  We investigate the parameter estimation problem in a two-qubit system, in
which each qubit is independently interacting with its Markovian environment.
We study in detail the sensitivity of the estimation on the decoherence rate
$\gamma$ and the two-qubit interaction strength $v$. In particular, the
dynamics of quantum Fisher information are employed as a measure to quantify
the precision of the estimations. We find that the quantum Fisher information
with respect to the decoherence rate scales like $t^2/[\exp(\gamma t)-1]$ and
$\exp(-2\gamma t)$ in the unitary limit and the completely decoherent limit,
respectively. When we estimate the interaction strength $v$, the quantum Fisher
information shows oscillation behavior in term of time. In addition, our
results provide further evidence that the entanglement of the input state may
not enhance quantum metrology.
