  The coarse-graining approach to deriving the quantum Markovian master
equation is revisited, with close attention given to the underlying
approximations. It is further argued that the time interval over which the
coarse-graining is performed is a free parameter that can be given a physical
measurement-based interpretation. In the case of the damping of composite
systems to reservoirs of different temperatures, currently of much interest in
the study of quantum thermal machines with regard to the validity of `local'
and `global' forms of these equations, the coupling of the subsystems leads to
a further timescale with respect to which the coarse-graining time interval can
be chosen. Different choices lead to different master equations that correspond
to the local and global forms. These can be then understood as having different
physical interpretations based on the role of the coarse-graining, as well as
different limitations in application.
