  We present simulations of the superradiant dynamics of ensembles of atoms in
the presence of collective and individual atomic decay processes. We unravel
the density matrix with Monte-Carlo wave-functions and identify the quantum
jumps in a reduced Dicke state basis, which reflects the permutation symmetry
of the identical atoms. While the number of density matrix elements in the
Dicke representation increases polynomially with atom number, the quantum jump
dynamics populates only a single Dicke state at the time and thus efficient
simulations can be carried out for tens of thousands of atoms. The calculated
superradiance pulses from initially excited atoms agree quantitatively with
recent experimental results with strontium atoms but rapid atom loss in these
experiments does not permit steady-state superradiance. By introducing an
incident flux of new atoms, the system can maintain a large average atom
number, and our theoretical calculations predict lasing with millihertz
linewidth despite rapid atom number fluctuations.
