Civil Engineering Reference
In-Depth Information
is given by
While the values of and are obvious from earlier discussions‚ the others
have specific interpretations. The second central moment‚ corresponds to
the
variance
of the distribution‚ which is higher if has a wider “spread”
from its mean value. The third central moment‚ has an interpretation as the
skewness
of the distribution: its sign is positive [negative] if the mode‚ or the
maximum of is to the left [right] of the mean‚ and its magnitude measures
the distance from the mean. The fourth central moment is the
kurtosis
of the
distribution‚ which reflects the total area that corresponds to the tails of
The above parameters provide measures for the mean and the mode of the
distribution; however‚ the 50% delay that we are interested in is the
median
of
the distribution‚ and generally speaking‚ no simple closed-form formulas for the
median are available. However‚ it was proved in [GTP97] that for the impulse
response at any node of an RC tree‚ the Elmore delay‚ which corresponds
to the mean‚ is an upper bound on the median‚ or the 50% delay.
3.7.2 Delay metrics based on probabilistic interpretations
Several techniques have used the probabilistic interpretation of moments to
arrive at a delay metric‚ and these are surveyed here. All of the methods listed
here are stable in that they result in reasonable (i.e.‚ physically realizable)
values for the parameters that define the delay distributions‚ and therefore‚
physically reasonable delay values.
The PRIMO metric.
The PRIMO method [KP98] attacked the problem
by fitting the impulse response to the gamma distribution‚ given by
This distribution is completely described by two parameters‚ and and
therefore a gamma function approximation to can be found by fitting the
moments of the gamma function to the moments of
The first few central
moments of
are given by
