Tight Analysis of Priority Queuing Policy for Egress Traffic

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 {\em egress traffic}) on switches that support Quality of Service (QoS), and analyze the performance of one of the most fundamental scheduling policies {\em 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 $α_{j}$ ($1 \leq j \leq m$), where $α_{1} \leq α_{2} \leq \cdots \leq α_{m}$, $α_{1} = 1$, and $α_{m} = α$ 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 $2 - \min_{x \in [1, m-1] } \{ \frac{ α_{x+1} }{ \sum_{j = 1}^{x+1} α_{j} } \}$. 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 $1 + \frac{ α^3 + α^2 + α}{ α^4 + 4 α^3 + 3 α^2 + 4 α+ 1 }$.

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