Some well-behaved estimators for the M/M/1 queue

It is known, given the observed traffic intensity ρˆ < 1, the expected value of the estimator ρˆ/(1-ρˆ) for the average number of customers ρ/(1-ρ) in a stationary M/M/1 queueing model is infinite. In this paper we generalize the above findings to other system performance measures. Second, we show that, for the following four system performance measures: (a) mean waiting time in queue, (b) mean waiting time in system, (c) mean number of customers in queue and (d) mean number of customers in the system, estimators constructed by substituting parameter estimators for unknown parameters in the formula for the performance measure all have the undesirable properties that the expected value of the estimator does not exist and the estimator has infinite mean-squared error. Finally, we propose alternative estimators for these four system performance measures when ρ<ρ0, where ρ0<1 is a known constant, and show that these alternative estimators are strongly consistent, asymptotically unbiased and have finite variance and finite mean squared error.