  Until recently synthetic AGB models had not taken into account the break-down
of the core mass-luminosity (Mc-L) relation due to the occurrence of envelope
burning in the most massive (M > 3.5 Msun) and luminous (Mbol > -6) stars.
Marigo et al. (1998) made the first attempt to consistently include the related
over-luminosity effect (i.e. above the Mc-L relation) in synthetic TP-AGB
calculations. In this paper the reliability of the solution scheme is tested by
comparison with the results of complete evolutionary calculations for a 7 Msun
AGB star undergoing envelope burning (e.g. Bloecker & Schoenberner 1991).
Indeed, the method proves to be valid as it is able to reproduce with
remarkable accuracy several evolutionary features of the 7 Msun star. We
present extensive synthetic TP-AGB calculations for stars with initial masses
of 3.5, 4.0, 4.5, and 5.0 Msun, and three choices of the initial metallicity,
i.e. Z=0.019, Z=0.008, and Z=0.004. Three values of the mixing-length parameter
are used, i.e. alpha=1.68, 2.0, 2.5. We investigate the dependence of envelope
burning on such stellar parameters (M, Z, and alpha). The comparison between
different cases gives hints on the interplay between envelope burning
over-luminosity and mass loss, and related effects on TP-AGB lifetimes.
