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

It is known that, given the observed tra c intensity ̂¡ 1, the expected value of the estimator ̂=(1− ̂) for the average number of customers =(1− ) in a stationary M=M=1 queueing model is in nite (Schruben and Kulkarni, Oper. Res. Lett. 1 (1982) 75–78). In this paper we generalize the above ndings 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 in nite mean-squared error. Finally, we propose alternative estimators for these four system performance measures when ¡ 0, where 0¡ 1 is a known constant, and show that these alternative estimators are strongly consistent, asymptotically unbiased and have nite variance and nite mean squared error. c © 2000 Elsevier Science B.V. All rights reserved.