  We consider an overloaded multiclass nonidling first-in-first-out
single-server queue with abandonment. The interarrival times, service times,
and deadline times are sequences of independent and identically, but generally
distributed random variables. In prior work, Jennings and Reed studied the
workload process associated with this queue. Under mild conditions, they
establish both a functional law of large numbers and a functional central limit
theorem for this process. We build on that work here. For this, we consider a
more detailed description of the system state given by $K$ finite, nonnegative
Borel measures on the nonnegative quadrant, one for each job class. For each
time and job class, the associated measure has a unit atom associated with each
job of that class in the system at the coordinates determined by what are
referred to as the residual virtual sojourn time and residual patience time of
that job. Under mild conditions, we prove a functional law of large numbers for
this measure-valued state descriptor. This yields approximations for related
processes such as the queue lengths and abandoning queue lengths. An
interesting characteristic of these approximations is that they depend on the
deadline distributions in their entirety.
