  We address the non-perturbative calculation of the decay rate of inclusive
semi-leptonic $B_{(s)}$ mesons decays from lattice QCD. Precise theoretical
Standard Model predictions are key ingredients in searches for new physics.
This type of computation may eventually provide new insight into the
long-standing tension between the inclusive and exclusive determinations of the
CKM matrix elements $|V_{cb}|$ and $|V_{ub}|$. We perform a pilot lattice
computation for $B_s \rightarrow X_c \, l \nu_l$ and improve on existing
techniques. The valence-quark masses in our simulations are approximately
physical for the domain-wall strange and charm quarks as well as for the bottom
quark, for which we use a relativistic heavy quark effective action. We report
on our progress and discuss future plans towards a first study with fully
controlled systematic effects.
