Fluid Limits for Overloaded Multiclass FIFO Single-Server Queues with General Abandonment

TL;DR

This paper develops fluid limit models for overloaded multiclass FIFO queues with abandonment, providing detailed measure-valued descriptions and approximations that incorporate deadline distributions.

Contribution

It extends previous workload analysis by introducing a measure-valued state descriptor and proving a functional law of large numbers for it.

Findings

Fluid limit approximations for queue lengths and abandonments

Dependence of approximations on entire deadline distributions

Measure-valued process convergence under mild conditions

Abstract

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.