We point to the connection between a recently introduced class of
non-Markovian master equations and the general structure of quantum
collisional models. The basic construction relies on three basic
ingredients: a collection of time dependent completely positive maps, a
completely positive trace preserving transformation and a waiting time
distribution characterizing a renewal process. The relationship between
this construction and a Lindblad dynamics is clarified by expressing
the solution of a Lindblad master equation in terms of demixtures over
different stochastic trajectories for the statistical operator weighted
by suitable probabilities on the trajectory space.