  A particularly important substructure in modeling joint linear
chance-constrained programs with random right-hand sides and finite sample
space is the intersection of mixing sets with common binary variables (and
possibly a knapsack constraint). In this paper, we first revisit basic mixing
sets by establishing a strong and previously unrecognized connection to
submodularity. In particular, we show that mixing inequalities with binary
variables are nothing but the polymatroid inequalities associated with a
specific submodular function. This submodularity viewpoint enables us to unify
and extend existing results on valid inequalities and convex hulls of the
intersection of multiple mixing sets with common binary variables. Then, we
study such intersections under an additional linking constraint lower bounding
a linear function of the continuous variables. This is motivated from the
desire to exploit the information encoded in the knapsack constraint arising in
joint linear CCPs via the quantile cuts. We propose a new class of valid
inequalities and characterize when this new class along with the mixing
inequalities are sufficient to describe the convex hull.
