  Motivated by call center practice, we propose a tractable model for
$\mbox{GI}/\mbox{GI}/n+\mbox{GI}$ queues in the efficiency-driven (ED) regime.
We use a one-dimensional diffusion process to approximate the virtual waiting
time process that is scaled in both space and time, with the number of servers
and the mean patience time as the respective scaling factors. Using this
diffusion model, we obtain the steady-state distributions of virtual waiting
time and queue length, which in turn yield simple formulas for performance
measures such as the service level and the effective abandonment fraction.
These formulas are generally accurate when the mean patience time is several
times longer than the mean service time and the patience time distribution does
not change rapidly around the mean virtual waiting time. For practical
purposes, these formulas outperform existing results that rely on the
exponential service time assumption.
  To justify the diffusion model, we formulate an asymptotic framework by
considering a sequence of queues, in which both the number of servers and the
mean patience time go to infinity. We prove that the space-time scaled virtual
waiting time process converges in distribution to the one-dimensional diffusion
process. A fundamental result for proving the diffusion limit is a functional
central limit theorem (FCLT) for the superposition of renewal processes. We
prove that the superposition of many independent, identically distributed
stationary renewal processes, after being centered and scaled in space and
time, converges in distribution to a Brownian motion. As a useful technical
tool, this theorem characterizes the service completion process in heavy
traffic, allowing us to greatly simplify the many-server analysis when service
times follow a general distribution.
