model for other pipeline structures

Our focus – Analysis of Bubble Razor

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Mechanism

– 2-stage Bubble Razor Ring

Bubble Razor

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Clock Cycle Time (C)

– Cycle time of global clock

• Effective Clock Cycle Time (EC)

– Average time to process each instruction – EC > C because of pipelines stalls

• Performance of Bubble Razor

– Assume we can somehow find π(working)

• Probability of a latch processing an instruction

– Then, EC = C / π(working)

Quantifying Performance of Bubble Razor

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Examples No timing violations Timing violation every instruction

• π(working) = 1
• EC = C
• π(working) = 1/2
• EC = 2C

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Circuit State

– If modeled correctly, next circuit state depends on current state and probability of error p – System is then a Markov Chain

• Our approach

– Define circuit state as combination of latch states

Markov Chain Analysis

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Description of all latch states

Working A timing violation is detected? No W Yes E Stalling To whom this latch sends bubbles? Neither N Right Neighbor (RN) R Left Neighbor (LN) L Both B

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Circuit state transition rule

– Easily derived from latch state transition rule

• Latch state transition rule
• Leads to stationary distribution of circuit states, and thus

latch states [ π(working) = π(W) + π(E) ] – Expressed as a function of p and N

Markov Chain Analysis

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LN RN Next State W L, B L E L (annihilation) B, R N (annihilation) B, R W, E, N, R R E B N, L W W W with probability of (1-p) E with probability of p

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

Markov Chain Analysis Results

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• Closed-form formulas

0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 N = 1 MC Model N = 2 MC Model N = 3 MC Model N = 4 MC Model

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Transition Probability Matrix (T)

– Product of 2 transition matrices

• Each represents one clock phase

– Can be reduced

• The problem

– Not feasible for large N

Markov Chain– State Explosion Problem

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N Theoretical size = 6 ^ (2N) Optimized size (impossible states deleted) Final size (unreachable states deleted) 1 36 7 5 2 1,296 45 21 3 46,656 301 95 4 1,679,616 2017 449 Size of T

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Stalling is related to all other latches

– But, can assume they are independent – 2N independent possible sources of a stall

• Thus

– Probability (Stalling) = 1 - ( 1 - p ) ^ 2N – EC = C + C * Probability (Stalling) = C [ 2 – ( 1 – p ) ^ 2N ]

Our Solution: Simplified Analysis

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• EC ~ C, N, p

– Simplified model is conservative – Results for small p are close

Markov Chain versus Simplified Models

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0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 N = 1 MC Model N = 1 Simplified Model N = 4 MC Model N = 4 Simplified Model

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Delay of pipeline stage (d) is a random variable

– p = Probability (d > C/2) – Two kinds of delay distribution

• Normal distribution
• Log-normal distribution [Zhai 2005, Chandrakasan 2005]

– μ: mean of a delay – σ: standard deviation of a delay

• Final results

– EC is a function of

• C:

Clock cycle time

• N:

Number of stages

• σ/μ: represents delay variance

Delay Distribution

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• How do N, σ/μ, distribution type (normal / log-normal)

and model type (Markov Chain / Simplified) influence EC? – EC ~ C, N, σ/μ

• Is Bubble Razor better than traditional counterparts?

– Condition

• Constrained with same Systematic Error Rate

– For example, 0.1% – Performance metric

• EC(Bubble Razor) / EC(Traditional circuit)
• Set μ = 0.5

Systematic Error Rate

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Vary pipeline depth N

– Set σ/μ = 0.4 – Assume Log-normal distribution

• Results

– BR always better than sync – As N↑, benefits drop slightly (2%)

Performance Analysis Results (1 of 3)

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0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 4 N = 1 N = 2 N = 3 N = 4

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Vary σ/μ

– Assume N = 4 – Assume log-normal distribution

• Results

– σ/μ↑, benefit improves – For σ/μ = 0.5, EC is reduced by 40.2%

Performance Analysis Results (2 of 3)

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0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5

σ/µ = 20% σ/µ = 30% σ/µ = 40% σ/µ = 50%

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Vary distribution and

model – Assume N = 4 – Assume σ/μ = 0.25

• Results

– Distribution type (line color) impacts benefits – Simplified model tracks well ( < 5% )

• p is usually < 25%

Performance Analysis Results (3 of 3)

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0.5 1 1.5 2 0.5 1 1.5 2 2.5 Log-Normal, MC Model Log-Normal, Simplified Model Normal, MC Model Normal, Simplified Model

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Point A

– Constrained by SER – Easy to realize

• Increase f until PoFF
• Point B

– Local Minimum Effective Cycle (EC) time

Optimizing Strategy

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0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5

σ/µ = 20% σ/µ = 30% σ/µ = 40% σ/µ = 50%

Better than traditional sync Optimal (for every N) Point A σ/μ ≥ 16% (moderate variance) σ/μ ≥ 31% (high variance) Point B σ/μ ≥ 3% σ/μ ≤ 28%

Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

• Proposed Analytical Methods for Analyzing Bubble Razor

– Markov Chain & Simplified Model – The latter is a conservative and a close approximation

• Bubble Razor indeed has better performance than

traditional circuits, especially under high delay variance

• Setting clock cycle time as short as possible is often

efficient

Summary and Conclusions

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Guowei Zhang, Tsinghua University and Peter A. Beerel, USC

Thank you!

26-Mar-14 20

Guowei Zhang

Department of Microelectronics and Nanoelectronics Tsinghua University, China

Peter A. Beerel

Ming Hsieh Electrical Engineering Department

• Univ. of Southern California, USA