The maximum likelihood estimation technique is used to estimate the average service rate for the E/M/1 queueing model. The observation method primarily considered is sampling the number of customers in the system at periodic time intervals. The small sample size properties of the estimator are investigated, and the estimator is shown to he "biased and inefficient. A simplification of the formula for the Cramer-Rao "bound for parameter estimation in.Markov chains is derived, and it is shown that the prohler: is similar to computing the cost function in a Markov chain with the costs associated with state occupancy. The result is used to examine the asymptotic error variance for the periodic sampling estimator. When the period of observation is fixed, and the sampling rate is varied (0 -*»), it is shown that the accuracy of the estimator converges to the accuracy of estimators for which analytical results are available. The rate of convergence of the actual error variance to the Cramer-Rao "bound is briefly examined, and the results suggest that the Cramer bound closely approximates the actual error variance for relatively small sample sizes.