  An important question in the theory of double parton scattering is how to
incorporate the possibility of the parton pairs being generated perturbatively
via $1 \to 2$ splitting into the theory, whilst avoiding double counting with
single parton scattering loop corrections. Here, we describe a consistent
approach for solving this problem, which retains the notion of double parton
distributions (DPDs) for individual hadrons. Further, we discuss the
construction of appropriate model DPDs in our framework, and the use of these
to compute the DPS part, presenting DPS 'luminosities' from our model DPDs for
a few sample cases.
