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.

Strategic Model for Yellow Hydrogen Production Using the Metalog Family of Probability Distributions

“…Metalog is a flexible probability distribution that can be used to model a wide range of density functions using only a small number of parameters obtained from experts. Scientists prefer using the metalog family of distributions to describe processes in various fields of science such as theology [124], mathematics [125], and electronics [101].…”

“…Only then are the subsequent links in the logistics chain activated, which are related to the charging, storage, transport, and refueling of hydrogen [100]. The aim of the article is to support the decision-making processes related to the production of yellow hydrogen using a strategic model which applies the metalog family of probability distributions [101]. The model allows its users to calculate, with accuracy regarding the probability distribution, the amount of energy produced by photovoltaic systems with a specific peak power.…”

A Method of Assessing the Selection of Carport Power for an Electric Vehicle Using the Metalog Probability Distribution Family

“…They also run a website on which the principles of the operation of the software that is created on the basis of this method are described in detail [53]. In addition, other scientists who have used the Metalog distribution families present in their articles a mathematical description of the algorithms used [54].…”

“…Not only the owners of carports and other photovoltaic installations can access large amounts of real data from various photovoltaic systems installed around the world, but some manufacturers of photovoltaic inverters also have large cloud resources and make them publicly available. An example is the SolarEdge platform [54], which allows access to the monitoring systems of many installations that use the company's inverter. Figure 15 shows the monthly generated graphs of the 3 kWp system installed in Randburg, South Africa.…”

Strategic Model for Charging a Fleet of Electric Vehicles with Energy from Renewable Energy Sources

“…Metalog is a flexible probability distribution that can be used to model a wide range of density functions using only a small number of parameters obtained from experts. Scientists prefer using the Metalog family of distributions to describe processes in various fields of science such as theology [41], mathematics [42], and electronics [43]. The authors have already used Metalog probability distribution families to select the peak power of a photovoltaic carport for an electric vehicle [44].…”