  We characterize heavy-traffic process and steady-state limits for systems
staffed according to the square-root safety rule, when the service requirements
of the customers are perfectly correlated with their individual patience for
waiting in queue. Under the usual many-server diffusion scaling, we show that
the system is asymptotically equivalent to a system with no abandonment. In
particular, the limit is the Halfin-Whitt diffusion for the $M/M/n$ queue when
the traffic intensity approaches its critical value $1$ from below, and is
otherwise a transient diffusion, despite the fact that the prelimit is positive
recurrent. To obtain a refined measure of the congestion due to the
correlation, we characterize a lower-order fluid (LOF) limit for the case in
which the diffusion limit is transient, demonstrating that the queue in this
case scales like $n^{3/4}$. Under both the diffusion and LOF scalings, we show
that the stationary distributions converge weakly to the time-limiting behavior
of the corresponding process limit.
