Robust Interconnect Robust Interconnect Communication Capacity - PowerPoint PPT Presentation

Robust Interconnect Robust Interconnect Communication Capacity Algorithm Communication Capacity Algorithm by Geometric Programming by Geometric Programming Jifeng Chen, Jin Sun and Janet Wang Department of Electrical and Computer Engineering University of Arizona, Tucson, AZ 85721 E-mail: { chenjf, sunj, wml} @ece.arizona.edu Speaker : Jin Sun

Talk Outline Talk Outline � Motivation � Proposed Methodology � A robust model for channel capacity considering parameter variations. � A robust capacity optimization procedure by Geometric Programming (GP) � Numerical Results � Conclusion

Motivation Motivation � Network-on-chip (NoC) is one of the most important features in today’s computer architecture. � Modeling and characterization of global interconnect is very important. � The deep submicron (DSM) technologies make it possible to build high speed and high density global buses. � Shrinking feature size leads to severe random variations in circuit parameters. � The goal of this paper is to optimize the interconnect capacity by Geometric Programming (GP) under parameters variations.

Talk Outline Talk Outline � Motivation � Proposed Methodology � A robust model for channel capacity considering parameter variations. � A robust capacity optimization procedure by Geometric Programming (GP) � Numerical Results � Conclusion

Global Bus Structure (1/2) Global Bus Structure (1/2) � The global bus is modeled as a RC network: � The calculation of R, Cc and Cg follows PTM (Predictive Technology Model) [1] . [1]. Predictive Technology Model Website. [Online]. Available: http://www.eas.asu.edu/~ptm

Global Bus Structure (2/2) Global Bus Structure (2/2) � Deterministic transfer function: − = + 1 ( ) ( ) H s G sC B � G : conductance matrix � C : capacitance matrix � B : input-output relationship � Considering process variations, G and C matrices are random with deviations from their nominal values: ° ° ° − = + 1 ( ) ( ) H s G sC B

Deterministic Capacity Model Deterministic Capacity Model � Output signal: � BER (Bit Error Rate): = = > = = < (1| 0) ( ) p p P V V (0 |1) ( ) p p P V V 0 0 e d 1 1 e d � Channel Capacity: ( ) = Γ @ , max ( , ) C p p I X Y 1 0 e e � The capacity is deterministic.

Robust Capacity Model Robust Capacity Model � Considering parameter variations: ~ ~ ~ ~ ~ ~ pdf = + − = + − 1 1 ( ) ( ) ( ) ( ) H s G sC B H s G sC B 1 1 1 1 1 1 ~ ~ ~ = + − 1 ( ) ( ) H s G sC B 2 2 2 ~ ~ ~ = + − 1 ( ) ( ) H s G sC B ( ) ( ) f 0 x f 1 x 2 2 2 ~ ~ ~ = + − 1 ( ) ( ) H s G sC B ~ ~ ~ 3 3 3 = + − 1 ( ) ( ) H s G sC B 3 3 3 ~ ~ ~ = + − ~ ~ ~ 1 ( ) ( ) H s G sC B = + − 1 0 0 0 ( ) ( ) H s G sC B 0 0 0 x V V V 1 0 d p p 1 0 e e � Pe ’s are variational: = ( , , , , , ) p f w d t h W L e i i i i eff eff � Pe ’s are not deterministic, so is the channel capacity: = Γ ( , , , , , ) C w d t h W L i i i i eff eff

Talk Outline Talk Outline � Motivation � Proposed Methodology � A robust model for channel capacity considering parameter variations. � A robust capacity optimization procedure by Geometric Programming (GP) � Numerical Results � Conclusion

Robust Optimization Model Robust Optimization Model � Communication capacity optimization problem: maximize C ≤ Subject to p P e spec ≤ ≤ X X X min max = variables ( , , , , , ) X w t d h W L eff eff � finds the optimal nominal parameter values. � maximizes the capacity with parameter variations. � a BER constraint specified by the designers. � upper and lower bounds for parameter variation. � Standard GP formulation is required.

Standard GP Form Standard GP Form � Standard GP optimization problem: minimize ( ) f x o ≤ = Subject to ( ) 1, 1,2,..., f x i m i = = ( ) 1, 1,2,..., h x l p l variables x ( ) f x � The objective is posynomial. o � The inequality constraints are posynomials. � The equality constraints are monomials. (1) (2) ( ) n ⋅ a a a L d x x x K 1 2 ∑ n (1) (2) ( ) n ⋅ a a a L d x x x k k k 0.017 0.008 1 2 w l k n = = C 1 k 0.355 0.344 t h posynomial monomial

Objective Formulation in GP Objective Formulation in GP � Formulates the objective into GP form: ⇔ maximize minimize - C C � Converting a signomial objective into posynomial form: � separates positive items and negative items: = − minimize - ( ) ( ) C f X f X 01 02 � introduces two slack variables: − ≤ ( ) ( ) f X f X u 01 02 1 ≤ ≤ + ( ) ( ) f X u f X u 01 2 02 1 � translates into GP forms: minimize u 1 − − + ≥ 1 1 Subject to ( ) 1 u f X u 2 02 2 − ≤ 1 ( ) 1 u f X 2 01

Constraint Formulation in GP (1/3) Constraint Formulation in GP (1/3) � Constraint function with parameter variations: K ^ posy fitting ∑ = ≈ ≤ g a b m n p q ( ) p f X c w d t h W L P ` k k k k k k e k eff eff spec = 1 k ^ ^ ⇒ + δ ⇒ + δ X X X ( ) ( ) f X f X X � Use first-order Taylor expansion to further expand: ^ ^ ^ + δ = + ∇ + δ − g ( ) ( ) ( ) ( ) f X X f X f X X X X ^ ∂ ( ) ^ ∑ f X = + δ ≤ ( ) f X X T ∂ i spec X i i � We need to formulate the variational constraint into deterministic function.

Constraint Formulation in GP (2/3) Constraint Formulation in GP (2/3) � Variational constraint function: ^ ^ + < ∇ δ > ≤ ( ) ( ), f X f X X P spec { } ^ ^ + < ∇ δ > ≤ ( ) max ( ), f X f X X P spec x � UE (Uncertainty Ellipsoid) : 2 { } % = + ≤ 1/2 1 X X P u u o � P is the covariance matrix. u is 2-norm of u. � X o % δ = − = 1/ 2 X X X P u 0 x 1

Constraint Formulation in GP (3/3) Constraint Formulation in GP (3/3) � How to get rid of u : { } ^ ^ + < ∇ > ≤ 1/2 ( ) max ( ), f X f X P u P spec u = x 1 2 Cauchy-Schwarz Inequality < > < ⋅ , a b a b X o ^ ^ + ∇ ≤ 1/ 2 ( ) f X f P u P spec x 1 � More slack variables to eliminate the item. 1/2 P

Resulting Model in Standard GP Form Resulting Model in Standard GP Form Minimize Minimize Minimize u u u 1 1 1 − − − − − − − − − − − − − − − − − − 1 1 c c a c a c b c b c m m c c n c n c c 1 a c b c m c n c Subject to Su bject to Subject to e e w w d d t t h h e 0 0 w 0 0 0 0 d 0 0 0 0 t 0 0 0 0 h 0 0 0 0 0 0 0 0 0 0 0 0 0 − − − − − − + + ≤ ≤ − − − + ≤ 1 1 1 1 g g p p c c q q c c c c 1 g p c q c 1 c 1 1 W W L L u u u u 1 W 0 0 0 0 L 0 0 0 0 u u 0 0 0 0 0 0 0 1 1 2 2 1 2 + + + + ≤ ≤ + + ≤ g g a a b b m m n n p p q q g a b m n p q c w d t h W c w d t h W L L r r r r P P c w d t h W L r r P 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 s spec pec spec φ φ φ φ − − ≤ ≤ − φ φ ≤ 2 2 T T T 2 1 1 P r P r 1 P r 1 1 1 1 1 1 1 1 1 φ φ φ φ − − ≤ ≤ − φ φ ≤ 2 2 T T 2 T 1 1 P r P r 1 P r 2 2 2 2 2 2 2 2 2 ≤ ≤ ≤ ≤ ≤ ≤ X X X X X X X X X min min max max min max = = = variables variables ( , , , , ( , , , , , ), , ), , , , , , , variables X X ( , , , , w t d h W L u u r r w t d h W L u u r r , ), , , , X w t d h W L u u r r 1 1 2 2 1 1 2 2 1 2 1 2 Robust GP formulation for p e constraint Robust GP formulation for objective

A Simple Example (1/2) A Simple Example (1/2) � For this circuit, the channel capacity is: 0.017 0.008 w l = C 0.355 0.344 t h � Thus, the optimization problem is formulated as: 0.017 0.008 w l − Minimize 0.355 0.344 t h 0.6021 0.5506 0.1415 0.1415 w w + ≤ Subject to T spec 0.4089 0.2307 0.0319 0.4129 0.2339 0.0435 t h l t h l