  We study the concepts of complete positivity, positivity and non-Markovianity
in a two-level open quantum system whose dynamics are governed by a time-local
quantum master equation. We establish necessary and sufficient conditions on
the time-dependent relaxation rates to ensure complete positivity and
positivity of the dynamical map. We discuss their relations with the
non-Markovian behavior of the open system. We also analyze the long-time
asymptotic behavior of the dynamics as a function of the rates. We show under
which conditions on the rates the system tends to the equilibrium state.
Different examples illustrate this general study.
