The statistics of gravitationally lensed arcs was recognised earlier as a potentially powerful cosmological probe. However, while fully numerical models find orders of magnitude difference between the arc probabilities in different cosmological models, analytic models tend to find markedly different results. We introduce in this paper an analytic cluster lens model which improves upon existing analytic models in four ways. (1) We use the more realistic Navarro-Frenk-White profile instead of singular isothermal spheres, (2) we include the effect of cosmology on the compactness of the lenses, (3) we use elliptical instead of axially symmetric lenses, and (4) we take the intrinsic ellipticity of sources into account. While these improvements to the analytic model lead to a pronounced increase of the arc probability, comparisons with numerical models of the same virial mass demonstrate that the analytic models still fall short by a substantial margin of reproducing the results obtained with numerical models. We attribute the remaining discrepancy to substructure inside clusters and tidal fields contributed by the cluster surroundings, effects that cannot reasonably and reliably be mimicked in analytic models.