  Upcoming wide-area surveys in the submillimetre regime will allow the
construction of complete galaxy cluster samples through their thermal
Sunyaev-Zel'dovich effect. We propose an analytic method to predict the number
of gravitationally lensed giant arcs produced by the cluster sample expected to
be detectable for the Planck satellite. The statistics of lensed arcs can be
used to constrain cosmological parameters. This computation implies the choice
of an analytic cluster model which has to reproduce the lensing properties of
realistic galaxy clusters. Comparing the results of analytic computations and
numerical ray-tracing simulations, we show that spherical models with the
Navarro-Frenk-White density profile fail by far to produce as many giant arcs
as realistic galaxy clusters. This confirms the great importance of asymmetries
and substructures in the lensing matter distribution. By elliptically
distorting the lensing potential of spherical NFW density profiles, we show
that the discrepancy between analytic and numerical computations can almost
completely be removed.
