The design of optical buffers is an important issue in all-optical packet switching. One of the most general types of buffering schemes is priority queues, which includes first-in first-out (FIFO) queues and last-in first-out (LIFO) queues as special cases (where the packet arrival times are used for the assignment of packet priorities). Recently, it was shown in our previous work that an optical priority queue with buffer size $2^{O(\sqrt{{\alpha}M})}$ can be implemented by using an optical (M + 2) × (M + 2) (bufferless) crossbar switch and M fiber delay lines under a simple priority-based routing policy, where α is a constant that depends on the parameters used in the constructions. This achieved buffer size $2^{O(\sqrt{{\alpha}M})}$ (which is exponential in $\sqrt{M}$ ) is the best result currently known in the literature and significantly improves on all previous results (all of which are only polynomial in M). In this paper, we focus on our previous constructions of optical priority queues. The first contribution of this paper is to derive a closed-form expression for the maximum buffer size that can be achieved in our previous constructions. Such an expression is of sufficient theoretical interest itself and can be used to directly compute the maximum buffer size (in contrast, the maximum buffer size has to be computed recursively in our previous work). The second contribution of this paper is to use the closed-form expression to show that in the regime that s ≥ 2, k ≥ 2s + 1, and m ≥ 2, where s, k, and m are parameters used in the constructions, the maximum buffer size U k is given by $U_{k}=2^{O(\sqrt{M \log_{2}(2s+2)\log_{2}\ m/((2s+1)m)})}$ under a mild constraint that is applicable in practical scenarios. This result can be regarded as a complement to the approximate result $U_{k} \approx 2^{O(\sqrt{M \log_{2}(2s+2)\log_{2}\ m/((2s+1)m)})}$ in our previous work.

中文翻译：

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论一类光优先级队列结构所能达到的最大缓冲区大小

光缓冲器的设计是全光分组交换中的一个重要问题。最常见的缓冲方案类型之一是优先级队列，其中包括先进先出 (FIFO) 队列和后进先出 (LIFO) 队列作为特殊情况（其中数据包到达时间用于分配数据包优先级）。最近，我们之前的工作表明，具有缓冲区大小的光学优先级队列$2^{O(\sqrt{{\alpha}M})}$可以通过在简单的基于优先级的路由策略下使用光学 (M + 2) × (M + 2) （无缓冲）交叉开关和 M 条光纤延迟线来实现，其中 α 是一个常数，取决于在建筑。这样就达到了缓冲区大小$2^{O(\sqrt{{\alpha}M})}$（这是指数$\sqrt{M}$ ）是目前文献中已知的最佳结果，并且显着改进了之前的所有结果（所有结果都只是 M 中的多项式）。在本文中，我们重点讨论之前的光学优先队列的构造。本文的第一个贡献是导出了我们之前的构造中可以实现的最大缓冲区大小的封闭式表达式。这样的表达式本身就具有足够的理论意义，可以用来直接计算最大缓冲区大小（相反，在我们之前的工作中必须递归计算最大缓冲区大小）。本文的第二个贡献是使用封闭式表达式来表明，在 s ≥ 2、k ≥ 2s + 1 和 m ≥ 2 的情况下，其中 s、k 和 m 是构造中使用的参数，最大缓冲区大小U k是（谁）给的$U_{k}=2^{O(\sqrt{M \log_{2}(2s+2)\log_{2}\ m/((2s+1)m)})}$在适用于实际场景的温和约束下。这个结果可以看作是对近似结果的补充$U_{k} \约2^{O(\sqrt{M \log_{2}(2s+2)\log_{2}\ m/((2s+1)m)})}$在我们之前的工作中。