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

Otis B Jennings,Amber L Puha

doi:10.1287/12-ssy085

Otis B Jennings, Amber L Puha

Open Access

https://doi.org/10.1287/12-ssy085

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Journal: Stochastic Systems	Publication Date: Jun 1, 2013
Citations: 2	License type: cc-by

Affiliation: California State University, San Marcos

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.

Highlights

We consider a single-server queue fed by K arrival streams, each corresponding to a distinct job class
We prove a functional law of large numbers for this measure-valued state descriptor
We develop a fluid approximation for this measure-valued state descriptor, which yields fluid approximations for the queue-length vector and other functionals of interest

Summary

Stochastic Systems

Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org. Fluid Limits for Overloaded Multiclass FIFO Single-Server Queues with General Abandonment. Puha (2013) Fluid Limits for Overloaded Multiclass FIFO Single-Server Queues with General Abandonment. Full terms and conditions of use: https://pubsonline.informs.org/Publications/Librarians-Portal/PubsOnLine-Terms-andConditions. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. INFORMS provides unique networking and learning opportunities for individual professionals, and organizations of all types and sizes, to better understand and use O.R. and analytics tools and methods to transform strategic visions and achieve better outcomes. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org

California State University San Marcos

Introduction

Then the deadline process is given by

Virtual Sojourn Time

Note that c

Then it follows that

Proof of

Then δ

Fix such a κ and set δ

Then t