Fluid limits for overloaded multiclass FIFO single-server queues with general abandonment

Abstract

We consider an overloaded multiclass nonidling first-in-first-outsingle-server queue with abandonment. The interarrival times, servicetimes, and deadline times are sequences of independent and identically,but generally distributed random variables. In prior work, Jennings andReed studied the workload process associated with this queue. Undermild conditions, they establish both a functional law of large numbersand a functional central limit theorem for this process. We build on thatwork here. For this, we consider a more detailed description of thesystem state given by K finite, nonnegative Borel measures on thenonnegative quadrant, one for each job class. For each time and jobclass, the associated measure has a unit atom associated with each jobof that class in the system at the coordinates determined by what arereferred to as the residual virtual sojourn time and residual patiencetime of that job. Under mild conditions, we prove a functional law oflarge numbers for this measure-valued state descriptor. This yieldsapproximations for related processes such as the queue lengths andabandoning queue lengths. An interesting characteristic of theseapproximations is that they depend on the deadline distributions intheir entirety.

Keywords

• Overloaded queue
• abandonment
• first-in-first-out
• multiclass queue
• measure-valued state descriptor
• queue-length vector
• fluid limits
• fluid model
• invariant states