  Assuming a Markovian time evolution of a quantum sensing system, we provide a
general characterization of the optimal sensitivity scalings with time, under
the most general quantum control protocols. We allow the estimated parameter to
influence both the Hamiltonian as well as the dissipative part of the quantum
master equation. We focus on the asymptotic-time as well as the short-time
sensitivity scalings, and investigate the relevant time scales on which the
transition between the two regimes appears. This allows us to characterize, via
simple algebraic conditions (in terms of the Hamiltonian, the jump operators as
well as their parameter derivatives), the four classes of metrological models
that represent: quadratic-linear, quadratic-quadratic, linear-linear and
linear-quadratic time scalings. We also provide universal numerical methods to
obtain quantitative bounds on sensitivity that are the tightest that exist in
the literature.
