  We establish heavy-traffic stochastic-process limits for waiting times in
many-server queues with customer abandonment. If the system is asymptotically
critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a
bounding argument shows that the abandonment does not affect waiting-time
processes. If instead the system is overloaded, as in the efficiency-driven
(ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh
Annual Allerton Conference on Communication, Control and Computing (1999)
1095--1104], we treat customer abandonment by studying the limiting behavior of
the queueing models with arrivals turned off at some time $t$. Then, the
waiting time of an infinitely patient customer arriving at time $t$ is the
additional time it takes for the queue to empty. To prove stochastic-process
limits for virtual waiting times, we establish a two-parameter version of
Puhalskii's invariance principle for first passage times. That, in turn,
involves proving that two-parameter versions of the composition and inverse
mappings appropriately preserve convergence.
