Indirect Estimation via L = λW

P. W. Glynn and W. Whitt

Operations Research, Vol. 37, 82-103 (1989)

For a large class of queueing systems, Little's law (L = λW) helps provide a variety of statistical estimators for the longrun time-average queue length L and the long-run customer-average waiting time W. We apply central limit theorem versions of Little's law to investigate the asymptotic efficiency of these estimators. We show that an indirect estimator for L using the natural estimator for W plus the known arrival rate λ is more efficient than a direct estimator for L, provided that the interarrival and waiting times are negatively correlated, thus extending a variance-reduction principle for the GI/G/s model due to A. M. Lawa nd J. S. Carson. We also introduce a general framework for indirect estimation which can be applied to other problems besides L = λW . We show that the issue of indirect-versus-direct estimation is related to estimation using nonlinear control variables. We also show, under mild regularity conditions, that any nonlinear control-variable scheme is equivalent to a linear control-variable scheme from the point of view of asymptotic efficiency. Finally, we show that asymptotic bias is typically asymptotically negligible compared to asymptotic efficiency.