Fluid Limits for Overloaded Multiclass FIFO Single-Server Queues with   General Abandonment
  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. 
Otis B. Jennings and Amber L. Puha
math.PR