h-gamma: an RC delay metric based on a gamma distribution approximation of the homogeneous response

Recently a probability interpretation of moments was proposed as a compromise between the Elmore delay and higher order moment matching for RC timing estimation (Kay and Pileggi, 1998). By modeling RC impulses as time-shifted incomplete gamma distribution function, the delays could be obtained via table lookup using a gamma integral table and the first three moments of the impulse response. However, while this approximation works well for many examples, it struggles with responses when the metal resistance becomes dominant, and produces results with impractical time shift values In this paper the probability interpretation is extended to the circuit homogeneous response, without requiring the time shift parameter. The gamma distribution is used to characterize the normalized homogeneous portion of the step response. For a generalized RC interconnect model (RC tree or mesh), the stability of the homogeneous-gamma distribution model is guaranteed. It is demonstrated that when a table model is carefully constructed, the h-gamma approximation provides for excellent improvement over the Elmore delay in terms of accuracy, with very little additional cost in terms of CPU time.