This paper addresses two problems of interest in service system analysis: (a) that of making statistical, data-driven estimates of the long-run probability of a long delay, and (b) the assessment of rate of approach to a long-run system performance measure such as expected delay, the rate being characterized by a simple exponential, at least initially. Both are illustrated by reference to M/G/1 and related systems. Keywords: Estimation of virtual waiting time distribution; Terminating renewal process; Bootstrap; Jacknife; Transient behavior of queueing systems. (Author)