It is well known that the Heitler model and its extensions (like the Heitler-Matthews model) describe qualitatively many fundamental properties of extensive air showers initiated in the atmosphere by the high-energy cosmic rays. Typically, only the secondary particle multiplicity and the fraction of neutral-to-charged pions are considered as analytically treatable parameters of hadronic interactions. Other relevant parameters of hadronic interactions such as inelasticity and the lifetime of the leading particle are taken into account only in detailed Monte Carlo simulations. Here we present an extension of the Heitler-Matthews model that includes the leading particle effect and analytically derive the number of muons produced in air showers in a self-consistent way. In a second step we apply this model to calculate the dependence of the depth of the shower center-of-gravity on various model parameters. In this way, we obtain predictions for the depth of shower maximum, which is a well known observable sensitive to the mass of primary particles and is regularly used in cosmic-ray research.