  The subject of this paper is the problem of estimating service time
distribution of the $M/G/\infty$ queue from incomplete data on the queue. The
goal is to estimate $G$ from observations of the queue--length process at the
points of the regular grid on a fixed time interval. We propose an estimator
and analyze its accuracy over a family of target service time distributions.
The original $M/G/\infty$ problem is closely related to the problem of
estimating derivatives of the covariance function of a stationary Gaussian
process. We consider the latter problem and derive lower bounds on the minimax
risk. The obtained results strongly suggest that the proposed estimator of the
service time distribution is rate optimal.
