  A wide class of non-Markovian completely positive master equations can be
formulated on the basis of quantum collisional models. In this phenomenological
approach the dynamics of an open quantum system is modeled through an ensemble
of stochastic realizations that consist in the application at random times of a
(collisional) completely positive transformation over the system state. In this
paper, we demonstrate that these kinds of models can be embedded in bipartite
Markovian Lindblad dynamics consisting of the system of interest and an
auxiliary one. In contrast with phenomenological formulations, here the
stochastic ensemble dynamics an the inter-event time interval statistics are
obtained from a quantum measurement theory after assuming that the auxiliary
system is continuously monitored in time. Models where the system
inter-collisional dynamics is non-Markovian [B. Vacchini, Phys. Rev. A 87,
030101(R) (2013)] are also obtained from the present approach. The formalism is
exemplified through bipartite dynamics that leads to non-Markovian system
effects such as an environment-to-system back flow of information.
