  We present a general method for estimating the number of particles impinging
on a segmented counter or, in general, on a counter with sub-units. We account
for unresolved particles, i.e., the effect of two or more particles hitting the
same sub-unit almost simultaneously. To achieve full time resolution we account
for the dead time that occurs after the first time-bin of a particle signal.
This general counting method can be applied to counting muons in existing
detectors like the Underground Muon Detector of the Pierre Auger Observatory.
We therefore use the latter as a study case to test the performance of our
method and to compare it to other methods from literature. Our method proves to
perform with little bias, and also provides an estimate of the number of
particles as a function of time (as seen by the detector) to a single time-bin
resolution. In this context, the new method can be useful for reconstructing
parameters sensitive to cosmic ray mass, which are key to unveiling the origin
of cosmic rays.
