  The Fermi satellite has been reporting the detailed temporal properties of
gamma-ray bursts (GRBs) in an extremely broad spectral range, 8 keV - 300 GeV,
in particular, the unexpected delays of the GeV emission onsets behind the MeV
emission of some GRBs. We focus on GRB 080916C, one of the Fermi-LAT GRBs for
which the data of the delayed high-energy emission are quite extensive, and we
show that the behavior of the high-energy emission of this burst can be
explained by a model in which the prompt emission consists of two components:
one is the MeV component due to the synchrotron-self-Compton radiation of
electrons accelerated in the internal shock of the jet and the other is the
high-energy component due to inverse Compton scattering of the photospheric
X-ray emission of the expanding cocoon off the same electrons in the jet. Such
an external inverse Compton effect could be important for other Fermi-LAT GRBs,
including short GRBs as well. In this model, the delay timescale is directly
linked to the physical properties of GRB progenitor.
