  For atomic frequency standards in which fluctuations of the local oscillator
(LO) frequency are the dominant noise source, we examine the role of the servo
algorithm that predicts and corrects these frequency fluctuations. We derive
the optimal linear prediction algorithm, showing how to measure the relevant
spectral properties of the noise and optimise servo parameters while the
standard is running, using only the atomic error signal. We find that, for
realistic LO noise spectra, a conventional integrating servo with a properly
chosen gain performs nearly as well as the optimal linear predictor. Using
simple analytical models and numerical simulations, we establish optimum probe
times as a function of clock atom number and of the dominant noise type in the
local oscillator. We calculate the resulting LO-dependent scaling of achievable
clock stability with atom number for product states as well as for
maximally-correlated states.
