  We develop a statistical theory describing the operation of multiplexed
single-photon sources equipped with photon-number-resolving detectors that
includes the potential use of different input mean photon numbers in each of
the multiplexed units. This theory accounts for all relevant loss mechanisms
and allows for the maximization of the single-photon probabilities under
realistic conditions by optimizing the different input mean photon numbers
unit-wise and the detection strategy that can be defined in terms of actual
detected photon numbers. We apply this novel description to analyze periodic
single-photon sources based on asymmetric spatial multiplexing realized with
general asymmetric routers. We show that optimizing the different input mean
photon numbers results in maximal single-photon probabilities higher than those
achieved by using optimal identical input mean photon numbers in this setup. We
identify the parameter ranges of the system for which the enhancement in the
single-photon probability for the various detection strategies is relevant. An
additional advantage of the unit-wise optimization of the input mean photon
numbers is that it can result in the decrease of the optimal system size needed
to maximize the single-photon probability. We find that the highest
single-photon probability that our scheme can achieve in principle when
realized with state-of-the-art bulk optical elements is 0.935. This is the
highest one to our knowledge that has been reported thus far in the literature
for experimentally realizable single-photon sources.
