Prediction of Interconnect Fan-out Distribution Using Rent’s Rule

1. Prediction of Interconnect Fan-out Distribution Using Rent’s Rule Payman Zarkesh-Ha, Jeffrey A. Davis, William Loh*, and James D. Meindl Georgia Institute of Technology Microelectronics Research Center * LSI Logic CorporationDevice Technology Group

2. Outline • Motivation • Derivation of Fan-out Distribution • Comparison with a Previous Model • Applications of Fan-out Distribution • Conclusion

3. Motivation How can a closed-form fan-out distribution model be useful? - Prediction of fan-out distribution • Characterization of the interconnect structure • Prediction of the average fan-out • Prediction of the total number of nets - Estimation of Rent’s exponent • Easy estimation with no clustering - Heterogeneous netlist information • Fast computation for approximation

4. Fo=2 Fo=1 ··· ··· What is the fan-out distribution?

5. System of N gates Derivation of Fan-out Distribution Rent’s Rule: Underlying assumption for prediction of a priori fan-out distribution T = # of IO’s k and p are empirical constants

6. Conservation of I/O’s in a random logic network Conservation of I/O’s for three blocks Conservation of I/O’s for two blocks

7. Derivation of fan-out distribution Conservation of I/O’s for m block: Applying Rent’s rule: Setting up the recursive equation: The solution:

8. The closed-form fan-out distribution model Substituting m by Fo+1 gives the fan-out distribution (number of nets versus fan-out) Where Ng is the total number of gates and k and p are the Rent’s parameters

9. Model verification - systems with no internal net The case with k=0 The case with p=1 p=1 Net(Fo)=0 for all fan-out k=0

10. How does the fan-out distribution look like? [Stroobandt and Kurdahi GLVLSI’98] Ng=15,000, k=2.0, p=0.6

11. Comparison with Previous Model Actual data Numerically evaluated model [Stroobandt and Kurdahi GLVLSI’98] New closed-form model Ng=23,815, k=2.41, p=0.28

12. Applications of Fan-out Distribution - Prediction of fan-out distribution • Characterization of the interconnect structure • Prediction of the average fan-out • Prediction of the total number of nets - Estimation of Rent’s exponent • Easy estimation with no clustering - Heterogeneous netlist information • Fast computation for approximation

13. Characterization of the interconnect structure Maximum Fan-out: Total Number of Nets: Average Fan-out: Where:

14. Prediction of the interconnect structure Data from ISCAS benchmark in [Stroobandt and Kurdahi GLVLSI’98]

15. Variation of the average fan-out as a function of p, k and Ng

16. Ng=44,803, k=3.36, p=0.6 Estimation of Rent’s exponent

17. ~ ~ Ng=20, K=738.4, P=0.34 90 sec <0.1 sec Heterogeneous netlist information

18. Conclusion  A closed-form model for fan-out distribution is derived based on Rent’s rule  The closed-form model is verified through comparison with actual data from ISCAS  Applications of the closed-form fan-out distribution model are presented