  Detectors inherently capable of resolving photon numbers have undergone a
significant development recently, and this is expected to affect multiplexed
periodic single-photon sources where such detectors can find their
applications. We analyze various spatially and time-multiplexed periodic
single-photon source arrangements with photon-number-resolving detectors,
partly to identify the cases when they outperform those with threshold
detectors. We develop a full statistical description of these arrangements in
order to optimize such systems with respect to maximal single-photon
probability, taking into account all relevant loss mechanisms. The model is
suitable for the description of all spatial and time multiplexing schemes. Our
detailed analysis of symmetric spatial multiplexing identifies a particular
range of loss parameters in which the use of the new type of detectors leads to
an improvement. Photon number resolution opens an additional possibility for
optimizing the system in that the heralding strategy can be defined in terms of
actual detected photon numbers. Our results show that this kind of optimization
opens an additional parameter range of improved efficiency. Moreover, this
higher efficiency can be achieved by using less multiplexed units, i.e.,
smaller system size as compared to threshold-detector schemes. We also extend
our investigation to certain time-multiplexed schemes of actual experimental
relevance. We find that the highest single-photon probability is 0.907 that can
be achieved by binary bulk time multiplexers using photon-number-resolving
detectors.
