# Tight Analysis of Priority Queuing for Egress Traffic

• Jun Kawahara
• Koji M. Kobayashi
• Tomotaka Maeda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

## Abstract

Recently, the problems of evaluating performances of switches and routers have been formulated as online problems, and a great amount of results have been presented. In this paper, we focus on managing outgoing packets (called egress traffic) on switches that support Quality of Service (QoS), and analyze the performance of one of the most fundamental scheduling policies Priority Queuing ($$PQ$$) using competitive analysis. We formulate the problem of managing egress queues as follows: An output interface is equipped with $$m$$ queues, each of which has a buffer of size $$B$$. The size of a packet is unit, and each buffer can store up to $$B$$ packets simultaneously. Each packet is associated with one of $$m$$ priority values $$\alpha _{j}$$ ($$1 \le j \le m$$), where $$\alpha _{1} \le \alpha _{2} \le \cdots \le \alpha _{m}$$, $$\alpha _{1} = 1$$, and $$\alpha _{m} = \alpha$$ and the task of an online algorithm is to select one of $$m$$ queues at each scheduling step. The purpose of this problem is to maximize the sum of the values of the scheduled packets.

For any $$B$$ and any $$m$$, we show that the competitive ratio of $$PQ$$ is exactly $$2 - \min _{x \in [1, m-1] } \{ \frac{ \alpha _{x+1} }{ \sum _{j = 1}^{x+1} \alpha _{j} } \}$$. That is, we conduct a complete analysis of the performance of $$PQ$$ using worst case analysis. Moreover, we show that no deterministic online algorithm can have a competitive ratio smaller than $$1 + \frac{ \alpha ^3 + \alpha ^2 + \alpha }{ \alpha ^4 + 4 \alpha ^3 + 3 \alpha ^2 + 4 \alpha + 1 }$$.

## Notes

### Acknowledgments

We would like to deeply thank Associate Professor Shuichi Miyazaki for a lot of advice on an earlier version of this paper. This work was supported by JSPS KAKENHI Grant Number 26730008 and Cyber Physical System Integrated IT Platform project.

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Jun Kawahara
• 1
• Koji M. Kobayashi
• 2
• Tomotaka Maeda
• 3
1. 1.Nara Institute of Science and TechnologyNaraJapan
2. 2.National Institute of InformaticsChiyodaJapan
3. 3.Academic Center for Computing and Media StudiesKyoto UniversityKyotoJapan