Most models for on-chip wire length estimation involve two tasks: the enumeration of two-terminal net placement sites, and the population of these sites using a function derived from Rent’s rule. Rent’s rule estimates the number of boundary-crossing wires using the number of cells enclosed by the boundary. As the number of cells becomes comparable to the total number of cells in the circuit, the estimation becomes worse.

The author of this paper has derived a first-order differential equation to describe the relation between the number of boundary-crossing wires and the number of cells enclosed by the boundary, by considering the processes that generate and terminate boundary-crossing wires. The differential equation model accurately captures the “non-Rentian” deviations displayed in experimental data from a commercial placement tool. Using the differential model in wire length estimation also shows a better fit to the experimental data in region II of Rent’s graph.

The main contribution of this paper is the presentation of a method for using a first-order differential equation to describe the process of wires being generated or terminated through the boundary. A second contribution is the author’s demonstration that Rent’s rule is a special solution to the differential equation, given the condition that the number of cells in the boundary box is small compared to the total number of cells, and that the effect of pins is neglected. The proposed model helps to provide a better estimation for on-chip wire lengths.