  This work considers a many-server queueing system in which impatient
customers with i.i.d., generally distributed service times and i.i.d.,
generally distributed patience times enter service in the order of arrival and
abandon the queue if the time before possible entry into service exceeds the
patience time. The dynamics of the system is represented in terms of a pair of
measure-valued processes, one that keeps track of the waiting times of the
customers in queue and the other that keeps track of the amounts of time each
customer being served has been in service. Under mild assumptions, essentially
only requiring that the service and reneging distributions have densities, as
both the arrival rate and the number of servers go to infinity, a law of large
numbers (or fluid) limit is established for this pair of processes. The limit
is shown to be the unique solution of a coupled pair of deterministic integral
equations that admits an explicit representation. In addition, a fluid limit
for the virtual waiting time process is also established. This paper extends
previous work by Kaspi and Ramanan, which analyzed the model in the absence of
reneging. A strong motivation for understanding performance in the presence of
reneging arises from models of call centers.
