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We characterize heavy-traffic process and steady-state limits for systemsstaffed according to the square-root safety rule, when the service requirementsof the customers are perfectly correlated with their individual patience forwaiting in queue. Under the usual many-server diffusion scaling, we show thatthe system is asymptotically equivalent to a system with no abandonment. Inparticular, the limit is the Halfin-Whitt diffusion for the $M/M/n$ queue whenthe traffic intensity approaches its critical value $1$ from below, and isotherwise a transient diffusion, despite the fact that the prelimit is positiverecurrent. To obtain a refined measure of the congestion due to thecorrelation, we characterize a lower-order fluid (LOF) limit for the case inwhich the diffusion limit is transient, demonstrating that the queue in thiscase scales like $n^{3/4}$. Under both the diffusion and LOF scalings, we showthat the stationary distributions converge weakly to the time-limiting behaviorof the corresponding process limit.

Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times