Extreme Path Delay Estimation of Critical Paths in Within-Die Process Fluctuations Using Multi-Parameter Distributions

Abstract

Two multi-parameter distributions, namely the Pearson type IV and metalog distributions, are discussed and suggested as alternatives to the normal distribution for modelling path delay data that determines the maximum clock frequency (FMAX) of a microprocessor or other digital circuit. These distributions outperform the normal distribution in goodness-of-fit statistics for simulated path delay data derived from a fabricated microcontroller, with the six-term metalog distribution offering the best fit. Furthermore, 99.7% confidence intervals are calculated for some extreme quantiles on each dataset using the previous distributions. Considering the six-term metalog distribution estimates as the golden standard, the relative errors in single paths vary between 4 and 14% for the normal distribution. Finally, the within-die (WID) variation maximum critical path delay distribution for multiple critical paths is derived under the assumption of independence between the paths. Its density function is then used to compute different maximum delays for varying numbers of critical paths, assuming each path has one of the previous distributions with the metalog estimates as the golden standard. For 100 paths, the relative errors are at most 14% for the normal distribution. With 1000 and 10,000 paths, the corresponding errors extend up to 16 and 19%, respectively.