## A Central-Limit-Theorem Version of L = λW

P. W. Glynn and W. Whitt

Queueing Systems:  Theory and Applications, Vol. 2, 191-215 (1986)

Underlying the fundamental queueing formula L = λW is a relation between cumulative processes in continuous time (the integral of the queue length process) and in discrete lime (the sum of the waiting times of successive customers). Except for remainder terms which usually are asymptotically negligible, each cumulative process is a random time-transformation of the other. As a consequence, in addition to the familiar relation between the with-probability-one limits of the averages, roughly speaking, the customer-average wait obeys a central limit theorem if and only if the time-average queue length obeys a central limit theorem, in which case both averages, properly normalized, converge in distribution jointly, and the individual limiting distributions are simply related. This relation between the central limit theorems is conveniently expressed in terms of functional central limit theorems, using the continuous mapping theorem and related arguments. The central limit theorems can be applied to compare the asymptotic efficiency of different estimators of queueing parameters. For example, when the arrival rate h is known and the interarrival times and waiting times are negatively correlated, it is more asymptotically efficient to estimate the long-run time-average queue length L indirectly by the sample-average of the waiting times, invoking L = λW, than it is to estimate it by the sample-average of the queue length. This variance-reduction principle extends a corresponding result for the standard GI/G/smodel established by Carson and Law [2].

[2] J.S. Carson and A.M. Law, Conservation equations and variance reduction in queueing simulations, Oper. Res. 28(1980)535.