论文标题
Buslaev控制二进制链的最佳竞争解决规则
Optimal Competition Resolution Rule for Buslaev Controlled Binary Chain
论文作者
论文摘要
研究了一种称为二进制封闭轮廓链的动力系统。 Dynamica系统属于Buslaev网络类。该系统包含$ n $ {\ it轮廓。}每个轮廓中都有两个单元格和一个粒子。每个轮廓都有两个相邻轮廓。相邻轮廓有一个共同点。这个共同点称为节点。该节点位于单元格之间。在系统的确定性版本中，在任何离散时刻，如果没有延迟，每个粒子都会移至轮廓的另一个单元格。延迟是由于两个粒子可能不会同时通过公共节点。如果两个粒子试图跨越相同的节点，则会发生一个{\ it竞争}，并且这些粒子中只有一个按照规定的竞争分辨率规则移动。在系统的随机版本中，每个粒子以概率$ 1- \ varepsilon的概率移动，如果系统处于状态，则在确定性系统的相同状态下，该粒子会移动。其中$ \ varepsilon $是一个小价值。我们已经获得了一项竞争解决规则，以使系统产生一个状态，使所有粒子在当前和将来（自由流动的状态）不延误，并且该系统在最短时间内导致自由流动的状态。每一个时间单元的$ i $ th粒子过渡数的期望称为{\ IT平均该粒子的速度，} $ v_i，$ $ $ i = 1，\ dots，n。$n。$ n。$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $n。$在$ n = 3，$的假设下，我们已经证明了以下内容。对于最佳规则，粒子的平均速度等于$ v_1 = v_2 = 1-2 \ varepsilon+o（\ varepsilon）$ $（\ varepsilon \ to 0）。$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $（$ $ v = v_1 = v_2 = \ frac {6} {7}+o（\ sqrt {\ varepsilon}）。$
A dynamical system, called a binary closed chain of contours, is studied. The dynamica system belongs to the class of Buslaev networks. The system contains $N$ {\it contours.} There two cells and a particle in each contour. There two adjacent contours for each contour. There is a common point of adjacent contours. This common point is called a node. The node is located between the cells. In the deterministic version of the system, at any discrete moment, each particles moves to the other cell of the contour if there is no delay. The delays are due to that two particles may not pass through the common node simultaneously. If two particles try to cross the same node, then a {\it competition} occurs, and only one of these particles moves in accordance with a prescribed competition resolution rule. In the stochastic version of the system, each particle moves with the probability $1-\varepsilon,$ if the system is in the state such that, in the same state of the deterministic system, this particle moves. where $\varepsilon$ is a small value. We have obtained a competition resolution rule such that the system results in a state such that all particles move without delays in present time and in the future (a state of free movement), and the system results in the state of free movement over a minimum time. The expectation of the number of the $i$th particle transitions per a time unit is called the {\it average velocity of this particle,} $v_i,$ $i=1,\dots,N.$ For the stochastic version of the system, under the assumption that $N=3,$ we have proved the following. For the optimal rule, the average velocity of particles is equal to $v_1=v_2=1-2\varepsilon+o(\varepsilon)$ $(\varepsilon\to 0).$ For the left-priority rule, which is studied earlier, the average velocity of particles equals $v=v_1=v_2=\frac{6}{7}+o(\sqrt{\varepsilon}).$
