  We study systems with two classes of impatient customers who differ across
the classes in their distribution of service times and patience times. The
customers are served on a first-come, first served basis (FCFS) regardless of
their class. Such systems are common in customer call centers, which often
segment their arrivals into classes of callers whose requests differ in their
complexity and criticality. We first consider an $M$/$G$/1 + $M$ queue and then
analyze the $M$/$M$/$k$ + $M$ case. Analyzing these systems using a queue
length process proves intractable as it would require us to keep track of the
class of each customer at each position in queue. Consequently, we introduce a
virtual waiting time process where the service times of customers who will
eventually abandon the system are not considered. We analyze this process to
obtain performance measures such as the percentage of customers receiving
service in each class, the expected waiting times of customers in each class,
and the average number of customers waiting in queue. We use our
characterization to perform a numerical analysis of the $M$/$M$/$k$ + $M$
system, and find several managerial implications of administering a FCFS system
with multiple classes of impatient customers. Finally, we compare the
performance a system based on data from a call center with the steady-state
performance measures of a comparable $M$/$M$/$k$ + $M$ system. We find that the
performance measures of the $M$/$M$/$k$ + $M$ system serve as good
approximations of the system based on real data.
