New models of star-forming filamentary clouds are presented in order to quantify their properties and to predict their evolution. These 2D axisymmetric models describe filaments that have no core, one low-mass core, and one cluster-forming core. They are based on Plummer-like cylinders and spheroids that are bounded by a constant-density surface of finite extent. In contrast to 1D Plummer-like models, they have specific values of length and mass, they approximate observed column density maps, and their distributions of column density (N-pdfs) are pole-free. Each model can estimate the star-forming potential of a core-filament system by identifying the zone of gas dense enough to form low-mass stars and by counting the number of enclosed thermal Jeans masses. This analysis suggests that the Musca central filament may be near the start of its star-forming life, with enough dense gas to make its first ∼3 protostars, while the Coronet filament is near the midpoint of its star formation, with enough dense gas to add ∼8 protostars to its ∼20 known stars. In contrast, L43 appears to be near the end of its star-forming life, since it lacks enough dense gas to add any new protostars to the two young stellar objectsalready known.