System Level Power And Thermal Modeling and Analysis by Orthogonal Polynomial based Response Surface

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System Level Power And Thermal Modeling and
Analysis by Orthogonal Polynomial based Response
Surface Approach (OPRS)
Authors Janet M. Wang, Bharat Srinivas,
Dongsheng Ma1 Charlie Chung Ping Chen2 and Jun
Li3 Speaker Vineet Agarwal1
1 University of Arizona, Tucson. AZ 2 University
of Wisconsin, Madison, WI 3 Self-employed, Santa
Clara. CA
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Outline

• Orthogonal Polynomial Response Surface method
  (OPRS).
• Recursive OPRS Framework.
• Leakage Power estimation using OPRS.
• Applications
• Experimental Results.
• Conclusion

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Advantages of OPRS

• Complexity doesnt increase dramatically with
  number of variables.
• Gives us a simple equation for the estimation of
  leakage power with changes in all variation
  sources (Vdd, Vth,W,L,..)
• Recursive application of OPRS gives us the
  system-level leakage power with gate-level
  resolution.
• Works equally well with deterministic and
  statistical variables.
• NO additional information about the topology of
  gates or functional blocks required.

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An Introduction of OPRS
Black Box (Gate, Block, System)
Parameters
performance
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Deterministic OPRS Method Traditional approach
(1)

• Consider a simple model
• The performance is approximated as
• This is called a Nth Order model

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Deterministic OPRS Method Traditional approach
(2)

• Residual Error
• Aim To minimize residual error
• 4 different ways of minimizing this error
• Collocation Method
• Sub-domain Method
• Method of Moments (or Galerkin)
• Least Square Method

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Deterministic OPRS Method Traditional approach
(3)

• Least Square Method
• Heavily used
• Accurate Results
• Sample point affect accuracy
• Sub-domain and Moments method
• Most Accurate
• Efficiency issues
• Collocation Method
• Most Efficient
• Least Accurate
• Orthogonal Polynomial based Collocation Method
• As accurate as sub-domain and moments method
• Good Efficiency

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Deterministic OPRS Method (1)

• The performance is approximated as
• where are orthogonal polynomials
• N1 is order of approximation

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Deterministic OPRS Method (2)

• The coefficients ai can be calculated by
  requiring that residual and each member of
  should be
  orthogonal
• Gaussian Approximation

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Deterministic OPRS Method (3)

• If vj and gi(x) are of same sign and are not
  zero, then Ri0
• Ri is minimum
• OPRS doesnt need complete information about
  residual function
• The sampling points are zeros of next higher
  order orthogonal polynomial, than the order of
  approximation
• Lets denote these zeros by

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Solving Coefficients for OPRS

• Given and with known the above
  equation becomes a linear equation with unknown
  coefficients

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Deterministic OPRS Method (5)

• To solve this linear equation we require N1
  points. (Note number of sample points decided by
  order of approximation)
• Complexity doesnt increase dramatically with
  variables

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Statistical OPRS Method

• If ? is a random variable then the orthogonal
  relationship is transformed to probabilistic
  space using JPDF
• At collocation points

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How to choose Collocation Points (2)

• The total number of collocation points for Nth
  order orthogonal polynomial with q number of
  variables is (N1)q.
• No. of required collocation points (Nc) No. of
  unknowns in the Polynomial approximation.
• Nc is always greater than the No. of unknowns in
  the system.
• The collocation points are selected from the
  roots and the highest probability region.

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How to Choose Collocation Points (1) - for Normal
distribution
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Leakage Power estimation with DOPRS SOPRS

• Leakage Power is estimated as
• Leakage Power is modeled using
• Use OPRS to find the function f

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DOPRS based Power Model

• Gate-parameters - Vdd, Vth, Temperature (T)
• Consider a Legendre Polynomial
• The analytical equation for leakage power is
• where is a vector of input variables such
  as Vdd, Vth, T

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SOPRS Model Power Model

• Critical Length (L) and Gate-oxide thickness
  (tox) process variation parameters, assuming a
  Gaussian distribution
• Assuming a Hermite Polynomial for finding the
  leakage power with Process Variations
• The Leakage power with process variations is
  modeled as

random variable vector is L N(µ1,s1) lt-
N(0,1) and tox N(µ2,s2) lt- N(0,1)
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Transformation of N(0,1) to other distributions
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Recursive OPRS Framework

• Analytical OPRS model built for gate library
  gives the Component level estimation and there by
  System level estimation of the variable.
• The system level estimate of power is in terms of
  gate level variables such as L, tox, W,
  Temperature.
• When any of the variable changes the system level
  estimate can be predicted without full chip
  analysis.
• Recursive pseudo-code

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Application Joint Power Optimization Technique
W. Hung, Y. Xie, N. Vijayakrishnan, M. Kandemir,
M.J. irwin and Y. Tsai, Total Power
Optimization Through Simultaneously Multiple Vdd,
Multiple Vth Assignment and Device Sizing with
Stack Forcing, in ISLPED 2004
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Application Power Management of Network-on-Chip
The total energy for a work load of ? is
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Results for DOPRS modeling
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Comparison of GA based DOPRS Approach HSPICE
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Comparison of Mean and Std. Deviation
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Comparison of SOPRS HSPICE
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PDFs of SOPRS and Monte Carlo Analysis
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Power divisions for power optimized Circuits
without considering process variations
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Conclusion

• A new Orthogonal Polynomial based Response
  Surface Method to provide back-end Leakage power
  and thermal estimation
• A novel recursive strategy to create analytical
  power and thermal equations for system level
• Applications for NOC temperature control and
  power optimization.

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Thank you

• Contact Information
• Janet Wang wml_at_ece.arizona.edu
• Questions