This thesis is primarily concerned with the efficient estimation of mean delay, d, and time average number in queue, Q. For a certain class of queueing systems, it is analytically shown to be more efficient to estimate Q by multiplying an estimate of d by the arrival rate of customers, than to estimate Q directly. This relationship is empirically verified for a much larger class of queueing systems. For single queueing systems, an estimator is introduced which is more efficient than the normal estimator of d. The efficient estimation of mean waiting time, w, and time average amount of work in system, E(V), is also considered. It is seen to be more efficient to estimate w and E(V) from an estimate of d, than to estimate them directly.