The $GI/G/1$ queueing model is regarded as a functional that maps the service and interarrival time distribution functions onto the stationary waiting time distribution function. By considering the output of the functional when it is applied to nonparametric estimators of the input distribution functions, we obtain a nonparametric estimator of the stationary waiting time distribution function. Using appropriate continuity and differentiability properties of the functional, we show that statistical properties of the input estimators carry over to corresponding properties for the stationary waiting time distribution function estimator.
"Nonparametric Estimation of the Stationary Waiting Time Distribution Function for the $GI/G/1$ Queue." Ann. Statist. 22 (3) 1428 - 1446, September, 1994. https://doi.org/10.1214/aos/1176325635