  Active optical frequency standards provide interesting alternatives to their
passive counterparts. Particularly, such a clock alone continuously generates
highly-stable narrow-line laser radiation. Thus a local oscillator is not
required to keep the optical phase during a dead time between interrogations as
in passive clocks, but only to boost the active clock's low output power to
practically usable levels with the current state of technology. Here we
investigate the spectral properties and the stability of active clocks,
including homogeneous and inhomogeneous broadening effects. We find that for
short averaging times the stability is limited by photon shot noise from the
limited emitted laser power and at long averaging times by phase diffusion of
the laser output. Operational parameters for best long-term stability were
identified. Using realistic numbers for an active clock with $^{87}$Sr we find
that an optimized stability of $\sigma_y(\tau) \approx
4\times10^{-18}/\sqrt{\tau [\mathrm{s}]}$ is achievable.
