Utilizing the approach recently proposed for the inclusive semi-leptonic decay rate on the lattice, we compute the differential decay rate of a $B_s$ meson for various kinematical channels. The results are compared with the contributions from the ground states ($D$ and $D^*$) as well as from the orbitally excited states ($D^{**}$'s). The computation so far is carried out with an unphysically light bottom quark and strange spectator quark.