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

It is known that, given the observed traffic intensity <(rho)over cap> < 1, the expected value of the estimator <(rho)over cap>/(1 - <(rho)over cap>) for the average number of customers rho/(1 - rho) in a stationary M/M/1 que ueing model is infinite (Schruben and Kulkarni, Oper. Res. Lett. 1 (1982) 7 5-78). In this paper we generalize the above findings to other system perfo rmance measures. Second, we show that, for the following four system perfor mance measures: (a) mean waiting time in queue, (b) mean waiting time in sy stem, (c) mean number of customers in queue and (d) mean number of customer s in the system, estimators constructed by substituting parameter estimator s for unknown parameters in the formula for the performance measure all hav e the undesirable properties that the expected value of the estimator does not exist and the estimator has infinite mean-squared error. Finally, we pr opose alternative estimators for these four system performance measures whe n rho < rho(0), where rho(0) < 1 is a known constant, and show that these a lternative estimators are strongly consistent, asymptotically unbiased and have finite variance and finite mean squared error. (C) 2000 Elsevier Scien ce B.V. All rights reserved.