  This paper studies many-server limits for multi-server queues that have a
phase-type service time distribution and allow for customer abandonment. The
first set of limit theorems is for critically loaded $G/Ph/n+GI$ queues, where
the patience times are independent and identically distributed following a
general distribution. The next limit theorem is for overloaded $G/ Ph/n+M$
queues, where the patience time distribution is restricted to be exponential.
We prove that a pair of diffusion-scaled total-customer-count and
server-allocation processes, properly centered, converges in distribution to a
continuous Markov process as the number of servers $n$ goes to infinity. In the
overloaded case, the limit is a multi-dimensional diffusion process, and in the
critically loaded case, the limit is a simple transformation of a diffusion
process. When the queues are critically loaded, our diffusion limit generalizes
the result by Puhalskii and Reiman (2000) for $GI/Ph/n$ queues without customer
abandonment. When the queues are overloaded, the diffusion limit provides a
refinement to a fluid limit and it generalizes a result by Whitt (2004) for
$M/M/n/+M$ queues with an exponential service time distribution. The proof
techniques employed in this paper are innovative. First, a perturbed system is
shown to be equivalent to the original system. Next, two maps are employed in
both fluid and diffusion scalings. These maps allow one to prove the limit
theorems by applying the standard continuous-mapping theorem and the standard
random-time-change theorem.
