In this paper a discrete-time GI/G/1 (General Independent arrivals/General service times/single server) queueing model is considered. It is useful for design and performance evaluation of asynchronous communication systems, in particular for ATM (Asynchronous Transfer Mode) systems. The model assumes an infinite waiting time room (buffer). The attention is focused on the equilibrium distribution of the waiting time experienced by the information units (ATM cells) in the buffer, and in particular on the tail probability. In fact, it is equivalent to the overflow probability, and hence closely related to the cell loss probability. A non-parametric estimate of an upper bound of the waiting time tail probability is first given. It is then extended to cases where a special dependence between inter-arrival and service times is allowed. Asymptotic confidence intervals are also studied, as well as confidence intervals obtained via bootstrap methods. Applications to real data are also considered. The data come from measurements made by Telecom Italia in the framework of the European ATM Pilot Project. The applications considered are videoconference and transport of routing information between IP (Internet Protocol) network routers. The available measurements are the sequences of the exact cell arrival times. The obtained estimate of the upper bound of the waiting time tail probability is evaluated on the available data to assess the performance of an output queueing Cell Switch Router (an IP router employing ATM as data link and switching technology).