  The mass composition of ultra-high energy cosmic rays can be studied from the
distributions of the depth of shower maximum and/or the muon shower size. Here,
we study the dependence of the mean muon shower size on the depth of shower
maximum in detail. Air showers induced by protons and iron nuclei were
simulated with two models of hadronic interactions already tuned with LHC data
(run I-II). The generated air showers were combined to obtain various types of
mass composition of the primary beam. We investigated the shape of the
functional dependence of the mean muon shower size on the depth of shower
maximum and its dependency on the composition mixture. Fitting this dependence
we can derive the primary fractions and the muon rescaling factor with a
statistical uncertainty at a level of few percent. The difference between the
reconstructed primary fractions is below 20% when different models are
considered. The difference in the muon shower size between the two models was
observed to be around 6%.
