Optimal Competition Resolution Rule for Buslaev Controlled Binary Chain

Abstract

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 $v=v_1=v_2=\frac{6}{7}+o(\sqrt{\varepsilon}).$