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Stem and precursor cells play a critical role in tissue development, maintenance, and repair throughout the life. Often, experimental limitations prevent direct observation of the stem cell compartment, thereby posing substantial challenges to the analysis of such cellular systems. Two-type age-dependent branching processes with immigration are proposed to model populations of progenitor cells and their differentiated progenies. Immigration of cells into the pool of progenitor cells is formulated as a non-homogeneous Poisson process. The asymptotic behavior of the process is governed by the largest of two Malthusian parameters associated with embedded Bellman-Harris processes. Asymptotic approximations to the expectations of the total cell counts are improved by Markov compensators. [ABSTRACT FROM AUTHOR]