## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 11, Number 1 (2001), 1-48.

### Largest Weighted Delay First Scheduling: Large Deviations and Optimality

Kavita Ramanan and Alexander L. Stolyar

#### Abstract

We consider a single server system with *N* input flows. We assume that each flow has stationary increments and satisfies a sample path large deviation principle, and that the system is stable. We introduce the largest weighted delay first (LWDF) queueing discipline associated with any given weight vector α=(α_{1},...,α_{N}). We show that under the LWDF discipline the sequence of scaled stationary distributions of the delay \(\hat{w}_{i}\) of each flow satisfies a large deviation principle with the rate function given by a finite- dimensional optimization problem. We also prove that the LWDF discipline is optimal in the sense that it maximizes the quantity $$\min_{i= 1,\space ...,\space N}\left[α_i \lim_{n\to\infty}\frac{−1}{n}\log P(\hat{w}_i>n)\right],$$ within a large class of work conserving disciplines.

#### Article information

**Source**

Ann. Appl. Probab., Volume 11, Number 1 (2001), 1-48.

**Dates**

First available in Project Euclid: 27 August 2001

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/998926986

**Digital Object Identifier**

doi:10.1214/aoap/998926986

**Mathematical Reviews number (MathSciNet)**

MR1825459

**Zentralblatt MATH identifier**

1024.60012

**Subjects**

Primary: 60F10: Large deviations 90B12 60K25: Queueing theory [See also 68M20, 90B22]

**Keywords**

queueing theory queueing delay large deviations rate function optimality fluid limit control scheduling quality of service (Qos) earliest deadline first (EDF) LWDF

#### Citation

Stolyar, Alexander L.; Ramanan, Kavita. Largest Weighted Delay First Scheduling: Large Deviations and Optimality. Ann. Appl. Probab. 11 (2001), no. 1, 1--48. doi:10.1214/aoap/998926986. https://projecteuclid.org/euclid.aoap/998926986