  We analyze the performance of quantum parameter estimation in the presence of
the most general Gaussian dissipative reservoir. We derive lower bounds on the
precision of phase estimation and a closely related problem of frequency
estimation. For both problems we show that it is impossible to achieve the
Heisenberg limit asymptotically in the presence of such a reservoir. However,
we also find that for any fixed number of probes used in the setup there exists
a Gaussian dissipative reservoir, which, in principle, allows for the
Heisenberg-limited performance for that number of probes. We discuss a
realistic implementation of a frequency estimation scheme in the presence of a
Gaussian dissipative reservoir in a cavity system.
